{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "Forward-Backward method on the dual problem\n", "===========================\n", "\n", "This numerical tour is an introduction to convex duality with an\n", "application to total variation denoising.\n", "\n", "$\\newcommand{\\dotp}[2]{\\langle #1, #2 \\rangle}$\n", "$\\newcommand{\\enscond}[2]{\\lbrace #1, #2 \\rbrace}$\n", "$\\newcommand{\\pd}[2]{ \\frac{ \\partial #1}{\\partial #2} }$\n", "$\\newcommand{\\umin}[1]{\\underset{#1}{\\min}\\;}$\n", "$\\newcommand{\\umax}[1]{\\underset{#1}{\\max}\\;}$\n", "$\\newcommand{\\umin}[1]{\\underset{#1}{\\min}\\;}$\n", "$\\newcommand{\\uargmin}[1]{\\underset{#1}{argmin}\\;}$\n", "$\\newcommand{\\norm}[1]{\\|#1\\|}$\n", "$\\newcommand{\\abs}[1]{\\left|#1\\right|}$\n", "$\\newcommand{\\choice}[1]{ \\left\\{ \\begin{array}{l} #1 \\end{array} \\right. }$\n", "$\\newcommand{\\pa}[1]{\\left(#1\\right)}$\n", "$\\newcommand{\\diag}[1]{{diag}\\left( #1 \\right)}$\n", "$\\newcommand{\\qandq}{\\quad\\text{and}\\quad}$\n", "$\\newcommand{\\qwhereq}{\\quad\\text{where}\\quad}$\n", "$\\newcommand{\\qifq}{ \\quad \\text{if} \\quad }$\n", "$\\newcommand{\\qarrq}{ \\quad \\Longrightarrow \\quad }$\n", "$\\newcommand{\\ZZ}{\\mathbb{Z}}$\n", "$\\newcommand{\\CC}{\\mathbb{C}}$\n", "$\\newcommand{\\RR}{\\mathbb{R}}$\n", "$\\newcommand{\\EE}{\\mathbb{E}}$\n", "$\\newcommand{\\Zz}{\\mathcal{Z}}$\n", "$\\newcommand{\\Ww}{\\mathcal{W}}$\n", "$\\newcommand{\\Vv}{\\mathcal{V}}$\n", "$\\newcommand{\\Nn}{\\mathcal{N}}$\n", "$\\newcommand{\\NN}{\\mathcal{N}}$\n", "$\\newcommand{\\Hh}{\\mathcal{H}}$\n", "$\\newcommand{\\Bb}{\\mathcal{B}}$\n", "$\\newcommand{\\Ee}{\\mathcal{E}}$\n", "$\\newcommand{\\Cc}{\\mathcal{C}}$\n", "$\\newcommand{\\Gg}{\\mathcal{G}}$\n", "$\\newcommand{\\Ss}{\\mathcal{S}}$\n", "$\\newcommand{\\Pp}{\\mathcal{P}}$\n", "$\\newcommand{\\Ff}{\\mathcal{F}}$\n", "$\\newcommand{\\Xx}{\\mathcal{X}}$\n", "$\\newcommand{\\Mm}{\\mathcal{M}}$\n", "$\\newcommand{\\Ii}{\\mathcal{I}}$\n", "$\\newcommand{\\Dd}{\\mathcal{D}}$\n", "$\\newcommand{\\Ll}{\\mathcal{L}}$\n", "$\\newcommand{\\Tt}{\\mathcal{T}}$\n", "$\\newcommand{\\si}{\\sigma}$\n", "$\\newcommand{\\al}{\\alpha}$\n", "$\\newcommand{\\la}{\\lambda}$\n", "$\\newcommand{\\ga}{\\gamma}$\n", "$\\newcommand{\\Ga}{\\Gamma}$\n", "$\\newcommand{\\La}{\\Lambda}$\n", "$\\newcommand{\\si}{\\sigma}$\n", "$\\newcommand{\\Si}{\\Sigma}$\n", "$\\newcommand{\\be}{\\beta}$\n", "$\\newcommand{\\de}{\\delta}$\n", "$\\newcommand{\\De}{\\Delta}$\n", "$\\newcommand{\\phi}{\\varphi}$\n", "$\\newcommand{\\th}{\\theta}$\n", "$\\newcommand{\\om}{\\omega}$\n", "$\\newcommand{\\Om}{\\Omega}$\n" ] }, { "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": [ "\n", "\n", "Convex Duality\n", "--------------\n", "Given a convex, proper, and lower semi-continuous function $f$ defined on $\\RR^N$, its\n", "_conjugate_ $f^*$ is the convex function defined as\n", "$$ \\forall u \\in \\RR^N, \\quad f^*(u) = \\umax{x \\in \\RR^N} \\dotp{x}{u} - f(x). $$\n", "\n", "\n", "One has $(f^*)^*=f$.\n", "\n", "One can show that if $f(x) = \\frac{1}{2} \\norm{A x - b}^2$, where $A \\in \\RR^{N \\times N}$\n", "is an invertible matrix, then\n", "$$f^*(u) = \\frac{1}{2} \\norm{(A^*)^{-1} u + b}^2- \\frac{1}{2}\\norm{b}^2$$\n", "\n", "One can show that in the case of $\\ell^p$ norms\n", "$$ f(x) = \\norm{x}_p = \\pa{ \\sum_{i=1}^N \\abs{x_i}^p }^{1/p}, $$\n", "with the usual extension to $p=+\\infty$\n", "$$ \\norm{x}_\\infty = \\umax{1 \\leq i \\leq N} \\abs{x_i},$$\n", "then one has\n", "$$ f^* = \\iota_{\\norm{\\cdot}_q \\leq 1}\n", " \\qwhereq \\frac{1}{p}+\\frac{1}{q}=1, $$\n", "where $\\iota_{\\Cc}$ is the indicator function of the convex set\n", "$\\Cc$.\n", "\n", "Forward-backward on the Dual Problem\n", "------------------------------------------\n", "We are concerned with the minimization of composite problems of the form\n", "$$ \\umin{x \\in \\RR^N} f(x) + g(Lx) $$\n", "where $ L \\in \\RR^{P \\times N} $ is a linear operator (a matrix),\n", " $f : \\RR^N \\rightarrow \\RR $\n", "and $g : \\RR^P \\rightarrow \\RR $\n", "are convex functionals.\n", "\n", "\n", "We now assume that $f$ is $\\beta$-strongly convex.\n", "Then, $f^*$ is differentiable and\n", "its gradient is $\\beta$-Lipschitz continuous.\n", "\n", "\n", "In this case, the Fenchel-Rockafellar theorem shows that one can solve\n", "the following _dual problem_\n", "$$ \\umin{u \\in \\RR^P} f^*( -L^* u ) + g^*(u) $$\n", "and recover the unique solution $x^\\star$\n", "of the primal problem from a (non-necessarily unique) solution $u^\\star$\n", "of the dual problem, as\n", "$$ x^\\star = \\nabla f^*( -L^* u^\\star ). $$\n", "\n", "Moreover, the optimal primal and dual costs satisfy\n", "$$ E^\\star= f(x^\\star) + g(Lx^\\star) =\n", " - f^*( -L^* u^\\star ) - g^*(u^\\star). $$\n", "\n", "\n", "Denoting $F(u) = f^*( -L^* u )$ and $G(u) = g^*(u)$, one thus needs\n", "to solve the problem\n", "$$ \\umin{u \\in \\RR^P} F(u) + G(u). $$\n", "\n", "\n", "We assume that the function $g$ is simple, in the sense that\n", "one can apply the proximity operator $\\mathrm{prox}_{\\ga g}$\n", "for any $\\ga > 0$.\n", "\n", "Then, $G=g^*$ is simple as well, \n", "because of the\n", "Moreau identity\n", "$$ x = \\text{prox}_{\\ga g^*}(x) + \\ga \\text{prox}_{g/\\ga}(x/\\ga). $$\n", "\n", "\n", "Since $F$ is smooth and $G$ is simple, one can apply the\n", "forward-backward algorithm, which reads,\n", "after initilizing $u^{(0)} \\in \\RR^P$,\n", "$$ u^{(k+1)} = \\text{prox}_{\\ga G}\\pa{ u^{(k)} - \\ga \\nabla F( u^{(k)} ) }. $$\n", "\n", "\n", "\n", "The primal iterates are defined as\n", "$$ x^{(k)} = \\nabla f^*( -L^* u^{(k)} ). $$\n", "\n", "So, we can write the algorithm in the primal-dual form\n", "$$\n", "\\left|\\begin{array}{l}\n", "x^{(k)} = \\nabla f^*( -L^* u^{(k)} )\\\\\n", "u^{(k+1)} = \\text{prox}_{\\ga g^*}\\pa{ u^{(k)} + \\ga Lx^{(k)}}.\n", "\\end{array}\\right.\n", "$$\n", "\n", "If $0< \\ga < 2/(\\beta\\|L\\|^2)$, $x^{(k)}$ converges to a primal solution $x^\\star$ and $u^{(k)}$ converges to a dual solution $u^\\star$.\n", "\n", "\n", "Total Variation\n", "---------------\n", "\n", "Like in lab 1, we are interested in imaging inverse problems: we want to estimate the unknown sharp image $s^\\sharp$ from the data $y=Ax^\\sharp\\ +\\mbox{ noise}$, where $A$ is the linear observation operator. \n", "\n", "In lab 1, we\n", "defined the 'discrete gradient' $D$ and we used as regularizer the Tikhonov functional $\\frac{1}{2}\\|Dx\\|^2_{2,2}=\\frac{1}{2}\\sum_{n_1,n_2} \\|(Dx)_{n_1,n_2}\\|^2_2=\\frac{1}{2}\\sum_{n_1,n_2} (Dx)_{n_1,n_2,v}^2+(Dx)_{n_1,n_2,h}^2$.\n", "\n", "A better regularizer for imaging problems is the _total variation_, which corresponds to taking the $\\ell_1$ norm instead of the squared $\\ell_2$ norm of the discrete gradient:\n", "\n", "$$\\mathrm{TV}(x)=\\|Dx\\|_{1,2}=\\sum_{n_1,n_2} \\|(Dx)_{n_1,n_2}\\|_2=\\sum_{n_1,n_2} \\sqrt{ (Dx)_{n_1,n_2,v}^2+(Dx)_{n_1,n_2,h}^2}.$$\n", "\n", "The total variation was first introduced in L. I. Rudin, S. Osher, E. Fatemi,\n", "\"Nonlinear total variation based noise removal algorithms,\" _Physica D_, vol. 60, 1992.\n", "\n", "The difficulty is that there is no way to compute the proximity operator of the total variation in 2-D (fast exact algorithms exist in 1-D). So, we must view it as the simple functional $\\lambda \\|\\cdot\\|_{1,2}$ composed with the linear operator $D$.\n", "Indeed, the proximity operator of $\\lambda \\|\\cdot\\|_{1,2}$ is simply elementwise soft-thresholding of the magnitude of the vector field. \n", "\n", "So, if $A$ is invertible, the dual forward-backward algorithm is appropriate to solve least-squares problems regularized with the total-variation.\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Denoising Lena\n", "\n", "In the following, we consider the image denoising problem, i.e. $A=\\mathrm{Id}$, applied to the Lena image. So, given the noisy image $y$, we want to estimated the clean Lena image $x^\\sharp$ by the image $x^\\star$ solution to the strongly convex primal problem\n", "\n", "$$\\umin{x \\in \\RR^{N_1\\times N_2}} E_p(x)= f(x)+g(Dx)=\\frac{1}{2}\\|x-y\\|^2+\\lambda\\|Dx\\|_{1,2}$$\n", "\n", "The dual problem is\n", "\n", "$$\\umin{u \\in \\RR^{N_1\\times N_2\\times 2}} E_d(u)= f^*(-D^*u)+g^*(u)=\\frac{1}{2}\\|-D^*u+y\\|^2- \\frac{1}{2}\\norm{y}^2+\\imath_{\\|\\cdot\\|_{\\infty,2}\\leq\\lambda}(u)$$\n", "\n", "Chambolle Dual Algorithm \n", "-------------------------\n", "\n", "We will solve the total-variation regularized denoising problem by applying the forward-backward algorithm on the dual problem. This is sometimes called the _Chambolle Dual Algorithm_, because this idea was initially proposed in\n", "A. Chambolle,\n", "\"An Algorithm for Total Variation Minimization and Applications\",\n", "_Journal of Mathematical Imaging and Vision_, 20(1-2), 2004.\n", "\n", "Note that since $g^*$ is an indicator function, the forward-backward algorithm reverts to projected gradient descent.\n", "\n", "For a more general framework, see\n", "P. L. Combettes, D. Dung, and B. C. Vu,\n", "\"Dualization of signal recovery problems\",\n", "_Set-Valued and Variational Analysis_, vol. 18, 2010\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We create the noisy image $y$." ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false }, "outputs": [], "source": [ "from scipy import misc\n", "xsharp = misc.lena()\n", "xsharp = xsharp.astype(float32)" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAAhoAAAIhCAYAAAAIBBNIAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzsvVuIbVua5/XNdY11iYgVEfvsc05lZXkyqypT0vKGVmE9\n", "tC1dhS10UxYF4gURQUoFfRDqofVB7BcVH6RV+qWQBm9YrQhCt0Vr0Q8liYJ2CgoN3VYXVlZezr5F\n", "xIoV6x7rMn2I8xvrN8de+2QezdORXWcO2OyIWGvOOcY3vsv/+39jjFmUZRl1q1vd6la3utWtbp9H\n", 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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": 7, "metadata": { "collapsed": true }, "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": [ "The gradient of $f^*$ is simply $\\nabla f^*:x\\mapsto x+y$." ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def grad_f_conj (x,y) :\n", " return x + y" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The proximity operator $\\text{prox}_{\\ga g^*}$ does not depend on $\\gamma$ and is such that $$\\big(\\text{prox}_{\\ga g^*}(u)\\big)_{n_1,n_2}=\\frac{1}{\\max(\\|u_{n_1,n_2}\\|_2/\\lambda,1)}u_{n_1,n_2}\\in\\mathbb{R}^2,\\quad\\forall (n_1,n_2).$$" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def prox_g_conj (u, Lambda) :\n", " return u/tile(maximum(sqrt(sum(u**2,2,keepdims=True))/Lambda,1),(1,1,2)) # soft-thresholding" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

\n", " \n", "__Exercise: write the code of the dual forward-backward iteration for denoising.__

" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false }, "outputs": [], "source": [ "Lambda = 25\n", "gamma = 1.9/8 # we must have 0 < gamma < 2/8\n", "nbiter = 400\n", "(N1,N2) = y.shape\n", "u = zeros([N1,N2,2])\n", "Ep_array = zeros(nbiter) # array for the primal energy E_p\n", "Ed_array = zeros(nbiter) # array for the dual energy E_d\n", "sqnormy = norm(y)**2/2\n", "for iter in range(nbiter): \n", " # put your code here\n", " Ep_array[iter] = norm(x-y)**2/2 + Lambda*sum(sqrt(sum(D(x)**2,2))) \n", " Ed_array[iter] = norm(y-Dadj(u))**2/2 - sqnormy\n", "xdenoised = x" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We display the denoised image." ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": [ 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"plot(Ep_array)\n", "plot(-Ed_array)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We plot the difference in log-scale." ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "[]" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": [ "iVBORw0KGgoAAAANSUhEUgAAA28AAAGoCAYAAADRvd3ZAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz\n", "AAALEgAACxIB0t1+/AAAIABJREFUeJzt3Xm0ZWV97vvvAwUijSAKRUGBoCD2dEoQk1gxNkAipjtB\n", "7znGw9HEYWJ3Helv7pHkJCeJ43iO0UQuQ2OCHjUxqCTe4A2KlhqNKFaBSN83JVAoPQTpfvePOTd7\n", "1a7dVq295pp7fT9jvGOv2dSav+0ck6rH953vm6pCkiRJkjTedui6AEmSJEnSwgxvkiRJktQDhjdJ\n", "kiRJ6gHDmyRJkiT1gOFNkiRJknrA8CZJkiRJPbBgeEvye0kuSXJxkk8kecKM4+uS3J1kY9v+YPnK\n", "lSRJkqTJtGq+g0kOBn4VeHZV/SjJ3wOvBc6ccepXqurkZalQkiRJkjR/eAPuAR4Gdk3yKLArsGmW\n", "8zLswiRJkiRJ0+YdNllVdwDvBW4Evg/cVVVfnHkacHySi5Kck+Q5y1OqJEmSJE2uecNbkmcA7wQO\n", 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"H/wfq6rz2vvlM9S9ue7PR32GxtbU/95D+3vIRbolSZIkqQeWfZFuSZIkSdL2M7xJkiRJUg8Y3iRJ\n", "kiSpBwxvkiRJktQDhjdJkiRJ6gHDmyRJkiT1gOFNkiRJknrA8CZJkiRJPfD/A2CnX+WoOTH8AAAA\n", "AElFTkSuQmCC\n" ], "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "figsize(15,7)\n", "plot(log10(Ep_array+Ed_array))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We observe that the convergence is slow: $E_d$ decreases like $O(1/k)$ " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

\n", " \n", "__Exercise: like in the lab 2, write the code with the forward-backward algorithm replaced by the FISTA-like accelerated forward-backward algorithm.__

" ] }, { "cell_type": "code", "execution_count": 91, "metadata": { "collapsed": false }, "outputs": [], "source": [ "gamma = # put the right value here\n", "a = 3 # try different values\n", "Lambda = 25\n", "nbiter = 400\n", "(N1,N2) = y.shape\n", "Ep_array_fista = zeros(nbiter) # array for the primal energy E_p\n", "Ed_array_fista = zeros(nbiter) # array for the dual energy E_d\n", "sqnormy = norm(y)**2/2\n", "# put your code here\n", "for iter in range(nbiter):\n", " # put your code here\n", " Ep_array_fista[iter] = norm(x-y)**2/2 + Lambda*sum(sqrt(sum(D(x)**2,2))) \n", " Ed_array_fista[iter] = norm(y-Dadj(u))**2/2 - sqnormy\n", "xdenoised = x" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We compare the convergence speed of the two algorithms." ] }, { "cell_type": "code", "execution_count": 93, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "[]" ] }, "execution_count": 93, "metadata": {}, "output_type": "execute_result" }, { "data": { 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Here, the image obtained with 50 iterations of the accelerated forward-backward algorithm is visually identical to the one obtained with much more iterations." ] } ], "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": "2.7.9" } }, "nbformat": 4, "nbformat_minor": 0 }