SMI.m

classdef SMI
    % SMI: Standard Model Imaging class
    %
    % SMI is an implementation of the Standard Model framework for
    % modeling white matter microstructure with diffusion MRI (supports
    % diffusion relaxometry data too)
    %
    %
    % Compulsory inputs:
    %       - dwi (4D array: [nx x ny x nz x N]) with diffusion-weighted data
    %       - b [1 x N] array with b-values
    %       - dirs [3 x N] array with b-vectors
    %       - sigma (3D array: [nx x ny x nz]) with the noise map
    %
    % Outputs:
    %       [out] = SMI.fit(dwi,options);
    %
    %       out.kernel has a [nx x ny x nz x 6] array (if input data had a
    %       fixed TE)
    %       with the kernel parameters in the following order:
    %       (f, Da, Deparallel, Deperpendicular, fw, p2)
    %       if 2 compartments were fit (IAS + EAS) fw output is simply 0.
    %
    %       out.kernel has a [nx x ny x nz x 8] array (if input data had a
    %       multiple TE)
    %       with the kernel parameters in the following order:
    %       (f, Da, Deparallel, Deperpendicular, fw, T2a, T2e, p2)
    %       if 2 compartments were fit (IAS + EAS) fw output is simply 0.
    %
    %       out.shells is a [4 x Nshells] array: [b;beta;#directions;TE]
    %       one column per shell.
    %       This matrix sorts all the shells in the protocol by columns.
    %       Nshells is the total number of 'shells' (shell: unique
    %       combination of {b,beta,TE} )
    %
    %       out.RotInvs.S0 contains a [nx x ny x nz x Nshells] array with
    %       the zeroth order rotational invariant at each shell
    %       out.RotInvs.S2 contains a [nx x ny x nz x Nshells] array with
    %       the second order rotational invariant at each shell
    %       out.RotInvs.S0/S2 are ordered like the shells in out.shells
    %
    % =====================================================================
    % Minimal usage:
    %
    % % % Add SMI.m to the path, e.g.:
    % addpath('/Documents/SantiagoCoelho/Git/SMI')
    % 
    % % % Load dwi, protocol, and mask
    % 
    % % % Specify protocol information
    % options.b    = bval;
    % options.dirs = dirs;
    % options.MergeDistance = []; % If []: default is 0.05 [ms/um^2], this 
    % is the threshold for considering different b-values as the same shell
    % 
    % % % Specify mask and noise map
    % options.mask  = logical(mask);
    % options.sigma = sigma;
    % 
    % % % Run SM fitting (dwi is a 4D array)
    % [out] = SMI.fit(dwi,options);
    % 
    % =====================================================================
    % More advanced usage:
    %
    % % % Add SMI.m to the path, e.g.:
    % addpath('/Documents/SantiagoCoelho/Git/SMI')
    % 
    % % % Load dwi, protocol, and mask
    % 
    % % % Specify protocol information
    % options.b    = bval;
    % options.beta = beta;
    % options.dirs = dirs;
    % options.TE   = TE;
    % 
    % % % Specify mask and noise map
    % options.mask  = logical(mask);
    % options.sigma = sigma;
    % 
    % % % Specify options for the fit
    % options.compartments = {'IAS','EAS','FW'}; % The order does not matter
    % options.NoiseBias    = 'Rician'; % use 'None' for zero-mean noise
    % 
    % % % Run SM fitting (dwi is a 4D array)
    % [out] = SMI.fit(dwi,options);
    %
    % =====================================================================
    %
    % % % ADVANCED OPTIONS:
    %
    % % % The protocol is defined by
    % options.b
    % options.dirs
    % options.beta
    % options.TE
    %
    % % % Select compartments (FW can be removed)
    % options.compartments = {'IAS','EAS','FW'};
    %
    % % % Bounds for training data sampled from a uniform distribution       
    % options.MLTraining.Ntraining is the number of training samples (default is 40000)
    % lb_training = options.MLTraining.bounds % [2 x 8] array of lower and upper bounds
    % 
    % The order is: [f, Da, Depar, Deperp, fw, T2a, T2e, p2] (If data has
    % fixed TE then the T2a and T2e priors are simply ignored)
    %
    % default is options.MLTraining.bounds = [0.05, 1, 1, 0.1, 0, 50, 50, 0.05; 0.95, 3, 3, 1.2, 0.5, 150, 120, 0.99];
    %
    % If options.MLTraining does not exist then the training data is generated with the default options
    %
    % options.Nlevels divides the range of noise levels into Nlevels. This
    % performs a separate regression for each noise level and applies to each
    % voxel the corresponding one depending on its SNR (default is 10 levels)
    %
    % % % masking
    % options.mask contains a 3D logical array with the values where the fit
    % should be applied
    %
    % % % Noise bias
    % options.NoiseBias = 'None' (default) the spherical harmonics are fit to the
    %                     DWI
    % options.NoiseBias = 'Rician' a Rician bias correction step is applied to 
    %                     the DWI before the spherical harmonics fitting
    %
    % % % Noise level
    % options.sigma contains a 3D array with the noise level, this can be
    % computed using MPPCA (see github documentation for more details)
    %
    % Free water compartment
    % options.D_FW this diffusivity is by default fixed at 3
    % micrometer^2/ms which is the water diffusivity at body temperature.
    % If working with ex vivo samples at lower temperature it can be set to
    % a different value
    %
    % =====================================================================
    %
    %  Authors: Santiago Coelho (santiago.coelho@nyulangone.org), Jelle Veraart, Els Fieremans, Dmitry Novikov
    %  Copyright (c) 2022 New York University
    %              
    %   Permission is hereby granted, free of charge, to any non-commercial entity ('Recipient') obtaining a 
    %   copy of this software and associated documentation files (the 'Software'), to the Software solely for
    %   non-commercial research, including the rights to use, copy and modify the Software, subject to the 
    %   following conditions: 
    % 
    %     1. The above copyright notice and this permission notice shall be included by Recipient in all copies
    %     or substantial portions of the Software. 
    % 
    %     2. THE SOFTWARE IS PROVIDED 'AS IS', WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT
    %     NOT LIMITED TO THE WARRANTIESOF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.
    %     IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BELIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,
    %     WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF ORIN CONNECTION WITH THE
    %     SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE. 
    % 
    %     3. In no event shall NYU be liable for direct, indirect, special, incidental or consequential damages
    %     in connection with the Software. Recipient will defend, indemnify and hold NYU harmless from any 
    %     claims or liability resulting from the use of the Software by recipient. 
    % 
    %     4. Neither anything contained herein nor the delivery of the Software to recipient shall be deemed to
    %     grant the Recipient any right or licenses under any patents or patent application owned by NYU. 
    % 
    %     5. The Software may only be used for non-commercial research and may not be used for clinical care. 
    % 
    %     6. Any publication by Recipient of research involving the Software shall cite the references listed
    %     below.
    %
    %  REFERENCES:
    %  - Coelho, S., Baete, S., Lemberskiy, G., Ades-Aron, B., Barrol, G., Veraart, J., Novikov, D.S., Fieremans, E., 2022 (ArXiv)
    %  - Novikov, D.S., Veraart, J., Jelescu, I.O., Fieremans, E., 2018. Rotationally-invariant mapping of scalar and orientational metrics of neuronal microstructure with diffusion MRI. NeuroImage 174, 518 – 538
    %  - Reisert, M., Kellner, E., Dhital, B., Hennig, J., Kiselev, V.G., 2017. Disentangling micro from mesostructure by diffusion MRI: A Bayesian approach. NeuroImage 147, 964 – 975.
    %

    methods     ( Static = true )
        % =================================================================
        function [out] = fit(dwi,options)
            % [out] = fit(dwi,options)
            %
            % This function is a wrapper that fits the and the SM parameters from the
            % input dwi (4D array) using the information in 'options'.
            %
            % First, the rotational invariants of the diffusion signal are estimated.
            % Then, from these the SM kernel parameters are estimated.
            
            if (~isfield(options,'b'))||(~isfield(options,'dirs'))
                error('b and dirs are compulsory arguments for SMI to run')
            end
            dirs = options.dirs;
            if max(options.b)<100
                b_micro_units=options.b;
            else
                b_micro_units=options.b/1000;
            end
            
            if ~isfield(options,'MergeDistance')
                MergeDistance = [];
            else
                MergeDistance = options.MergeDistance;
            end
            if ~isfield(options,'beta')
                beta = [];
            else
                beta = options.beta;
            end
            if ~isfield(options,'TE')
                TE = [];
            else
                TE = options.TE;
            end
            if isempty(TE)||length(unique(TE))==1
                fit_T2=0;
            else
                fit_T2=1;
            end
            if ~isfield(options,'CS_phase') 
                CS_phase = 1;
            else
                CS_phase = options.CS_phase;
            end
            if ~isfield(options,'D_FW') 
                D_FW = 3;
            else
                D_FW = options.D_FW;
            end
            if ~isfield(options,'flag_fit_fODF') 
                flag_fit_fODF = 0;
            else
                flag_fit_fODF = options.flag_fit_fODF;
            end
            if ~isfield(options,'flag_rectify_fODF') 
                flag_rectify_fODF = 0;
            else
                flag_rectify_fODF = options.flag_rectify_fODF;
            end
            if ~isfield(options,'flag_freeze_seeds') 
                flag_freeze_seeds = 1;
            else
                flag_freeze_seeds = options.flag_freeze_seeds;
            end
            if flag_freeze_seeds
                rng default
            end


            % Grab compartments data from options (default is IAS + EAS + FW)
            flag_compartments=[0 0 0 0];
            if ~isfield(options,'compartments')
                flag_compartments=[1 1 0 0]; % Default is 2 compartments
            else
                id_IAS = find(strcmp(options.compartments, 'IAS'));
                id_EAS = find(strcmp(options.compartments, 'EAS'));
                id_FW  = find(strcmp(options.compartments, 'FW' ));
                id_DOT = find(strcmp(options.compartments, 'DOT'));
                if id_IAS
                    flag_compartments(1)=1;
                end
                if id_EAS
                    flag_compartments(2)=1;
                end
                if id_FW
                    flag_compartments(3)=1;
                end
                if id_DOT
                    flag_compartments(4)=1;
                    error('Current version does not support a DOT compartment (update should be ready soon)')
                end
            end
                        
            if ~isfield(options,'Nlevels')
                Nlevels = 10;
            else
                Nlevels = options.Nlevels;
            end

            if ~isfield(options,'Lmax')
                Lmax = SMI.GetDefaultLmax(b_micro_units,beta,TE,MergeDistance);
            else
                Lmax = options.Lmax;
            end

            if ~isfield(options,'RotInv_Lmax')&&(max(Lmax)>=4)
                RotInv_Lmax = 4;
            elseif ~isfield(options,'RotInv_Lmax')
                RotInv_Lmax = 2;
            else
                RotInv_Lmax = options.RotInv_Lmax;
            end

            if ~isfield(options,'Degree_Kernel_PR')
                Degree_Kernel_PR = 3;
            else
                Degree_Kernel_PR = options.Degree_Kernel_PR;
            end

            if ~isfield(options,'run_maximumlikelihood')
                run_maximumlikelihood = 0;
            else
                run_maximumlikelihood = options.run_maximumlikelihood;
            end

            if run_maximumlikelihood
                flag_fit_fODF = 1; % This is needed as an initial condition for the search
            end

            % Generate or read priors
            if any(any(isnan(options.MLTraining.bounds)))&&isfield(options.MLTraining,'prior')
                prior = options.MLTraining.prior;
                if ~fit_T2
                    % Add fake T2s (will be ignored since TE is fixed and T2 will not be estimated)
                    prior=[prior(:,1:5), 100*ones(size(prior,1),2), prior(:,6:end)];
                end
%             elseif isfield(options.MLTraining,'prior')
%                 prior=options.MLTraining.prior;
            else
                if ~isfield(options.MLTraining,'Ntraining')
                    if fit_T2
                        Ntraining = 2e5;
                    else
                        Ntraining = 1e5;
                    end
                else
                    Ntraining = options.MLTraining.Ntraining;
                end
                Lmax_training=6;
                if ~isfield(options,'MLTraining')
                    lb_training = [0.05, 1, 1, 0.1,   0,  50,  50, 0.05];
                    ub_training = [0.95, 3, 3, 1.2, 0.5, 150, 120, 0.99];
                else
                    lb_training = options.MLTraining.bounds(1,:);
                    ub_training = options.MLTraining.bounds(2,:);
                end
                [f,Da,Depar,Deperp,f_FW,T2a,T2e,p2,plm_train] = SMI.Get_uniformly_distributed_SM_prior(Ntraining,lb_training,ub_training,Lmax_training);
                if RotInv_Lmax==2
                    prior=[f,Da,Depar,Deperp,f_FW,T2a,T2e,p2];
                elseif RotInv_Lmax==4
                    p4=sqrt(sum(plm_train(:,6:14).^2,2));
                    prior=[f,Da,Depar,Deperp,f_FW,T2a,T2e,p2,p4];
                elseif RotInv_Lmax==6
                    p4=sqrt(sum(plm_train(:,6:14).^2,2));
                    p6=sqrt(sum(plm_train(:,15:27).^2,2));
                    prior=[f,Da,Depar,Deperp,f_FW,T2a,T2e,p2,p4,p6];
                end
            end

            if (RotInv_Lmax==2&&size(prior,2)<8)||(RotInv_Lmax==4&&size(prior,2)<9)||(RotInv_Lmax==6&&size(prior,2)<10)
                error('Inconsistency between desired Lmax for ML fitting and prior distribution')
            end


            if ~isfield(options,'mask')
                sz = size(dwi);
                mask = true(sz(1:3));
            else
                mask = options.mask;
            end
            
            % Apply Rician bias correction if needed
            if ~isfield(options,'NoiseBias')||strcmp(options.NoiseBias,'None')
                flag_Rician_bias = 0;
            elseif strcmp(options.NoiseBias,'Rician')
                flag_Rician_bias = 1;
            end
            
            if ~isfield(options,'sigma')||isempty(options.sigma)
                if isempty(TE)
                    b0_for_sigma_flag=(b_micro_units<0.1);
                else
                    b0_for_sigma_flag=(b_micro_units<0.1)&(TE==min(TE));
                end
                b0s=dwi(:,:,:,b0_for_sigma_flag);
                % sigma = smooth3(std(b0s,[],4),'gaussian',5,1);
                sigma = double(smooth3(SMI.PatchBasedSigma(b0s,[3 3 3]),'gaussian',3,1));
                warning('sigma was not an input (not recommended)')
            else
                sigma = double(options.sigma);
            end
            
            if flag_Rician_bias
                ndwi = size(dwi, 4);
                dwi=sqrt(abs(dwi.^2 - repmat(sigma.^2,[1 1 1 ndwi])));
            end
            
            % Spherical harmonics fit
            %%%%%[~,Sl,~,table_4D_sorted] = SMI.Fit2D4D_LLS_RealSphHarm_wSorting_norm_varL(dwi,mask,b_micro_units,dirs,beta,TE,Lmax,MergeDistance);
            % Spherical harmonics fit with rank 1 denoising on Slm to compute Sl
            [Slm,Sl,~,table_4D_sorted] = SMI.Fit2D4D_LLS_RealSphHarm_wSorting_norm_varL_rank1Sl(dwi,mask,b_micro_units,dirs,beta,TE,Lmax,MergeDistance);

            % Concatenate rotational invariants
            out.RotInvs.S0=squeeze(Sl(:,:,:,1,:));
            if RotInv_Lmax==0
                RotInvs=squeeze(Sl(:,:,:,1,:));
            elseif RotInv_Lmax==2
                RotInvs=cat(4,squeeze(Sl(:,:,:,1,:)),squeeze(Sl(:,:,:,2,:)));
                out.RotInvs.S2=squeeze(Sl(:,:,:,2,:));         
            elseif RotInv_Lmax==4
                RotInvs=cat(4,squeeze(Sl(:,:,:,1,:)),squeeze(Sl(:,:,:,2,:)),squeeze(Sl(:,:,:,3,:)));
                out.RotInvs.S2=squeeze(Sl(:,:,:,2,:));         
                out.RotInvs.S4=squeeze(Sl(:,:,:,3,:));         
            elseif RotInv_Lmax==6
                RotInvs=cat(4,squeeze(Sl(:,:,:,1,:)),squeeze(Sl(:,:,:,2,:)),squeeze(Sl(:,:,:,3,:)),squeeze(Sl(:,:,:,4,:)));
                out.RotInvs.S2=squeeze(Sl(:,:,:,2,:));         
                out.RotInvs.S4=squeeze(Sl(:,:,:,3,:));         
                out.RotInvs.S6=squeeze(Sl(:,:,:,4,:));         
            end
            out.shells=table_4D_sorted;
            
            % Run polynomial regression training and fitting
            sigma_norm_limits = [0 0.2];
            KERNEL = SMI.StandardModel_MLfit_RotInvs(RotInvs,mask,sigma,b_micro_units,beta,TE,prior,Nlevels,sigma_norm_limits,flag_compartments,MergeDistance,RotInv_Lmax,Lmax,D_FW,Degree_Kernel_PR);
            out.kernel = KERNEL;
            out.sigma = sigma;
            if flag_fit_fODF
                % [plm,pl] = SMI.get_plm_from_Slm_and_kernel(Slm,Lmax,KERNEL,mask,table_4D_sorted,D_FW);
                if ~fit_T2
                    s0 = Sl(:,:,:,1,1); % This does not work for variable TE data! (this is not S0)
                else % Fit s0 if variable TE data (not the b0)
                    logb0s = SMI.vectorize(log(dwi(:,:,:,b_micro_units<0.1)),mask);
                    TE_b0s = TE(b_micro_units<0.1);
                    coeffs = logb0s'*pinv([ones(length(TE_b0s),1) TE_b0s(:)]');
                    s0 = SMI.vectorize(exp(coeffs(:,1))',mask);
                end
                [plm,pl] = SMI.get_plm_from_S_and_kernel(dwi./s0,Lmax,KERNEL,mask,b_micro_units,beta,TE,dirs,CS_phase,D_FW);
                out.plm = plm;
                out.pl = pl;
                out.CS_phase=CS_phase;
                out.Lmax=Lmax;
            end
            if flag_rectify_fODF
                plm=SMI.vectorize(plm,mask);
                [EPSILON,~,~] = SMI.Compute_eps_ODF_rectification(plm(2:end,:),CS_phase);
                out.epsilon=SMI.vectorize(EPSILON',mask);
            end

            if run_maximumlikelihood
                flag_FW = single(id_FW);
                if flag_FW || fit_T2
                    kernel_from_PR = cat(4,out.kernel(:,:,:,1:7),out.plm(:,:,:,1:5));
                elseif flag_FW || ~fit_T2
                    kernel_from_PR = cat(4,out.kernel(:,:,:,1:5),out.plm(:,:,:,1:5));
                elseif ~flag_FW || fit_T2
                    kernel_from_PR = cat(4,out.kernel(:,:,:,[1:4 6:7]),out.plm(:,:,:,1:5));
                elseif ~flag_FW || ~fit_T2
                    kernel_from_PR = cat(4,out.kernel(:,:,:,1:4),out.plm(:,:,:,1:5));
                end
                KernelfODFInitializations = cat(4,kernel_from_PR,out.plm(:,:,:,1:5));
                [KERNEL_maxlik,p2m_maxlik,RESNORM,Niterations] = SMI.MaxLikfit_on_Slm(Slm,Lmax,MergeDistance,KernelfODFInitializations,mask,b_micro_units,beta,TE,flag_FW,D_FW,CS_phase);
                toc
                out.KERNEL_maxlik = KERNEL_maxlik;
                out.p2_maxlik = sqrt(sum(p2m_maxlik.^2,4));
                out.p2m_maxlik = p2m_maxlik;
                out.RESNORM_maxlik = RESNORM;
                out.Niterations_maxlik = Niterations;
            end

            % =================================================================
            % Write log file with all fitting details
            if ~isfield(options,'path_log')
                path_log = pwd;
            else
                path_log = options.path_log;
            end
            if ~isfield(options,'filename_log')
                randomstr = char(randi([65 90],1,10));
                filename_log = ['log_SMI_',randomstr,'.txt'];
            else
                filename_log = options.filename_log;
            end
            if ~strcmp(filename_log(end-3:end),'.txt')
                filename_log = [filename_log,'.txt'];
            end

            file_log = 'SMI fitting parameters:\n';
            Todays_date = datetime('now','TimeZone','local','Format','d-MMM-y HH:mm:ss Z');
            file_log = [file_log sprintf('Performed on: %s\n',Todays_date)];
            file_log = [file_log sprintf('- Noise bias: %s \n',options.NoiseBias)];
            file_log = [file_log,'- Compartments:'];
            for ii=1:length(options.compartments)
                if ii==length(options.compartments)
                    file_log = [file_log,' ',options.compartments{ii},'\n'];
                else
                    file_log = [file_log,' ',options.compartments{ii}];
                end
            end
            Nshells = size(out.shells,2);
            file_log = [file_log sprintf('- Sigma (normalized) ranges: [%2f %2f] split into %d intervals\n',sigma_norm_limits(1),sigma_norm_limits(2),Nlevels)];
            file_log = [file_log sprintf('- Shells merging factor: %f\n',options.MergeDistance)];
            file_log = [file_log sprintf(['- Final shells Lmax:   ',repmat('%d ',1,Nshells),' \n'],Lmax)];
            file_log = [file_log sprintf(['- Final shells bval:   ',repmat('%.2f ',1,Nshells),' \n'],out.shells(1,:))];
            file_log = [file_log sprintf(['- Final shells bshape: ',repmat('%.2f ',1,Nshells),' \n'],out.shells(2,:))];
            file_log = [file_log sprintf(['- Final shells Ndirs:  ',repmat('%d  ',1,Nshells),' \n'],out.shells(3,:))];
            if length(unique(out.shells(4,:)))==1
                file_log = [file_log '- Fixed TE data\n'];
            else
                file_log = [file_log sprintf(['- Final shells TE: ',repmat('%.1f ',1,Nshells),' \n'],out.shells(4,:))];
            end
            file_log = [file_log sprintf('- Training samples: %d (for [ f, Da, Depar, Deperp, f_w, T2a, T2e, p2])\n',Ntraining)];
            file_log = [file_log sprintf(['- Lower bounds for training (uniform distribution):  ',repmat('%.2f  ',1,length(lb_training)),' \n'],lb_training)];
            file_log = [file_log sprintf(['- Upper bounds for training (uniform distribution):  ',repmat('%.2f  ',1,length(ub_training)),' \n'],ub_training)];
            file_log = [file_log sprintf('- MAX S_l used for kernel polynomial regression: %d \n',RotInv_Lmax)];
            file_log = [file_log sprintf('- Degree used for kernel polynomial regression: %d \n',Degree_Kernel_PR)];
            file_log = [file_log sprintf('- Free water diffusivity used: %.2f um^2/ms \n',D_FW)];

            fid = fopen(fullfile(path_log,filename_log),'wt');
            fprintf(fid, file_log);
            fclose(fid);
        end
        % =================================================================
        function [plm,pl] = get_plm_from_S_and_kernel(dwi_norm,Lmax,kernel,mask,b,beta,TE,dirs,CS_phase,D_FW)
            % [plm,pl] = get_plm_from_S_and_kernel(dwi_norm,Lmax,kernel,mask,b,beta,TE,dirs,CS_phase,D_FW) 
            %
            % This function does not assume shells for plm estimation,
            % joint plm estimation from DWI+kernel
            %
            % PLM ARE NORMALIZED
            %
            if ~exist('beta', 'var') || isempty(beta)
                beta = ones(size(b));
            end
            if ~exist('TE', 'var') || isempty(TE)
                TE = zeros(size(b));
            end

            LMAX=max(Lmax);
            if LMAX==0, error('Lmax needs to be larger than 0 for fitting the fODF'), end
            L_all    = repelem(0:2:LMAX,2*(0:2:LMAX)+1);
            N_l      = sqrt((2*L_all+1)*(4*pi));
            Y_LM_matrix = SMI.get_even_SH(dirs,LMAX,CS_phase);
            if LMAX==0
                plm=nan;
                fprintf('~~~ plm will not be estimated since only S0 is estimated\n')
            else
                warning('off','MATLAB:rankDeficientMatrix')
                sz=size(kernel);
                if isempty(mask)
                    mask=true(sz(1:3));
                end
                dwi_norm=SMI.vectorize(dwi_norm,mask);

                % Grab kernel
                kernel=SMI.vectorize(kernel,mask);
                f=kernel(1,:);
                Da=kernel(2,:);
                Depar=kernel(3,:);
                Deperp=kernel(4,:);
                fw=kernel(5,:);

                Ndwi=length(b); % No shells are assumed, we compute Rotational invariants for each dwi
                Nvoxels=length(f);

                if length(unique(TE))>1
                    T2a=kernel(6,:);
                    T2e=kernel(7,:);
                else
                    T2a=ones(size(f));
                    T2e=ones(size(f));
                end

                x=[f; Da; Depar; Deperp; fw; T2a; T2e]';
                Kell=zeros(Nvoxels,Ndwi,LMAX/2+1);
                Kell(:,:,1) = SMI.RotInv_Kell_wFW_b_beta_TE_numerical(0,b,beta,TE,x,D_FW);
                if LMAX>=2
                    Kell(:,:,2) = SMI.RotInv_Kell_wFW_b_beta_TE_numerical(2,b,beta,TE,x,D_FW);
                end
                if LMAX>=4
                    Kell(:,:,3) = SMI.RotInv_Kell_wFW_b_beta_TE_numerical(4,b,beta,TE,x,D_FW);
                end
                if LMAX>=6
                    Kell(:,:,4) = SMI.RotInv_Kell_wFW_b_beta_TE_numerical(6,b,beta,TE,x,D_FW);
                end
                if LMAX>=8
                    Kell(:,:,5) = SMI.RotInv_Kell_wFW_b_beta_TE_numerical(8,b,beta,TE,x,D_FW);
                end
                Kell=permute(Kell,[3 2 1]);

                % plm=zeros(LMAX*(LMAX+3)/2+1,Nvoxels);
                plm=zeros(LMAX*(LMAX+3)/2,Nvoxels);
                Nlm=2*(0:2:LMAX)+1;
                for ii=1:Nvoxels
                    KltimesYlm=repelem(Kell(:,:,ii),Nlm,1)'.*(Y_LM_matrix.*N_l);
                    % plm(:,ii)=KltimesYlm\dwi_norm(:,ii);
                    dwi_norm_minus_spherical_mean = dwi_norm(:,ii)-KltimesYlm(:,1);
                    plm(:,ii)=KltimesYlm(:,2:end)\dwi_norm_minus_spherical_mean;
                end

                plm=SMI.vectorize(plm,mask);
                NormEll=ones(1,4);%4*pi*(2*(2:2:8)+1);
                if LMAX==2
                    pl=sqrt(sum(plm(:,:,:,1:5).^2,4)/NormEll(1));
                elseif LMAX==4
                    pl=cat(4,sqrt(sum(plm(:,:,:,1:5).^2,4)/NormEll(1)),sqrt(sum(plm(:,:,:,6:14).^2,4)/NormEll(2)));
                elseif LMAX==6
                    pl=cat(4,sqrt(sum(plm(:,:,:,1:5).^2,4)/NormEll(1)),sqrt(sum(plm(:,:,:,6:14).^2,4)/NormEll(2)),sqrt(sum(plm(:,:,:,15:27).^2,4)/NormEll(3)));
                elseif LMAX==8
                    pl=cat(4,sqrt(sum(plm(:,:,:,1:5).^2,4)/NormEll(1)),sqrt(sum(plm(:,:,:,6:14).^2,4)/NormEll(2)),sqrt(sum(plm(:,:,:,15:27).^2,4)/NormEll(3)),sqrt(sum(plm(:,:,:,28:44).^2,4)/NormEll(4)));
                end                    
            end
        end
        % =================================================================
        function [plm,pl] = get_plm_from_Slm_and_kernel(Slm,Lmax,kernel,mask,table_shells,D_FW)
            % [plm,pl] = get_plm_from_Slm_and_kernel(Slm,Lmax,kernel,mask,table_shells,D_FW)

            % x=rand(1000,7);
            % b=[0 1 2 5 6];
            % beta=[1 1 1 1 1];
            % TE=[0 0 0 0 0];f=x(:,1);
            
            LMAX=max(Lmax);
            if LMAX==0
                plm=nan;
                fprintf('~~~ plm will not be estimated since only S0 is estimated\n')
            else
                warning('off','MATLAB:illConditionedMatrix')
                sz=size(kernel);
                if isempty(mask)
                    mask=true(sz(1:3));
                end
                % Grab b0 fit
                b0=SMI.vectorize(Slm(:,:,:,1,1),mask);

                % Grab kernel
                kernel=SMI.vectorize(kernel,mask);
                f=kernel(1,:);
                Da=kernel(2,:);
                Depar=kernel(3,:);
                Deperp=kernel(4,:);
                fw=kernel(5,:);

                b=table_shells(1,:);
                beta=table_shells(2,:);
                TE=table_shells(4,:);

                Nshells=length(b);
                Nvoxels=length(f);

                if length(unique(TE))>1
                    T2a=kernel(6,:);
                    T2e=kernel(7,:);
                else
                    T2a=ones(size(f));
                    T2e=ones(size(f));
                    TE=zeros(size(TE));
                end

                x=[f; Da; Depar; Deperp; fw; T2a; T2e]';
                
                Kell=zeros(Nvoxels,Nshells,4);
                Kell(:,:,1) = SMI.RotInv_Kell_wFW_b_beta_TE_numerical(2,b,beta,TE,x,D_FW);
                if LMAX>=4
                    Kell(:,:,2) = SMI.RotInv_Kell_wFW_b_beta_TE_numerical(4,b,beta,TE,x,D_FW);
                end
                if LMAX>=6
                    Kell(:,:,3) = SMI.RotInv_Kell_wFW_b_beta_TE_numerical(6,b,beta,TE,x,D_FW);
                end
                if LMAX>=8
                    Kell(:,:,4) = SMI.RotInv_Kell_wFW_b_beta_TE_numerical(8,b,beta,TE,x,D_FW);
                end
                Kell=permute(Kell,[2 1 3]);

                % Kell=zeros(Nshells,Nvoxels,4);
                
                Slm_all=[];Kl_all=[];ells_all=[];shells_all=[];ells_all_unique=[];
                Nlm=Lmax.*(Lmax+3)/2+1;
                for jj=1:Nshells
                    if Lmax(jj)==0
                        continue
                    end
                    Slm_current=SMI.vectorize(Slm(:,:,:,2:Nlm(jj),jj),mask);
                    current_ells=(2:2:Lmax(jj))/2;
                    current_ms=(2:2:Lmax(jj))*2+1;
                    % Kl_current=repelem(squeeze(permute(Kell(jj,:,current_ells),[2 1 3]))',current_ms,1);
                    Kl_current=squeeze(permute(Kell(jj,:,current_ells),[2 1 3]))';
                    ells_all=[ells_all repelem(current_ells*2,1,current_ms)];
                    ells_all_unique=[ells_all_unique current_ells*2];
                    % ms_all=[ms_all repelem(current_ells,1,current_ms)];
                    shells_all=[shells_all jj*ones(1,size(Slm_current,1))];
                    Slm_all=[Slm_all;Slm_current];
                    Kl_all=[Kl_all;Kl_current];
                end
                
                % Normalize Slm with b0 (This step is not valid for
                % variable TE data)
                Slm_all=Slm_all./b0;

                % Getting plm
                plm=zeros(max(Nlm)-1,Nvoxels);
                plm_limits=[1 (2:2:8).*((2:2:8)+3)/2+1];
                for currentL=2:2:LMAX
                    current_lm_ids=plm_limits(currentL/2):(plm_limits(currentL/2+1)-1);
                    Kell_current=Kl_all(ells_all_unique==currentL,:);
                    Sellm_current=Slm_all(ells_all==currentL,:);
                    NshellsCurrentL=size(Kell_current,1);
                    for ii=1:Nvoxels
                        Kellmatrix=[];
                        for jj=1:NshellsCurrentL
                            Kellmatrix=[Kellmatrix;Kell_current(jj,ii)*eye(2*currentL+1)];
                        end
                        plm(current_lm_ids,ii)=Kellmatrix\Sellm_current(:,ii);
                    end
                end
                plm=SMI.vectorize(plm,mask);
                NormEll=4*pi*(2*(2:2:8)+1);
                NormEll = [1 1 1 1];
                if LMAX==2
                    pl=sqrt(sum(plm(:,:,:,1:5).^2,4)/NormEll(1));
                elseif LMAX==4
                    pl=cat(4,sqrt(sum(plm(:,:,:,1:5).^2,4)/NormEll(1)),sqrt(sum(plm(:,:,:,6:14).^2,4)/NormEll(2)));
                elseif LMAX==6
                    pl=cat(4,sqrt(sum(plm(:,:,:,1:5).^2,4)/NormEll(1)),sqrt(sum(plm(:,:,:,6:14).^2,4)/NormEll(2)),sqrt(sum(plm(:,:,:,15:27).^2,4)/NormEll(3)));
                elseif LMAX==8
                    pl=cat(4,sqrt(sum(plm(:,:,:,1:5).^2,4)/NormEll(1)),sqrt(sum(plm(:,:,:,6:14).^2,4)/NormEll(2)),sqrt(sum(plm(:,:,:,15:27).^2,4)/NormEll(3)),sqrt(sum(plm(:,:,:,28:44).^2,4)/NormEll(4)));
                end                    
            end
        end
        % =================================================================
        % =================================================================
        % =================================================================
        % Maximum likelihood implementation on | Slm(b,beta,TE) - plm*Kl(b,beta,TE) |
        function [KERNEL,plm,RESNORM,Niterations] = MaxLikfit_on_Slm(Slm,Lmax,MergeDistance,KernelfODFInitializations,mask,bval,beta,TE,flag_FW,D_FW,CS_phase)
            % 
            % 
            % [KERNEL,plm,RESNORM,Niterations] = MaxLikfit_on_Slm(Slm,Lmax,MergeDistance,KernelfODFInitializations,mask,bval,beta,TE,flag_FW,D_FW,CS_phase)
            %
            % Output: KERNEL = [s0 f Da Depar Deperp f_FW T2a T2e p2m]; (as 4D array)
            % Output: KERNEL = [s0 f Da Depar Deperp T2a T2e p2m]; (as 4D array)
            % Output: KERNEL = [s0 f Da Depar Deperp f_FW p2m]; (as 4D array)
            % Output: KERNEL = [s0 f Da Depar Deperp f_FW p2m]; (as 4D array)
            if ~isvector(bval)
                error('bval should be a vector')
            else
                bval=bval(:)';
            end
            if isempty(beta)
                beta=zeros(size(bval))+1;
            else
                beta=beta(:)';
            end
            if isempty(TE)
                TE=zeros(size(bval));
            else
                TE=TE(:)';
            end
            if ~exist('D_FW', 'var') || isempty(D_FW)
                D_FW = 3;
            end
            if ~exist('CS_phase', 'var') || isempty(CS_phase)
                CS_phase=1;
            else
                CS_phase=0;
            end
            if length(unique(TE))==1
                do_not_fit_T2=1;
            else
                do_not_fit_T2=0;
            end
            [table_4D,~,~] = SMI.Group_dwi_in_shells_b_beta_TE(bval,beta,TE,MergeDistance);
            Nshells = size(table_4D,2);
            sz_Slm=size(Slm);
            if length(sz_Slm)==5
                flag_4D_output=1;
            elseif length(sz_Slm)==4
                flag_4D_output=0;
            else
                error('Spherical Harmonics must be a 3D or 5D array')
            end
            
            if flag_4D_output % Slm are 5D
                if isempty(mask)
                    mask=true(sz_Slm(1:3));
                end
                SlmNotNormalized = [];
                for ii=1:Nshells
                    SlmNotNormalized = [SlmNotNormalized ; SMI.vectorize(squeeze(Slm(:,:,:,:,ii)),mask)];
                end
            else
                SlmNotNormalized = Slm;
            end       
            
            max_ellm = 1/2 * (max(Lmax) + 1) * (max(Lmax) + 2);
            id_00 = 1:max_ellm:max_ellm*Nshells;
            if max_ellm>=2
                id_2m = [2:max_ellm:max_ellm*Nshells ; 3:max_ellm:max_ellm*Nshells ; 4:max_ellm:max_ellm*Nshells ; 5:max_ellm:max_ellm*Nshells ; 6:max_ellm:max_ellm*Nshells]';
                id_2m = id_2m(:);
                L2_used = repmat(Lmax>=2,1,5);
                id_pick = [id_00 id_2m(L2_used)'];
            else
                id_pick = id_00;
            end
            SlmNotNormalized2fit = SlmNotNormalized(id_pick,:);
            ell = [zeros(1,Nshells) , 2*ones(1,5*sum(Lmax>=2))];
            table_4D_rep =repmat(table_4D,1,6);
            if do_not_fit_T2
                table_Klm = [[table_4D(1:2,:), table_4D_rep(1:2,repmat(Lmax>=2,1,5))];ell];
            else
                table_Klm = [[table_4D([1:2 4],:), table_4D_rep([1:2 4],repmat(Lmax>=2,1,5))];ell];
            end
        
            if length(unique(TE))==1
                do_not_fit_T2=1;
                s0_initialization = SlmNotNormalized(1,:);
            else
                do_not_fit_T2=0;
                ids = 1:max_ellm*Nshells;
                bval_S00 = repelem(table_4D(1,:),1,max_ellm);
                TE_S00 = repelem(table_4D(4,:),1,max_ellm);
                keep_b0s = ismember(ids, id_00) & (bval_S00<0.1);
                logb0s = log(abs(SlmNotNormalized(keep_b0s,:)));
                TE_b0s = TE_S00(keep_b0s);
                coeffs = logb0s'*pinv([ones(length(TE_b0s),1) TE_b0s(:)]');
                s0_initialization = exp(coeffs(:,1))';
                inf_s0 = ~isfinite(s0_initialization);
                s0_initialization(inf_s0) = 2 * SlmNotNormalized(1,inf_s0);
            end
        
            if ~exist('KernelfODFInitializations', 'var') || isempty(KernelfODFInitializations)
                if flag_FW
                    kernel_default_x0 = [0.5 2 2 1 0.05];
                else
                    kernel_default_x0 = [0.5 2 2 1];
                end
                if do_not_fit_T2
                    KernelfODFInitializations = repmat(kernel_default_x0,length(s0_initialization),1);       
                else
                    KernelfODFInitializations = repmat([kernel_default_x0 90 50],length(s0_initialization),1);       
                end
            else
                KernelfODFInitializations = SMI.vectorize(KernelfODFInitializations,mask);
            end     
            NkernelParam = 5 + flag_FW + 2*double(~do_not_fit_T2);
            NvoxelsMasked=size(SlmNotNormalized,2);
            if flag_FW
                parfor ii=1:NvoxelsMasked
                    init = [s0_initialization(ii) ; KernelfODFInitializations(:,ii)];
                    Slm_not_normalized = SlmNotNormalized2fit(:,ii);
                    [parameter_hat(:,ii),RESNORM(1,ii),~,exitflag(1,ii),Niterations(1,ii)] = SMI.MaxLik_Slm_fit_SM_withFW_wrapper(Slm_not_normalized, table_Klm, init);
                end
            else
                parfor ii=1:NvoxelsMasked
                    init = [s0_initialization(ii) ; KernelfODFInitializations(:,ii)];
                    Slm_not_normalized = SlmNotNormalized2fit(:,ii);
                    [parameter_hat(:,ii),RESNORM(1,ii),~,exitflag(1,ii),Niterations(1,ii)] = SMI.MaxLik_Slm_fit_SM_withoutFW_wrapper(Slm_not_normalized, table_Klm, init);
                end
            end
            if flag_4D_output % 
                KERNEL=SMI.vectorize(parameter_hat(1:NkernelParam,:),mask);
                plm=SMI.vectorize(parameter_hat((NkernelParam+1):end,:),mask);
                RESNORM=SMI.vectorize(RESNORM,mask);
                Niterations=SMI.vectorize(Niterations,mask);
            end
        end
        % =================================================================
        function [parameter_hat,RESNORM,residual,exitflag,Niterations] = MaxLik_Slm_fit_SM_withFW_wrapper(Slm_not_normalized, table_Klm, init)
        %
        % Optimiser for SM nonlinear fitting (using Slm energy function)
        %
        %
        % The cost function is currently Least Squares
        
        % Gradient Descent stage
        % No gradient specified (finite difference)
        % h=optimoptions('lsqnonlin','Algorithm','trust-region-reflective', 'MaxIter',3000,'MaxFunEvals',10000,'TolX',1e-4,'TolFun',1e-5, 'FunValCheck','on', 'Display', 'off','UseParallel',false);
        h=optimoptions('lsqnonlin','Algorithm','trust-region-reflective', 'MaxIter',10000,'MaxFunEvals',20000,'TolX',1e-14,'TolFun',1e-14, 'FunValCheck','on', 'Display', 'off','UseParallel',false);
        % h=optimoptions('lsqnonlin','Algorithm','levenberg-marquardt', 'MaxIter',10000,'MaxFunEvals',20000,'TolX',1e-20,'TolFun',1e-20, 'FunValCheck','on', 'Display', 'off','UseParallel',false);
        
        LowerBounds=[];%[0 0 0 0 0 0 0];
        Dmax=3.5;
        UpperBounds=[];%;[2 1 Dmax Dmax Dmax 1 1];
        [parameter_hat,RESNORM,residual,exitflag,output,~,~] = lsqnonlin(@(x)SMI.MaxLik_Slm_fit_SM_withFW_fobj(x, table_Klm, Slm_not_normalized), init, LowerBounds,UpperBounds,h);
        Niterations=output.iterations;
        end
        % =================================================================
        function residuals = MaxLik_Slm_fit_SM_withFW_fobj(s0_kernel_fODF,table_Klm,Slm)
        %
        % 
        D_FW = 3;
        ell = table_Klm(end,:);
        S0=s0_kernel_fODF(1); Da = s0_kernel_fODF(3); Depar = s0_kernel_fODF(4); Deperp = s0_kernel_fODF(5); %p2 = s0_kernel_fODF(7);
        f = s0_kernel_fODF(2);
        f_FW = s0_kernel_fODF(6);
        if size(table_Klm,1)==3
            TEs = zeros(1,size(table_Klm,2));
            T2a = 100; T2e = 100;
            if length(s0_kernel_fODF)==6
                plm = [];
            else
                plm = s0_kernel_fODF(7:end);
            end
        else
            TEs = table_Klm(3,:);
            T2a = s0_kernel_fODF(7); T2e = s0_kernel_fODF(8);
            if length(s0_kernel_fODF)==8
                plm = [];
            else
                plm = s0_kernel_fODF(9:end);
            end
        end
        K0_all = SMI.RotInv_Kell_wFW_b_beta_TE_numerical(0,table_Klm(1,:),table_Klm(2,:),TEs,[f, Da, Depar, Deperp, f_FW, T2a, T2e],D_FW);
        K2_all = SMI.RotInv_Kell_wFW_b_beta_TE_numerical(2,table_Klm(1,:),table_Klm(2,:),TEs,[f, Da, Depar, Deperp, f_FW, T2a, T2e],D_FW);
        
        p2m = plm(1:5);
        NshellsL0 = sum(ell==0);
        NshellsL2 = sum(ell==2)/5;
        S00_pred = S0*K0_all(ell==0);
        K2m = K2_all(ell==2);
        S2m_pred = S0*K2m.*repelem(p2m(:)',1,NshellsL2);
        Slm_pred = [ S00_pred S2m_pred ];
        
        % No weights - plain Slm differences
        residuals =  (Slm(:)' - Slm_pred)';
        residuals(~isfinite(residuals))=1e5;
        end
        % =================================================================
        function [parameter_hat,RESNORM,residual,exitflag,Niterations] = MaxLik_Slm_fit_SM_withoutFW_wrapper(Slm_not_normalized, table_Klm, init)
        %
        % Optimiser for SM nonlinear fitting (using Slm energy function)
        %
        %
        % The cost function is currently Least Squares
        
        % Gradient Descent stage
        % No gradient specified (finite difference)
        % h=optimoptions('lsqnonlin','Algorithm','trust-region-reflective', 'MaxIter',3000,'MaxFunEvals',10000,'TolX',1e-4,'TolFun',1e-5, 'FunValCheck','on', 'Display', 'off','UseParallel',false);
        h=optimoptions('lsqnonlin','Algorithm','trust-region-reflective', 'MaxIter',10000,'MaxFunEvals',20000,'TolX',1e-14,'TolFun',1e-14, 'FunValCheck','on', 'Display', 'off','UseParallel',false);
        % h=optimoptions('lsqnonlin','Algorithm','levenberg-marquardt', 'MaxIter',10000,'MaxFunEvals',20000,'TolX',1e-20,'TolFun',1e-20, 'FunValCheck','on', 'Display', 'off','UseParallel',false);
        
        LowerBounds=[];%[0 0 0 0 0 0 0];
        Dmax=3.5;
        UpperBounds=[];%;[2 1 Dmax Dmax Dmax 1 1];
        [parameter_hat,RESNORM,residual,exitflag,output,~,~] = lsqnonlin(@(x)SMI.MaxLik_Slm_fit_SM_withoutFW_fobj(x, table_Klm, Slm_not_normalized), init, LowerBounds,UpperBounds,h);
        Niterations=output.iterations;
        
        end
        % =================================================================
        function residuals = MaxLik_Slm_fit_SM_withoutFW_fobj(s0_kernel_fODF,table_Klm,Slm)
        %
        % 
        D_FW = 3;
        ell = table_Klm(end,:);
        S0=s0_kernel_fODF(1); f = s0_kernel_fODF(2); Da = s0_kernel_fODF(3); Depar = s0_kernel_fODF(4); Deperp = s0_kernel_fODF(5); %p2 = s0_kernel_fODF(7);
        if size(table_Klm,1)==3
            TEs = zeros(1,size(table_Klm,2));
            T2a = 100; T2e = 100;
            if length(s0_kernel_fODF)==5
                plm = [];
            else
                plm = s0_kernel_fODF(6:end);
            end
        else
            TEs = table_Klm(3,:);
            T2a = s0_kernel_fODF(6); T2e = s0_kernel_fODF(7);
            if length(s0_kernel_fODF)==7
                plm = [];
            else
                plm = s0_kernel_fODF(8:end);
            end
        end
        
        K0_all = SMI.RotInv_Kell_wFW_b_beta_TE_numerical(0,table_Klm(1,:),table_Klm(2,:),TEs,[f, Da, Depar, Deperp, 0*f, T2a, T2e],D_FW);
        K2_all = SMI.RotInv_Kell_wFW_b_beta_TE_numerical(2,table_Klm(1,:),table_Klm(2,:),TEs,[f, Da, Depar, Deperp, 0*f, T2a, T2e],D_FW);
        
        p2m = plm(1:5); % Normalized plm are estimated here
        NshellsL0 = sum(ell==0);
        NshellsL2 = sum(ell==2)/5;
        S00_pred = S0*K0_all(ell==0);
        K2m = K2_all(ell==2);
        S2m_pred = S0*K2m.*repelem(p2m(:)',1,NshellsL2);
        Slm_pred = [ S00_pred S2m_pred ];
        
        % No weights - plain Slm differences
        residuals =  (Slm(:)' - Slm_pred)';
        residuals(~isfinite(residuals))=1e5;
        end
        % =================================================================
        % =================================================================
        % =================================================================
        function [EPSILON,plm_rect,pl_rect] = Compute_eps_ODF_rectification(plm,CS_phase)
            % [EPSILON,plm_rect,pl_rect] = Compute_eps_ODF_rectification(plm,CS_phase)
            %
            % plm must be [Nlm x Nreps]
            if nargin==1 || isempty(CS_phase) || ~exist('CS_phase','var') || CS_phase
                CS_phase=1;
            end
            Nodfs=size(plm,2);
            
            %%%%%%% leb = getLebedevSphere(4334);
            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           
            Nlm=size(plm,1);
            Lmax=sqrt(2*Nlm+9/4)-3/2;
            Ylm_n = SMI.get_even_SH(dirs,Lmax,CS_phase);
            EPSILON=zeros(size(plm,2),1);
            plm_rect_all=zeros(Nlm,Nodfs);
            options=optimset('MaxIter',1000,'TolFun',1e-5);
            for ii=1:Nodfs
                % Get epsilon from a 1D optimization
                ObjFun=@(epsilon)abs(leb_weights*(abs(Ylm_n*[1/sqrt(4*pi);plm(:,ii)]-epsilon)-Ylm_n*[1/sqrt(4*pi);plm(:,ii)]-epsilon));
                EPSILON(ii) = fminsearch(ObjFun,0.1,options);
                % Get rectified plm and pl
                rect_odf=1/2*(abs(Ylm_n*[1/sqrt(4*pi);plm(:,ii)]-EPSILON(ii))+Ylm_n*[1/sqrt(4*pi);plm(:,ii)]-EPSILON(ii));
                plm_rect=leb_weights*(rect_odf.*Ylm_n);
                plm_rect=plm_rect(2:end);
                plm_rect_all(:,ii)=plm_rect(:);
            end
            plm_rect=plm_rect_all;
            % ObjFun(EPSILON)
            % 1/2*leb.w'*(abs(Ylm_n*[1/sqrt(4*pi);plm]-EPSILON)+Ylm_n*[1/sqrt(4*pi);plm]-EPSILON)
            %             % Get rectified plms
            %             rect_odf=1/2*(abs(Ylm_n*[1/sqrt(4*pi);plm]-EPSILON)+Ylm_n*[1/sqrt(4*pi);plm]-EPSILON);
            %             plm_rect=leb.w'*(rect_odf.*Ylm_n);
            %             plm_rect=plm_rect(2:end);
            %             plm_rect=plm_rect(:);
            if Lmax==2
                pl_rect=sqrt(sum(plm_rect(1:5,:).^2))';
            elseif Lmax==4
                pl_rect=[sqrt(sum(plm_rect(1:5,:).^2))' sqrt(sum(plm_rect(6:14,:).^2))'];
            elseif Lmax==6
                pl_rect=[sqrt(sum(plm_rect(1:5,:).^2))' sqrt(sum(plm_rect(6:14,:).^2))' sqrt(sum(plm_rect(15:27,:).^2))'];
            elseif Lmax==8
                pl_rect=[sqrt(sum(plm_rect(1:5,:).^2))' sqrt(sum(plm_rect(6:14,:).^2))' sqrt(sum(plm_rect(15:27,:).^2))' sqrt(sum(plm_rect(28:44,:).^2))'];
            elseif Lmax==10
                pl_rect=[sqrt(sum(plm_rect(1:5,:).^2))' sqrt(sum(plm_rect(6:14,:).^2))' sqrt(sum(plm_rect(15:27,:).^2))' sqrt(sum(plm_rect(28:44,:).^2))' sqrt(sum(plm_rect(45:65,:).^2))'];
            end
        end
        % =================================================================
        function KERNEL = StandardModel_MLfit_RotInvs(RotInvs,mask,sigma,bval,beta,TE,prior,Nlevels,sigma_norm_limits,flag_compartments,MergeDistance,RotInv_Lmax,Lmax,D_FW,Degree_Kernel)
            % KERNEL = StandardModel_MLfit_RotInvs(RotInvs,mask,sigma,bval,beta,TE,prior,Nlevels,sigma_norm_limits,flag_compartments,MergeDistance,RotInv_Lmax,Lmax,D_FW,Degree_Kernel)
            %
            % Output: KERNEL=[f_ML_fit Da_ML_fit Depar_ML_fit Deperp_ML_fit f_FW_ML_fit T2a_ML_fit T2e_ML_fit p2_ML_fit p4_ML_fit p6_ML_fit];
            if ~exist('Degree_Kernel', 'var') || isempty(Degree_Kernel)
                Degree_Kernel=3;
            end            

            sz_RotInvs=size(RotInvs);
            if length(sz_RotInvs)==4
                flag_4D=1;
            elseif length(sz_RotInvs)==2
                flag_4D=0;
            else
                error('RotInvs must be a 2D or 4D array')
            end

            if flag_4D
                if isempty(mask)
                    mask=true(sz_RotInvs(1:3));
                end
                RotInvsNormalized = SMI.vectorize(RotInvs,mask);
                SigmaNormalized = SMI.vectorize(sigma,mask);
            else
                RotInvsNormalized = RotInvs;
                SigmaNormalized = sigma;
            end

            s0_lowest_TE=RotInvsNormalized(1,:);

            RotInvsNormalized=(RotInvsNormalized./s0_lowest_TE)';
            SigmaNormalized=(SigmaNormalized./s0_lowest_TE)';

            % Normalize RotInvs and sigma with the b0 of the lowest TE
            if ~isvector(bval)
                error('bval should be a vector')
            else
                bval=bval(:)';
            end
            if isempty(beta)
                beta=zeros(size(bval))+1;
            else
                beta=beta(:)';
            end
            if isempty(TE)
                TE=zeros(size(bval));
            else
                TE=TE(:)';
            end

            if length(unique(TE))==1
                do_not_fit_T2=1;
            else
                do_not_fit_T2=0;
            end

            [table_4D,~,~] = SMI.Group_dwi_in_shells_b_beta_TE(bval,beta,TE,MergeDistance);
            Ndirs=table_4D(3,:);

            keep_non_zero_S0=true(1,size(table_4D,2));
            if RotInv_Lmax==0
                keep_RotInvs_kernel=keep_non_zero_S0;
            elseif RotInv_Lmax==2
                keep_non_zero_S2=(abs(table_4D(2,:))>0.2)&(table_4D(1,:)>0.1)&(Lmax>=2);  %remove |beta| <= 0.2 and b <= 0.1
                keep_RotInvs_kernel=[keep_non_zero_S0 keep_non_zero_S2];
            elseif RotInv_Lmax==4
                keep_non_zero_S2=(abs(table_4D(2,:))>0.2)&(table_4D(1,:)>0.1)&(Lmax>=2);  %remove |beta| <= 0.2 and b <= 0.1
                keep_non_zero_S4=(abs(table_4D(2,:))>0.2)&(table_4D(1,:)>0.1)&(Lmax>=4);  %remove |beta| <= 0.2 and b <= 0.1
                keep_RotInvs_kernel=[keep_non_zero_S0 keep_non_zero_S2 keep_non_zero_S4];
            elseif RotInv_Lmax==6
                keep_non_zero_S2=(abs(table_4D(2,:))>0.2)&(table_4D(1,:)>0.1)&(Lmax>=2);  %remove |beta| <= 0.2 and b <= 0.1
                keep_non_zero_S4=(abs(table_4D(2,:))>0.2)&(table_4D(1,:)>0.1)&(Lmax>=4);  %remove |beta| <= 0.2 and b <= 0.1
                keep_non_zero_S6=(abs(table_4D(2,:))>0.2)&(table_4D(1,:)>0.1)&(Lmax>=6);  %remove |beta| <= 0.2 and b <= 0.1
                keep_RotInvs_kernel=[keep_non_zero_S0 keep_non_zero_S2 keep_non_zero_S4 keep_non_zero_S6];
            end

            sigma_noise_norm_levels_edges=linspace(sigma_norm_limits(1),sigma_norm_limits(2),Nlevels+1);
            sigma_noise_norm_levels_ids = discretize(SigmaNormalized,sigma_noise_norm_levels_edges);
            sigma_noise_norm_levels_ids(SigmaNormalized<sigma_noise_norm_levels_edges(1))=1;
            sigma_noise_norm_levels_ids(SigmaNormalized>sigma_noise_norm_levels_edges(end))=Nlevels;
            sigma_noise_norm_levels_mean=1/2*(sigma_noise_norm_levels_edges(2:end)+sigma_noise_norm_levels_edges(1:end-1));

            X_fit_norm = SMI.Compute_extended_moments(RotInvsNormalized(:,keep_RotInvs_kernel),Degree_Kernel);

            NvoxelsMasked=size(RotInvsNormalized,1);
            f_ML_fit=zeros(NvoxelsMasked,1);
            Da_ML_fit=zeros(NvoxelsMasked,1);
            Depar_ML_fit=zeros(NvoxelsMasked,1);
            Deperp_ML_fit=zeros(NvoxelsMasked,1);
            f_FW_ML_fit=zeros(NvoxelsMasked,1);
            T2a_ML_fit=zeros(NvoxelsMasked,1);
            T2e_ML_fit=zeros(NvoxelsMasked,1);
            
            f=prior(:,1);
            Da=prior(:,2);
            Depar=prior(:,3);
            Deperp=prior(:,4);
            f_FW=prior(:,5);
            T2a=prior(:,6);
            T2e=prior(:,7);

            
            Ntraining=length(f);
            if ~flag_compartments(1)
                f=zeros(Ntraining,1);
            end
            if ~flag_compartments(3)
                f_FW=zeros(Ntraining,1);
            end
            if ~flag_compartments(2)
                f=1-f_FW;
            end            
                        
            if RotInv_Lmax==0
                p2=prior(:,8);
                RotInvs_train = SMI.Generate_SM_wFW_b_beta_TE_ws0_training_data(bval,beta,TE,[0*f+1,f,Da,Depar,Deperp,f_FW,T2a,T2e,p2],MergeDistance,D_FW);
                p2_ML_fit=zeros(NvoxelsMasked,1);  
                sigma_Ndirs_factor=sqrt(Ndirs);
                RotInvs_train = RotInvs_train(:,1:end/2);
            end
            if RotInv_Lmax==2
                p2=prior(:,8);
                RotInvs_train = SMI.Generate_SM_wFW_b_beta_TE_ws0_training_data(bval,beta,TE,[0*f+1,f,Da,Depar,Deperp,f_FW,T2a,T2e,p2],MergeDistance,D_FW);
                p2_ML_fit=zeros(NvoxelsMasked,1);  
                sigma_Ndirs_factor=sqrt([Ndirs Ndirs*5]);
            end
            if RotInv_Lmax==4
                p2=prior(:,8);
                p4=prior(:,9);
                RotInvs_train = SMI.Generate_SM_wFW_b_beta_TE_ws0_training_data(bval,beta,TE,[0*f+1,f,Da,Depar,Deperp,f_FW,T2a,T2e,p2,p4],MergeDistance,D_FW);
                p2_ML_fit=zeros(NvoxelsMasked,1);            
                p4_ML_fit=zeros(NvoxelsMasked,1);            
                sigma_Ndirs_factor=sqrt([Ndirs Ndirs*5 Ndirs*9]);
            end
            if RotInv_Lmax==6
                p2=prior(:,8);
                p4=prior(:,9);
                p6=prior(:,10);
                RotInvs_train = SMI.Generate_SM_wFW_b_beta_TE_ws0_training_data(bval,beta,TE,[0*f+1,f,Da,Depar,Deperp,f_FW,T2a,T2e,p2,p4,p6],MergeDistance,D_FW);
                p2_ML_fit=zeros(NvoxelsMasked,1);            
                p4_ML_fit=zeros(NvoxelsMasked,1);       
                p6_ML_fit=zeros(NvoxelsMasked,1);      
                sigma_Ndirs_factor=sqrt([Ndirs Ndirs*5 Ndirs*9 Ndirs*13]);
            end
            RotInvs_train_norm=RotInvs_train./RotInvs_train(:,1);


            for ii=1:Nlevels
                flag_current_noise_level=sigma_noise_norm_levels_ids==ii;
                if ~any(flag_current_noise_level)
                    % fprintf('no need for PR training on noise level %d/%d\n',ii,Nlevels)
                    continue
                end
                % Add noise to training data
                % constant sigma for noise level range
                sigma_RotInvs_training=sigma_noise_norm_levels_mean(ii)./sigma_Ndirs_factor;
                meas_RotInvs_train=RotInvs_train_norm+sigma_RotInvs_training.*randn(size(RotInvs_train_norm));
                % =========================================================================
                % Performing PR on RotInvs to estimate the kernel
                X_train = SMI.Compute_extended_moments(meas_RotInvs_train(:,keep_RotInvs_kernel),Degree_Kernel);
                Pinv_X=pinv(X_train);
                coeffs_f=Pinv_X*f;
                coeffs_Da=Pinv_X*Da;
                coeffs_Depar=Pinv_X*Depar;
                coeffs_Deperp=Pinv_X*Deperp;
                coeffs_f_FW=Pinv_X*f_FW;
                coeffs_T2a=Pinv_X*T2a;
                coeffs_T2e=Pinv_X*T2e;
                
                f_ML_fit(flag_current_noise_level)=X_fit_norm(flag_current_noise_level,:)*coeffs_f;
                Da_ML_fit(flag_current_noise_level)=X_fit_norm(flag_current_noise_level,:)*coeffs_Da;
                Depar_ML_fit(flag_current_noise_level)=X_fit_norm(flag_current_noise_level,:)*coeffs_Depar;
                Deperp_ML_fit(flag_current_noise_level)=X_fit_norm(flag_current_noise_level,:)*coeffs_Deperp;
                f_FW_ML_fit(flag_current_noise_level)=X_fit_norm(flag_current_noise_level,:)*coeffs_f_FW;
                T2a_ML_fit(flag_current_noise_level)=X_fit_norm(flag_current_noise_level,:)*coeffs_T2a;
                T2e_ML_fit(flag_current_noise_level)=X_fit_norm(flag_current_noise_level,:)*coeffs_T2e;
                
                if RotInv_Lmax>=2
                    coeffs_p2=Pinv_X*p2;
                    p2_ML_fit(flag_current_noise_level)=X_fit_norm(flag_current_noise_level,:)*coeffs_p2;
                end
                if RotInv_Lmax>=4
                    coeffs_p4=Pinv_X*p4;
                    p4_ML_fit(flag_current_noise_level)=X_fit_norm(flag_current_noise_level,:)*coeffs_p4;
                end
                if RotInv_Lmax>=6
                    coeffs_p6=Pinv_X*p6;
                    p6_ML_fit(flag_current_noise_level)=X_fit_norm(flag_current_noise_level,:)*coeffs_p6;
                end

                % fprintf('Done PR training on noise level %d/%d\n',ii,Nlevels)
            end
            f_ML_fit(f_ML_fit<0)=0;f_ML_fit(f_ML_fit>1)=1;
            Da_ML_fit(Da_ML_fit<0)=0;Da_ML_fit(Da_ML_fit>3)=3;
            Depar_ML_fit(Depar_ML_fit<0)=0;Depar_ML_fit(Depar_ML_fit>3)=3;
            Deperp_ML_fit(Deperp_ML_fit<0)=0;Deperp_ML_fit(Deperp_ML_fit>3)=3;
            f_FW_ML_fit(f_FW_ML_fit<0)=0;f_FW_ML_fit(f_FW_ML_fit>1)=1;
            T2a_ML_fit(T2a_ML_fit<30)=30;T2a_ML_fit(T2a_ML_fit>150)=150;
            T2e_ML_fit(T2e_ML_fit<30)=30;T2e_ML_fit(T2e_ML_fit>150)=150;
            if RotInv_Lmax==0
                p2_ML_fit=[];
                p4_ML_fit=[];
                p6_ML_fit=[];
            end
            if RotInv_Lmax==2
                p2_ML_fit(p2_ML_fit<0)=0;p2_ML_fit(p2_ML_fit>1)=1;
                p4_ML_fit=[];
                p6_ML_fit=[];
            end
            if RotInv_Lmax==4
                p2_ML_fit(p2_ML_fit<0)=0;p2_ML_fit(p2_ML_fit>1)=1;
                p4_ML_fit(p4_ML_fit<0)=0;p4_ML_fit(p4_ML_fit>1)=1; 
                p6_ML_fit=[];
            end
            if RotInv_Lmax==6
                p2_ML_fit(p2_ML_fit<0)=0;p2_ML_fit(p2_ML_fit>1)=1;
                p4_ML_fit(p4_ML_fit<0)=0;p4_ML_fit(p4_ML_fit>1)=1; 
                p6_ML_fit(p6_ML_fit<0)=0;p6_ML_fit(p6_ML_fit>1)=1;
            end
            if do_not_fit_T2
                KERNEL=[f_ML_fit Da_ML_fit Depar_ML_fit Deperp_ML_fit f_FW_ML_fit p2_ML_fit p4_ML_fit p6_ML_fit]';
            else
                KERNEL=[f_ML_fit Da_ML_fit Depar_ML_fit Deperp_ML_fit f_FW_ML_fit T2a_ML_fit T2e_ML_fit p2_ML_fit p4_ML_fit p6_ML_fit]';
            end
            if flag_4D
                KERNEL=SMI.vectorize(KERNEL,mask);
            end
        end        
        % =================================================================
        function RotInvs = Generate_SM_wFW_b_beta_TE_ws0_training_data(b,beta,TE,kernel_params,MergeDistance,D_FW)
            % RotInvs = Generate_SM_wFW_b_beta_TE_ws0_training_data(b,beta,TE,kernel_params,MergeDistance,D_FW)
            %
            % kernel_params = [s0 f Da Depar Deperp f_FW T2a T2e p2 p4 p6]
            %
            % By: Santiago Coelho
            s0=kernel_params(:,1);
            f=kernel_params(:,2);
            Da=kernel_params(:,3);
            Depar=kernel_params(:,4);
            Deperp=kernel_params(:,5);
            f_FW=kernel_params(:,6);
            f_extra=1-f-f_FW;
            T2a=kernel_params(:,7);
            T2e=kernel_params(:,8);
            [table_4D,~,~] = SMI.Group_dwi_in_shells_b_beta_TE(b,beta,TE,MergeDistance);                
            % NShells=size(table_4D,2);
            
            K0_all = SMI.RotInv_Kell_wFW_b_beta_TE_numerical(0,table_4D(1,:),table_4D(2,:),table_4D(4,:),[f, Da, Depar, Deperp, f_FW, T2a, T2e],D_FW);
            if size(kernel_params,2)==9
                p2=kernel_params(:,9);
                K2_all = SMI.RotInv_Kell_wFW_b_beta_TE_numerical(2,table_4D(1,:),table_4D(2,:),table_4D(4,:),[f, Da, Depar, Deperp, f_FW, T2a, T2e],D_FW);
                RotInvs=s0.*[K0_all p2.*abs(K2_all)];
            elseif size(kernel_params,2)==10
                p2=kernel_params(:,9);
                K2_all = SMI.RotInv_Kell_wFW_b_beta_TE_numerical(2,table_4D(1,:),table_4D(2,:),table_4D(4,:),[f, Da, Depar, Deperp, f_FW, T2a, T2e],D_FW);
                p4=kernel_params(:,10);
                K4_all = SMI.RotInv_Kell_wFW_b_beta_TE_numerical(4,table_4D(1,:),table_4D(2,:),table_4D(4,:),[f, Da, Depar, Deperp, f_FW, T2a, T2e],D_FW);
                RotInvs=s0.*[K0_all p2.*abs(K2_all) p4.*abs(K4_all)];
            elseif size(kernel_params,2)==11
                p2=kernel_params(:,9);
                K2_all = SMI.RotInv_Kell_wFW_b_beta_TE_numerical(2,table_4D(1,:),table_4D(2,:),table_4D(4,:),[f, Da, Depar, Deperp, f_FW, T2a, T2e],D_FW);
                p4=kernel_params(:,10);
                K4_all = SMI.RotInv_Kell_wFW_b_beta_TE_numerical(4,table_4D(1,:),table_4D(2,:),table_4D(4,:),[f, Da, Depar, Deperp, f_FW, T2a, T2e],D_FW);
                p6=kernel_params(:,11);
                K6_all = SMI.RotInv_Kell_wFW_b_beta_TE_numerical(6,table_4D(1,:),table_4D(2,:),table_4D(4,:),[f, Da, Depar, Deperp, f_FW, T2a, T2e],D_FW);
                RotInvs=s0.*[K0_all p2.*abs(K2_all) p4.*abs(K4_all) p6.*abs(K6_all)];
            end
        end
        % =================================================================
        function [Kell] = RotInv_Kell_wFW_b_beta_TE_numerical(ell,b,beta,TE,kernel,D_FW)
            %
            % [Kell] = RotInv_Kell_wFW_b_beta_TE_numerical(ell,b,beta,TE,kernel,D_FW)
            %
            % kernel must have: [f,Da,Depar,Deperp,f_w,T2a,T2e]
            % with size: [Nvoxels x 5] or [Nvoxels x 7]
            %
            % This function evaluates the SM kernel projected on the ell-th order
            % Legendre polynomial. The integral is computed numerically
            % using Gaussian quadrature with 200 points (more than enough accuracy).
            %
            % b is the b-value of the shell in um/ms2 kernel=[f,Da,Depar,Deperp,f_extra];
            %
            % By: Santiago Coelho
            Nvoxels=size(kernel,1);
            if ~isscalar(ell)
                error('Only scalar L values are supported')
            end
            if isvector(b)
                b=reshape(b,1,[]);
            else
                error('b must be a vector')
            end
            if isvector(beta)
                beta=reshape(beta,1,[]);
            else
                error('beta must be a vector')
            end
            if isvector(TE)
                TE=reshape(TE,1,[]);
            else
                error('TE must be a vector')
            end
            if ~exist('D_FW', 'var') || isempty(D_FW)
                D_FW = 3;
            end
            f=kernel(:,1);
            Da=kernel(:,2);
            Depar=kernel(:,3);
            Deperp=kernel(:,4);
            Deltae=Depar-Deperp;
            fw=kernel(:,5);
            f_extra=1-f-fw;
            if size(kernel,2)==7
                T2a=kernel(:,6);
                T2e=kernel(:,7);
            else
                T2a=1000*ones(size(f));
                T2e=1000*ones(size(f));
            end
            T2_FW=1500;
            % I=w*fun(x)
            % n=100;
            % xx=[0.000143136613279400;0.000754024680202100;0.00185243263343740;0.00343753148127830;0.00550780237850410;0.00806122964697150;0.0110953207565408;0.0146071121181468;0.0185931728720922;0.0230496085372541;0.0279720649318720;0.0333557324784602;0.0391953509273330;0.0454852145087352;0.0522191775146365;0.0593906603074908;0.0669926557514177;0.0750177360602043;0.0834580600557996;0.0923053808304118;0.101551053804843;0.111186045175252;0.121200940740146;0.131585955098990;0.142330941213472;0.153425400322099;0.164858492198430;0.176619045742935;0.188695569898146;0.201076264876411;0.213749033689310;0.226701493967453;0.239920990059119;0.253394605395905;0.267109175113321;0.281051298913984;0.295207354160849;0.309563509187685;0.324105736813789;0.338819828049735;0.353691405980764;0.368705939814248;0.383848759077513;0.399105067952132;0.414459959730698;0.429898431381943;0.445405398209969;0.460965708593282;0.476564158789204;0.492185507789229;0.507814492210772;0.523435841210796;0.539034291406718;0.554594601790031;0.570101568618057;0.585540040269302;0.600894932047868;0.616151240922487;0.631294060185752;0.646308594019236;0.661180171950265;0.675894263186211;0.690436490812315;0.704792645839151;0.718948701086016;0.732890824886679;0.746605394604095;0.760079009940881;0.773298506032547;0.786250966310691;0.798923735123589;0.811304430101854;0.823380954257065;0.835141507801571;0.846574599677901;0.857669058786528;0.868414044901010;0.878799059259854;0.888813954824748;0.898448946195157;0.907694619169588;0.916541939944200;0.924982263939796;0.933007344248582;0.940609339692509;0.947780822485363;0.954514785491265;0.960804649072667;0.966644267521540;0.972027935068128;0.976950391462746;0.981406827127908;0.985392887881853;0.988904679243459;0.991938770353029;0.994492197621496;0.996562468518722;0.998147567366563;0.999245975319798;0.999856863386721];
            % w=[0.000367317245252900,0.000854696326759100,0.00134196268577670,0.00182798060066320,0.00231222503171110,0.00279421400193280,0.00327347422542270,0.00374953662773240,0.00422193573483450,0.00469020982684720,0.00515390128743450,0.00561255701159300,0.00606572883148980,0.00651297394648580,0.00695385535185940,0.00738794226372060,0.00781481053877300,0.00823404308807260,0.00864523028416180,0.00904797036106400,0.00944186980668740,0.00982654374721760,0.0102016163231047,0.0105667210562638,0.0109215012081237,0.0112656101281681,0.0115987115926271,0.0119204801329841,0.0122306013539785,0.0125287722407898,0.0128147014551040,0.0130881096197728,0.0133487295917855,0.0135963067232884,0.0138305991103962,0.0140513778295506,0.0142584271611975,0.0144515448005626,0.0146305420553191,0.0147952440299563,0.0149454897966664,0.0150811325525846,0.0152020397632274,0.0153080932919902,0.0153991895155763,0.0154752394252455,0.0155361687137833,0.0155819178481049,0.0156124421274247,0.0156277117269317,0.0156277117269317,0.0156124421274247,0.0155819178481049,0.0155361687137833,0.0154752394252455,0.0153991895155763,0.0153080932919902,0.0152020397632274,0.0150811325525846,0.0149454897966664,0.0147952440299563,0.0146305420553191,0.0144515448005626,0.0142584271611975,0.0140513778295506,0.0138305991103962,0.0135963067232884,0.0133487295917855,0.0130881096197728,0.0128147014551040,0.0125287722407898,0.0122306013539785,0.0119204801329841,0.0115987115926271,0.0112656101281681,0.0109215012081237,0.0105667210562638,0.0102016163231047,0.00982654374721760,0.00944186980668740,0.00904797036106400,0.00864523028416180,0.00823404308807260,0.00781481053877300,0.00738794226372060,0.00695385535185940,0.00651297394648580,0.00606572883148980,0.00561255701159300,0.00515390128743450,0.00469020982684720,0.00422193573483450,0.00374953662773240,0.00327347422542270,0.00279421400193280,0.00231222503171110,0.00182798060066320,0.00134196268577670,0.000854696326759100,0.000367317245252900];
            n=200;
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            % 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.0031316386894657,0.0031302795813513,0.0031287971419441,0.0031271914296513,0.0031254625077371,0.0031236104443201,0.0031216353123706,0.0031195371897076,0.0031173161589961,0.0031149723077434,0.0031125057282960,0.0031099165178358,0.0031072047783762,0.0031043706167582,0.0031014141446464,0.0030983354785240,0.0030951347396888,0.0030918120542481,0.0030883675531140,0.0030848013719979,0.0030811136514053,0.0030773045366304,0.0030733741777503,0.0030693227296189,0.0030651503518610,0.0030608572088661,0.0030564434697815,0.0030519093085063,0.0030472549036838,0.0030424804386951,0.0030375861016514,0.0030325720853869,0.0030274385874511,0.0030221858101010,0.0030168139602932,0.0030113232496755,0.0030057138945789,0.0029999861160090,0.0029941401396371,0.0029881761957913,0.0029820945194479,0.0029758953502215,0.0029695789323560,0.0029631455147148,0.0029565953507709,0.0029499286985973,0.0029431458208563,0.0029362469847896,0.0029292324622077,0.0029221025294787,0.0029148574675181,0.0029074975617773,0.0029000231022327,0.0028924343833738,0.0028847317041919,0.0028769153681684,0.0028689856832626,0.0028609429618997,0.0028527875209585,0.0028445196817590,0.0028361397700493,0.0028276481159935,0.0028190450541580,0.0028103309234987,0.0028015060673477,0.0027925708333992,0.0027835255736968,0.0027743706446185,0.0027651064068636,0.0027557332254380,0.0027462514696398,0.0027366615130450,0.0027269637334926,0.0027171585130698,0.0027072462380969,0.0026972272991120,0.0026871020908559,0.0026768710122561,0.0026665344664116,0.0026560928605766,0.0026455466061445,0.0026348961186320,0.0026241418176625,0.0026132841269495,0.0026023234742800,0.0025912602914977,0.0025800950144859,0.0025688280831504,0.0025574599414022,0.0025459910371396,0.0025344218222313,0.0025227527524979,0.0025109842876944,0.0024991168914917,0.0024871510314589,0.0024750871790441,0.0024629258095566,0.0024506674021475,0.0024383124397915,0.0024258614092671,0.0024133148011383,0.0024006731097343,0.0023879368331310,0.0023751064731305,0.0023621825352421,0.0023491655286617,0.0023360559662523,0.0023228543645234,0.0023095612436109,0.0022961771272563,0.0022827025427866,0.0022691380210929,0.0022554840966099,0.0022417413072947,0.0022279101946057,0.0022139913034810,0.0021999851823175,0.0021858923829484,0.0021717134606224,0.0021574489739812,0.0021430994850377,0.0021286655591538,0.0021141477650184,0.0020995466746245,0.0020848628632471,0.0020700969094202,0.0020552493949143,0.0020403209047133,0.0020253120269914,0.0020102233530903,0.0019950554774954,0.0019798089978125,0.0019644845147447,0.0019490826320681,0.0019336039566085,0.0019180490982170,0.0019024186697467,0.0018867132870277,0.0018709335688436,0.0018550801369064,0.0018391536158328,0.0018231546331189,0.0018070838191157,0.0017909418070046,0.0017747292327719,0.0017584467351842,0.0017420949557628,0.0017256745387589,0.0017091861311279,0.0016926303825040,0.0016760079451746,0.0016593194740545,0.0016425656266603,0.0016257470630845,0.0016088644459691,0.0015919184404800,0.0015749097142804,0.0015578389375049,0.0015407067827327,0.0015235139249612,0.0015062610415796,0.0014889488123419,0.0014715779193406,0.0014541490469792,0.0014366628819457,0.0014191201131855,0.0014015214318739,0.0013838675313893,0.0013661591072857,0.0013483968572653,0.0013305814811510,0.0013127136808587,0.0012947941603701,0.0012768236257043,0.0012588027848907,0.0012407323479403,0.0012226130268185,0.0012044455354166,0.0011862305895239,0.0011679689067991,0.0011496612067425,0.0011313082106676,0.0011129106416722,0.0010944692246106,0.0010759846860644,0.0010574577543143,0.0010388891593115,0.0010202796326482,0.0010016299075298,0.0009829407187452,0.00096421280263830,0.0009454468970787,0.00092664374143300,0.00090780407653520,0.00088892864465800,0.00087001818948320,0.00085107345607250,0.00083209519083810,0.00081308414151370,0.00079404105712440,0.00077496668795750,0.00075586178553320,0.00073672710257460,0.00071756339297820,0.00069837141178430,0.00067915191514730,0.00065990566030550,0.000640633405552,0.0006213359102042,0.0006020139345746,0.0005826682399402,0.000563299588513,0.0005439087434101,0.00052449646862360,0.00050506352899080,0.0004856106901643,0.0004661387185821,0.0004466483814379,0.0004271404466515,0.00040761568283950,0.00038807485928630,0.00036851874591540,0.00034894811326190,0.00032936373244640,0.00030976637515070,0.0002901568135969,0.0002705358205337,0.0002509041692319,0.0002312626335004,0.0002116119877392,0.0001919530070667,0.0001722864676061,0.0001526131471336,0.0001329338266317,0.0001132492943548,9.35603579253000e-05,7.38678873878e-05,5.41730143455000e-05,3.44785364134000e-05,1.48118222427000e-05];
            X=repmat(xx,1,length(b));
            BVAL=repmat(b,n,1);
            BSHAPE=repmat(beta,n,1);
            TE_rep=repmat(TE,n,1);
            if ell==0
                Pell_xi=0*X+1; 
            elseif ell==2
                Pell_xi=(3/2*X.^2-1/2);
            elseif ell==4
                Pell_xi=(35*X.^4-30*X.^2+3)/8;
            elseif ell==6
                Pell_xi=(231*X.^6-315*X.^4+105*X.^2-5)/16;
            elseif ell==8
                Pell_xi=(6435*X.^8-12012*X.^6+6930*X.^4-1260*X.^2+35)/128;
            end
            Kell=zeros(Nvoxels,length(b));
            for ii=1:Nvoxels
                Kell(ii,:)=w*((f(ii).*exp(-BVAL.*BSHAPE.*Da(ii).*X.^2-BVAL.*Da(ii)/3.*(1-BSHAPE)-TE_rep/T2a(ii))+...
                               f_extra(ii).*exp(-BVAL.*BSHAPE.*Deltae(ii).*X.^2-BVAL.*Deltae(ii)/3.*(1-BSHAPE)-BVAL.*Deperp(ii)-TE_rep/T2e(ii))+...
                               fw(ii)*exp(-BVAL*D_FW-TE_rep/T2_FW)).*Pell_xi);
            end  
            abs_beta=abs(beta);
            if ell>0
                Kell(:,b<1e-6)=0;
                Kell(:,abs_beta<1e-6)=0;
            else
                Kell(:,b<1e-6)=f.*exp(-TE(b<1e-6)./T2a)+f_extra.*exp(-TE(b<1e-6)./T2e)+(1-f-f_extra).*exp(-TE(b<1e-6)./T2_FW);

                Kell(:,abs_beta<1e-6)=f.*exp(-TE(abs_beta<1e-6)./T2a).*exp(-b(abs_beta<1e-6).*Da/3)+...
                                      f_extra.*exp(-TE(abs_beta<1e-6)./T2e).*exp(-b(abs_beta<1e-6).*Deltae/3-b(abs_beta<1e-6).*Deperp)+...
                                      fw.*exp(-TE(abs_beta<1e-6)./T2_FW).*exp(-b(abs_beta<1e-6).*D_FW);
            end
        end
        % =================================================================
        function [ signal ] = SM_wFW_b_beta_TE_RealSphHarm_quadInt(f,Da,Depar,Deperp,f_extra,T2a,T2e,plm,b,bvec,beta,TE,NlebPoints,CS_phase,epsilon_rectify,D_FW)
        %
        % [ signal ] = SM_wFW_b_beta_TE_RealSphHarm_quadInt(f,Da,Depar,Deperp,f_extra,T2a,T2e,plm,b,bvec,beta,TE,NlebPoints,CS_phase,epsilon_rectify,D_FW)
        %
        % This function computes the signal from applying a multidimensional diffusion
        % encoding in 3D while the fiber bundle (composed of 2 non-exchanging
        % compartments) is oriented according to a Watson distribution with main
        % axis [phiS,thetaS,1]. Intracelullar compartment is modelled with sticks.
        %
        %
        % This functions assumes the ODF is real, thus, it uses a real basis for
        % the spherical harmonics expansion as defined in 
        % https://cs.dartmouth.edu/~wjarosz/publications/dissertation/appendixB.pdf
        %
        %
        % This signal model considers the anisotropic medium has the same main
        % orientation as the sticks (Axially symmetric DT).
        %
        % Valid for all B-tensors (the input is a matrix of 6xN, where N is the
        % total number of measurements)
        %
        % B=[Bxx;Byy;Bzz;Bxy;Bxz;Byz]; for all columns.
        %
        % It allows for each measurement to have a different b value, and different
        % tensor.
        %
        % [ signal ] = SM_B_inSH_quad_efficient(f,Da,Depar,Deperp,plm,B)
        %
        % Da (intracellular) and Depar, Deperp (extracellular, parallel and
        % perpendicular) in [um^2/ms]. In this units D_{water} = 3
        %
        %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
        % IMPORTANT: Unlike the other tissue models, Da Depar and Deperp here are
        % the LOCAL DIFFUSIVITIES of the kernel.
        %~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
        % 
        % Only valid for expansions of even order in SH: plm with lmax=2,4,6,..
        % plm is thus, a column vector of size [5x1] or [14x1] or [27x1]
        %
        %
        % f: T2-weighted Water Fraction [0,1]
        % B is the non-normalised B-tensor (trace(B)=b)
        % b is in [ms/um^2] (typical values are ~1,2,5)
        %
        % The output is a COLUMN VECTOR [Nx1]
        %
        % author: Santiago Coelho, 27/03/2020
        %
        %==========================================================================

        if ~exist('CS_phase','var') || isempty(CS_phase) || CS_phase
            CS_phase=1;
        end
        if ~exist('epsilon_rectify','var') || isempty(epsilon_rectify) || ~epsilon_rectify
            flag_rectify=0;
        else
            flag_rectify=1;
        end
        if ~exist('D_FW', 'var') || isempty(D_FW)
            D_FW = 3;
        end

        [B,~] = SMI.ConstructAxiallySymmetricBset(b,beta,bvec);
        B(:,b<1e-3)=0; % To avoid NaN due to normalisation of the b-tensor
        Ndirs=length(b);
        %==========================================================================
        % SH expansion of the signal
        % The output is a column vector ~ Quadrature Integral
        % % signal=zeros(Ndirs,1); % Now this is vectorized
        %==========================================================================
        % Quadrature integration
        % leb = getLebedevSphere(Nq);
        % Angular grid choices: 6, 14, 26, 38, 50, 74, 86, 110, exact integration up to order L (3,5,7,9,11,13,15,17)
        % Loading pre-computed directions and weights for speeding up

        Nlm=size(plm,2);
        Lmax=sqrt(2*Nlm+9/4)-3/2;
        if ~any(Lmax==2:2:10)
            error('Lmax should be 2, 4, 6, 8, or 10')
        end

        % This lines rescales back the plm from being between 0,1 to being between
        % 0 and N_l as it is defined for SH
        L_unique=2:2:Lmax;
        L_all=repelem(L_unique,2*L_unique+1);
        N_l=sqrt((2*L_all+1)/(4*pi));
        plm_not_normalized=plm.*N_l;

        Nq=NlebPoints;
        % 266, 302, 350, 434, 590, 770, 974, 1202 clc,clear, L=getLebedevSphere(770);
        if Nq==266
            lebx=[1;-1;0;0;0;0;0;0;0;0;0.707106781186548;0.707106781186548;-0.707106781186548;-0.707106781186548;0.707106781186548;-0.707106781186548;0.707106781186548;-0.707106781186548;0.577350269189626;-0.577350269189626;0.577350269189626;0.577350269189626;-0.577350269189626;0.577350269189626;-0.577350269189626;-0.577350269189626;0.703937339158548;-0.703937339158548;0.703937339158548;0.703937339158548;-0.703937339158548;-0.703937339158548;0.703937339158548;-0.703937339158548;-0.703937339158548;0.703937339158548;0.703937339158548;-0.703937339158548;-0.703937339158548;0.703937339158548;-0.703937339158548;0.703937339158548;0.0945750764037131;-0.0945750764037131;0.0945750764037131;0.0945750764037131;-0.0945750764037131;-0.0945750764037131;0.0945750764037131;-0.0945750764037131;0.101252624857241;-0.101252624857241;0.101252624857241;0.101252624857241;-0.101252624857241;-0.101252624857241;0.101252624857241;-0.101252624857241;-0.101252624857241;0.101252624857241;0.101252624857241;-0.101252624857241;-0.101252624857241;0.101252624857241;-0.101252624857241;0.101252624857241;0.989694807462905;-0.989694807462905;0.989694807462905;0.989694807462905;-0.989694807462905;-0.989694807462905;0.989694807462905;-0.989694807462905;0.464744872642054;-0.464744872642054;0.464744872642054;0.464744872642054;-0.464744872642054;-0.464744872642054;0.464744872642054;-0.464744872642054;-0.464744872642054;0.464744872642054;0.464744872642054;-0.464744872642054;-0.464744872642054;0.464744872642054;-0.464744872642054;0.464744872642054;0.753673939250816;-0.753673939250816;0.753673939250816;0.753673939250816;-0.753673939250816;-0.753673939250816;0.753673939250816;-0.753673939250816;0.327742065497163;-0.327742065497163;0.327742065497163;0.327742065497163;-0.327742065497163;-0.327742065497163;0.327742065497163;-0.327742065497163;-0.327742065497163;0.327742065497163;0.327742065497163;-0.327742065497163;-0.327742065497163;0.327742065497163;-0.327742065497163;0.327742065497163;0.886098344997499;-0.886098344997499;0.886098344997499;0.886098344997499;-0.886098344997499;-0.886098344997499;0.886098344997499;-0.886098344997499;0.662033866369997;-0.662033866369997;0.662033866369997;0.662033866369997;-0.662033866369997;-0.662033866369997;0.662033866369997;-0.662033866369997;-0.662033866369997;0.662033866369997;0.662033866369997;-0.662033866369997;-0.662033866369997;0.662033866369997;-0.662033866369997;0.662033866369997;0.351315128564633;-0.351315128564633;0.351315128564633;0.351315128564633;-0.351315128564633;-0.351315128564633;0.351315128564633;-0.351315128564633;0.850650808352040;-0.850650808352040;0.850650808352040;-0.850650808352040;0.525731112119134;-0.525731112119134;0.525731112119134;-0.525731112119134;0.850650808352040;-0.850650808352040;0.850650808352040;-0.850650808352040;0.525731112119134;-0.525731112119134;0.525731112119134;-0.525731112119134;0;0;0;0;0;0;0;0;0.323348454269290;-0.323348454269290;0.323348454269290;0.323348454269290;-0.323348454269290;0.323348454269290;-0.323348454269290;-0.323348454269290;0.115311201100970;-0.115311201100970;0.115311201100970;0.115311201100970;-0.115311201100970;0.115311201100970;-0.115311201100970;-0.115311201100970;0.939227929749916;-0.939227929749916;0.939227929749916;0.939227929749916;-0.939227929749916;0.939227929749916;-0.939227929749916;-0.939227929749916;0.939227929749916;-0.939227929749916;0.939227929749916;0.939227929749916;-0.939227929749916;0.939227929749916;-0.939227929749916;-0.939227929749916;0.323348454269290;-0.323348454269290;0.323348454269290;0.323348454269290;-0.323348454269290;0.323348454269290;-0.323348454269290;-0.323348454269290;0.115311201100970;-0.115311201100970;0.115311201100970;0.115311201100970;-0.115311201100970;0.115311201100970;-0.115311201100970;-0.115311201100970;0.231479015871260;-0.231479015871260;0.231479015871260;0.231479015871260;-0.231479015871260;0.231479015871260;-0.231479015871260;-0.231479015871260;0.524493924092237;-0.524493924092237;0.524493924092237;0.524493924092237;-0.524493924092237;0.524493924092237;-0.524493924092237;-0.524493924092237;0.819343388819120;-0.819343388819120;0.819343388819120;0.819343388819120;-0.819343388819120;0.819343388819120;-0.819343388819120;-0.819343388819120;0.819343388819120;-0.819343388819120;0.819343388819120;0.819343388819120;-0.819343388819120;0.819343388819120;-0.819343388819120;-0.819343388819120;0.231479015871260;-0.231479015871260;0.231479015871260;0.231479015871260;-0.231479015871260;0.231479015871260;-0.231479015871260;-0.231479015871260;0.524493924092237;-0.524493924092237;0.524493924092237;0.524493924092237;-0.524493924092237;0.524493924092237;-0.524493924092237;-0.524493924092237];
            leby=[0;0;1;-1;0;0;0.707106781186548;-0.707106781186548;0.707106781186548;-0.707106781186548;0;0;0;0;0.707106781186548;0.707106781186548;-0.707106781186548;-0.707106781186548;0.577350269189626;0.577350269189626;-0.577350269189626;0.577350269189626;-0.577350269189626;-0.577350269189626;0.577350269189626;-0.577350269189626;0.703937339158548;0.703937339158548;-0.703937339158548;0.703937339158548;-0.703937339158548;0.703937339158548;-0.703937339158548;-0.703937339158548;0.0945750764037131;-0.0945750764037131;0.0945750764037131;-0.0945750764037131;0.0945750764037131;-0.0945750764037131;-0.0945750764037131;0.0945750764037131;0.703937339158548;0.703937339158548;-0.703937339158548;0.703937339158548;-0.703937339158548;0.703937339158548;-0.703937339158548;-0.703937339158548;0.101252624857241;0.101252624857241;-0.101252624857241;0.101252624857241;-0.101252624857241;0.101252624857241;-0.101252624857241;-0.101252624857241;0.989694807462905;-0.989694807462905;0.989694807462905;-0.989694807462905;0.989694807462905;-0.989694807462905;-0.989694807462905;0.989694807462905;0.101252624857241;0.101252624857241;-0.101252624857241;0.101252624857241;-0.101252624857241;0.101252624857241;-0.101252624857241;-0.101252624857241;0.464744872642054;0.464744872642054;-0.464744872642054;0.464744872642054;-0.464744872642054;0.464744872642054;-0.464744872642054;-0.464744872642054;0.753673939250816;-0.753673939250816;0.753673939250816;-0.753673939250816;0.753673939250816;-0.753673939250816;-0.753673939250816;0.753673939250816;0.464744872642054;0.464744872642054;-0.464744872642054;0.464744872642054;-0.464744872642054;0.464744872642054;-0.464744872642054;-0.464744872642054;0.327742065497163;0.327742065497163;-0.327742065497163;0.327742065497163;-0.327742065497163;0.327742065497163;-0.327742065497163;-0.327742065497163;0.886098344997499;-0.886098344997499;0.886098344997499;-0.886098344997499;0.886098344997499;-0.886098344997499;-0.886098344997499;0.886098344997499;0.327742065497163;0.327742065497163;-0.327742065497163;0.327742065497163;-0.327742065497163;0.327742065497163;-0.327742065497163;-0.327742065497163;0.662033866369997;0.662033866369997;-0.662033866369997;0.662033866369997;-0.662033866369997;0.662033866369997;-0.662033866369997;-0.662033866369997;0.351315128564633;-0.351315128564633;0.351315128564633;-0.351315128564633;0.351315128564633;-0.351315128564633;-0.351315128564633;0.351315128564633;0.662033866369997;0.662033866369997;-0.662033866369997;0.662033866369997;-0.662033866369997;0.662033866369997;-0.662033866369997;-0.662033866369997;0.525731112119134;0.525731112119134;-0.525731112119134;-0.525731112119134;0.850650808352040;0.850650808352040;-0.850650808352040;-0.850650808352040;0;0;0;0;0;0;0;0;0.850650808352040;-0.850650808352040;0.850650808352040;-0.850650808352040;0.525731112119134;-0.525731112119134;0.525731112119134;-0.525731112119134;0.115311201100970;0.115311201100970;-0.115311201100970;0.115311201100970;-0.115311201100970;-0.115311201100970;0.115311201100970;-0.115311201100970;0.323348454269290;0.323348454269290;-0.323348454269290;0.323348454269290;-0.323348454269290;-0.323348454269290;0.323348454269290;-0.323348454269290;0.323348454269290;0.323348454269290;-0.323348454269290;0.323348454269290;-0.323348454269290;-0.323348454269290;0.323348454269290;-0.323348454269290;0.115311201100970;0.115311201100970;-0.115311201100970;0.115311201100970;-0.115311201100970;-0.115311201100970;0.115311201100970;-0.115311201100970;0.939227929749916;0.939227929749916;-0.939227929749916;0.939227929749916;-0.939227929749916;-0.939227929749916;0.939227929749916;-0.939227929749916;0.939227929749916;0.939227929749916;-0.939227929749916;0.939227929749916;-0.939227929749916;-0.939227929749916;0.939227929749916;-0.939227929749916;0.524493924092237;0.524493924092237;-0.524493924092237;0.524493924092237;-0.524493924092237;-0.524493924092237;0.524493924092237;-0.524493924092237;0.231479015871260;0.231479015871260;-0.231479015871260;0.231479015871260;-0.231479015871260;-0.231479015871260;0.231479015871260;-0.231479015871260;0.231479015871260;0.231479015871260;-0.231479015871260;0.231479015871260;-0.231479015871260;-0.231479015871260;0.231479015871260;-0.231479015871260;0.524493924092237;0.524493924092237;-0.524493924092237;0.524493924092237;-0.524493924092237;-0.524493924092237;0.524493924092237;-0.524493924092237;0.819343388819120;0.819343388819120;-0.819343388819120;0.819343388819120;-0.819343388819120;-0.819343388819120;0.819343388819120;-0.819343388819120;0.819343388819120;0.819343388819120;-0.819343388819120;0.819343388819120;-0.819343388819120;-0.819343388819120;0.819343388819120;-0.819343388819120];
            lebz=[0;0;0;0;1;-1;0.707106781186548;0.707106781186548;-0.707106781186548;-0.707106781186548;0.707106781186548;-0.707106781186548;0.707106781186548;-0.707106781186548;0;0;0;0;0.577350269189626;0.577350269189626;0.577350269189626;-0.577350269189626;0.577350269189626;-0.577350269189626;-0.577350269189626;-0.577350269189626;0.0945750764037131;0.0945750764037131;0.0945750764037131;-0.0945750764037131;0.0945750764037131;-0.0945750764037131;-0.0945750764037131;-0.0945750764037131;0.703937339158548;0.703937339158548;-0.703937339158548;0.703937339158548;-0.703937339158548;-0.703937339158548;-0.703937339158548;0.703937339158548;0.703937339158548;0.703937339158548;0.703937339158548;-0.703937339158548;0.703937339158548;-0.703937339158548;-0.703937339158548;-0.703937339158548;0.989694807462905;0.989694807462905;0.989694807462905;-0.989694807462905;0.989694807462905;-0.989694807462905;-0.989694807462905;-0.989694807462905;0.101252624857241;0.101252624857241;-0.101252624857241;0.101252624857241;-0.101252624857241;-0.101252624857241;-0.101252624857241;0.101252624857241;0.101252624857241;0.101252624857241;0.101252624857241;-0.101252624857241;0.101252624857241;-0.101252624857241;-0.101252624857241;-0.101252624857241;0.753673939250816;0.753673939250816;0.753673939250816;-0.753673939250816;0.753673939250816;-0.753673939250816;-0.753673939250816;-0.753673939250816;0.464744872642054;0.464744872642054;-0.464744872642054;0.464744872642054;-0.464744872642054;-0.464744872642054;-0.464744872642054;0.464744872642054;0.464744872642054;0.464744872642054;0.464744872642054;-0.464744872642054;0.464744872642054;-0.464744872642054;-0.464744872642054;-0.464744872642054;0.886098344997499;0.886098344997499;0.886098344997499;-0.886098344997499;0.886098344997499;-0.886098344997499;-0.886098344997499;-0.886098344997499;0.327742065497163;0.327742065497163;-0.327742065497163;0.327742065497163;-0.327742065497163;-0.327742065497163;-0.327742065497163;0.327742065497163;0.327742065497163;0.327742065497163;0.327742065497163;-0.327742065497163;0.327742065497163;-0.327742065497163;-0.327742065497163;-0.327742065497163;0.351315128564633;0.351315128564633;0.351315128564633;-0.351315128564633;0.351315128564633;-0.351315128564633;-0.351315128564633;-0.351315128564633;0.662033866369997;0.662033866369997;-0.662033866369997;0.662033866369997;-0.662033866369997;-0.662033866369997;-0.662033866369997;0.662033866369997;0.662033866369997;0.662033866369997;0.662033866369997;-0.662033866369997;0.662033866369997;-0.662033866369997;-0.662033866369997;-0.662033866369997;0;0;0;0;0;0;0;0;0.525731112119134;0.525731112119134;-0.525731112119134;-0.525731112119134;0.850650808352040;0.850650808352040;-0.850650808352040;-0.850650808352040;0.525731112119134;0.525731112119134;-0.525731112119134;-0.525731112119134;0.850650808352040;0.850650808352040;-0.850650808352040;-0.850650808352040;0.939227929749916;0.939227929749916;0.939227929749916;-0.939227929749916;0.939227929749916;-0.939227929749916;-0.939227929749916;-0.939227929749916;0.939227929749916;0.939227929749916;0.939227929749916;-0.939227929749916;0.939227929749916;-0.939227929749916;-0.939227929749916;-0.939227929749916;0.115311201100970;0.115311201100970;0.115311201100970;-0.115311201100970;0.115311201100970;-0.115311201100970;-0.115311201100970;-0.115311201100970;0.323348454269290;0.323348454269290;0.323348454269290;-0.323348454269290;0.323348454269290;-0.323348454269290;-0.323348454269290;-0.323348454269290;0.115311201100970;0.115311201100970;0.115311201100970;-0.115311201100970;0.115311201100970;-0.115311201100970;-0.115311201100970;-0.115311201100970;0.323348454269290;0.323348454269290;0.323348454269290;-0.323348454269290;0.323348454269290;-0.323348454269290;-0.323348454269290;-0.323348454269290;0.819343388819120;0.819343388819120;0.819343388819120;-0.819343388819120;0.819343388819120;-0.819343388819120;-0.819343388819120;-0.819343388819120;0.819343388819120;0.819343388819120;0.819343388819120;-0.819343388819120;0.819343388819120;-0.819343388819120;-0.819343388819120;-0.819343388819120;0.524493924092237;0.524493924092237;0.524493924092237;-0.524493924092237;0.524493924092237;-0.524493924092237;-0.524493924092237;-0.524493924092237;0.231479015871260;0.231479015871260;0.231479015871260;-0.231479015871260;0.231479015871260;-0.231479015871260;-0.231479015871260;-0.231479015871260;0.524493924092237;0.524493924092237;0.524493924092237;-0.524493924092237;0.524493924092237;-0.524493924092237;-0.524493924092237;-0.524493924092237;0.231479015871260;0.231479015871260;0.231479015871260;-0.231479015871260;0.231479015871260;-0.231479015871260;-0.231479015871260;-0.231479015871260];
            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        elseif Nq==350
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        elseif Nq==4334
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        else
            error('Precomputed options for NlebPoints are 266, 350, 434, 590, 770, 1202, or 4334')
        end

        n_quad_1x6=[lebx.*lebx,leby.*leby,lebz.*lebz,2*lebx.*leby,2*lebx.*lebz,2*leby.*lebz];
        %==========================================================================
        % Adding p00 and enforcing that the p_{l-m} are linked to p_{lm} because
        % the ODF is real.
        % Evaluating the SH in Lebedev directions

        % Computing Spherical Harmonics
        dirs_Leb=[lebx,leby,lebz];
        Ylm_n = SMI.get_even_SH(dirs_Leb,Lmax,CS_phase);


        T2_FW=1500;
        M=length(f);

        Pn_Quad_SH_ALL_ODF=zeros(Nq,M);
        
        if flag_rectify
            for ii=1:M
                Pn_Quad_SH_ALL_ODF(:,ii)=1/2*(abs(Ylm_n*[1/sqrt(4*pi);plm_not_normalized(ii,:)']-epsilon_rectify(ii))+Ylm_n*[1/sqrt(4*pi);plm_not_normalized(ii,:)']-epsilon_rectify(ii));
            end
        else
            for ii=1:M
                Pn_Quad_SH_ALL_ODF(:,ii)=Ylm_n*[1/sqrt(4*pi);plm_not_normalized(ii,:)'];
            end
        end
        signal=zeros(M,Ndirs);
        Bij_ninj_all=zeros(Nq,Ndirs);
        for ii=1:Ndirs
            % psi_squared is B:(n x n) for all n's in n_quad.
            Bij_ninj_all(:,ii)=dot(n_quad_1x6,repmat(B(:,ii)',Nq,1),2);
        end
        b_matrix=repmat(b,Nq,1);
        TE_matrix=repmat(TE,Nq,1);
        LebW=lebw';
        parfor jj=1:M
        %     Current_Pn_Quad_SH=Pn_Quad_SH_ALL_ODF(:,jj);
            Current_Pn_Quad_SH=repmat(Pn_Quad_SH_ALL_ODF(:,jj),1,Ndirs);
            Quad_Kernel=f(jj)*exp(-Da(jj).*Bij_ninj_all-TE_matrix/T2a(jj))+f_extra(jj)*exp(-b_matrix*Deperp(jj)-(Depar(jj)-Deperp(jj)).*Bij_ninj_all-TE_matrix/T2e(jj))+(1-f(jj)-f_extra(jj))*exp(-b_matrix*D_FW-TE_matrix/T2_FW);
        %     S2=repmat(Current_Pn_Quad_SH,1,Ndirs).*Quad_Kernel;
            S2=Current_Pn_Quad_SH.*Quad_Kernel;
            signal(jj,:)=LebW*S2;% Here I perform the integral of the ODF and the signal
        end
        signal=signal';

        end
        % =================================================================
        function Moments_Extended = Compute_extended_moments(Moments,degree)
            % Moments_Extended = Compute_extended_moments(Moments,degree)
            %
            % This function computes the extended matrix of moments for polynomial
            % interpolation in multiple dimensions.
            %
            % Moments is [NxD] where N is the number of samples and D the dimension of
            % the problem (the number of different moments)
            %
            % e.g. in 1D this extended matrix equals the Vandermonde matrix
            % Moments_Extended = [1 X X^2 ... X^degree]
            %
            %
            % By: Santiago Coelho
            N=size(Moments,1);
            N_moments=size(Moments,2);
            % Preparing data points for polynomial fitting
            x_0=ones(N,1);
            if degree>=1
                x_1=Moments;
                X=[x_0 x_1];
            else
                X=x_0;
            end
            if degree>=2
                for current_degree=2:degree
                    combinations=nchoosek(1:(N_moments+current_degree-1),current_degree);
                    NtotalCombs=size(combinations,1);
                    flags=zeros(NtotalCombs,N_moments+current_degree-1);
                    sums=zeros(NtotalCombs,N_moments+current_degree-1);
                    which=zeros(NtotalCombs,current_degree);
                    for ii=1:NtotalCombs
                        flags(ii,combinations(ii,:))=1;
                        sums(ii,:)=cumsum(~flags(ii,:))+1;
                        which(ii,:)=sums(ii,combinations(ii,:));
                    end
                    Currentx=ones(N,NtotalCombs);
                    for ii=1:NtotalCombs
                        for jj=1:current_degree
                            current_id=which(ii,jj);
                            Currentx(:,ii)=Currentx(:,ii).*Moments(:,current_id);
                        end
                    end
                    X=[X Currentx];
                end
            end
            Moments_Extended=X;
        end
        % =================================================================
        function [Bset,B3x3xN] = ConstructAxiallySymmetricBset(b,beta,dirs)
            % [Bset,B3x3xN] = ConstructAxiallySymmetricBset(b,beta,dirs)
            %
            % This function computes a set of axially symmetric
            % b-tensors from their b-value, b-tensor shape, and b-vectors.
            %
            % By: Santiago Coelho
            Ndwi=length(b);
            Bset=zeros(6,Ndwi);
            B3x3xN=zeros(3,3,Ndwi);
            if size(dirs,1)~=3
                dirs=dirs';
            end
            if size(dirs,2)~=Ndwi || length(beta)~=Ndwi
                error('dirs should have the same number of columns as elements in b and beta')
            end

            for ii=1:Ndwi
                B_aux= beta(ii)*dirs(:,ii)*dirs(:,ii)' + 1/3*(1-beta(ii))* eye(3);
                B_aux=B_aux/trace(B_aux);
                if any(isnan(B_aux(:)))
                    B_aux=zeros(3);
                else
                    B_aux=b(ii)*B_aux;
                end
                Bset(:,ii)=[B_aux(1,1),B_aux(2,2),B_aux(3,3),B_aux(1,2),B_aux(1,3),B_aux(2,3)]';
                B3x3xN(:,:,ii)=B_aux;
            end
        end
        % =================================================================
        function [Slm,Sl,ids_clusters_all,table_4D_sorted] = Fit2D4D_LLS_RealSphHarm_wSorting_norm_varL(DWI,mask,bval,bvec,beta,TE,Lmax,MergeDistance,CS_phase)
            % [Slm,Sl,ids_clusters_all,table_4D_sorted] = Fit2D4D_LLS_RealSphHarm_wSorting_norm_varL(DWI,mask,bval,bvec,beta,TE,Lmax,MergeDistance,CS_phase)
            % 
            % This function performs the linear fit of Slm on each shell using real spherical
            % harmonics as defined in https://cs.dartmouth.edu/~wjarosz/publications/dissertation/appendixB.pdf
            %
            % The Condon-Shortley phase is used in this definition
            %
            % Inputs: 
            % DWI is assumed to be either 2D [Nmeasurements x Nvoxels] or 4D [Nx x Ny x Nz x Nmeasurements]
            % bval, bvec, beta, TE indicate the acquisition
            % If beta or TE are constant throughout the acquisition, these can be left
            % empty
            % Lmax is the maximum order considered for the fitting of Slms
            % (this can be a different order in all clusters but input order is the same as given by Group_dwi_in_shells_b_beta_TE)
            %
            % Outputs: Slm is size [N_SH_coeffs x Nvoxels x Nclusters]
            %          Sl is size [Lmax/2+1 x Nvoxels x Nclusters]
            %          table_4D_sorted has a size [4 x Nshells], 
            %          The order of the Rotational Invariants is given by Group_dwi_in_shells_b_beta_TE
            %          and it is provided in table_4D_sorted
            %
            % By: Santiago Coelho
            
            if ~exist('CS_phase', 'var') || isempty(CS_phase)
                CS_phase=1;
            else
                CS_phase=0;
            end
            
            sz_DWI=size(DWI);
            if length(sz_DWI)==4
                flag_4D=1;
            elseif length(sz_DWI)==2
                flag_4D=0;
            else
                error('DWI must be a 2D or 4D array')
            end

            if flag_4D
                if isempty(mask)
                    mask=true(sz_DWI(1:3));
                end
                dwi_2D = SMI.vectorize(DWI,mask);
            else
                dwi_2D = DWI;
            end

            if ~isvector(bval)
                error('bval should be a vector')
            else
                bval=bval(:)';
            end
            if isempty(beta)
                beta=zeros(size(bval))+1;
            else
                beta=beta(:)';
            end
            if isempty(TE)
                TE=zeros(size(bval));
            else
                TE=TE(:)';
            end

            Nvoxels=size(dwi_2D,2);
            Nmeas=size(dwi_2D,1);

            if length(bval)~=Nmeas
                error('bval should be a vector with the same length as the number of DWI')
            end

            [bb,~,~] = SMI.Group_dwi_in_shells_b_beta_TE(bval,beta,TE,MergeDistance);


            if isempty(Lmax)
                Lmax = SMI.GetDefaultLmax(bval,beta,TE,MergeDistance);
            end

            Nshells=size(bb,2);
            if isscalar(Lmax)
                Lmax=Lmax*ones(1,Nshells);
            end

            % Hard-wiring Lmax=0 for b0s and STE data
            Lmax(bb(1,:)<0.025)=0;
            Lmax(abs(bb(2,:))<0.025)=0;

            if max(bval)<500
                flag_microstructure_units=1;
                % make all b0s LTE
                beta(bval<0.05)=1;
            else
                flag_microstructure_units=0;
                % make all b0s LTE
                beta(bval<50)=1;
            end

            if isempty(MergeDistance)
                if flag_microstructure_units
                    MergeDistance=0.1;
                else
                    MergeDistance=100;
                end
            end


            N_SH_coeffs=1+Lmax.*(Lmax+3)/2;
            if size(bvec,1)==Nmeas
                dirs=bvec;
            else
                dirs=bvec';
            end
            l=0:2:max(Lmax);
            l_all=[];
            m_all=[];
            for ii=1:length(l)
                l_all=[l_all, l(ii)*ones(1,2*l(ii)+1)];
                m_all=[m_all -l(ii):l(ii)];
            end
            N_l_all=sqrt((2*l_all+1)*(4*pi));
            N_l_unique=sqrt((2*l+1)*(4*pi));
            
            Y_LM_matrix = SMI.get_even_SH(dirs,max(Lmax),CS_phase);

            S_l_clusters_all=zeros(max(Lmax)/2+1,Nvoxels,Nshells);
            S_lm_clusters_all=zeros(max(N_SH_coeffs),Nvoxels,Nshells);
            id_measurements=1:Nmeas;
            ids_current_cluster_all=[];
            for ii=1:Nshells
                ids_current_cluster=id_measurements((abs(bb(1,ii)-bval)<MergeDistance)&(abs(bb(2,ii)-beta)<MergeDistance)&(abs(bb(4,ii)-TE)<1e-1));
                if abs(length(ids_current_cluster)-bb(3,ii))>0.1
                    error('count of elements in current cluster failed, check bb and B inputs')
                end
                if (abs(bb(2,ii))>0.045)&&(bb(1,ii)>0.05)
                    ids_current_lm=1:N_SH_coeffs(ii);
                    S_lm_current_cluster=Y_LM_matrix(ids_current_cluster,ids_current_lm)\dwi_2D(ids_current_cluster,:);
                    S_l_current_cluster=zeros(Lmax(ii)/2+1,Nvoxels);
                    S_l_current_cluster(1,:)=abs(S_lm_current_cluster(1,:));
                    for jj=2:2:Lmax(ii)
                        ids_currentl_m=(1/2*(jj+1).*(jj+2)-jj)+((-jj):(jj));
                        S_l_current_cluster(jj/2+1,:)=sqrt(sum(S_lm_current_cluster(ids_currentl_m,:).^2,1));
                    end
                else
                    S_lm_current_cluster=zeros(N_SH_coeffs(ii),Nvoxels);
                    S_l_current_cluster=zeros(Lmax(ii)/2+1,Nvoxels);
                    S_lm_current_cluster(1,:)=sqrt(4*pi)*mean(dwi_2D(ids_current_cluster,:),1);
                    S_l_current_cluster(1,:)=abs(S_lm_current_cluster(1,:));
                end
                S_lm_clusters_all(1:N_SH_coeffs(ii),:,ii)=S_lm_current_cluster./repmat(N_l_all(1:N_SH_coeffs(ii))',1,Nvoxels);
                S_l_clusters_all(1:(Lmax(ii)/2+1),:,ii)=S_l_current_cluster./repmat(N_l_unique(1:(Lmax(ii)/2+1))',1,Nvoxels);
                ids_current_cluster_all=[ids_current_cluster_all,[ids_current_cluster*0+ii;ids_current_cluster]];
            end  
            Slm=S_lm_clusters_all;
            Sl=S_l_clusters_all;
            if nargout>=3
                ids_clusters_all=ids_current_cluster_all;
            end
            if nargout==4
                table_4D_sorted=bb;
            end

            if flag_4D
                % reshape Sl and Slm
                Sl=zeros(sz_DWI(1),sz_DWI(2),sz_DWI(3),max(Lmax)/2+1,Nshells);
                Slm=zeros(sz_DWI(1),sz_DWI(2),sz_DWI(3),max(N_SH_coeffs),Nshells);
                for ii=1:Nshells
                    Sl(:,:,:,:,ii)=SMI.vectorize(S_l_clusters_all(:,:,ii),mask);
                    Slm(:,:,:,:,ii)=SMI.vectorize(S_lm_clusters_all(:,:,ii),mask);
                end
            end
        end
        % =================================================================
        function [Slm,Sl,ids_clusters_all,table_4D_sorted] = Fit2D4D_LLS_RealSphHarm_wSorting_norm_varL_rank1Sl(DWI,mask,bval,bvec,beta,TE,Lmax,MergeDistance,CS_phase)
            % [Slm,Sl,ids_clusters_all,table_4D_sorted] = Fit2D4D_LLS_RealSphHarm_wSorting_norm_varL_rank1Sl(DWI,mask,bval,bvec,beta,TE,Lmax,MergeDistance,CS_phase)
            % 
            % Identical to Fit2D4D_LLS_RealSphHarm_wSorting_norm_varL but it
            % removes non-central xi bias in Sl by assuming S_lm(b) is rank-1 for all b
            %
            % This function performs the linear fit of Slm on each shell using real spherical
            % harmonics as defined in https://cs.dartmouth.edu/~wjarosz/publications/dissertation/appendixB.pdf
            %
            % The Condon-Shortley phase is used in this definition
            %
            % Inputs: 
            % DWI is assumed to be either 2D [Nmeasurements x Nvoxels] or 4D [Nx x Ny x Nz x Nmeasurements]
            % bval, bvec, beta, TE indicate the acquisition
            % If beta or TE are constant throughout the acquisition, these can be left
            % empty
            % Lmax is the maximum order considered for the fitting of Slms
            % (this can be a different order in all clusters but input order is the same as given by Group_dwi_in_shells_b_beta_TE)
            %
            % Outputs: Slm is size [N_SH_coeffs x Nvoxels x Nclusters]
            %          Sl is size [Lmax/2+1 x Nvoxels x Nclusters]
            %          table_4D_sorted has a size [4 x Nshells], 
            %          The order of the Rotational Invariants is given by Group_dwi_in_shells_b_beta_TE
            %          and it is provided in table_4D_sorted
            %
            % By: Santiago Coelho
            
            if ~exist('CS_phase', 'var') || isempty(CS_phase)
                CS_phase=1;
            end
            
            sz_DWI=size(DWI);
            if length(sz_DWI)==4
                flag_4D=1;
            elseif length(sz_DWI)==2
                flag_4D=0;
            else
                error('DWI must be a 2D or 4D array')
            end

            if flag_4D
                if isempty(mask)
                    mask=true(sz_DWI(1:3));
                end
                dwi_2D = SMI.vectorize(DWI,mask);
            else
                dwi_2D = DWI;
            end

            if ~isvector(bval)
                error('bval should be a vector')
            else
                bval=bval(:)';
            end
            if isempty(beta)
                beta=zeros(size(bval))+1;
            else
                beta=beta(:)';
            end
            if isempty(TE)
                TE=zeros(size(bval));
            else
                TE=TE(:)';
            end

            Nvoxels=size(dwi_2D,2);
            Nmeas=size(dwi_2D,1);

            if length(bval)~=Nmeas
                error('bval should be a vector with the same length as the number of DWI')
            end

            [bb,~,~] = SMI.Group_dwi_in_shells_b_beta_TE(bval,beta,TE,MergeDistance);


            if isempty(Lmax)
                Lmax = SMI.GetDefaultLmax(bval,beta,TE,MergeDistance);
            end

            Nshells=size(bb,2);
            if isscalar(Lmax)
                Lmax=Lmax*ones(1,Nshells);
            end

            % Hard-wiring Lmax=0 for b0s and STE data
            Lmax(bb(1,:)<0.025)=0;
            Lmax(abs(bb(2,:))<0.025)=0;

            if max(bval)<500
                flag_microstructure_units=1;
                % make all b0s LTE
                beta(bval<0.05)=1;
            else
                flag_microstructure_units=0;
                % make all b0s LTE
                beta(bval<50)=1;
            end

            if isempty(MergeDistance)
                if flag_microstructure_units
                    MergeDistance=0.1;
                else
                    MergeDistance=100;
                end
            end


            N_SH_coeffs=1+Lmax.*(Lmax+3)/2;
            if size(bvec,1)==Nmeas
                dirs=bvec;
            else
                dirs=bvec';
            end
            l=0:2:max(Lmax);
            l_all=[];
            m_all=[];
            for ii=1:length(l)
                l_all=[l_all, l(ii)*ones(1,2*l(ii)+1)];
                m_all=[m_all -l(ii):l(ii)];
            end
            N_l_all=sqrt((2*l_all+1)*(4*pi));
            N_l_unique=sqrt((2*l+1)*(4*pi));
            
            Y_LM_matrix = SMI.get_even_SH(dirs,max(Lmax),CS_phase);

            S_l_clusters_all=zeros(max(Lmax)/2+1,Nvoxels,Nshells);
            S_lm_clusters_all=zeros(max(N_SH_coeffs),Nvoxels,Nshells);
            id_measurements=1:Nmeas;
            ids_current_cluster_all=[];
            for ii=1:Nshells
                ids_current_cluster=id_measurements((abs(bb(1,ii)-bval)<MergeDistance)&(abs(bb(2,ii)-beta)<MergeDistance)&(abs(bb(4,ii)-TE)<1e-1));
                if abs(length(ids_current_cluster)-bb(3,ii))>0.1
                    error('count of elements in current cluster failed, check bb and B inputs')
                end
                if (abs(bb(2,ii))>0.045)&&(bb(1,ii)>0.05)
                    ids_current_lm=1:N_SH_coeffs(ii);
                    S_lm_current_cluster=Y_LM_matrix(ids_current_cluster,ids_current_lm)\dwi_2D(ids_current_cluster,:);
                    S_l_current_cluster=zeros(Lmax(ii)/2+1,Nvoxels);
                    S_l_current_cluster(1,:)=abs(S_lm_current_cluster(1,:));
                    for jj=2:2:Lmax(ii)
                        ids_currentl_m=(1/2*(jj+1).*(jj+2)-jj)+((-jj):(jj));
                        S_l_current_cluster(jj/2+1,:)=sqrt(sum(S_lm_current_cluster(ids_currentl_m,:).^2,1));
                    end
                else
                    S_lm_current_cluster=zeros(N_SH_coeffs(ii),Nvoxels);
                    S_l_current_cluster=zeros(Lmax(ii)/2+1,Nvoxels);
                    S_lm_current_cluster(1,:)=sqrt(4*pi)*mean(dwi_2D(ids_current_cluster,:),1);
                    S_l_current_cluster(1,:)=abs(S_lm_current_cluster(1,:));
                end
                S_lm_clusters_all(1:N_SH_coeffs(ii),:,ii)=S_lm_current_cluster./repmat(N_l_all(1:N_SH_coeffs(ii))',1,Nvoxels);
                S_l_clusters_all(1:(Lmax(ii)/2+1),:,ii)=S_l_current_cluster./repmat(N_l_unique(1:(Lmax(ii)/2+1))',1,Nvoxels);
                ids_current_cluster_all=[ids_current_cluster_all,[ids_current_cluster*0+ii;ids_current_cluster]];
            end  
            Slm=S_lm_clusters_all;
            Sl=S_l_clusters_all;
            Sl_dn=S_l_clusters_all;
            Slm_dn=S_lm_clusters_all;
            if nargout>=3
                ids_clusters_all=ids_current_cluster_all;
            end
            if nargout==4
                table_4D_sorted=bb;
            end
            
            id_shells = 1:Nshells;
            for ll=2:2:max(Lmax)
                if sum(Lmax >= ll) > 1
                    id_useful_shells = id_shells(Lmax>=ll);
                    id_current_band = l_all == ll;
                    for kk = 1:Nvoxels                        
                        Slm_voxel_raw = squeeze(Slm(id_current_band,kk,id_useful_shells));
                        Slm_voxel_dn = SMI.LowRankDenoising(Slm_voxel_raw,1);
                        Slm_dn(id_current_band,kk,id_useful_shells) = Slm_voxel_dn;
                        Sl_dn(ll/2+1,kk,id_useful_shells) = sqrt(sum(Slm_voxel_dn.^2,1));
                    end
                end
            end

            if flag_4D
                % reshape Sl and Slm
                Sl=zeros(sz_DWI(1),sz_DWI(2),sz_DWI(3),max(Lmax)/2+1,Nshells);
                Slm=zeros(sz_DWI(1),sz_DWI(2),sz_DWI(3),max(N_SH_coeffs),Nshells);
                for ii=1:Nshells
                    Sl(:,:,:,:,ii)=SMI.vectorize(Sl_dn(:,:,ii),mask);
                    Slm(:,:,:,:,ii)=SMI.vectorize(Slm_dn(:,:,ii),mask);
                end
            end
        end
        % =================================================================
        function Xdn = LowRankDenoising(X,p)
            % Xdn = LowRankDenoising(X,p)
            % 
            % Fixed p low-rank denoising, does svd and keeps p largest singular values
            [U,S,V] = svd(X);
            Xdn= U(:,1:p)*S(1:p,1:p)*V(:,1:p)';
        end
        % =================================================================
        function Ylm_n = get_even_SH(dirs,Lmax,CS_phase)
            % Ylm_n = get_even_SH(dirs,Lmax,CS_phase)
            %
            % if CS_phase=1, then the definition uses the Condon-Shortley phase factor
            % of (-1)^m. Default is CS_phase=0 (so this factor is ommited)
            %
            % By: Santiago Coelho
            if size(dirs,2)~=3
                dirs=dirs';
            end
            Nmeas=size(dirs,1);
            [PHI,THETA,~]=cart2sph(dirs(:,1),dirs(:,2),dirs(:,3)); THETA=pi/2-THETA;
            l=0:2:Lmax;
            l_all=[];
            m_all=[];
            for ii=1:length(l)
                l_all=[l_all, l(ii)*ones(1,2*l(ii)+1)];
                m_all=[m_all -l(ii):l(ii)];
            end
            K_lm=sqrt((2*l_all+1)./(4*pi) .* factorial(l_all-abs(m_all))./factorial(l_all+abs(m_all)));
            if nargin==2 || isempty(CS_phase) || ~exist('CS_phase','var') || ~CS_phase
                extra_factor=ones(size(K_lm));
                extra_factor(m_all~=0)=sqrt(2);
            else
                extra_factor=ones(size(K_lm));
                extra_factor(m_all~=0)=sqrt(2);
                extra_factor=extra_factor.*(-1).^(m_all);
            end
            P_l_in_cos_theta=zeros(length(l_all),Nmeas);
            phi_term=zeros(length(l_all),Nmeas);
            id_which_pl=zeros(1,length(l_all));
            for ii=1:length(l_all)
                all_Pls=legendre(l_all(ii),cos(THETA));
                P_l_in_cos_theta(ii,:)=all_Pls(abs(m_all(ii))+1,:);
                id_which_pl(ii)=abs(m_all(ii))+1;
                if m_all(ii)>0
                    phi_term(ii,:)=cos(m_all(ii)*PHI);
                elseif m_all(ii)==0
                    phi_term(ii,:)=1;
                elseif m_all(ii)<0
                    phi_term(ii,:)=sin(-m_all(ii)*PHI);
                end
            end
            Y_lm=repmat(extra_factor',1,Nmeas).*repmat(K_lm',1,Nmeas).*phi_term.*P_l_in_cos_theta;
            Ylm_n=Y_lm';
        end
        % =================================================================
        function sigma = PatchBasedSigma(data_4D,kernel)
        % sigma = PatchBasedSigma(data_4D,kernel)
        %
        %
            [Nx,Ny,Nz,Nmeas] = size(data_4D);
            residuals = data_4D-repmat(mean(data_4D,4),[1 1 1 Nmeas]);
            if isempty(kernel)||kernel(1)==1
                sigma = 1.4826*mad(residuals, 1,4);% use mad(,0) for mean instead of median
            else
                residuals=padarray(single(residuals),floor(kernel./2),'circular');
                sigma=single(zeros([Nx,Ny,Nz]));
                kh=floor(kernel/2);
                sz_res=size(residuals);
                for x=1:Nx
                    Res_x=residuals(kh(1)+((x-kh(1)):(x+kh(1))),:,:,:);
                    Res_x=reshape(permute(Res_x,[2 3 4 1]),sz_res(2),sz_res(3),[]); % Now it is size Ny x Nz x (kx x Nd)
                    if kernel(1)==3
                        Res_x_rep=cat(3,Res_x(1:Ny,  1:Nz,:),Res_x(1:Ny,  2:Nz+1,:),Res_x(1:Ny,  3:Nz+2,:),...
                                        Res_x(2:Ny+1,1:Nz,:),Res_x(2:Ny+1,2:Nz+1,:),Res_x(2:Ny+1,3:Nz+2,:),...
                                        Res_x(3:Ny+2,1:Nz,:),Res_x(3:Ny+2,2:Nz+1,:),Res_x(3:Ny+2,3:Nz+2,:));
                    elseif kernel(1)==5
                        Res_x_rep=cat(3,Res_x(1:Ny,  1:Nz,:),Res_x(1:Ny,  2:Nz+1,:),Res_x(1:Ny,  3:Nz+2,:),Res_x(1:Ny,  4:Nz+3,:),Res_x(1:Ny,  5:Nz+4,:),...
                                        Res_x(2:Ny+1,1:Nz,:),Res_x(2:Ny+1,2:Nz+1,:),Res_x(2:Ny+1,3:Nz+2,:),Res_x(2:Ny+1,4:Nz+3,:),Res_x(2:Ny+1,5:Nz+4,:),...
                                        Res_x(3:Ny+2,1:Nz,:),Res_x(3:Ny+2,2:Nz+1,:),Res_x(3:Ny+2,3:Nz+2,:),Res_x(3:Ny+2,4:Nz+3,:),Res_x(3:Ny+2,5:Nz+4,:),...
                                        Res_x(4:Ny+3,1:Nz,:),Res_x(4:Ny+3,2:Nz+1,:),Res_x(4:Ny+3,3:Nz+2,:),Res_x(4:Ny+3,4:Nz+3,:),Res_x(4:Ny+3,5:Nz+4,:),...
                                        Res_x(5:Ny+4,1:Nz,:),Res_x(5:Ny+4,2:Nz+1,:),Res_x(5:Ny+4,3:Nz+2,:),Res_x(5:Ny+4,4:Nz+3,:),Res_x(5:Ny+4,5:Nz+4,:));
                    else
                        error('kernel for sigma estimation can only be [1 1 1], [3 3 3], or [5 5 5]')
                    end
                    % Now Res_x_rep is size Ny x Nz x (kx xky x ky x Nd)
            %         sigma_yz=std(Res_x_rep,[],3);
                    sigma_yz = 1.4826 * mad(Res_x_rep,1,3);
                    sigma(x,:,:) = sigma_yz;
                end
            end
        end
        % =================================================================
        function [table_4D,sorted_ids,shell_ids] = Group_dwi_in_shells_b_beta_TE(bval,beta,TE,MergeDistance)
            % [table_4D,sorted_ids,shell_ids] = Group_dwi_in_shells_b_beta_TE(bval,beta,TE,MergeDistance)
            %
            % table_4D contains [bval;beta;Ndirs;TE] for each shell (N shells -> N columns)
            %                   Shells are in ascending b-value, beta, TE
            % shell_ids contains [shell numberof each input in the old order
            % sorted_ids contains original input order for the ordered shells
            %
            % If the input protocol has equal TE on all DWI, or equal beta on all DWI,
            % then this function returns the shells ordered in increasing b-value.
            %
            % By: Santiago Coelho
            if ~isvector(bval)
                error('bval should be a vector')
            else
                bval=bval(:)';
            end
            if isempty(beta)
                beta=zeros(size(bval))+1;
            else
                beta=beta(:)';
            end
            if isempty(TE)
                TE=zeros(size(bval));
            else
                TE=TE(:)';
            end

            if max(bval)<500
                flag_microstructure_units=1;
                % make all b0s LTE
                beta(bval<0.05)=1;
            else
                flag_microstructure_units=0;
                % make all b0s LTE
                beta(bval<50)=1;
            end

            if isempty(MergeDistance)
                if flag_microstructure_units
                    MergeDistance=0.1;
                else
                    MergeDistance=100;
                end
            end
            Clusters=[bval;beta;TE]';
            id_cluster=clusterdata(Clusters,'Criterion','distance','Cutoff',MergeDistance);
            sorted_ids_intermediate=[];
            shell_ids_intermediate=[];
            id_original_SortParam=1:length(id_cluster);
            table_4D=zeros(4,length(unique(id_cluster)));
            for ii=1:length(unique(id_cluster))
                table_4D(:,ii)=[mean(bval(id_cluster==ii));mean(beta(id_cluster==ii));sum(id_cluster==ii);mean(TE(id_cluster==ii))];
                sorted_ids_intermediate=[sorted_ids_intermediate id_original_SortParam(id_cluster==ii)];
                shell_ids_intermediate=[shell_ids_intermediate ii*ones(1,sum(id_cluster==ii))];
            end

            % Now I order them in ascending b-value, beta, TE
            table_aux=table_4D;
            table_aux(2,:)=round(table_aux(2,:),2);
            if flag_microstructure_units
               table_aux(1,:)=round(table_aux(1,:),2);
            else
               table_aux(1,:)=round(table_aux(1,:),-1);
            end
            [table_aux_sorted,index_sorted2]=sortrows(table_aux',[1 2 4]);
            table_aux_sorted=table_aux_sorted';
            sorted_ids=[];
            shell_ids=sorted_ids_intermediate*0;
            for ii=1:size(table_aux,2)
                aux=sorted_ids_intermediate(shell_ids_intermediate==index_sorted2(ii));
                sorted_ids=[sorted_ids aux];
                shell_ids(aux)=ii;
            end
            table_4D=table_aux_sorted;
        end
        % =================================================================
        function [f,Da,Depar,Deperp,f_FW,T2a,T2e,p2,plm] = Get_uniformly_distributed_SM_prior(M,lb,ub,Lmax,flag_Depar_Deperp)
            % [f,Da,Depar,Deperp,f_FW,T2a,T2e,p2,plm] = Get_uniformly_distributed_SM_prior(M,lb,ub,Lmax,flag_Depar_Deperp)
            %
            % lower and upper bounds on {'$f$','$D_a$','$D_e^{||}$','$D_e^{\perp}$','$f_{FW}$','$T_a^2$','$T_e^2$','$p_2$'};
            if nargin<=4||isempty(flag_Depar_Deperp)
                flag_Depar_Deperp=0;
            end
            % Water fractions
            f=rand(5*M,1)*(ub(1)-lb(1))+lb(1);
            f_FW=rand(5*M,1)*(ub(5)-lb(5))+lb(5);
            keep=1>(f+f_FW);
            f=f(keep);f=f(1:M);
            f_FW=f_FW(keep);f_FW=f_FW(1:M);
            f_extra=1-f-f_FW;
            % Diffusivities
            Da = rand(M,1)*(ub(2)-lb(2))+lb(2);
            if flag_Depar_Deperp
                Depar = rand(5*M,1)*(ub(3)-lb(3))+lb(3);
                Deperp = rand(5*M,1)*(ub(4)-lb(4))+lb(4);
                keep=Depar>Deperp;
                Depar=Depar(keep);Depar=Depar(1:M);
                Deperp=Deperp(keep);Deperp=Deperp(1:M);
            else
                Depar = rand(M,1)*(ub(3)-lb(3))+lb(3);
                Deperp = rand(M,1)*(ub(4)-lb(4))+lb(4);
            end
            % T2 values
            T2a=rand(M,1)*(ub(6)-lb(6))+lb(6);
            T2e=rand(M,1)*(ub(7)-lb(7))+lb(7);
            % p2m
            if (nargin<=3)||isempty(Lmax)||(Lmax==2)
                p2 = rand(M,1)*(ub(8)-lb(8))+lb(8);
                norm_p2l=@(p2l)sqrt(sum(p2l.^2));
                DIM=5;
                x=randn(DIM,M);
                initial_norm=norm_p2l(x);
                x_normalised=x./repmat(initial_norm,DIM,1);
                x_newnorm=x_normalised.*repmat(p2',DIM,1);
                p2m=x_newnorm;
                plm=p2m';
            else
                p2=rand(1,M)*(ub(8)-lb(8))+lb(8);
                p4=rand(1,M).*p2*0.9; 
                p6=rand(1,M).*p4*0.9; 
                p8=rand(1,M).*p6*0.9;    

                norm_p2l=@(p2l)sqrt(sum(p2l.^2));
                norm_p4l=@(p4l)sqrt(sum(p4l.^2));
                norm_p6l=@(p6l)sqrt(sum(p6l.^2));
                norm_p8l=@(p8l)sqrt(sum(p8l.^2));
                if Lmax>=2
                    DIM=5;
                    x=randn(DIM,M);
                    initial_norm=norm_p2l(x);
                    x_normalised=x./repmat(initial_norm,DIM,1);
                    x_newnorm=x_normalised.*repmat(p2,DIM,1);
                    p2m=x_newnorm;
                    plm=p2m';
                end
                if Lmax>=4
                    DIM=9;
                    x=randn(DIM,M);
                    initial_norm=norm_p4l(x);
                    x_normalised=x./repmat(initial_norm,DIM,1);
                    x_newnorm=x_normalised.*repmat(p4,DIM,1);
                    p4m=x_newnorm;
                    plm=[p2m;p4m]';
                end
                if Lmax>=6
                    DIM=13;
                    x=randn(DIM,M);
                    initial_norm=norm_p6l(x);
                    x_normalised=x./repmat(initial_norm,DIM,1);
                    x_newnorm=x_normalised.*repmat(p6,DIM,1);
                    p6m=x_newnorm;
                    plm=[p2m;p4m;p6m]';
                end
                if Lmax>=8
                    DIM=17;
                    x=randn(DIM,M);
                    initial_norm=norm_p8l(x);
                    x_normalised=x./repmat(initial_norm,DIM,1);
                    x_newnorm=x_normalised.*repmat(p8,DIM,1);
                    p8m=x_newnorm;
                    plm=[p2m;p4m;p6m;p8m]';
                end
                p2=p2';
            end
        end
        % =================================================================
        function Lmax = GetDefaultLmax(b,beta,TE,MergeDistance)
            % Lmax = GetDefaultLmax(b,beta,TE,MergeDistance)
            %
            % This function returns a 'good' Lmax based on the b-value of
            % the shell
            [table_4D,~,~] = SMI.Group_dwi_in_shells_b_beta_TE(b,beta,TE,MergeDistance);
            b=table_4D(1,:);
            beta=table_4D(2,:);
            Ndirs=table_4D(3,:);
            Lmax=zeros(1,size(table_4D,2));
            if max(b)<100
                b_micro_units=b;
            else
                b_micro_units=b/1000;
            end
            for ii=1:length(b_micro_units)
                if (abs(beta(ii))<0.1)||(b_micro_units(ii)<0.1)
                    Lmax(ii)=0;
                else
                    if b_micro_units(ii)<1.2
                        Lmax(ii)=2;
                    elseif b_micro_units(ii)<=2.5
                        Lmax(ii)=4;
                    elseif b_micro_units(ii)<6.5
                        Lmax(ii)=6;
                    else
                        Lmax(ii)=8;
                    end
                    NfreeParam=Lmax(ii)*(Lmax(ii)+3)/2+1;
                    while NfreeParam>=(Ndirs(ii)/1.3)
                        Lmax(ii)=Lmax(ii)-2;
                        NfreeParam=Lmax(ii)*(Lmax(ii)+3)/2+1;
                    end
                end
            end
        end
        % =================================================================
        function plotSlices(ARRAY_4D, slice,clims,names,Nrows,positions,nanTransparent,colorbar_flag)
            %
            sz=size(ARRAY_4D);
            if length(sz)==4
                Nplots=sz(4);
            elseif length(sz)==3
                Nplots=sz(3);
            end

            if ~exist('positions', 'var') || isempty(positions)
                positions=1:Nplots;
            end
            if ~exist('nanTransparent', 'var') || isempty(nanTransparent)
                nanTransparent=0;
            end
            if ~exist('colorbar_flag', 'var') || isempty(colorbar_flag)
                colorbar_flag=1;
            end

            if isvector(clims)
                clims=repmat(clims,Nplots,1);
            end

            for ii=1:Nplots
                subplot(Nrows,ceil(Nplots/Nrows),positions(ii))
                if ~nanTransparent
                    if length(sz)==4
                        imagesc(ARRAY_4D(:,:,slice,ii),clims(ii,:))
                    elseif length(sz)==3
                        imagesc(ARRAY_4D(:,:,ii),clims(ii,:))
                    end
                else
                    if length(sz)==4
                        h=pcolor(ARRAY_4D(:,:,slice,ii)); caxis(clims(ii,:)),set(h, 'EdgeColor', 'none');
                    elseif length(sz)==3
                        h=pcolor(ARRAY_4D(:,:,ii)); caxis(clims(ii,:)),set(h, 'EdgeColor', 'none');
                    end
                end

                if isempty(names)
                    title(['case ',num2str(ii)],'interpreter','latex')
                else
                    title(names{ii},'interpreter','latex')
                end
                set(gca,'FontSize',30), axis off, grid off

                if colorbar_flag
                cb=colorbar('south'); 
            %     cb_pos =  cb.Position;cb_pos(3)=0.03;
            %     set(cb,'position',cb_pos)
                cb.Ticks=clims(ii,:);    
                end
            end
        end        
        % =================================================================
        function [s, mask] = vectorize(S, mask)
            if nargin == 1
               mask = ~isnan(S(:,:,:,1));
            end
            if ismatrix(S)
                n = size(S, 1);
                [x, y, z] = size(mask);
                s = NaN([x, y, z, n], 'like', S);
                for i = 1:n
                    tmp = NaN(x, y, z, 'like', S);
                    tmp(mask(:)) = S(i, :);
                    s(:,:,:,i) = tmp;
                end
            else
                for i = 1:size(S, 4)
                   Si = S(:,:,:,i);
                   s(i, :) = Si(mask(:));
                end
            end
        end
        % =================================================================
    end
end