{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Gradient Descent" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Populating the interactive namespace from numpy and matplotlib\n" ] } ], "source": [ "from __future__ import division\n", "%pylab inline\n", "%load_ext autoreload\n", "%autoreload 2" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$\\newcommand{\\dotp}[2]{\\langle #1, #2 \\rangle}$\n", "$\\newcommand{\\umin}[1]{\\underset{#1}{\\min}\\;}$\n", "$\\newcommand{\\RR}{\\mathbb{R}}$\n", "$\\newcommand{\\pd}[2]{ \\frac{ \\partial #1}{\\partial #2} }$\n", "$\\newcommand{\\norm}[1]{\\|#1\\|}$\n", "$\\newcommand{\\pa}[1]{\\left(#1\\right)}$\n", "\n", "We consider the problem of finding a minimizer of a convex smooth function $\\;f : \\RR^d \\rightarrow \\RR$; that is, we want to solve\n", "$$\\umin{x \\in \\RR^d} f(x)$$\n", "\n", "Note that the minimizer is not necessarily unique.\n", "\n", "The simplest method is gradient descent, which iteratively computes\n", "$$ x^{(k+1)} = x^{(k)} - \\tau \\nabla f(x^{(k)}), $$\n", "where $\\nabla f(x) \\in \\RR^d$ is the gradient of $f$ at $x$, $x^{(0)} \\in \\RR^d$ is an arbitrary initial point, and the stepsize $\\tau$ must satisfy \n", "$$0<\\tau<2/\\beta,$$\n", "to have convergence, where $\\beta$ is a Lipschitz constant of $\\nabla f$; that is, \n", "$$ \\forall (x,x') \\in \\RR^N \\times \\RR^N, \\quad\n", " \\norm{\\nabla f(x) - \\nabla f(x')} \\leq \\beta \\norm{x-x'}. $$\n", "\n", "For instance, if $f$ is of class $C^2$, $$\\beta= \\sup_x \\norm{Hf(x)},$$\n", "where $Hf(x) \\in \\RR^{d \\times d}$ is the Hessian of $f$ at $x$ and $\\norm{\\cdot}$ is the spectral operator norm (largest eigenvalue). " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Gradient descent in 2-D" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We consider a simple problem, corresponding to the minimization of a 2-D ($d=2$) quadratic form\n", "$$ f(x) = \\frac{1}{2} \\pa{ x_1^2 + \\eta x_2^2 } ,$$\n", "where $\\eta>0$ controls the anisotropy, and hence the difficulty, of the problem." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can note that minimizing a strongly convex quadratic function is equivalent to solving a positive definite linear system. Gradient descent is then equivalent to the Richardson iteration applied to this linear system." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We define the anisotropy parameter $\\eta$:" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false }, "outputs": [], "source": [ "eta = 8" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We define the function $f$:" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [], "source": [ "f = lambda x : (x[0]**2 + eta*x[1]**2) / 2" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Note: a 'lambda' function is a one-line function definition; \n", "in Matlab, this would be `f=@(x)(x(1)^2+eta*x(2)^2)/2;`\n", "\n", "An example of function evaluation:" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "38.0" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "f([2,3])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We display the function using a contourplot:" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAXcAAAFrCAYAAAAq1MkeAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJztnX3MJld53q+btWlw+Vi7xF5Yr43BJjiRQsHCGOKWTcGK\n", "gdSOUhXqJCUKRLUiOaDSJHaChLZ/tIotVbiUCBHbodBUMchE1LQU4oRsEK0NxjHQwNp4MYbdpd7F\n", "YANpLAUrd/94nvHOzs73zJn7us/cP2m1z8e8z7nOmTO/c94z88wrqoogCIIgL55iHSAIgiCYn5B7\n", "EARBhoTcgyAIMiTkHgRBkCEh9yAIggwJuQdBEGTIKdYBCkQkrskMgiAYgapK9TUauQPAu1veu3Di\n", "Z38QwJsm/PzU8qfyHwD8mx7b7X5+6iTD2fddYN8Z1inGE/nn58iD/bft2/dTcWDiz091T1f5b214\n", "nUruFgK1lnYTVUk/87vAbrID1JTzFixLFyzv6wuVY8yQSUi57w8ZFOairyOmDgJTy69CJfc5qTbI\n", 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linspace(-.7,.7,101)\n", "[u,v] = meshgrid(t,t)\n", "F = (u**2 + eta*v**2) / 2\n", "contourf(t,t,F,35)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We define the gradient of $f$:" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [], "source": [ "Grad_f = lambda x : array([x[0], eta*x[1]])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "An example of evaluation:" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "array([ 1, 16])" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Grad_f([1,2])" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Since $f$ is quadratic, its Hessian is the constant matrix $\\left(\\begin{array}{cc}1&0\\\\0&\\eta\\end{array}\\right)$. Its spectral norm, which is the Lipschitz constant of `Grad_f`, is $\\beta=\\max(1,\\eta)$. \n", "\n", "The stepsize $\\tau$ must satisfy $0< \\tau < 2/\\beta$." ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "0.225" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "tau = 1.8/max(eta,1); tau" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now we implement the gradient descent method: given the initial estimate $x^{(0)}$ of the solution, the stepsize $\\tau$ and the number $k$ of iterations, we compute $x^{(k)}$:" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "array([ 0.07816584, 0.10737418])" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "nbiter = 10\n", "x = [1,1] # initial estimate of the solution\n", "for iter in range(nbiter): # iter goes from 0 to nbiter-1\n", " x = x - tau*Grad_f(x)\n", "x # to display x, like in Matlab. Use print(x) if this is not the last command of the cell, else nothing is displayed" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Note: there is no 'end' for the loops in Python; instead, the content of the loop is determined by the indentation: here four blank spaces before `x=...` " ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "0.049171809817584886" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "f(_) # _ is the current variable, like Matlab's ans" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We encapsulate the code in a function called `GradientDescent`." ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def GradDescent(Grad_f, x0, nbiter, tau):\n", " x = x0;\n", " for iter in range(nbiter): # iter goes from 0 to nbiter-1\n", " x = x - tau*Grad_f(x) # x has type 'array'. Like in C, one can also write x-=tau*Grad_f(x)\n", " return x" ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "array([ 0.07816584, 0.10737418])" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" } ], "source": [ "GradDescent(Grad_f,[1,1],10,tau)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We define a function called `GradDescentArray` which puts the iterates in a 'matrix' (a 2-D array in fact); they are first put in a list and the list is converted to an array at the end (the + operation on lists concatenates them, whereas on arrays this is the classical elementwise addition)." ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "collapsed": false }, "outputs": [], "source": [ "def GradDescentArray(Grad_f, x0, nbiter, tau):\n", " x = x0;\n", " xlist=[list(x0)]; # the cast is just in case x0 would be an array\n", " for iter in range(nbiter): \n", " x = x - tau*Grad_f(x) \n", " xlist = xlist + [list(x)]\n", " return array(xlist)" ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "array([[ 0.6 , 0.6 ],\n", " [ 0.465 , -0.48 ],\n", " [ 0.360375 , 0.384 ],\n", " [ 0.27929063, -0.3072 ],\n", " [ 0.21645023, 0.24576 ],\n", " [ 0.16774893, -0.196608 ],\n", " [ 0.13000542, 0.1572864 ],\n", " [ 0.1007542 , -0.12582912],\n", " [ 0.07808451, 0.1006633 ],\n", " [ 0.06051549, -0.08053064],\n", " [ 0.04689951, 0.06442451]])" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "xarray=GradDescentArray(Grad_f,[0.6,0.6],10,0.225)\n", "xarray" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can access the elements of an array like in Matlab. Be careful: like in C, the first element has index 0! Some examples:" ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "0.46499999999999997" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "xarray[1,0]" ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "array([ 0.465, -0.48 ])" ] }, "execution_count": 16, "metadata": {}, "output_type": "execute_result" } ], "source": [ "xarray[1,:]" ] }, { "cell_type": "code", "execution_count": 17, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "array([ 0.6 , 0.465 , 0.360375 , 0.27929063, 0.21645023,\n", " 0.16774893, 0.13000542, 0.1007542 , 0.07808451, 0.06051549,\n", " 0.04689951])" ] }, "execution_count": 17, "metadata": {}, "output_type": "execute_result" } ], "source": [ "xarray[:,0]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can apply the function $f$ to every iterate $x^{(k)}$ of the list `xarray`, using the command `map`:" ] }, { "cell_type": "code", "execution_count": 18, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "[1.6199999999999999,\n", " 1.0297125000000003,\n", " 0.65475907031250036,\n", " 0.41648898660644562,\n", " 0.26501726238049639,\n", " 0.16868867468928564,\n", " 0.10740675137733219,\n", " 0.06840757437694138,\n", " 0.043580991731946392,\n", " 0.027771796276949725,\n", " 0.017701851534330564]" ] }, "execution_count": 18, "metadata": {}, "output_type": "execute_result" } ], "source": [ "map(f,xarray)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We plot the cost function $f(x^{(k)})$ as a function of $k$, in log-scale:" ] }, { "cell_type": "code", "execution_count": 19, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "[]" ] }, "execution_count": 19, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": [ 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Also, we can note that gradient descent is monotonic: $f(x^{(k)})$ is decreased at every iteration." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We plot the iterates above the contourplot of $f$:" ] }, { "cell_type": "code", "execution_count": 20, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "[]" ] }, "execution_count": 20, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAecAAAHaCAYAAAA66YEnAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzsvXtwJdd93/k9uHhjgAEwA2DwuA9SJEXqGZGUSFmWNJZk\n", "StbD3lRKMjW243Kc7Ky3lHXtercs21sp+g+7LFWceB1vpUaykjhOrSSXnJLlkiWZtkTLDknxIYkU\n", "xRmT1BAXF28MMIPBAIP32T8aPWhc9LtPd5/f6d+nisUBbuP275zbfT73d14tpJRgGIZhGEYfWvIO\n", "gGEYhmGYo7CcGYZhGEYzWM4MwzAMoxksZ4ZhGIbRDJYzwzAMw2hGa94B2AgheNo4wzAMUyiklMLt\n", "99rIGQD+MMHf3pPw3P8VwD9P+B5RSBqvSn4fwK9ldK7x2zM6UQQeWQEeGcw7ivzg8utX/pnL2Z0r\n", 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\n", " \n", "__Exercise: re-run the previous cells with different values of tau \n", "(try 0.25,0.225,0.2,0.125,0.05).__

\n", "\n", "

__Also try different initial estimates.__

\n", "\n", "

__Also try different values of the anisotropy eta__.\n", "

" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Gradient descent in large dimension: image restoration" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Local differential operators like the discrete gradient are fundamental for variational image processing." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We load the image Lena in the array $x^\\sharp$ (we choose the symbol # - 'sharp', because the image is clean, not degraded)." ] }, { "cell_type": "code", "execution_count": 27, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "(512, 512)\n", "The size of the image is 512 x 512.\n", "The range of the pixel values is [25,245].\n" ] } ], "source": [ "from scipy import misc\n", "xsharp = misc.lena()\n", "print(xsharp.shape) # like Matlab's size(xsharp). Given as a tuple.\n", "print(\"The size of the image is %s x %s.\" % (xsharp.shape[0],xsharp.shape[1]))\n", "print(\"The range of the pixel values is [%s,%s].\" % (xsharp.min(),xsharp.max()))\n", "xsharp = xsharp.astype(float32) # like Matlab's xsharp=double(xsharp) so that the pixel values are floating point numbers" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We display the image." ] }, { "cell_type": "code", "execution_count": 28, "metadata": { "collapsed": false, "scrolled": false }, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 28, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAlwAAAIWCAYAAACC31O6AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzsvWuMbWl63/VfVXtX1d61q06dqtPdM7E99MSODQ4BohAn\n", "ERoCsUmCEtlWJBSCIoSEDIggLrJECBJKvhDkDyQBRUgRisQlikMUBEqIgiUkgmwsJfGIBEJsFMv4\n", "Mt19Tp86ddt7165dVXsvPlT/3vqtt6tnpsM5U+Oe9UqlqtqXtd7Lc/k//+d539W0bZu+9a1vfetb\n", 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"title('This is Lena')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We define the 'discrete gradient' $D$, which is the linear operator which maps an image to an image whose values are pairs of vertical and horizontal finite differences:\n", "\n", "$$(D x)_{n_1,n_2} = \\big((D x)_{n_1,n_2,v},(D x)_{n_1,n_2,h}\\big)=( x_{n_1+1,n_2}-x_{n_1,n_2}, x_{n_1,n_2+1}-x_{n_1,n_2} ) \\in \\RR^2,$$\n", "where $n_1$ and $n_2$ are the row and column numbers, respectively (the origin $n_1=n_2=0$ corresponds to the pixel at the top left corner) and where Neumann boundary conditions are assumed: a difference accross the boundary is zero." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us define the operator $D$ as a function:" ] }, { "cell_type": "code", "execution_count": 29, "metadata": { "collapsed": false }, "outputs": [], "source": [ "def D(x):\n", " vdiff = r_[diff(x,1,0), zeros([1,x.shape[1]])] # the r_ command concatenates along the rows\n", " hdiff = c_[diff(x,1,1), zeros([x.shape[0],1])] # the c_ command concatenates along the columns\n", " return concatenate((vdiff[...,newaxis], hdiff[...,newaxis]), axis=2) # combination along a third dimension\n", "# An alternative, more compact, definition:\n", "#D = lambda x : c_['2,3',r_[diff(x,1,0), zeros([1,x.shape[1]])],c_[diff(x,1,1), zeros([x.shape[0],1])]]" ] }, { "cell_type": "code", "execution_count": 30, "metadata": { "collapsed": false }, "outputs": [], "source": [ "v = D(xsharp)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We display the two components as two grayscale images." ] }, { "cell_type": "code", "execution_count": 31, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 31, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAA74AAAHCCAYAAADIARWSAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", 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"4XA4HA6Hw+Fw7Gu44+twOBwOh8PhcDgcjn0Nd3wdDofD4XA4HA6Hw7Gv8f+RuJK0QXFDbQAAAABJ\n", "RU5ErkJggg==\n" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "from mpl_toolkits.axes_grid1 import make_axes_locatable\n", "fig, (axv,axh) = subplots(1,2,figsize=(16,7)) # one figure with two horizontal subfigures\n", "suptitle('The two images of vertical and horizontal finite differences')\n", "imv=axv.imshow(v[:,:,0], cmap=cm.gray)\n", "imh=axh.imshow(v[:,:,1], cmap=cm.gray)\n", "axv.set_title('Image of vertical finite differences')\n", "axh.set_title('Image of horizontal finite differences')\n", "dividerv = make_axes_locatable(axv)\n", "caxv = dividerv.append_axes(\"right\", size=\"7%\", pad=0.1)\n", "colorbar(imv,cax=caxv)\n", "dividerh = make_axes_locatable(axh)\n", "caxh = dividerh.append_axes(\"right\", size=\"7%\", pad=0.1)\n", "colorbar(imh,cax=caxh)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let us display the image of the magnitudes $\\norm{(D x)_{n_1,n_2}}$, which are large near edges." ] }, { "cell_type": "code", "execution_count": 32, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAlwAAAILCAYAAAA9l0L/AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzsvUmMZdmWprXObez21ngbHhGvyVQ+oRSgFAyAEUwYIJCo\n", "IaoBEkhIqBCMGIBUICQQEyYMGAMSExgiIVFixOAhoRQkpRxVpeo1mRHuLzzczdzMbmN27bYMLL59\n", "v7PiRryol+5hL6PulkxmdptzdrOaf/1r7X2q7XYbh3Zoh3Zoh3Zoh3Zoh/bhWuOhO3Boh3Zoh3Zo\n", "h3Zoh/ZDbwfAdWiHdmiHdmiHdmiH9oHbAXAd2qEd2qEd2qEd2qF94HYAXId2aId2aId2aId2aB+4\n", "HQDXoR3aoR3aoR3aoR3aB24HwHVoh3Zoh3Zoh3Zoh/aB2wcBXFVV/WtVVf3Dqqr+UVVV/8mHuMeh\n", "HdqhHdqhHdqhHdrflFa973O4qqpqRsRfRMS/GhGvIuL/iYi/vd1u/8F7vdGhHdqhHdqhHdqhHdrf\n", "kNb6ANf8FyLiF9vt9i8jIqqq+l8i4m9FxAFwHdqhHdqhHdqhHdpvbVVVfS+nsm+32+r7uE/EhwFc\n", "n0TE5/r/ZUT8ix/gPod2aId2aId2aIf2A20f+kk4VfW9Ya2I+DCA67fO0PeFXA/t0A7t0A7t0A7t\n", 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to zero at most pixels. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Imaging inverse problems consist in estimating an unknown image $x^\\sharp$, having at our disposal only the data $y$, which contains partial degraded information about $x^\\sharp$: we have the forward model\n", "$$y=Ax^\\sharp\\ +\\mbox{ noise}$$\n", "where $A$ is a known linear observation/degradation operator. $A$ is not invertible in general, which makes the problem ill-posed. So, one must inject some prior information modeling what a natural image is, to regularize the problem and make it well-posed. Maybe the simplest prior one can use is that natural images are smooth; that is, their gradient has low energy (like for the Lena image above). This yields Tikhonov regularization: $x^\\sharp$ is estimated by $x^\\star$, which is a solution to \n", "$$\\umin{x \\in \\RR^d} f(x)$$\n", "where \n", "$$f(x)=\\frac{1}{2}\\|Ax-y\\|^2+\\frac{\\lambda}{2}\\|Dx\\|^2_{2,2},$$\n", "with $\\|Dx\\|^2_{2,2}=\\sum_{n_1,n_2} \\|(Dx)_{n_1,n_2}\\|^2_2=\\sum_{n_1,n_2} (Dx)_{n_1,n_2,v}^2+(Dx)_{n_1,n_2,h}^2$\n", "and some regularization parameter $\\lambda$, which depends on the noise level.\n", "\n", "Here again, the function $f$ to minimize is smooth (it is quadratic) and we can use gradient descent. We have:\n", "$$\\nabla f(x)=A^*(Ax-y)+\\lambda D^*Dx=(A^*A+\\lambda D^*D)x-A^*y.$$\n", "\n", "So, to compute $\\nabla f(x)$ for some image $x$, we must be able to apply the operator $A^*A$, where $A^*$ is the adjoint of $A$, and $D^*D$. For this, let us define $D^*$ as a function:" ] }, { "cell_type": "code", "execution_count": 33, "metadata": { "collapsed": false }, "outputs": [], "source": [ "Dadj = lambda v : r_['0,2',-v[0,:,0],-diff(v[:-1,:,0],1,0),v[-2,:,0]] + c_['1,2',-v[:,0,1],-diff(v[:,:-1,1],1,1),v[:,-2,1]]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "$D^*$ can be viewed as the opposite of a discrete divergence operator. Hence, $D^*\\circ D$ can be viewed as the opposite of a discrete Laplacian operator. In fact, $D^*\\circ D$ amounts to convolving the image with the filter $\\left(\\begin{array}{ccc}0&-1&0\\\\-1&4&-1\\\\0&-1&0\\end{array}\\right)$, with symmetric boundary conditions. \n", "\n", "So, $D^*\\circ D$ could be implemented more efficiently than here, but this is not the point, and we will need $D$ and $D^*$ separately later, for the total variation." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We check the relation $\\norm{D x}^2 - \\dotp{D^*Dx}{x}=0.$" ] }, { "cell_type": "code", "execution_count": 34, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "0.0" ] }, "execution_count": 34, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(D(xsharp)**2).sum() - (Dadj(D(xsharp))*xsharp).sum()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "##Image Denoising" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We consider the problem of denoising the image $y \\in \\RR^{N_1\\times N_2}$, which corresponds to the problem above with $A$ the Identity operator. So we want to solve $$\\umin{x \\in \\RR^d} f(x)$$\n", "where \n", "$$f(x)=\\frac{1}{2}\\|x-y\\|^2+\\frac{\\lambda}{2}\\|Dx\\|^2_{2,2}.$$\n", "Note we are now looking for an unknown $x$ which lives in a Hilbert space of dimension $d=N_1.N_2$, the number of pixels of the image; for Lena, $d=512^2=262144$, hence the term 'large-scale optimization'." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We add noise to the original image $x^\\sharp$, to simulate the noisy image $y$." ] }, { "cell_type": "code", "execution_count": 35, "metadata": { "collapsed": false }, "outputs": [], "source": [ "(N1,N2) = shape(xsharp)\n", "noiselevel = 20\n", "y = xsharp + noiselevel * randn(N1,N2)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We display the noisy image $y$." ] }, { "cell_type": "code", "execution_count": 36, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 36, "metadata": {}, "output_type": "execute_result" 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"iqIo+4wGXIqiKIqiKPuMBlyKoiiKoij7jAZciqIoiqIo+4wGXIqiKIqiKPuMBlyKoiiKoij7jAZc\n", "iqIoiqIo+4wGXIqiKIqiKPvMfwFfopPQz1JB3wAAAABJRU5ErkJggg==\n" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "figsize(11,11)\n", "imshow(y, interpolation='nearest', cmap=cm.gray, vmin=0, vmax=255)\n", "colorbar() \n", "subplots_adjust(top=0.75)\n", "title('This is a noisy version of Lena')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We define the gradient of $f$ as a function." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

\n", " \n", "__Exercise: write the code of the function Grad_f.__

" ] }, { "cell_type": "code", "execution_count": 37, "metadata": { "collapsed": false }, "outputs": [], "source": [ "def Grad_f(x, y, Lambda):\n", " # put your code here." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The Lipschitz constant $\\beta$ of $\\nabla f$ is $\\|Id+\\lambda D^*D\\|=1+8\\lambda$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

\n", " \n", "__Exercise: write the code of the function GradDescent.__

" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [], "source": [ "def GradDescent(Grad_f, y, Lambda, x0, nbiter, tau):\n", " # put your code here." ] }, { "cell_type": "code", "execution_count": 39, "metadata": { "collapsed": false }, "outputs": [], "source": [ "Lambda = 8; xdenoised = GradDescent(Grad_f, y, Lambda, y, 100, 1.9/(1+8*Lambda))" ] }, { "cell_type": "code", "execution_count": 40, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 40, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAlIAAAJZCAYAAABiGNuwAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzsvX2sbu121jXm3mut/Z7X2kIVDR5azpGP05YYE4OQGi0N\n", "GtIEaJMWQRRrgRijERLRpiWiKIlKFBUDRIMBAqgIidpAIkK1qWlVGjRGDPhFsK2lQCnSc953v/tr\n", "rTX9Y6+x9rV+67rGPfdrj2uf+ozkyTOfOe+PMcY9Pq553/ecz7bve53oRCc60YlOdKITnejt6dFD\n", 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So, this denoising process has bad performances. Good denoising processes are nonlinear. We will see in the next lab that using the total variation instead of Tikhonov regularization is much better." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Projected Gradient Descent" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Often, one wants to minimize a smooth convex cost function, but with the additional constraint that the solution lies in some closed convex subset $\\Omega$; that is, one wants to solve\n", "$$\\umin{x \\in \\Omega} f(x)\\quad\\equiv\\quad\\umin{x \\in \\RR^d} f(x)\\quad\\mbox{s.t.}\\quad x\\in\\Omega$$\n", "\n", "A natural extension of gradient descent is then _projected gradient descent_, which iteratively computes\n", "$$ x^{(k+1)} = P_\\Omega\\big(x^{(k)} - \\tau \\nabla f(x^{(k)})\\big), $$\n", "\n", "where $P_\\Omega$ is orthogonal projection onto $\\Omega$.\n", "\n", "Here also, convergence of the iterates $x^{(k)}$ to a solution $x^\\star$ is guaranteed, provided that $0<\\tau<2/\\beta$.\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Image Inpainting by Projected Gradient Descent\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We now consider inpainting; that is, reconstructing missing pixel values from a subset of available pixels. \n", "\n", "We keep about 10% of the pixels at random, the other ones are missing." ] }, { "cell_type": "code", "execution_count": 41, "metadata": { "collapsed": false }, "outputs": [], "source": [ "mask = rand(xsharp.shape[0],xsharp.shape[1])>0.9" ] }, { "cell_type": "code", "execution_count": 42, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 42, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAA6YAAAGxCAYAAABxzqvhAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzsfXl4VdW1+Np3vsm9ufdmupnuzTzPIYQkZA4QkgAJgRCZ\n", "BBQElEFAQAQUjIoKIvysTxwQBceq1VpttX3aSvvq47W1ra+1tvXxaqttfXV4rbbPVu36/XHO3u5z\n", 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subfigures\n", "subfig1.imshow(mask, cmap=cm.gray)\n", "subfig2.imshow(mask*xsharp, cmap=cm.gray)\n", "subfig1.set_title('The binary mask')\n", "subfig2.set_title('The available pixel values are displayed, the missing pixels are in black')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The degradation operator $A$ is simply pixelwise multiplication by the binary mask. We have $A^*=A$ and $A^*A=A$." ] }, { "cell_type": "code", "execution_count": 43, "metadata": { "collapsed": true }, "outputs": [], "source": [ "y = mask*xsharp" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The problem we consider is $$\\umin{x \\in \\Omega} f(x)\\quad\\equiv\\quad\\umin{x \\in \\RR^d} \\frac{1}{2}\\|Dx\\|^2\\quad\\mbox{s.t.}\\quad Ax=y.$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

\n", " \n", "__Exercise: write the code of the function Grad_f.__

" ] }, { "cell_type": "code", "execution_count": 44, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def Grad_f(x):\n", " # put your code here" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The Lipschitz constant $\\beta$ of $\\nabla f$ is 8." ] }, { "cell_type": "code", "execution_count": 45, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def Proj_Omega(x, y, mask):\n", " x[mask] = y[mask] # parameters of functions are given by reference, so the content of x can be modified\n", " return x" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

\n", " \n", "__Exercise: write the code of the function Grad_f.__

" ] }, { "cell_type": "code", "execution_count": 46, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def ProjGradDescent(Grad_f, Proj_Omega, y, mask, x0, nbiter, tau):\n", " # put your code here" ] }, { "cell_type": "code", "execution_count": 47, "metadata": { "collapsed": true }, "outputs": [], "source": [ "tau = 1.9 / 8\n", "nbiter = 300" ] }, { "cell_type": "code", "execution_count": 48, "metadata": { "collapsed": false }, "outputs": [], "source": [ "xrestored = ProjGradDescent(Grad_f, Proj_Omega, y, mask, y, nbiter, tau)" ] }, { "cell_type": "code", "execution_count": 49, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 49, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAlIAAAJZCAYAAABiGNuwAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzsvWuspduaFvSOue5VtWvX6bY94RwubQOnufwwMUgHo20n\n", 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