{ "metadata": { "name": "", "signature": "sha256:b061ba84d6aece5dc5e04755d89cf97777a467a9bcc41fa9ee70095a63622ec3" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Cooperative Computation of Stereo Disparity\n", "\n", "Here I will attempt to demonstrate how Marr and Poggio's (1976) cooperative algorithm computes stereo disparity. \n", "As usual, we want to solve the stereo correspondence problem - that is, we want to reject false matches to recover the correct disparity.\n", "\n", "To do so, Marr and Poggio used a recurrent network that incorporates two constraints:\n", "* Uniqueness: each element in the image can be assigned at most one disparity value\n", "* Continuity: disparity varies smoothly across the image space\n", "\n", "These constraints were implemented by, respectively:\n", "* Inhibition along units in the same line of sight (which therefore represent different disparities)\n", "* Local excitation along units with similar disparity\n", "\n", "The algorithm can be described by the equation below. The activity $C$ of a unit at the 3-dimensional location $(x,y,d)$ at iteration $n+1$ can be described as: \n", "\n", "$$ C_{(x,y,d)}^{n+1} = \\sigma \\left( \\sum C_{(x,y,d) \\in E}^{n} - \\epsilon \\sum C_{(x,y,d) \\in I}^{n} + C_{(x,y,d)}^{0} \\right) $$\n", "\n", "where $E$ and $I$ are the excitatory and inhibitory intputs to the unit, and epsilon is an inhibitory constant. Below is an implementation of the algorithm (highly inefficient, by the way).\n" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", "%pylab inline" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "Populating the interactive namespace from numpy and matplotlib\n" ] } ], "prompt_number": 1 }, { "cell_type": "code", "collapsed": false, "input": [ "def coop_stereo(limg, rimg, max_iter=5, plot_flag=0, n_layers=10):\n", "\n", " im_h = np.shape(limg)[0]\n", " im_w = np.shape(limg)[1]\n", "\n", " # initialization\n", " c = np.zeros((im_h, im_w, im_w, max_iter))\n", " for i in np.arange(im_h):\n", " c[i,:,:,0] = np.dot(limg[i,:].reshape(-1,1),rimg[i,:].reshape(1,-1))\n", " c[c<0] = 0.\n", "\n", " # depth layers\n", " d_layers = np.arange(n_layers+1)-np.floor(n_layers/2.)\n", "\n", " # layer mask\n", " l_mask = np.zeros((im_w,im_w))\n", " for d in d_layers:\n", " for i in np.arange(im_w):\n", " if (i+d>=0) & (i+d=0) & ((r+h)=0) & ((i+xe)=0) \n", " & ((i+d+xe)=0)):\n", " \n", " # integrate the excitatory input\n", " c_e[i,i+d] += c[r+h, i+xe, i+d+xe, n-1]\n", " \n", " \n", " ### update activity\n", " act = c_e - epsilon*c_i + c[r,:,:,0]\n", "\n", " # threshold function (we could also use a sigmoid activation function here)\n", " c[r,:,:,n] = (act>=theta) \n", " \n", " d_map[r,:,n] = np.arange(im_w) - np.argmax(act, axis=1)\n", " d_map[d_mapnp.amax(d_layers)] = 0\n", "\n", " if plot_flag == 1:\n", " plt.imshow(d_map[:,:,n], interpolation='nearest', cmap='gray', origin='lower')\n", " plt.colorbar()\n", " plt.title('Iteration ' + str(n))\n", " plt.show()\n", "\n", " # next iteration\n", " n += 1\n" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 5 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us demonstrate the algorithm on a simple random dot stereogram depicting a central square in depth." ] }, { "cell_type": "code", "collapsed": false, "input": [ "# 2d image\n", "d = 4. # px\n", "im_size = 80.\n", "sq_size = 40.\n", "sq_start = (im_size - sq_size)/2.\n", "\n", "# generate 2d image\n", "cimg = np.random.randint(0,2,(im_size,im_size))*2-1\n", "limg = np.array(cimg[:,:])\n", "rimg = np.array(cimg[:,:])\n", "\n", "sq_img = np.random.randint(0,2,(sq_size,sq_size))*2-1\n", "limg[sq_start:(sq_start+sq_size), (sq_start-d/2):(sq_start+sq_size-d/2)] = sq_img\n", "rimg[sq_start:(sq_start+sq_size), (sq_start+d/2):(sq_start+sq_size+d/2)] = sq_img\n", "\n", "plt.subplot(2,2,3)\n", "plt.imshow(rimg, interpolation='nearest', cmap='gray')\n", "plt.axis('off')\n", "plt.subplot(2,2,4)\n", "plt.imshow(limg, interpolation='nearest', cmap='gray')\n", "plt.axis('off')\n", "plt.show()" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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"text": [ "" ] } ], "prompt_number": 14 }, { "cell_type": "code", "collapsed": false, "input": [ "coop_stereo(limg, rimg, max_iter=5, plot_flag=1, n_layers=10)" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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"text": [ "" ] } ], "prompt_number": 15 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now let us try a more complex stereogram, similar to those used in Marr and Poggio's original paper." ] }, { "cell_type": "code", "collapsed": false, "input": [ "# 2d image\n", "im_size = 80.\n", "sq_size = [50., 30.]\n", "d = [2., 4.] # px\n", "\n", "# generate 2d image\n", "cimg = np.random.randint(0,2,(im_size,im_size))*2-1\n", "limg = np.array(cimg[:,:])\n", "rimg = np.array(cimg[:,:])\n", "\n", "for ss in np.arange(np.shape(sq_size)[0]): \n", " sq_start = (im_size - sq_size[ss])/2.\n", " sq_img = np.random.randint(0,2,(sq_size[ss],sq_size[ss]))*2-1\n", " limg[sq_start:(sq_start+sq_size[ss]), (sq_start-d[ss]/2):(sq_start+sq_size[ss]-d[ss]/2)] = sq_img\n", " rimg[sq_start:(sq_start+sq_size[ss]), (sq_start+d[ss]/2):(sq_start+sq_size[ss]+d[ss]/2)] = sq_img\n", "\n", "plt.subplot(2,2,3)\n", "plt.imshow(rimg, interpolation='nearest', cmap='gray')\n", "plt.subplot(2,2,4)\n", "plt.imshow(limg, interpolation='nearest', cmap='gray')\n", "plt.show()" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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