{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Image statistics" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "##First-order" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [], "source": [ "# Load neccessary python modules\n", "from skimage import io, color, filter # skimage is an image processing library\n", "import matplotlib.pyplot as plt # matplotlib provides plot functions similar to MATLAB\n", "import numpy as np # numpy provides efficient matrix implementation in python\n", "from scipy import ndimage, stats, signal # scipy provides scientific computing functions\n", "from IPython.html.widgets.interaction import interact \n", "# interact function can create an interactive interface to allow user to manipulate parameters and see the result, \n", "# similar to \"Manipulate\" in Mathematica\n", "\n", "import math\n", "\n", "%matplotlib inline" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "##Load test image" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAQcAAAEACAYAAAC+rrMfAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzsvVmQXWd1BvrtM89jz5PUkro1W7MtydgWHinb4AHZRpAC\n", "4xCGkEpRVFLwkgQqRYrch/ASkwrkEnCIDTFEwTYg28GjPEgWtizJmqUeT/fp7jPP874P7W/pP9un\n", "ZQniiNzqv6qru8/eZ4//Wutb3xp+Tdd1LI7FsTgWh3GYrvQFLI7FsTj+MMeiclgci2NxtByLymFx\n", "LI7F0XIsKofFsTgWR8uxqBwWx+JYHC3HonJYHItjcbQcH4hy0DTtI5qmndQ07YymaV/7IM6xOBbH\n", "4vhgh/Y/neegaZoZwCkANwOIAHgDwB5d10/8j55ocSyOxfGBjg8COVwN4Kyu66O6rlcB/ATAXR/A\n", "eRbH4lgcH+D4IJRDL4AJ5f/Jdz9bHItjcfwfGh+EcljMx14ci+P/B8PyARwzAqBf+b8f8+hBhqZp\n", "iwpkcSyOKzR0XdcuZb8PQjkcAjCkadpSAFMAHgCwx7jTX//1X8NkMqFer6NUKiGdTmNwcBCf+9zn\n", "8Pbbb+MnP/kJTCYTCoUCarUaurq6UK/X4Xa7oes6Jicn0Wg0oOs6Vq9ejXw+D5PJhGKxiHvvvRcW\n", "iwWnT58GAGiahltvvVXO5fV6kc1m4XA4UKlUUCqVcOrUKRw9ehTFYhFutxv1eh02mw31eh2//e1v\n", 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}, "metadata": {}, "output_type": "display_data" } ], "source": [ "im = io.imread('../data/imgstat/alpine.png');\n", "# im = io.imread('../data/imgstat/lake.jpg')\n", "grayIm = color.rgb2gray(im)\n", "plt.figure(); io.imshow(grayIm)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "##Histogram of intensities" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAYAAAAEACAYAAAC6d6FnAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAE01JREFUeJzt3X+sZGddx/H3xwIJ2OpKakp/kVZZfizhR21oUTBco2kW\n", "/qBVEwoIojQErVDCP9LyB13/ENEEA8TQGC1QopQ0oFigFAr2BjTQTUl/LCwrXWVNd2kXQRAwTdjG\n", "r3/M6e50Onvv3HvnnvnxvF/JJGeeOTPz3HNnzud5nnOeM6kqJEnt+alZV0CSNBsGgCQ1ygCQpEYZ\n", "AJLUKANAkhplAEhSo9YMgCTnJrk9ydeTfC3JVV35niSHk9zV3V469JxrktyX5ECSS4bKL0yyr3vs\n", "vdv3J0mSJpG15gEkeQrwlKq6O8mpwFeBy4BXAD+qqr8cWX8X8BHgBcDZwOeBnVVVSfYCb6qqvUlu\n", 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bins=100)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "##Histogram of spatial derivatives of intensity\n", "\n", "Pick the radius of the filter (pixelradius), and the degree (nderivative) of the spatial derivative." ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAQcAAAEACAYAAAC+rrMfAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzsvVuobWtW79f6mGte12XfTKr23rUpRVQoOHgCuhXOg+dB\n", "QuCAkhdFCJFoICiJIeTBMg+5QnEUIiHnQXI5HjREY0E4Yh4k6oN4AtauqlAhgTonRLFAy7LKqqKq\n", "9pr3S8/Dmv8+f/0//l8ffcy1Vo25D7PBYNz66N/3ta99//Zv7buMru/7upd7uZd7cVlsugL3ci/3\n", "cjflHhzu5V7uJco9ONzLvdxLlHtwuJd7uZco9+BwL/dyL1HuweFe7uVeorwUcOi67t/ouu5fdF33\n", "/3Vd94svo4x7uZd7ebnSveh1Dl3XbVXV/1tVP1pVX6yqz1TVT/V9/89faEH3ci/38lLlZTCHd6vq\n", "z/q+/0Lf9+dV9b9U1Y+/hHLu5V7u5SXKywCHt6vqL/H+r64/u5d7uZcPkLwMcLhfj30v9/IvgTx4\n", 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"io.imshow(fgrayIm)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "##Plot the histogram and calculate its kurtosis" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Kurtosis 3.01058750335\n", "Kurtosis = 4.660904\n" ] }, { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAYoAAAEACAYAAACtVTGuAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAE21JREFUeJzt3X+sXOdd5/H3p/mhpjS7VlTk/FwlKxw1rkBtLWwkKuVW\n", "u5t6u7tJYEWSCoVIWAgUUBF/AHZBG/efblqtBCko+Wd/xKm2WVkqDUHJpnFDRrtCIm5p2poakxgw\n", "wm59g0pLi2CXZPvdP+6x7/hm7nNn7Pl177xf0shnnjlnzjmPz53PPM9zzplUFZIkrects94ASdJ8\n", "MygkSU0GhSSpyaCQJDUZFJKkJoNCktQ0VFAkOZXkq0leTnK0K7smyZEkryR5Psm2vvkPJHk1yYkk\n", "d/SV70pyrHvtkfHvjiRp3IZtURSwVFXvqardXdl+4EhV3Qq80D0nyU7gXmAnsBd4NEm6ZR4D9lXV\n", "DmBHkr1j2g9J0oSM0vWUNc/vBA5104eAu7vpu4Anq+r1qjoFnAT2JLkOuLqqjnbzPdG3jCRpTo3S\n", 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fisher=False)\n", "print 'Kurtosis = %f' % stats.kurtosis(fgrayIm.flatten(), fisher=False)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Exercise\n", "\n", "Try increasing the order of the derivative" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Kurtosis = 3.859692\n" ] }, { "data": { "text/plain": [ "" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAQcAAAEACAYAAAC+rrMfAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJztvVGsdUl1HvjVvf8POLFxCzE0dNOGfqAl2i9YlrAsJiYP\n", "FmNeTPJiBikKwp5RZM/EUWbEAJE8wX5ANlZQlJEmmoztCGfGJHgsEHlAA4wGySPZYYggBre77Za6\n", "bWigaRyssbFM//f+ex7uXff/7ne/tapqn3P/c35ylnR09q5du2pV7VpffWtV7XPasiw4yEEOchCV\n", "o10rcJCDHGQ/5QAOBznIQawcwOEgBzmIlQM4HOQgB7FyAIeDHOQgVg7gcJCDHMTKtYBDa+3HWmuP\n", 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"display_data" } ], "source": [ "from IPython.html.widgets import interact\n", "def manipulateGuassianOrder(order=0):\n", " pixelRadius = 16;\n", " fgrayIm = ndimage.gaussian_filter(grayIm, sigma = pixelRadius/2, order=(order, order))\n", " io.imshow(fgrayIm)\n", " plt.figure()\n", " plt.hist(fgrayIm.flatten(), bins = 100);\n", " print 'Kurtosis = %f' % stats.kurtosis(fgrayIm.flatten(), fisher=False)\n", "interact(manipulateGuassianOrder, order= (0, 3))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "##Second order statistics" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAQcAAAEACAYAAAC+rrMfAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzsvVmQXWd1BvrtM89jz5PUkro1W7MtydgWHinb4AHZRpAC\n", "4xCGkEpRVFLwkgQqRYrch/ASkwrkEnCIDTFEwTYg28GjPEgWtizJmqUeT/fp7jPP874P7W/pP9un\n", 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color.rgb2gray(im)\n", "img256 = LinearScale(grayIm, scaleRange=(0,255))\n", "whitenIm = img256 - img256.mean()\n", "\n", "io.imshow(whitenIm)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "##Autocorrelation" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Illustrates how nearby pixel intenstities tend to be correlated. Can you fit the autocorrelation with an exponential?" ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "[]" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAXUAAAEACAYAAABMEua6AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAF+1JREFUeJzt3X+MXeV54PHvgyHb0LAhNiu6AnfpBlOBU1iyikO3NNyU\n", "X+OKhsqulLjQtIFlrFW8JBtK7UERjHezuLQqS1xKNRA6W4UqTrAdSnbBhqS5IURtAAUcHGwYL0E7\n", "NikBul45aazaO8/+ca+dy/X9OXPHM/fc70ca5Z5z3nvu+yro8avnPO97IjORJBXDCXPdAUlS7xjU\n", "JalADOqSVCAGdUkqEIO6JBWIQV2SCqRtUI+Iv4iI1yLi+RZtNkbERETsiIgLe9tFSVKnOpmpjwND\n", "zS5GxK8DZ2fmEmAY+PMe9U2S1KW2QT0zvwn8nxZNPgT8ZbXtt4FTI+L03nRPktSNXuTUzwAma473\n", "Amf24L6SpC716kFp1B2794AkzYETe3CPfcDimuMzq+feIiIM9JI0DZlZP3Fuqhcz9YeBjwJExEXA\n", "/sx8rVHDU075BJs3byMzC/d32223zXkfHJ9jc3zF++tW25l6RHwBuAQ4LSImgduAkwAycywzH4mI\n", 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up kernels for the filters" ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAPwAAAD7CAYAAABOrvnfAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAEXhJREFUeJzt3W9sVNeZBvDnwWCwMX9kovDHxowdm1UKbFK2G5nsooZC\n", "JYiqpIpWaqJuShOpn7bbtFqlDam0GylfVq26TbVVPmybREFtUqk0ilIpsDhgNUSBhLRAzZ8Eg8Fg\n", "RwZi1hgDBgzvfpjBa7uesf3OPRfY8/wkpJm5973njM3jc+fOvffQzCAicZh0szsgIulR4EUiosCL\n", "RESBF4mIAi8SEQVeJCKTQ22YpL7vE7lJzIyjvR4s8ADw3HPPjfp6c3MzVq1aNeqySZN8Ox0lJSV5\n", "l23btg2rV692bTefq1ev5l1W6P1dvHjR1d7ly5fzLtu1axcaGxtHXTZ5su9XPGPGjLzLCr2/iooK\n", "V3uF+rl582asW7du1GWFfi6FXLp0Ke+yQu/P214+O3bswMqVK/Mu9/z+nn/++bzL3Lv0JNeS/Jhk\n", "K8kfeLcjIulxBZ5kCYCfA1gL4HMAHiN5d5IdE5HkeUf4+wAcMbPjZnYVwG8APDze4kwm42zWp7a2\n", "NtX20n5/1dXVqbaX9vurr69Ptb00319NTU1qbQH+wFcBODnkeUfutXFJO4B1dXWptpf2+0s78Gm/\n", "v4aGhlTbS/P9LVq0KLW2AH/gdQRe5DbkPUrfCWDhkOcLkR3lh2lubh58nMlkUh8ZRGJw/PhxtLe3\n", 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range(fsize))\n", " x = x - fsize/2\n", " y = y - fsize/2\n", " imFilter = np.exp((-x**2 -y**2) / (2 * sigma**2)) * func(2*math.pi * (fx * x + fy * y))\n", " return imFilter\n", "\n", "def SinGabor(fx, fy, sigma, fsize):\n", " return GaborFilter(fx, fy, sigma, fsize, func=np.sin)\n", "def CosGabor(fx, fy, sigma, fsize):\n", " return GaborFilter(fx, fy, sigma, fsize, func=np.cos)\n", "\n", "filter1 = SinGabor(1.0/8, -1.0/8, 4.0, 16)\n", "filter2 = SinGabor(1.0/8, 1.0/8, 4.0, 16)\n", "plt.figure(); io.imshow(filter1)\n", "plt.figure(); io.imshow(filter2)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "###Convolution" ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAQcAAAEACAYAAAC+rrMfAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzsvVmQXWd1BvrtM89jz5PUkro1W7MtydgWHinb4AHZRpAC\n", 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{}, "source": [ "The above filters are centered at the same place, but orthogonal to each other. We can also shift the filtered outputs relative to each other with the variable offset below." ] }, { "cell_type": "code", "execution_count": 17, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 17, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAP0AAAD+CAYAAADxoQNSAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzsvVmsJFd65/c7J5bc75Z3q1srq7izyGY3u4eiWhq1DGEs\n", "wMD4zYYAAwNjXg0YfpL84ueZeTJs+MXA2Jjxgz0DGBDmZWBoDLSs1rRarVaz2dUki2Tty92X3DNj\n", "OZ8fvohc7q26dYuslZ1/IHCXiIw4mRlfnO98y/9vRIQpppjitwf2eQ9giimmeLaYGv0UU/yWYWr0\n", "U0zxW4ap0U8xxW8ZpkY/xRS/ZZga/RRT/JbhGxm9MeaPjTGfG2O+NMb86ZMa1BRTTPH0YL5unt4Y\n", "4wFXgT8C7gE/B/5ERD57csObYoopnjS+yUz/D4CvROSmiMTA/wX8509mWFNMMcXTgv8NXnsauDP2\n", "913gw/EDjDHTcr8ppnhOEBHzoP9/E6M/oUH/AXATuDC2PU/8GPjRcx5Djh/z4owFXqzx/JgXZyxw\n", "eDy1s4bzf2y58J9azv+xBYFBU4ga0L4nrP9UWP+pY/0/OgaNpzsWxc1sy/GXD331NzH6e8DZsb/P\n", 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"metadata": {}, "source": [ "##Bow-tie: Normalize the columns to represent conditional probabilities - i.e. elements in a column sum to one" ] }, { "cell_type": "code", "execution_count": 18, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 18, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAP0AAAD+CAYAAADxoQNSAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzsvVmMJFt63/c7kXutXdX7vX3n9mwUSclcNCINSAZmZIwN\n", "QZDGb7YJ2BgYfDUg2IDBkR9s8MEAyRcbhgHDNmRjTBgyCRgaE/CDNKZxB5YfDFEkIdHkkLwzczl3\n", "6a7u2iv3zIjjh+98FV+eity6a6/4A4GsyoyIjIg8//Pt33Hee0qUKHF3kFz1BZQoUeJyUZK+RIk7\n", "hpL0JUrcMZSkL1HijqEkfYkSdwwl6UuUuGN4K9I75/6Wc+77zrk/d879ynldVIkSJS4O7k3j9M65\n", "CvCnwNeBT4F/BvyS9/5Pzu/ySpQocd54G0n/i8CH3vuPvPcj4H8F/q3zuawSJUpcFKpvcey7wMfm\n", "/0+Af9Xu4Jwr0/1KlLgieO9d0ftvQ/oFCf1V4CPgudmuEh8AX7via1B8wPW5Frhe1/MB1+da4Oz1\n", 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P. (2001). Natural signal statistics and sensory gain control. Nature Neuroscience, 4(8), 819–825. doi:10.1038/90526" ] } ], "metadata": { "kernelspec": { "display_name": "Python 2", "language": "python", "name": "python2" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython2", "version": "2.7.8" } }, "nbformat": 4, "nbformat_minor": 0 }