{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Gibbs sampling\n", "\n", "If you find this ipython notebook is unclear or contains bugs, please contact qiuwch@gmail.com \n", "If there's an error says \"something is undefined\", please run cell which contains the definition or use \"menu -> cell -> run all above\"" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Foreground/background classification.\n", "\n", "Here we consider a model for foreground/background classification so that it can include spatial context. Intuitively neighboring pixels in the image are likely to belong to the same class, i.e. are likely to be either all background or all foreground. This is a form of prior knowledge, or natural statistic, which can be learnt by analyzing natural images.\n", "\n", "For pixel i, the foreground label is $ S_i = 1 $, and background label is $ S_i $ = -1.\n", "\n", "The prior term encourages neighbouring pixels to have the same intensity: \n", "$ E_P[S] = \\gamma \\sum_{i} \\sum_{j \\in N(i)} { - S_i S_j} $ \n", "\n", "The data term is defined as:\n", "\n", "$ E_d[S, I] = \\eta \\sum_{i} (I_i - S_i)^2 $\n", "\n", "\n", "These two terms are combined to get the energy.\n", "\n", "$ E(S) = E_p[S] + E_d[S, I] $\n", "\n", "Then the posterior of the labeling S given the image I is\n", "\n", "$ P(S|I) = \\frac{1}{Z} exp( - \\frac{E[S]}{T} ) $" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The block of code below initializes the ipython notebook" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [], "source": [ "# Initiialization code\n", "%matplotlib inline\n", "import numpy as np\n", "# from pylab import imshow, show, get_cmap, imread, figure, subplots, title, subplot\n", "import matplotlib.pyplot as plt\n", "from numpy import random" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The block of code below loads an image and convert it to range (-1, 1)." ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "original range: 0.0 0.846330682811\n", "remaped range: -1.0 1.0\n" ] }, { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAW8AAAEACAYAAAB8nvebAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzsvclvXVl25vud2/cd72VPkaL6zAgpQk6n7TBcgA0DfoZh\n", "Gw7AfvDAI9s1evM3Tf8RNajBQwGFMvAAG1VA1qDgQmVmJaoiHb0shRTqKFEUJTaXt+952zegf0v7\n", "3qQiIsvOpzSKGxAkkbc5Z5+91/rWt761tjeZTHQ2zsbZOBtn41/W8L3pCzgbZ+NsnI2z8bOPM+N9\n", "Ns7G2Tgb/wLHmfE+G2fjbJyNf4HjzHifjbNxNs7Gv8BxZrzPxtk4G2fjX+A4M95n42ycjbPxL3D8\n", 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skimage.color.rgb2gray(tinyim)\n", "\n", " # Linear map the data to -1, 1\n", " scale = grayim.max() - grayim.min()\n", " data = 2 * (grayim - grayim.min()) / scale - 1\n", " if debug:\n", " print 'original range:', grayim.min(), grayim.max()\n", " print 'remaped range:', data.min(), data.max()\n", "\n", " return [data, tinyim]\n", "\n", "[data, im] = preproc_data(im, debug=True) # data is normalized image" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The block of code below defines neighborhood structure for gibbs sampler." ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": false }, "outputs": [], "source": [ "def getneighor(y, x, h, w): # get 4-side neighbor\n", " n = []\n", " if (x != 0): n.append((y, x-1))\n", " if (x != w-1): n.append((y, x+1))\n", " if (y != 0): n.append((y-1, x))\n", " if (y != h-1): n.append((y+1, x))\n", " return n\n", "\n", "def poslist(h,w):\n", " '''Get point list of a grid'''\n", " pos = []\n", " for x in range(w):\n", " for y in range(h):\n", " pos.append((y, x))\n", " return pos" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Define a utility function to compute energy." ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "collapsed": false }, "outputs": [], "source": [ "def energy_prior(state, gamma):\n", " total = 0\n", " (h, w) = state.shape\n", " pos = poslist(h, w)\n", " for p in pos:\n", " neighbor = getneighor(p[0], p[1], h, w) # compute neighbor\n", " \n", " for n in neighbor:\n", " total += state[p[0]][p[1]] * state[n[0]][n[1]]\n", " E = - gamma * total / 2 # gamma is a global variable\n", " return E/2\n", " \n", "def energy_data(state, data, eta):\n", " E = eta * sum((data - state)**2)\n", " return E\n", "\n", "def energy(state, data, gamma, eta):\n", " return energy_prior(state, gamma) + energy_data(state, data, eta)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Define the gibbs sampler." ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "collapsed": false }, "outputs": [], "source": [ "def gibbs_sampler(state, data, gamma, eta, debug=False): # 0/1 state\n", " (h, w) = state.shape\n", " new_state = state.copy()\n", " pos = poslist(h, w)\n", " for p in pos:\n", " neighbor_pos = getneighor(p[0], p[1], h, w)\n", " neighbor_value = [new_state[n[0]][n[1]] for n in neighbor_pos]\n", "\n", " tmp1 = -gamma * -1 * sum(neighbor_value) # x_i = -1\n", " tmp2 = -gamma * 1 * sum(neighbor_value) # x_i = 1\n", " \n", " # add data term\n", " v = data[p[0]][p[1]]\n", " tmp1 += eta * (v - (-1))**2 # x_i = -1\n", " tmp2 += eta * (v - 1)**2 # x_i = 1\n", " \n", " tmp1 = np.exp(-tmp1)\n", " tmp2 = np.exp(-tmp2)\n", "\n", " p1 = tmp1 / (tmp1 + tmp2)\n", " prob = random.uniform() # roll a dice\n", " \n", " if (debug): print p1\n", " if (prob > p1):\n", " new_state[p[0]][p[1]] = 1\n", " else:\n", " new_state[p[0]][p[1]] = -1\n", " return new_state" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Animation: sample with data term included\n", "Run this demo below" ] }, { "cell_type": "code", "execution_count": 22, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAWcAAAD9CAYAAABgMrYcAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAADUZJREFUeJzt3V2oZeV9x/HvT42Yt2awwig6sHMRwUrbaMCkjW3GMAYT\n", "ivXKVGiRIr1KieQiOOlFB69ictMUSm9aI4MXFknaQUnTOLHO0NxYJU59zyQ0B9R2joG8kDaEJvjv\n", "xV7TbsfzNufsffZ/n/39wGLWevbb/8HxNw/rWc9aqSokSb1cMO8CJElvZThLUkOGsyQ1ZDhLUkOG\n", "syQ1ZDhLUkM7CucktyR5Ocl3k9wzraIkadllu9c5J7kQ+A5wCHgNeAq4o6peml55krScLtrBZ28A\n", "vldVKwBJ/g74feD/wjmJK1wkaQNVlbXadxLOVwKvTBy/CnzwrW87ApwADu7gp7o6QYd+HeHeqX3X\n", "CTr0aLpOYJ8WwQm216d7OTLdQqbqBBv3av3/d3cSzlscFZ8AVoY/R8MmSctoZdg2t5Nwfg04MHF8\n", "gPHo+RwH2Zv/zkvS+Rrx5gHqyXXfuZOrNZ4G3pdklORi4JPAI+sXtBeN5l3A1I3mXcAMjOZdwAyM\n", "5l3ADIzmXcBMjLb9yW2PnKvql0n+FPgGcCFw//pXaoy2+zPNjeZdwNSN5l3ADIzmXcAMjOZdwAyM\n", 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clear_output\n", "import time\n", "\n", "random_seed = 1 # Change this in your experiment\n", "random.seed(random_seed)\n", "\n", "(h, w) = data.shape\n", "mat = random.random((h,w))\n", "mat[mat>0.5] = 1\n", "mat[mat<=0.5] = -1\n", "random_state = mat\n", "init_state = random_state\n", "\n", "new_state = random_state.copy()\n", "E = [energy(init_state, data, gamma=20, eta=1)]\n", "\n", "f, ax = plt.subplots() # prepare animation\n", "for i in range(60):\n", " clear_output(wait=True)\n", " new_state = gibbs_sampler(new_state, data, gamma=20, eta=1)\n", "\n", " # time.sleep(1)\n", " myimshow(new_state)\n", " display(f)\n", "\n", "plt.title(\"Foreground\")\n", "mask = (new_state==1)\n", "fg = im.copy()\n", "for i in range(3): \n", " fg[:,:,i] = fg[:,:,i] * mask\n", "plt.imshow(fg, cmap='gray', interpolation='nearest')\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Exercise 1. Gibbs sampler\n", "Set random_seed to a different value\n", "1. Try different values of $ \\gamma $, $ \\eta $, including special case that only contains prior term. What happens when the parameters change?\n", "2. Run with different images, plot your result. Find two or three images from web or your image collection.\n", "3. When does the sampler converge for the Einstein image with $ \\gamma = 20 $ and $ \\eta = 1 $ and how do you know it?" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Run with different images and plot results." ] }, { "cell_type": "code", "execution_count": 23, "metadata": { "collapsed": false }, "outputs": [], "source": [ "def runGibbsSampler(data, gamma, eta, showAnimation=True):\n", " (h, w) = data.shape\n", " mat = random.random((h,w))\n", " mat[mat>0.5] = 1\n", " mat[mat<=0.5] = -1\n", " random_state = mat\n", " init_state = random_state\n", "\n", " new_state = random_state.copy()\n", " E = [energy(init_state, data, gamma=gamma, eta=eta)]\n", " \n", " if showAnimation:\n", " f, ax = subplots() # prepare animation\n", " \n", " for i in range(60):\n", " clear_output(wait=True)\n", " new_state = gibbs_sampler(new_state, data, gamma, eta)\n", " E.append(energy(new_state, data, gamma=gamma, eta=eta))\n", " # time.sleep(1)\n", " if showAnimation:\n", " myimshow(new_state)\n", " display(f)\n", " else:\n", " print 'iteration %d' % i\n", "\n", " return new_state\n", "\n", "def fgBgSegmentation(filename):\n", " im = plt.imread(filename)\n", " [data, im] = preproc_data(im, scale=0.5)\n", " state = runGibbsSampler(data, eta=1, gamma=1, showAnimation=False)\n", "\n", " mask = (state==1)\n", " fg = im.copy()\n", " for i in range(3): \n", " fg[:,:,i] = fg[:,:,i] * mask\n", "\n", " plt.subplot(1,3,1)\n", " plt.imshow(im, interpolation='nearest'); plt.axis('off');\n", " plt.subplot(1,3,2)\n", " plt.imshow(state); plt.axis('off');\n", " plt.subplot(1,3,3)\n", " plt.imshow(fg, interpolation='nearest'); plt.axis('off');" ] }, { "cell_type": "code", "execution_count": 21, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "iteration 59\n" ] }, { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAXIAAAB7CAYAAABpXAt1AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", 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"output_type": "stream", "text": [ "iteration 59\n" ] }, { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAXQAAABeCAYAAAAzI++3AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzsvXm8JEd15/uNiFxquWv37VZ3q7vVLYHUFtqFsCUhISF2\n", "DNieGWzZlp8fnjHMY/AYeGxmMI/x88cYPw94vM3YY2Mb22O8gAcvICO2h5/NYpCNJNSotTSi9+67\n", "1Z5LRLw/IrNu1b117627d7fq9/nkp6qyMjIj8+Q5GXnid84R1loGGGCAAQa48CG3ugMDDDDAAAOs\n", "DwYGfYABBhjgIsHAoA8wwAADXCQYGPQBBhhggIsEA4M+wAADDHCRYGDQBxhggAEuEqzJoAshXiaE\n", "OCyEOCKEeMd6dWqArcVArhcnBnK9+CFWy0MXQijgW8CLgOPAV4F7rbWPrl/3BthsDOR6cWIg12cG\n", "1jJCfx7wuLX2qLU2Af4EeM36dGuALcRArhcnBnJ9BmAtBv1S4Dsdv49l6wa4sDGQ68WJgVyfAViL\n", "QR/kDLg4MZDrxYmBXJ8B8NbQ9jiwr+P3PtxTvw0hxOAmOk9grRV9bjqQ6wWEgVwvXqxAtm2sxaD/\n", "E/BsIcQB4ATwg8C98zfyR7YjhGgvAEIohBBIKRFCoJRCyLmuNCuTlMd2ZP977e3yT6Q3165j31On\n", "jrJ99wGEmHvxEEJgrcVa2/HdgNHt9VOnv8PYjkux1hILC9aATsFasBpfKwyW1Gi3j+wyV6fOUBrd\n", "3j6WtRaMBsAY03Ud8snnZm2GQnl0wfpe23Yiqs9SLA712La3zPNzi5pVkrjec5tF0Jdc4b0r2WeG\n", 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