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sphere_tracing.py
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import torch
import torch.nn as nn
class SphereTracing(torch.autograd.Function):
@staticmethod
def forward(
ctx,
signed_distance_function,
ray_positions,
ray_directions,
num_iterations,
convergence_threshold,
foreground_masks=None,
*parameters,
):
h, w, ch = ray_positions.shape
ray_positions = ray_positions.reshape(h * w, ch)
ray_directions = ray_directions.reshape(h * w, ch)
if foreground_masks is None:
foreground_masks = torch.all(torch.isfinite(ray_positions), dim=-1, keepdim=True)
compute_mask = foreground_masks
compute_mask = compute_mask.squeeze()
signed_distances = torch.ones((h * w, 1), device=ray_directions.device)
# Determine the divergence threshold
max_dist = ray_positions.norm(dim=-1).max()
diverge_thresh = max_dist * 2
# vanilla sphere tracing
with torch.no_grad():
for i in range(num_iterations):
if i > 0:
compute_mask = compute_mask & ~(converged.squeeze()) & ~(diverged.squeeze())
ray_pos_recompute = ray_positions[compute_mask]
signed_distances[compute_mask] = signed_distance_function(ray_pos_recompute)
ray_positions = torch.where(compute_mask.unsqueeze(dim=-1),
ray_positions + ray_directions * signed_distances, ray_positions)
# Determine convergence and divergence
converged = torch.abs(signed_distances) < convergence_threshold
diverged = ray_positions.norm(dim=-1) > diverge_thresh
if torch.all(~compute_mask):
break
# save tensors for backward pass
ctx.save_for_backward(ray_positions, ray_directions, foreground_masks, converged)
ctx.signed_distance_function = signed_distance_function
ctx.parameters = parameters
ray_positions = ray_positions.reshape(h, w, ch)
converged = converged.reshape(h, w, 1)
return ray_positions, converged
@staticmethod
def backward(ctx, grad_outputs, _):
# restore tensors from forward pass
ray_positions, ray_directions, foreground_masks, converged = ctx.saved_tensors
signed_distance_function = ctx.signed_distance_function
parameters = ctx.parameters
# compute gradients using implicit differentiation
# NOTE: Differentiable Volumetric Rendering: https://arxiv.org/abs/1912.07372
with torch.enable_grad():
ray_positions = ray_positions.detach()
ray_positions.requires_grad_(True)
signed_distances = signed_distance_function(ray_positions)
grad_positions, = torch.autograd.grad(
outputs=signed_distances,
inputs=ray_positions,
grad_outputs=torch.ones_like(signed_distances),
retain_graph=True,
)
grad_outputs_dot_directions = torch.sum(grad_outputs * ray_directions, dim=-1, keepdim=True)
grad_positions_dot_directions = torch.sum(grad_positions * ray_directions, dim=-1, keepdim=True)
# NOTE: avoid division by zero
grad_positions_dot_directions = torch.where(
grad_positions_dot_directions > 0,
torch.max(grad_positions_dot_directions, torch.full_like(grad_positions_dot_directions, +1e-6)),
torch.min(grad_positions_dot_directions, torch.full_like(grad_positions_dot_directions, -1e-6)),
)
grad_outputs = -grad_outputs_dot_directions / grad_positions_dot_directions
# NOTE: zero gradients of unconverged points
grad_outputs = torch.where(converged, grad_outputs, torch.zeros_like(grad_outputs))
grad_parameters = torch.autograd.grad(
outputs=signed_distances,
inputs=parameters,
grad_outputs=grad_outputs,
retain_graph=True,
)
return None, None, None, None, None, None, None, *grad_parameters
def sphere_tracing(
signed_distance_function,
ray_positions,
ray_directions,
num_iterations,
convergence_threshold,
foreground_masks=None,
bounding_radius=None,
):
return SphereTracing.apply(
signed_distance_function,
ray_positions,
ray_directions,
num_iterations,
convergence_threshold,
foreground_masks,
bounding_radius,
)
def compute_shadows(
signed_distance_function,
surface_positions,
surface_normals,
light_directions,
num_iterations,
convergence_threshold,
foreground_masks=None,
bounding_radius=None,
):
surface_positions, converged = sphere_tracing(
signed_distance_function=signed_distance_function,
ray_positions=surface_positions + surface_normals * 1e-3,
ray_directions=light_directions,
num_iterations=num_iterations,
convergence_threshold=convergence_threshold,
foreground_masks=foreground_masks,
bounding_radius=bounding_radius,
)
return foreground_masks & converged
def compute_normal(
signed_distance_function,
surface_positions,
finite_difference_epsilon=None,
):
if finite_difference_epsilon:
finite_difference_epsilon_x = surface_positions.new_tensor([finite_difference_epsilon, 0.0, 0.0])
finite_difference_epsilon_y = surface_positions.new_tensor([0.0, finite_difference_epsilon, 0.0])
finite_difference_epsilon_z = surface_positions.new_tensor([0.0, 0.0, finite_difference_epsilon])
surface_normals_x = signed_distance_function(
surface_positions + finite_difference_epsilon_x) - signed_distance_function(
surface_positions - finite_difference_epsilon_x)
surface_normals_y = signed_distance_function(
surface_positions + finite_difference_epsilon_y) - signed_distance_function(
surface_positions - finite_difference_epsilon_y)
surface_normals_z = signed_distance_function(
surface_positions + finite_difference_epsilon_z) - signed_distance_function(
surface_positions - finite_difference_epsilon_z)
surface_normals = torch.cat((surface_normals_x, surface_normals_y, surface_normals_z), dim=-1)
else:
create_graph = surface_positions.requires_grad
surface_positions.requires_grad_(True)
with torch.enable_grad():
signed_distances = signed_distance_function(surface_positions)
surface_normals, = torch.autograd.grad(
outputs=signed_distances,
inputs=surface_positions,
grad_outputs=torch.ones_like(signed_distances),
create_graph=create_graph,
)
surface_normals = nn.functional.normalize(surface_normals, dim=-1)
return surface_normals
def phong_shading(
surface_normals,
view_directions,
light_directions,
light_ambient_color,
light_diffuse_color,
light_specular_color,
material_ambient_color,
material_diffuse_color,
material_specular_color,
material_emission_color,
material_shininess,
):
surface_normals = nn.functional.normalize(surface_normals, dim=-1)
view_directions = nn.functional.normalize(view_directions, dim=-1)
light_directions = nn.functional.normalize(light_directions, dim=-1)
reflected_directions = 2 * surface_normals * torch.sum(light_directions * surface_normals, dim=-1,
keepdim=True) - light_directions
diffuse_coefficients = nn.functional.relu(torch.sum(light_directions * surface_normals, dim=-1, keepdim=True))
specular_coefficients = nn.functional.relu(
torch.sum(reflected_directions * view_directions, dim=-1, keepdim=True)) ** material_shininess
images = torch.clamp(
material_emission_color +
material_ambient_color * light_ambient_color +
material_diffuse_color * light_diffuse_color * diffuse_coefficients +
material_specular_color * light_specular_color * specular_coefficients,
0.0, 1.0
)
return images
def cube_mapping(
surface_normals,
view_directions,
positive_x_images,
negative_x_images,
positive_y_images,
negative_y_images,
positive_z_images,
negative_z_images,
):
surface_normals = nn.functional.normalize(surface_normals, dim=-1)
view_directions = nn.functional.normalize(view_directions, dim=-1)
reflected_directions = 2 * surface_normals * torch.sum(view_directions * surface_normals, dim=-1,
keepdim=True) - view_directions
max_indices = torch.argmax(torch.abs(reflected_directions), dim=-1, keepdim=True)
max_values = torch.gather(reflected_directions, -1, max_indices)
texcoords_x, texcoords_y, texcoords_z = torch.unbind(reflected_directions / torch.abs(max_values), dim=-1)
# OpenGL specifications
# Chapter 8.13: Cube Map Texture Selection
# https://www.khronos.org/registry/OpenGL/specs/gl/glspec46.core.pdf
images = torch.stack((
positive_x_images,
negative_x_images,
positive_y_images,
negative_y_images,
positive_z_images,
negative_z_images,
), dim=-3)
def linear_mapping(inputs, in_min, in_max, out_min, out_max):
inputs = (inputs - in_min) / (in_max - in_min) * (out_max - out_min) + out_min
return inputs
grids = torch.where(
max_indices == 0,
torch.where(
max_values > 0,
torch.stack((-texcoords_z, texcoords_y, linear_mapping(torch.full_like(texcoords_x, 0), 0, 5, -1, 1)),
dim=-1),
torch.stack((texcoords_z, texcoords_y, linear_mapping(torch.full_like(texcoords_x, 1), 0, 5, -1, 1)),
dim=-1),
),
torch.where(
max_indices == 1,
torch.where(
max_values > 0,
torch.stack((texcoords_x, -texcoords_z, linear_mapping(torch.full_like(texcoords_y, 2), 0, 5, -1, 1)),
dim=-1),
torch.stack((texcoords_x, texcoords_z, linear_mapping(torch.full_like(texcoords_y, 3), 0, 5, -1, 1)),
dim=-1),
),
torch.where(
max_values > 0,
torch.stack((texcoords_x, texcoords_y, linear_mapping(torch.full_like(texcoords_z, 4), 0, 5, -1, 1)),
dim=-1),
torch.stack((-texcoords_x, texcoords_y, linear_mapping(torch.full_like(texcoords_z, 5), 0, 5, -1, 1)),
dim=-1),
),
)
)
images = nn.functional.grid_sample(images, grids.unsqueeze(-4)).squeeze(-3)
images = images.permute(0, 2, 3, 1)
return images