fft_transform.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true,
    "slideshow": {
     "slide_type": "-"
    }
   },
   "source": [
    "# Fast Fourier Transform snippets\n",
    "\n",
    "## Documentation\n",
    "\n",
    "- Numpy implementation: http://docs.scipy.org/doc/numpy/reference/routines.fft.html\n",
    "- Scipy implementation: http://docs.scipy.org/doc/scipy/reference/fftpack.html"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "## Import directives"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from matplotlib import cm"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "## Make data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "pattern = np.zeros((4, 4))\n",
    "pattern[1:3,1:3] = 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 0.,  0.,  0.,  0.],\n",
       "       [ 0.,  1.,  1.,  0.],\n",
       "       [ 0.,  1.,  1.,  0.],\n",
       "       [ 0.,  0.,  0.,  0.]])"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pattern"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "signal = np.tile(pattern, (2, 2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.image.AxesImage at 0x7b25940>"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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QPgDAGMIHABhD+AAAYwgfAGAM4QMAjCF8AIAxhA8AMIbwAQDGED4AwBjCBwAY4+zSA0xQ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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7b250b8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure(figsize=(16.0, 10.0))\n",
    "ax = fig.add_subplot(111)\n",
    "ax.imshow(signal, interpolation='nearest', cmap=cm.gray)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "## Fourier transform with Numpy\n",
    "\n",
    "### Do the fourier transform"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "transformed_signal = np.fft.fft2(signal)\n",
    "#transformed_signal"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.image.AxesImage at 0x8050fd0>"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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MIXwAgDGEDwAwRnX39jet2v6mAADn0d21bo0zPgDAGMIHABhD+AAAYwgfAGAM4QMAjCF8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\nAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x4479b00>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure(figsize=(16.0, 10.0))\n",
    "ax = fig.add_subplot(111)\n",
    "ax.imshow(abs(transformed_signal), interpolation='nearest', cmap=cm.gray)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "### Filter"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "max_value = np.max(abs(transformed_signal))\n",
    "filtered_transformed_signal = transformed_signal * (abs(transformed_signal) > max_value*0.5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.image.AxesImage at 0x7d6ab38>"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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whvABAMYQPgDAGMIHABhD+AAAYwgfAGAM4QMAjCF8AIAxhA8AMIbwAQDGED4AwBjCBwAY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\nxhA+AMAYwgcAGOP/AJPLMc9jQbOKAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7d72940>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "#filtered_transformed_signal[6, 6] = 0\n",
    "#filtered_transformed_signal[2, 2] = 0\n",
    "#filtered_transformed_signal[2, 6] = 0\n",
    "#filtered_transformed_signal[6, 2] = 0\n",
    "#filtered_transformed_signal[1, 6] = 0\n",
    "#filtered_transformed_signal[6, 1] = 0\n",
    "#filtered_transformed_signal[1, 2] = 0\n",
    "#filtered_transformed_signal[2, 1] = 0\n",
    "\n",
    "#filtered_transformed_signal\n",
    "\n",
    "fig = plt.figure(figsize=(16.0, 10.0))\n",
    "ax = fig.add_subplot(111)\n",
    "ax.imshow(abs(filtered_transformed_signal), interpolation='nearest', cmap=cm.gray)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "### Do the reverse transform"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "filtered_signal = np.fft.ifft2(filtered_transformed_signal)\n",
    "#filtered_signal"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.image.AxesImage at 0x7f2d4a8>"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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QPgDAGMIHABhD+AAAYwgfAGAM4QMAjCF8AIAxhA8AMIbwAQDGED4AwBjCBwAY4+zSA0xQ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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f445f8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = plt.figure(figsize=(16.0, 10.0))\n",
    "ax = fig.add_subplot(111)\n",
    "ax.imshow(abs(filtered_signal), interpolation='nearest', cmap=cm.gray)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "#shifted_filtered_signal = np.fft.ifftshift(transformed_signal)\n",
    "#shifted_filtered_signal"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "#shifted_transformed_signal = np.fft.fftshift(transformed_signal)\n",
    "#shifted_transformed_signal"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.5.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}