matlab/pde_4_wavelet_compression.ipynb

{"nbformat_minor": 0, "worksheets": [{"cells": [{"source": ["Wavelet Compression of Integral Operators\n", "=========================================\n", "\n*Important:* Please read the [installation page](http://gpeyre.github.io/numerical-tours/installation_matlab/) for details about how to install the toolboxes.\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"], "metadata": {}, "cell_type": "markdown"}, {"source": ["This numerical tour explores approximation of integral operators\n", "over wavelets bases."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 2, "cell_type": "code", "language": "python", "metadata": {}, "input": ["addpath('toolbox_signal')\n", "addpath('toolbox_general')\n", "addpath('solutions/pde_4_wavelet_compression')"]}, {"source": ["Representation of Operators in Bases\n", "------------------------------------\n", "We consider 1-D signals $f \\in \\Hh=L^2([0,1]) $ (we assume periodic boundary\n", "conditions).\n", "We consider an orthogonal basis $ \\Bb = (\\psi_m)_{m \\in \\Om} $ of $\\Hh$.\n", "The forward (decomposition) operator is\n", "$$ \\Psi : f \\in \\Hh \\rightarrow (\\dotp{f}{\\psi_m})_{m \\in \\Om} \\in\n", " \\Vv = \\RR^\\Om. $$\n", "The backaward  (recomposition) operator is\n", "$$ \\Psi^* : c \\in \\Vv \\rightarrow \\sum_{m \\in \\Om} c_m \\psi_m \\in \\Hh. $$\n", "\n", "\n", "Note that $\\Bb$ being orthonormal is equivalent to\n", "$$ \\Psi^* \\circ \\Psi = \\text{Id}_\\Hh \\qandq \\Psi \\circ \\Psi^* = \\text{Id}_\\Vv. $$\n", "\n", "\n", "A linear operator $T : \\Hh \\rightarrow \\Hh$\n", "can be represented as\n", "$$ T f(x) = \\int_0^1 K(x,y) f(y) d y. $$\n", "\n", "\n", "Here $K : [0,1]^2 \\rightarrow \\RR$ is a kernel function, that is\n", "sometimes assumed to be a distribution (for instance to define\n", "differential operators).\n", "\n", "\n", "$K$ is in some sense the representation of $T$ in the Dirac basis of $\\Hh$.\n", "One can represent the operator $T$ in the basis $\\Bb$\n", "as follow\n", "$$ T = \\Psi^* S \\Psi\n", "      \\qwhereq S = \\Psi T \\Psi^* \\in \\ell^2(\\Vv). $$\n", "Here the operator $S : \\Vv \\rightarrow \\Vv$ can be understood as an\n", "(possibly infinite) matrix whose entries can be computed as\n", "$$ \\forall (m,m') \\in \\Omega^2, \\quad\n", "      S_{m,m'} = \\dotp{T \\psi_m}{\\psi_{m'}}. $$\n", "\n", "\n", "In the following, in order to be able to perform\n", "numerical computations,\n", "we consider discretized signals $f \\in \\Hh = \\RR^N$\n", "and thus assume $\\Psi$ defines an orthonormal basis of $\\RR^N$.\n", "In this case $\\Vv = \\RR^N$, and both $K, S \\in \\RR^{N \\times N}$.\n", "\n", "\n", "The switch of representations\n", "$$ \\Theta : T \\in \\RR^{N \\times N}$\n", "  \\longmapsto S = \\Psi T \\Psi^* \\in \\RR^{N \\times N} $$\n", "is called the tensorial transform assiociated to $\\Psi$\n", "(also called the anisotropic transform in the case of Wavelet bases).\n", "\n", "\n", "It can be computed numerically by first applying $\\Psi$ to the rows\n", "of $T$ and then to its columns.\n", "\n", "Foveation Operator\n", "------------------\n", "We consider here a special case of integral operator, the so-called\n", "Foveation operator. It corresponds to a spacially varying convolution.\n", "\n", "\n", "Size $N$ of the signal."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 3, "cell_type": "code", "language": "python", "metadata": {}, "input": ["N = 1024;"]}, {"source": ["We can compute a signal and the action of the kernel on the signal."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 4, "cell_type": "code", "language": "python", "metadata": {}, "input": ["f = load_signal('piece-polynomial', N);"]}, {"source": ["Display it."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [{"metadata": {}, "png": 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"output_type": "display_data"}], "prompt_number": 5, "cell_type": "code", "language": "python", "metadata": {}, "input": ["clf; \n", "plot(f); axis tight;"]}, {"source": ["A spacially varying convolution is a blurring operator that is not\n", "translation invariant. For a Foveation operator, the blur has a spatially varying bandwidth\n", "parameterized by a scale $\\si(x)$, that dillates a\n", "given convolution profile $\\phi(x)$. It thus corresponds to\n", "$$ K(x,y) = \\frac{1}{\\si(x)} \\phi\\pa{ \\frac{x-y}{\\si(x)} }. $$\n", "\n", "\n", "We choose here a Gaussian kernel."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 6, "cell_type": "code", "language": "python", "metadata": {}, "input": ["phi = @(x)exp( -abs(x).^2 / 2 );"]}, {"source": ["One usually chose a foveation scale that increases linearly with the\n", "distance to some center point $c$ of Foveation. This reproduces the effect of\n", "human vision, that has a finer resolution in the center of the field of\n", "view. We thus choose\n", "$$ \\si(x) = \\al \\abs{x-c} + \\be $$\n", "for some constants $\\al,\\be > 0$.\n", "\n", "\n", "Center point $c$ of foveation"], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 7, "cell_type": "code", "language": "python", "metadata": {}, "input": ["c = 1/2;"]}, {"source": ["Rate of foveation and offset."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 8, "cell_type": "code", "language": "python", "metadata": {}, "input": ["alpha = .05; \n", "beta = 1e-3;"]}, {"source": ["Definition of $\\sigma$"], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 9, "cell_type": "code", "language": "python", "metadata": {}, "input": ["sigma = @(x)alpha*abs(x-c)+beta;"]}, {"source": ["Normalization operator (this is needed for the discrete kernel) the rows."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 10, "cell_type": "code", "language": "python", "metadata": {}, "input": ["normalize = @(K)K ./ repmat( sum(K,2), [1 N] );"]}, {"source": ["Definition of the kernel."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 11, "cell_type": "code", "language": "python", "metadata": {}, "input": ["t = (0:N-1)/N;\n", "[Y,X] = meshgrid(t,t);\n", "K = normalize( phi( (X-Y) ./ sigma(X)  ) );"]}, {"source": ["Display a few lines of the kernel."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [{"metadata": {}, "png": 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G7LrJwHta2sohwIbhsHe08Lc9/nbmevrLQHltLYANAmxWfJxu4A9lRdwK9LZD+6vNDMIE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\nRU5ErkJggg==\n", "output_type": "display_data"}], "prompt_number": 12, "cell_type": "code", "language": "python", "metadata": {}, "input": ["I = round( linspace(1,N,17) );\n", "clf;\n", "plot(K(:,I)); axis tight;\n", "axis([1 N -.005 .1]);"]}, {"source": ["Display the kernel as an image (using a change of contrast to better view\n", "the rows)."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [{"metadata": {}, "png": 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jZrNZtjrRrqYbOtvWYni7FHpZrUK5o0BEuKrzja65FCHkHqHAyK6AVsCZp1To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\n", "output_type": "display_data"}], "prompt_number": 13, "cell_type": "code", "language": "python", "metadata": {}, "input": ["clf;\n", "imageplot(K.^.1);"]}, {"source": ["Compute $f = T g$. In the discrete setting,\n", "this corresponds to the multiplication by the discretized kernel $K$\n", "(which is the representation of $T$ in the Dirac basis)."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 14, "cell_type": "code", "language": "python", "metadata": {}, "input": ["g = K*f;"]}, {"source": ["Display the action of the kernel."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [{"metadata": {}, "png": 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"output_type": "display_data"}], "prompt_number": 15, "cell_type": "code", "language": "python", "metadata": {}, "input": ["clf;\n", "subplot(2,1,1);\n", "plot(f); axis('tight');\n", "title('f');\n", "subplot(2,1,2);\n", "plot(g); axis('tight');\n", "title('g');"]}, {"source": ["For later use, store the average application time."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 16, "cell_type": "code", "language": "python", "metadata": {}, "input": ["t0 = 0;\n", "ntrials = 50;\n", "for i=1:ntrials\n", "    tic; g = K*f; t0 = t0 + toc();\n", "end\n", "t0 = t0 / ntrials;"]}, {"source": ["Representation of Operator in a Wavelet Basis\n", "---------------------------------------------\n", "Wavelet operator compression\n", "has been initially introduced for the compression of integral\n", "operator with a kernel $K$ that is\n", "sufficiently smooth [BeyCoifRokh91](#biblio).\n", "In this setting, this allows to reduce the storage of the discretized\n", "matrix $T \\in \\RR^{N \\times N}$ from\n", "$O(N^2)$ to $O(N\\log(N))$ by thresholding the entries of $S$.\n", "\n", "\n", "A more advanced representation, the so-called non-standard one, allows an\n", "even further compression to $O(N)$ entries. We do not detail here\n", "the non-standard representation, that is more involved.\n", "\n", "\n", "Representation of operator in wavelet bassis has also been applied to\n", "local differential operator (in which case the kernel $K$ is a\n", "distribution), see for instance [Beylkin92](#biblio). In this case, the advantage is not\n", "the storage of the matrix, but the ability to derive optimal\n", "pre-conditioner that are diagonal in the transformed domain, see for instance\n", "[DahmKun92](#biblio). This allows one to recover and extend\n", "classical multiscale preconditonners, such as for instance\n", "[BramPascXu90](#biblio). Such preconditionners are crucial to solve\n", "linear systems such as $T f = y$ (which typically\n", "corresponds to the resolution of a\n", "partial differential equation) using for instance\n", "the conjugate gradient method.\n", "\n", "\n", "An orthogonal wavelet basis of $\\Hh$ reads\n", "$$ \\forall m = (j,n), \\quad \\psi_{m}(x) = \\frac{1}{2^{j/2}}$\n", "      \\psi\\pa{ \\frac{x-2^j n}{2^j} }   $$\n", "for scales $j \\in \\ZZ$ and positions $ n = 0,\\ldots,2^{-j}-1 $.\n", "\n", "\n", "We use here the simplest wavelet, the Haar wavelets. Wavelet with a\n", "larger number of vanishing moments could be used as well."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 17, "cell_type": "code", "language": "python", "metadata": {}, "input": ["options.filter = 'haar';\n", "Jmin = 1;"]}, {"source": ["Define the forward transform $\\Psi$ and\n", "the backward transform $\\Psi^*$."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 18, "cell_type": "code", "language": "python", "metadata": {}, "input": ["options.separable = 0;\n", "Psi  = @(f)perform_wavelet_transf(f, Jmin, +1, options);\n", "PsiS = @(f)perform_wavelet_transf(f, Jmin, -1, options);"]}, {"source": ["We define the tensor product transform\n", "$$ \\Theta(T) = \\Psi T \\Psi^* $$\n", "where $T$ is represented through its kernel $K$."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 19, "cell_type": "code", "language": "python", "metadata": {}, "input": ["options.separable = 1;\n", "Theta = @(K)perform_wavelet_transf(K, Jmin, +1, options);"]}, {"source": ["Wavelet Foveation\n", "-----------------\n", "Computing foveation over wavelet bases has been initially proposed in\n", "[ChaMalYap00](#biblio). It extends an initial method proposed by [Burt88](#biblio).\n", "\n", "\n", "Display the wavelet transform $\\Psi f$ of the signal $f$."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [{"metadata": {}, "png": 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OXx1cSpn+edPl3joMj5d+FzcuWr/nensccw8p44xgMjm589eh84AB0CuXEAGIJGAARBIw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\n", "output_type": "display_data"}], "prompt_number": 20, "cell_type": "code", "language": "python", "metadata": {}, "input": ["clf;\n", "plot_wavelet(Psi(f), Jmin);"]}, {"source": ["compute $S = \\Theta(T)$."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 21, "cell_type": "code", "language": "python", "metadata": {}, "input": ["S = Theta(K);"]}, {"source": ["Display $S$."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [{"metadata": {}, "png": 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b1tHRUWS2rTFFNj2jlPI8X1NTU19fL0nSjh07Nm/e3N3dXVNTc+KJJ5533nkO\nh6Orq6u/vz994BYdMqTcmNQ2jSlgLlcKLpdL5xAUpbSpqamrq6u0rQIr1tbW9vnnnw8MDBATy4Bq\ndhmGmTFjxvbt299+++2LLrrooYceeuedd1QP0rTtNyHZXDSWZT0eT1tbG8dxu3fvfvvttz///PNA\nIHDGGWfccMMNsixv27ZtaGgo5VLjxUfKBAqY8SxYsCB9I6V0xYoVOmuglC5btuy9994rabv+YcWm\nT58+MDCwYcOG0lZeDlRTO2PGjE8//fT+++/ftGmT3+9fvXo1FEAzWjAZJY3n+ba2tjlz5mzdunXV\nqlX9/f379+8/5ZRT/H7/hAkTXn/9dXj1AVDGkJKDAmY86YPqKvofeJZl33333XIYCErp1KlTZ82a\n5ff7LWGAVAvL8/y9997r8/k+/PBDn8+3efNmghpWOlL0jGGY5ubmSy65ZMeOHT6fr6urq7m5+Ve/\n+pXP5+vo6Pjd734HPbrYr4iUEJwHZl68Xm9eM2x8Ph8sDlnyllBKGxsb77//fr/f7/V69TfJcKCp\nXq/X5/O9/vrrr7322k9+8pPceSaRgqFfne1w0kknnXjiiT6fb+rUqbW1tQ8//DD8ni+44IJFixZh\nfkWkeNADMy+UUr/fn1f5Bx98sHzt4TjO6/Xm1SSTAC6Cz+fz+Xzbtm3z+XzoAVQGbUCQz+eDF6BF\nixb5R/j888/xXiAFgwJmCl544YWM2xsaGvKqp9y9fKoMlO8QZYVSCtaTjDi4RrdozKF9mfD5fDNm\nzCCEwP955WNEEIJdiCbhiiuuyLj9tttuy7cqeM8tukWjYPXsQWBDVSWz9LlYHUqp1+slI69flNJ7\n770X8lvifUFygwJmCuhINsJ08pWKZcuW6clfVwzQkXj//ferSwBbEa3dJITA/9Y9HUsDlx38MK/X\nG4lEYMAMvoJfmrYkggAoYKYAogwyfuX3+/PqsmMYptxOGJj+KnBf1BAPWZbhdO655x5YMcu6J2Vp\n6MhSQeoTsXTp0hUrVqhiduONNzY3N6fvgoxNUMAMQ+tywet/SaCUnnXWWaeeemo5whG1R/F6vV1d\nXR9//PFxxx1naSNCKYX4lL6+vuHh4fHjx3d2dra2tqrfGtu8MQjVLERQW1sLYpZMJq+55poNGza8\n++67Ho/n+uuv7+zsXLhwocPhSN8XGSNgEIdhbNu2TU+x+++/P9+R7dNOO+3BBx+swHj4xIkTa2tr\nI5GI1Wf2gNVraGhobGzcvXv3sccee9RRR7399tuwYLHVz1f56tAAACAASURBVM6ipCQBYRhm2rRp\nl1122RNPPHHfffeJolhTU3PMMcfceOONgiBs3bq1v79fe7/wxo0FUMAMQ5s7LscYGMuywWAw38on\nTpx45MiRckckEkKmTJnS19cniiKx+Bxh1VBOnjz50KFDq1evnj179g9/+MOtW7fCQi1oEI1CK2OA\nzWYbN27cscceu2fPnpUrV3Z2djY3Nx933HHnnHNObW3tgQMHent74b2KaG4c3sHqAwXMME444QTt\nxxzPVW1tbV41U0qvv/76J598UpblCmhYW1vbgQMHXC6X1Q2EaiUh+cjEiRN/+tOfzpgx4+KLLz5w\n4EAoFIJvLX2O1iVdxmCjKIqtra2tra379u17/fXX9+7dW1dXN2/evBNOOGHSpEnxeLyvry8UCqUs\nUICSVh2ggBnG/fffr/24fPnyjMXAOSvgGTvxxBN1ruxcPI2NjQcPHvR4PGAmrGsRUkxkQ0ODKIov\nvPBCW1vb4sWLh4eHg8Fg+nu9QY0di6T0K2phGMbhcDQ0NNjt9s7Ozg8//HDHjh3JZHLy5Mlz5syZ\nMGECISQQCIRCIUmSUtbnREmzKChghvHKK69oP6asxKhl3759BdR//vnncxwXCAQq4IQRQtxu95Ej\nR2pqakA1rWsFUowjwzD19fWyLK9bt66hoWH+/PmyLAeDQa0FtO7JWpdsMkY063O63W5RFAcHB3fv\n3t3R0ZFIJFpbW6dOndrS0uJ0OiVJCofDsPJ4+u1DSbMEKGCG8dprr2k/Hjp0KFvJ9vb23bt351s/\npTQcDk+aNIlUyiUSRXFgYEDVsIodt+SkG0eWZWtqaiKRyCeffOJyuWbNmkUphXd5tQxausqTwyFT\nC7AsK4qiw+FgGGZoaKirq+vgwYPBYNDpdI4fP761tdXpdCqKEovF4vF4jmwgKGkmBAXMMN5//33t\nx0svvTRHYZZl83pg4JH2eDzHH398Dt+uJGiHjiiliURC21rrPufpZpHjOIfDMTg42N3d3dLSMnHi\nREVRotFoyjmigas8eqLnwTPjeZ7neUVRgsHgkSNHBgcHKaW1tbWNjY1Op5NSKkmSJEl68lqhmBkO\nCphhnHLKKdqPc+bMyVG44InJzz77LAQxVuYZ4zguGAxCune1k826T3i6WWQYxm63J5PJrq6unp6e\nhoaGlpaWZDIZi8VShlWIlU/ciozqjanFGIbhOE4QBI7jEolEIBDo7+8fHBxMJBIOh6Ourq6mpgY8\ntkQiIctyIpGAAP0c1aJ/ZggoYIYBnXsqU6dOzVE4JeJDP+PGjVuyZEkkEiHl1DCt1eA4LhwOC4LA\nsmxK6FeZjl5WMhpE6JViGKavr29gYMDj8TQ0NFBK4/F4Nhmz6OlbET0yppYEMeM4jmEYWZZDoRCE\n6sTjcY7jXC6X2+222+0cxymKkkgk9IgZgHpWATATh2GkPGO5H7n29vYjR47kW7+iKAzDvPfeex6P\np5Am5n84+J9l2UgkYrPZeJ6PRCJqThCLPsbZZukxDCMIQjKZHBwchN5FSmkkEonH4wzDpGekVCvB\nbBEVQL3IOn91WtlTtSoajcKthL5HQkgymQQNAxnT6fZlbAb+DIoHBcwCUEptNltKyhz9eDyeadOm\nffrppwXXUAAsy8ZiMZZlbTZbLBZjGGv7+jlmmoOBSyaTgUAAgt9sNls0GoVIloyJlS0q5BZFm5hK\n/y7pYkZGQkIYhoE01oqiJEdQNOtz6lQm/BkUDwqYZZg3b15hO3Ict3PnTrfbre3QKwcpVh7MuizL\noihWh4aR7EYHhCqZTIZCIRhfEUVRlmXoVCz3+gDIqBQgY+qOKWImyzJsUZ0z+FYrZsrIIhJ438uK\ntW3KmGL9+vX57qI+PIIgNDY2RqPRUjdq9AZAl4sgCNXxvpk7YhssWiKRiEQi0WiUUmq32yHPls6B\nE6Ss6O/xy7G7+joC72fxeFySJHC4WZYVBAECHVUvDYfBygd6YJahGA+G5/nu7m63213C9mQkvasN\nAuuhDZIkVcELaY7uRDJi48Bggd/JcZwoimDskslkMQYUKRUFO2QpNQBwuyF+h2oA5yxbNAf+DIoH\nBcwyFDmCRSmtgICRnBoGUcsVaEO5ya1hRDMWoiiKJEkwasLzvBoCgMbLDOQb6DFqPeSrcyfoV9EW\nQDErCShghpFXFKKeAtn2gueE5/m+vr4CaigJdGSCc1lXKaskOl/htUP9KTJWkWYiuijeIUupSkX7\nspJDzIo/7tgEBWwMkT5FqUxkdFBgPMzqoRwpjOqKEY1xhGEwGCeDmGzEVJRQxlLqBFStUjUM+5OL\nZPTHDykZ+EtFECQ3aJDzoapehxEEQZCxA3YhGobf5/P6fPAPbIGPPq+XZBnxggLLHnhA7YjLt/8B\nalixfHm2gMAKdGhAGx70+8lId02+k0zL065Ko979lF6mlH/KRNVcxqrBO2IHEP2gB2ZtCjNzMJcl\n474V61LWTvbMd8dytMcoUobxKzY6UmWXERmboIBZj+LjICCUgBhqxVTfqwBLXWXGNz2uWitj5T5u\n+epHkHKDAmY9YDWjYmpIJBJmSG6Ubq/HMtlkDF0xBMkGCpjpGNWgwESiYg4B8dwwJSv9cJW0aOmB\nxfp3LFujjMQQGavWi4lUPShgpmPUKa7FZ9VTFAWcsHwXei4HqGHpVF7GqvhiIlUMCpjpiMfjuQsU\nn9ldze2UzQmrMAVrWHVTYRkz/GeAIPmCAmY6IK11DiRJEkWxGHMD5g9WhTCDE0YK1TAztLzcwJXR\nut1wicoxijkWridSTaCAmQ5BEEYtY7PZ1GGwwowOrPohyzLkNEqpxBBDhn5YbrQypvphJZcxDE1E\nLAQKmOnweDy5C0D/YUkSGyYSiRxzwiqPVsP071LOFpkIdeKBSnqP4ti5GghCUMBMyMSJE3MXYBgm\nHA4Xb6royGqTxcfllxBVw8wQ6G8qqCYpsHbxXzqykKaqZMV7UeiHIZYABcx06OlCjEQisM5vMYDh\ni8fjHMcxDJPi0hlov1T3QqeGjSlTCyqlOmEgY0SzHnRpZaw0jUaQ8oACZjr279+vp5jH4ymJE6Yo\nSvFRISVH0Sw+QnUsWWKqxlcArYypi2QSjYyp2l+kjI21C4tYCxQw40mxEZ2dnaMaHY7jWJYdNV5R\nDwzDxGIxnuf16EQJGdW1Un0LMMfVakmLiVjROltaGYMZfizLlkTGsDsRMS0oYKaDYZhRlYlhmIGB\ngZIIGOhWLBaz2Wwpc6jLbbZyG25oGDRJj4ZZ2sgWM7srXcYSiQT0wZZWxgrbEUHKBwqY6XC73cFg\ncNRioVBIFEV4487XuKQYSpZlo9EoDKpV2E7lttopGlZhH9EQClayHDIG/nrxMlb1Fx+xHChgpkMQ\nhOHh4VGLcRzX0tIiSVJhR9GaSPg/Eok4nc5RE1mVA50axnFcbg2rJgtbmJJpZQzm+cmynEwmGYbh\nOA6idaBkYQnJqukKI1UACpgZaW5uzvEtWCie5zmOGzXvlE44jguHw3a7PeX1vGIGK4exBtGSZVlR\nlDGlYUABSqbdJZlMyrIsSVIymaSU8jzP87zaJVuAjFXfFUasCwqYGbnwwgtHLUMp7erqqqmpKfgo\nKWaRYZhAIOByuUoytFZwk3Jsh4ap8SZjzZLmq2RQWJ32LkmSJEmQxBlkTH0byFfGxtqVR0wLCpgZ\naW1t1VMsFArpLKkHlmXD4bBRvYgqo/phZLQcxFVvXgtzyMiINwa+bEqnYr4vBGPwBQIxIShgZuQv\nf/lL7gJgj2w2myiKReqNagrhbR2iOcypYTDbWpIklmXBeyhJPi2LkpdDRkdmhcO0v1gsFovFkskk\nx3GiKIJTq6b20N8G1DDEWFDAzMhf//pXPcVEUdy2bZuekEWd8Dzf29vb2tqqHVozxEhlM82gYbFY\njGVZURSzadiYMqz6lYyOTHMmhCQSCVXGeJ632+2CINCR7GL6HawxdakRs4ECZjoopePGjdNZWJIk\n6FUrxo5obR/P84FAoPhFn0tCRrsM0QexWExRFJvNRkqU17gK0KlkIGMsy1JKE4lEJBKJRCKJREIQ\nBIfDAa8FMCda54VFDUOMAgXMeNKDJpqbm0eNpAA7VVNTc/LJJxffBtXq8Tx/6NChqVOnRiKR4qst\nCdk0LBqNJpNJu91OMmnYWLaqOpVMTdiRTCYjkUgoFJJlWRRFl8sliiI4uxCFP+oRx/LVRgwEBcx4\nDh48mLKFYZgDBw7o2ZdhGJfLVRLzodo7h8PR29sLKaZgi+HmKaOGMQwTiUSSyaTD4YBQupQyhjfb\ncPQoGR1J2KEoSjgcHh4eliTJbrd7PB6Hw6GuG4fpKBETggJmPOvWrUvfuGXLFp27/8///E8Jh8EI\nIRzHdXd3H3fccaWttkjSDTFY3nA4nEwm3W63/i6vMcioSgYXk+M4QkgoFBocHAwGg4Ig1NXVuVwu\nSil0Vo+azQtlDKkkKGDGk/GZv/vuu0fdEexRfX19d3d3SVqiGrja2tp9+/Ydc8wxBWf6KBPpGsZx\nXDAYlGW5pqYmvb8L7WkKemQMghJjsVhfX9/Q0JAgCI2NjW63W1GUeDyuR8bK0HAEyQAKmPH85je/\nSd94xRVX6DQEHo/nuuuuK1VjwLoxDNPd3e1wOI4cOVKqmktFigmmlAqCEAwGJUmqr6+XJCnluqE9\nTYdqyPgty7KCIPA8H4/He3p6+vv7bTbb+PHjPR4PhC/mljG85khlQAEznnPOOSdlC6X061//up59\nwQA1NjaWvFWNjY1r16496aSTYE6Y2UxSiuUVBCEQCMRisaamJghQNKphliOHT8YwjCAINptNkqQv\nv/zy4MGDDodj8uTJNTU1kiRFIpH01wUVvAVIBUABMym7d+/WX/jHP/5xPB4vlckAc0YpnThxotvt\n7uzsLEm1JSfF8oqiGAwGw+Fwa2trJBIxJKOjdcntkEGEvaIoXV1d+/fvd7lcs2bNGj9+vCzLELuY\nsU4cEkPKDQqY8Sxbtix949lnn62/hnnz5n3xxRela9E/OxJfeOEFWGbFtGhtriAIg4ODPM87HA6z\njd5ZhRwyBvOdZVnet2/fkSNHampqpk+fLopiKBQq4fsTgugHBcx4li9fnr6xrq5O5+6U0oaGhoGB\ngZI26h/MmjXriSee0LO8i4GoBpdS6nK5Ojo6xo0bxzCMNooSzWteZHPIWJZ1Op0ul6uvr+/jjz/e\nt2/fnDlz5s+fL8vy4OBgxpcGvPJI+UABM57Fixdn3J7Xk//mm2+WqDn/gI4s2nLOOedATL+ZLZHW\n2rrd7s8++2zKlCnTp08PBALGNszqZFQyhmEcDkd9fT2ldMOGDVu2bDnxxBNPPvnkQCDQ19eX3qOI\nfYlImUABM56zzjor4/b3339ffyWjpl0oAKjzrLPOWrt2bakWHisr6kXweDw7duzYs2fP7NmzVT8M\nbWgxZFQynuebmpqcTufatWvXrVt34YUXnnXWWYcOHerr60v/weD1R0oOCphJoZSuXbtWf+EHHnig\nHB19YLPmzp376KOPlrzycqDVMIfD8dFHHx111FHhcNjYVlUT6UrGsmxzc3Nzc/Prr7/+xz/+8bvf\n/e4ll1zS19fX2dmZsqwBahhSWlDAjCeb85TX084wzOOPP16iFn0FSuk3v/lNiE23hAFSzSvDMC0t\nLWvWrJk5cyakxbJE+61CipIxDNPU1DR79uwXX3zx5z//+ezZs3/7299u27atq6sLI0KRMoECZl7O\nOecc/U87LfMixaeffvqDDz5YpsrLgWpe58yZ89xzz02YMKH4tP1IRlKUrK6ubvHixf39/RdccMHs\n2bOfeeaZl19++fDhw2p5vAVIqUABMy/vvPNOXuXLMQym1nzmmWda0e6ow3i/+MUvrOJBWhetkrEs\ne/LJJ9tstksvvbS5ufmpp57y+/1qTA3eC6QkoICZlwceeCCv8l6vt0wtIYRQSpcuXer3+8t3iDIB\n9vS73/3u448//vbbbxP0AMqPVskWLFiwZMmS733ve4SQRx991O/3q2GKeCOQIkEBMy+U0rwEI9/y\n+VJXV2fRF2cwpv/2b/+2Z88eK2qwdVGVrL293ev1/uQnPyGErFixwu/3w6QxK/6cEPOAAmZqrr32\n2rzKl9UcUEqPPvro8tVfbiilN910EyFk06ZNaDcrDMiYzWbzer0wXvvwww/D7Hu8F0jBoICZgmzP\n8HPPPZdXPeUbBgMuv/xyn89X1kOUFUqp1+tds2ZNNBo1ui1jEZAxn8/n8/mWLVv27//+7+AQo4Yh\nhYECZgqy9WvlK0gVUBefz2dpcwMG9Gc/+5mlz8LqUEoZhvH5fF6v1+/3+/1+vB1IAaCAmYJs8RcF\nxGWkTB0tOSAAZT1EBfD5fH6/3+piXAXAzwlkDG4H3hFEPyhgpiCbp1VAl2AFZmtde+21VWBlvF7v\ntddei+/+ZgC6du+9917wxoLBICoZogcUMFNQwgj4cg+DEUKmTp1aBb4LpXTKlCmzZ89GDTMDlFKe\n571e7+WXX/6HP/wBXGRlBKNbh5gUFDBTkCO2O9+n1+v1lvuBh6H4oaEhq1sWSukVV1zR0NCwZcsW\nNJSGA+9eRx111Pe///2bb76ZjPT0wrrPeIOQdFDATMGTTz6Z7auHHnoo39oqMNXJ6/U+/vjjyWSy\n3AeqALfccssnn3xSW1uLJtJw1P6DpqYmr9d75ZVXTp8+/aGHHvL7/YODgwRTeCBfBQXM7CxZsiTf\nXSrzhFNK//M//9Pq1gQs5nXXXffCCy/MnDlTfdk3ul1jF20f+OzZs6+66qpLL730O9/5zsqVK199\n9dUDBw6QERnDO4WggJkCmGCbEf2LqgDQv1eBB3vZsmVz584dHBysDiNy3nnnvfTSS/PmzYtEIgRn\nJhlKSpL7BQsWjB8//mtf+9qll166atWqnTt37tu3T11vDJVsLIMCZhg6H7kCgjLuuOOOcnfugYk5\n+uij29vbicXNvXqFjz322DVr1syfPx/WwESzaCzaXz7P88cdd1w8Hr/88stPOOGEhx56qL29ffv2\n7aFQSC2D92sMggJWhXg8ntWrV1fgQE1NTXfeeScstVUFUErr6+tDoZAsy+p0OrSJBpLy9maz2WbM\nmLFr165f/vKXbW1tN954Y19fX09Pj/Z1DWVsTIECZhg9PT16it17770FPJAvvPBCZR7jxYsXb9q0\niVjc0KuG0uVyffrpp0cddZQsy0NDQ7DR0qdmdVI0jOf5iRMn2u32NWvWvPzyyxdccMH5558/ODjY\n09OjJrkn2K84ZkABM4ynnnpKTzGe5wuovLu7u4C9CuDMM8/885//rNr6KsDtdu/evXvixIlz5swZ\nHh6GjWgNDSS9F10UxYaGhrq6uvXr17/33nsnnHDCkiVLJEnq6+tTx8YAVLLqBgXMMM444wz1/9xh\nF9pXS53cf//9EFBXWNv0oK5beMUVV0AvoqXNhNZKut3ujo6O7u7uGTNmwHgYYOkTtDQZR4J5nq+r\nqxNF8aOPPtqyZcusWbMWLlzIsuzg4GAsFksZBkYlq0pQwAzj1FNP1VnylVdeybfyJUuWrF+/Pt+9\nCmPOnDk//OEPjxw5Qixu4rVW0ul0SpLU09MzadKkUCiknhcaQaPIqGGUUkEQampqCCE7duzYs2dP\nW1vb3LlzRVEMh8PhcDg9NSgqWTWBAmYYgiDoLDljxoy8aoan+tFHHyXlXyGMECKK4sqVKxsaGuBY\n1WEa4BrGYrGBgYG2trZoNKo9r+o4R8uRI2WoKIpOp1OW5T179nR1dTU1Nc2cObOxsVGSpHA4nLEP\nA5WsCkABM4x58+ZpPx48eDBbyVdffbWAx6y3tzdHnaWlqanplFNO6e3trczhykeKiRQEIRqNDg8P\nt7S0xONxjHYznByzShiGEUXRbrfH4/HOzs7Ozk6WZSdOnNjY2JhMJiORiCzLGW8ZKpl1QQEzjMsu\nu0z78bXXXstYjFL6xz/+sYD6n332WYap0P11OBxvvvnmhAkTwMRXky0ADQuHww0NDYlEIn1kxaiG\njVm005zTARkTBEFRlN7e3sOHD1NKW1tb6+vrKaXxeDybjBFUMguCAmYWWltbs331s5/9rIAK29ra\nfvCDH5CK9CISQkRRnDZtGiSsK/dBy0q6ceQ4LhKJRKPR2traZDKZMTqggg1ECBltgj/LsjzP8zyf\nSCQGBgb6+/thkl9tbS3DMJIkSZKUY7I/KplVQAEzjLfeekv78fe//322kvPnzy+gfpfL9de//rWA\nCMbCcDgcmzdvbmtrq9gRy0eKcaSUchwXjUZjsZjb7VYUJd32obGrPKMmqWEYhud5juNgVh9M9qip\nqfF4PKIoJpNJSZISiUSOe4dKZnJQwAxj8+bN2o85VErtmssLSunnn39esXEpSmkymWxqagqHw7Cl\nmh57SinLsrFYTJIkp9NJCEENMwN6Eq0xDMOyLMuysiwHAgGYF2G3291utyiKhJBEIpFbxggqmVlB\nATMMtbdtVDiO0+Z804/T6YRHtDIPniiKhw4dcrlcqnGvpgcehl7i8biiKKIooh9mEvRoGKWUYRiG\nYSilkiRFIhGYuWiz2Ww2G8uycDf13D6UMVOBAmYYO3bs0H6EqPdsdHR05PXYwFPtdrtPPPHEwpqX\n77EIIRzH9ff3Q9C5+q1Fn/aMZhGCYqLRKKXUZrOB05lSBg1c5dGZ8JoZIZlMxmKxcDgcj8dZlnU4\nHHa7nWXZZDKZHqeTEXTITAIKmGHs379f+zF3qOHkyZMLOATDMNFotJKDUoIg9PX1OZ3O9AmkliOb\nhimKEovFYO4RWMP0YmjaKkzu0ERtsRQZi0Qi8XicYRibzSaKIsuyhBCQMf0OGd5uo0ABM4yWlhbt\nR5fLlaPwtddeW9hRdu3aBd2PlXnGeJ4fGhpqaGiQJEndaN3HO4eGxePxRCLB83wODbPuiVsUna4Y\nqB3LsnArJUmC0U01BJ/jOMjuplPGCCqZQaCAGcbMmTNzfExBzSqbL4lEAoIOKgbLsoFAANIiqBur\n7MEG0ZJlOZlM5tAwUnUnbn7yWj+PjgARiTBLDGIXC5MxgkpWWVDADKO+vl77saGhIUfhN998s7Cj\n2O329vb2cj9OWqvBcVwgEHC73elTpqxINoMI42GgYRzHQSBAxpJozipMYRoGMibLMvjWhBCO40DG\nwFHLS8YIKllFQAEzjJR1UnIvm6Jz8bB0WJaFaZukUt4ADDNEIhGHw1HFThgZMZSQ2UHtj8pWuPpO\n38zoHBJLKQ+7gIxJkqTeWY7jQMbg23xlCWWsfKCAGUb6bNkchT0eT8pCR/qPcvjw4QpPLmZZNhQK\n2e12QRC00RwWfYxz3Br4CmYR6dEwi14Bi5KXhpGvypiiKIlEQl2bG2QMbjEULkzG8AdQWlDArIHL\n5QoEAnntoj69sVisAsNgWmMB/8diMYfDkfLEWvQBzq1hYOySySSEt406JbYMDUQyk6+GkZwyxjBM\nkTJGUMlKCgqYNeA4ruDAdI7j6urqKjwcxbIsLEEC+egqeejKox3qVzUMuxNNQgEapu6olTE1tl5N\n7QEzo0mhvjUqWfGggFkAeJBSgj70A1NbKixgMCQej8chL3glD10mcttB0DA4U9Wu5aA6rolVKFjD\ntPsmR4Abrab2UAsUfE9RxgoGBcwy5LuspQrDMIFAoPJuEMMwMEUUchxU+OiVR/XDyEjSB6NbhPyT\nIjVM62mBK6beaNUV077EIJUBnzHLYLPZCt43HA7rXwC6YDKGpcTjccgcWO6jVwA9RlCrYcUYTaTk\nFHk7tBIFd1kNrNd6YyhjlQQFzDIUsLyytnPD5XJV+KGCeHpJkiDrwVhwwgDVxuUbzI2Um+LvSIo3\npsoY1eQL1o6NlabdSBZQwCxDAQKmAr0cle9FVGdK5Z7lZiF0mj/VD0MNMyEluSPat8MUGVP/4q0v\nNyhglkGbXTBfIKtvCRuTjfQnllIqy/JYc8IAeAFHQ2ZCSn5HUjoV4aZjN3K5QQGzDBzHFbwvpTQW\nixniBqnTpKD9VdCpot8kqWMhqGEmpBx3RJ1Nof7OUcbKCgqYZSgyCiOZTMLilpUHnDBCSI6EgdYC\nNaw6KNNNUdISJ6KMlQkUMMsA07kKBh6hUjUm30PDI82y7Bh8hjGmw+SU6aagN1YBUMAMI9/fcZG/\newgILKYGnWRsJ6U0kUgYKKIlJ6/bASYMNcy0lO+mKBrUY6mU6aBjhyqxJlVAmX7NarWU0soIWLZm\nqE6YUW0oOahh1URZb4p697X9yShjxYMCNoYwPAgwmUyO5cc1ZVDEwJYgGamAhpFMPwP8MRQMChhS\nerI9kNWXoaAA0wNOWDkagxRPuTUsxRUjX01+j+QLChhSISim2PlqR6LRbUEyU+5bk8MVQ/KFjnGD\nUlHwN4ogSG7QIOcDemAIgiCIJSk8uQNSJH6fz+vzwT+wBT76vF6SpUsBCniXLdPGFuZ1UKjhQb8/\nW0BHBboy1DaQgkbFqqazJeXuayl3v0jVXMNqwpvpl4DkBj0wBEEQxJKggFmPKhu2RG8gnYrFESCI\npUEBG4uYQTPUGZ2F7Vj1oIYhyKiggFmbgs1QjqlaRTQHyZscF9wM7xkIYmZQwKxH1dg17VxOY1ti\nLEa9NODLCmJ1UMCsR0nMPUwrLr6egilGvarP8mY7ozEu7QiSGxQw61F8QneTpJRNSahjbGMMxxAN\nq75XAWRMgQJmPRiGKdLuZExmYyCoXgD6YQiSFyhg1qMkHhgxh1nE7IgpVF7D8Poj1gUFzHSMalBK\n5YGZQcBU8u3SrGKzixqGIDpBATMdo1qTksRfqMNg6VVV2JzhMFg62JeIIHpAATMdoy47WRKBMZUT\nVpiGVbffUGENq+6LiVQrKGCmY1QBSyQSxR/FJIGIADphGUE/DEFygwJmOvQIWJEmTF1b0jymEDUs\nI5XUMHTCEMuBAmY6RnWwEolE8YGIJOcwWOXBXsRsrr+LKAAAIABJREFUZFu3FzUMQVDATMeoHlgy\nmWRZtvgDmc1aoROWDe28vXJrGIJYCBQw66EoCsdxJQlEJKWYVVYqzCaopiJFwwpe0VT/gRDE/JjF\neCEqehSFZdkis2lAz2EymcxoAQ1ML0tHMKQBZibljuMlQhAUMNMhimLuApTS4ucyA6aK4yAjiTlw\nGCwbKScLFwqdMGTMggJmOlwuV+4Cagxh8ccCD8wkcRyA6l6YSlnNA1wf9QdQJg1DEEuAAmY63G53\n7gKUUlmWSyU5iqKYahgsxQkzj7Kah4x9iaXVMLzsiCUwi+VCVPQImCRJxR8ITF4ymSxVh2Sp0Dph\nerxDUzW+MqgeGGoYMpZBATMdgiCMWkaSpFK5TWovYklqKxK1d9RUiUJMS8oUMbxiyFgDBcx0yLI8\naplEIsHzfPHHAsEw4TBYykiYedpmNlJGQ0urYXjZEZODAmY6AoGAnmJ6HDU9gAVM9+eMMl6qYmE0\nh05Qw5AxCwqY6ejv7x+1DKVUEIRSGRfITWUqU5WvE2aqxleesmoYgpgWFDDjSTG+wWAwfWMKDMOU\nSnLUXsTiqyoVWies6od2SnVq6TMrSlhzSepBkJKDAmY6QqHQqCZDDSAsyREhENFUw2CEEEgUMhai\nOcqhYVUv/AhCUMBMiJ45XgzDSJJUEgGj2ZdWKaue5bat6U7YqO0xlfrmS6nEpkx9iZa+tkgVgwJm\nOjiO06NM0Wi0VB4YISSRSGjzK1YGPbYVzhEcxPK3yGDMrGEIYkJQwEyHIAh6Iumj0WjBhillR0qp\nIQKW3pL0r6AjkYzkOK56V6AkelOOvsSqv/KIFUEBMx2CIMTj8VGLybKsLqpSpHGhlKrDYMXUU/DR\nc381ppwwADUMQfSAAmY6OI6LRqOjFqOU2u32UvUiqrPBTGiklJFlX0Z1wkzY+IIpRnLU15qU2Qg4\nKxypMlDATAfDMHo8MJZlHQ5HIpEo7CjpvYhal67C6HHCQF/HjhMGFHa+VJORK13DaBGrGaD4IaYC\nBcyM6DFbLMvabLaCBSz9iAYKGBntlPU7YdVHYa6YVqgyahgZY5cRqUpQwMxITU1Njm/pSOpxNcCh\neNRhsJTtZrBx6pkqisKybO75amZocDkopjuxtBpWrVcYsSIoYGakvr5+1DKU0ng8XkyXWsq+6qQr\nEzphoK/654RVJYX5YeSrg2EE/TCkikABMyPTp0/XU2x4eLhUKX2JCXoRSXYbDdsTiYTWCRuDlrdI\nDQN/PUXASP4aNgavPGJOUMDMyOTJk/UUCwaDo65+mRutQYTgEUEQSjg/uoSAEwb9nOZZQrryFDAk\nlqJh4MhCRIx6JVHDECsydg2BmbHZbHqKSZJUW1tbqoNSShOJhLEeGNHhhBFCWJYl2W3oWLCt+cqY\nqmHwEgCXCDSsYD8MQQwHBcyMfPnll3qKsSxrt9tLaHfUYbBSVVgYOTQsmUxC0pDcGjZGKEDDCCHJ\nEcjIsgaFadgYv/iIGUABMyNbtmzJXQAsDs/zxWdETO9F5HnetL2IZGTFao7jDIw3MQ+oYchYBgXM\njOzatUtPMY7jent7JUkq1XEZhonFYjabTStgppraDE6YLMssy0JvZ8bmjSnDmld3olaotBoGoTHq\nV+VoJ4KUHBQwMzI0NKSnGMMwfX19etJ25Ea1XBCIaJI4joxGGYw1rDjD8zzDMGZoqhkoRsMgmqMw\nDUO1QwwEBcx0UErtdrvOwvF4vIQeGBmZNVzCCoshm4aBE0Yp5TiOoA0dQb8rppYEAYPQGFXD1FSK\nZW0tghQPCpgZ0TORGRAEobRGnGXZSCTC87xJ7Fe6RYYtkiSBswh6ZkTTTEperpgaF6PVMMjprH+m\nnUl+KsgYBAXMeNKff5fLNapRADslimJTU1PxbVCtHsMw4XDY6XSWKstiOYCeQ0hEAlqbfrnGslXN\n1xVTFEXVMIjw1E4d01PPWL7aiIGggBlPugOhc1FmKNnc3Fza9iQSiZQ4DmPJ6ITBrLVEIsHzPEED\nmom8AhTVOc7kq3GJeGERM4MCZjzpURgsy8ZiMT37Mgwzfvz40rYHhvQhb1Npay6YdFsMKSRisRjH\ncRD3j05YOno0TDseJssy5DrhOA4mKpBML1gZwauNVB4UMOMZHh5O36gzEJEQEo/HS2I7VGPHcVwo\nFDJ5gB+oLIyE2Wy2sZkaUQ96uhO1GiZJUiKRKEzDEKTCoIAZT8a8G4cOHdK5++7du0trX1iWHRoa\ncrvdMGWYmOPlOqMTRimFkBOe5808aGc4BWgYIQQ0jIx0MI56FDP8TpAxBQqY8XR0dKRv7Onp0bn7\n3r17dfY36keSpLq6OlXATEJGDZMkKR6POxyOEq6OVpXo0TB4JwANg+kZ8HJAS7r4HIKUChQw48mo\nVS6Xa9Qd1dRKgUCgJC3R9iKqb99mBsINwuGwIAgZnTD0CbTo6U4sUsPwgiOVBAXMeBwOR/rGSy65\nROfuHo8nGAyWtEWE5/m+vr6GhgazvXSn21+WZSVJisViLpdLzY2E5ECnhhFCwLslhMD7AdEXWI8a\nhlQMFDDjWbhwYfrG2267TachcDqdety1vGAYpre3t6mpqbRpPkpCiv2llLIsGwgEbDYbjoTpZFRX\nTF10LR6PQwe1KIqweqqpwlORMQ4KmPF84xvfKGZ3juNOOeWUUjVGm7A8Ho9Ho9FS1Vw+WJaVZTkS\niXg8HkiTqP0WrW02cmuYOiQmSRL8DOAVgejQMLzmSGVAATOeK6+8MmVLvolZzzjjjNI2iRBis9m+\n+OILw9cGy0i6E8Zx3ODgoNPp5HnebLEnZia3K6ZqWDweD4fDiqI4HA5BECBzR+7eWtQwpAKggJmU\n3/72t/oLl8NYcBw3PDzs8XjKd4hiSDG74ITF43Gn04kCli96NAzGwyiloijCQjYZJ48jSCVBATMp\njzzyiP7Cr7zySgkPDeYM8rrOnDnTtHqgNbvghB05cqShoQH8AwMbZkVyaxisfx2JRMLhMM/zTqeT\nZVnIApxDw1DekHKDAmZSQqGQ/sJr1qwpx+uw0+m02WyRSKS01ZYJWJ86FovV1tamZOdCS6qHHN2J\n6so1kUgkFAoJguB2u0VRhHSUeHkRo0ABM54tW7akb1y0aJH+GhwORzlkhuO4rVu3ljxGv4SkOGGi\nKB44cGDcuHGQ08/AhlmX3BpGKQ2Hw4FAQBCEmpqaUTUMtQ0pKyhgxvPss8+mb5w1a5b+GiZPnnzg\nwIGSNUhjxbq7u2tqakpYc8nRGlye5yHcoL6+PhaLofUsjGyuGGgYwzChUGhoaEhRlPr6elEUZVnG\nPlvEEFDAjOf1119P35hxdnNGKKVOp/Pzzz8vaaP+gcfjOeOMM0zuzWitrcPh6OjomDp1ajQa1U5i\nQzHLl2waxvM8pHvu7e1NJpONjY2iKEJW5Yz14JVHygcKmPHs3bs3fWNe8euU0nnz5pWuRf/E5XK5\n3W6rDIORkaXUhoaGGhoa8hpHRNLJ5orBEjayLB8+fDiRSLS0tAiCEI/H0Q9DKgwKWJXwzDPPlLZC\nNRbx5ZdfPnz4MDH3q7TWzrpcru3btx977LGQZcrAVlUH2TRMFMVkMvnll18mEokJEyYIghCNRjNq\nmJl/OYilQQEznmxpD03y2B8+fBiiqE2OamdZluV5/vDhw4sWLcq41hqSLxldMYZhRFFUFKWzs1OW\n5UmTJnEcF4lEUMOQioECZjxHH310xu15hf99/etfL1Fz/gnYrPb29ltvvbXklZcVt9v9wQcfwKRm\n1Z6iDS2SjBpmt9sppR0dHZIkzZgxg2XZSCRi8kFTpGpAATMeURTTN1JKN27cqLMGSum7775b0kb9\nE7vd3tLSYom+ONXCMgzj8XjeeOON8847b3Bw0NhWVRPpGkYptdvtPM9/9tln8Xgc3sZCoVB6TAe+\nQCAlBwXMvOgXMELI4sWLy9eSJ554wioyoFpYl8sVi8WmTp06MDBgbJOqjIwaJooiy7J79uwRRXHS\npEmxWCwSiaBiIeUGBaxKmD9/fjkyx4O1+uijjz766KOSV15WGIaZMGHC73//+/nz51soitISpA+J\nMQzjdrsTicTGjRvb2tpmzZoVi8WCwSCuDICUFRQw8+L3+/UX5jhuw4YNZWrJCSeccM0115Sp8pKj\n2lZYlvN73/ueOlEBDWgJSZExSqnH42EY5u233542bdrZZ58dDAZxJgNSVlDAjCfblK9ly5blVc/6\n9etL0ZwM2Gy2W265hVhHANRLOnny5FtvvXXu3LmWGMOzIukaZrfbX3755d7e3ssuu6yvry9Fw6zy\nE0IsAQpY9ZAxp2LxgIU67rjjwuFwOeovKwzD8Dzv8/m2bdtmdFuqlpQ3MLfbXV9fv27dugMHDlx3\n3XXd3d3YhYuUCRQw8/Lzn/9c/+sqJOMo3+ttY2PjSy+9VKbKy4HawTV58uQTTzxx3LhxENuNHkA5\nSOlOdDqdU6ZMWbt27e7du2+//fbdu3fHYjH1W7wFSKlAATMvd999d17ly7d6MqX0lFNOOeqoo8pU\nf7k57rjjfvOb36ATVm60v0CWZWfMmLF27doPP/xwxYoVH3/8MaamREoOChiiC0rpm2++aXQr8gZM\namtrK8uyX3zxhdHNqX60rhjDMLNnz16/fv2bb7759NNPb9iwQZukAzUMKR4UMPNy33335VX+nnvu\nKVNLAEqpFY0O2NM77rhj8+bNHR0dRjdnTKBqGKV09uzZn3zyya9//etXX331nXfeseJPCDEtKGDm\nBdbA1Y8gCGVqCSGEUnrTTTetW7eufIcoKy6Xi1L63//93wTf/SuC1hWbMWPGoUOHbrnllvfff/8v\nf/mLWgZvBFIkKGCmJt/g73g8XqaWEEKampr+9re/WdHogCV94IEHFEUZGhoyujljCFXD2tvbPR7P\nMcccs3HjxnfeecfYViFVAwqYeaGUPv/883mV/9WvflW+9hArvzJTShmGoZSuXLnS6LaMLVQNa2pq\ncrvdhJD169erk8Os+4tCzAAKmKnZt29fXuXzSmCfL9mWN7QQ9913H1rMyqP+cs4999xLLrlEUZRf\n/vKXn376qdHtQiwPCpipyTcuo9zW+Uc/+lFZ6y8rlFKO4yilfr8fZazygIYdf/zx/+///T9FUR57\n7LEjR44QdMKQIkABMzUPP/xwXuXL7SE5nc68MjSaDUqp1+tFi2kU4Iq1tbXdfffdbW1tfX19ZR21\nRaoeFDBTYBWTatFI+hQopZ2dnUa3YuwCS4h5vd4XXnjhkUceIdb5/SNmAwXM1Ph8vrzKe73e8jTk\nn1h9GAycsN/97ndoNA0EXDH4uVrap0eMBQXMFGR7hvN9tiugLlXQBQeDYUa3AiFaDbP6jwoxBBQw\nU5DNcyrAoyq3IaCUPvroo2U9RAW4//77fT4fGk3DQQ1DigEFrNqowKh4IBAo9yHKDXRhodE0Ayka\nhncE0Q8KmCnI1vVXQJcgjIqXleoI5QCjCWusIIYDt6Onp0eW5Sr4dSGVAQWs2qjAw/+1r32t3Ieo\nDFOnTl2+fDmaS8OBF7W77767trb2oYceIhiXiOgDBcwU3HHHHUY3IQ/mz59f7pxVleHb3/429lmZ\nB5vN1t/fPzg46Pf79+/fj7cGGRUUMFPg8XhKVVVlwtz7+vqsblzgQlFKH3zwQaufSxWgrtx2/fXX\nK4ry7LPP/v3vfyfoiiE5QQEzBS+++GK2r/J9gOH5L7pFo7B3795yH6IyLFu2TFGU4eFhNJQmYe7c\nuWeffTal9M9//vNbb71FCEFXDMkGCpgpKGFi07a2tlJVlQ1K6fTp05PJpNXNCh3h8ccfJ/iybzRq\n58HixYtvvvnmefPmbdq06dVXX4WNeHeQdFDATIHlHs777rtv+fLlRreiNNxzzz2Konz++edGNwT5\nJ42NjYsWLbr88ss/+eSTP/3pT/CAoCuGpIACZgr+9re/ZftqzZo1+db2wgsvFNec0eE4rmpMiSAI\nlFIQsKo5KYuiHcFta2uz2+1Lly7dvHnzmjVrEokEbMd7hKiggJmCd999N9tXmzZtyre2Xbt2Fdcc\nvXzyySfVYU2WLl36wQcffPDBBwTto9FoNWzy5Ml9fX0+n++NN954//331Un66IohAAqY2bHb7fnu\nUplnm1L6pz/9qQIHKitgLmtra2fOnDl79myY14zG0SRQSqdNm7Z3795nnnnm6aef3rp1azQaVb/F\n24SggJmCHPPAwuFwXlWBRa7As33DDTdUkwU588wzH3vsMXDCEGPROmGU0ilTpuzYseO1115bvnx5\nR0eH9olAV2yMgwJmCko4D4xUaipYc3PzihUrJEmqDgvS2tp60UUXnXPOOdBPVR0nZV20v2GWZdvb\n27du3fq3v/3ttttuO3z4cDAY1N4gvFljFhQww7D6U0cpXbVq1f79+41uSGmglAqCcMcdd3R3d8MW\nq9+gaoLjuKampi1btgwPD5922ml9fX3BYFBbAG/W2AQFzOw0Nzfnu8s111xTjpZogRfk008//bPP\nPiv3scqN+rI/a9asX/ziF06nU5ZlY5uEkLSOBKfTWV9f/9prrzU0NJxyyinRaDRl+jl2J45BUMDM\nzpEjR/LdZfLkyaFQqByNSWHSpEkwAaA6DIfNZvv444/vuuuurq4u2FId51U1OJ1Op9P5zjvveDye\ns846K5lMpqdQwVs2pkABMztNTU357sKyLOTgKTcsy3Z3d1dgBbKKMXHixDfeeGP8+PHaiG1jmzSW\nSXHCKKUul0sQhL///e/d3d0nnXQSx3GBQCBlTRy8ZWMHFDDD0LkS1b/+678W8EAePny4Mo/xvffe\nWwV5EVVD6XA4tmzZcskllxw+fFjtSESDaB4opU6nk1La0dHR09Nz7LHHiqIYDAbVac4AdieOEVDA\nDOPgwYN6ihXggemvvHhsNhuEUFaHvaCU1tfXf/jhh5MmTRoYGDC6OUiGkFqGYZxOp81m279/f19f\n38yZM+12ezgcTtEwgjI2BkABM4y//OUveoqxLFtA5YXtVQATJky48cYbq8lMOJ3OTz/9dOHChXa7\nXZIk2FhNJ1gFMAwjiqIgCN3d3UNDQ1OmTHE6nZFIJGP0Dd67KgYFzDA6OzvV/0s+c+upp54qbYXp\nQJttNtvg4GBlYkYqA8MwDodjz549CxcuHB4eVrejHTSKjE8HTHtgWfbgwYOBQGDChAlOpzMWi6GG\njSlQwAyjvr5e+7G0z9iqVatKXmdGKKWvvfZab29vZQ5XPrR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"output_type": "display_data"}], "prompt_number": 22, "cell_type": "code", "language": "python", "metadata": {}, "input": ["clf;\n", "plot_wavelet(S, Jmin, options);"]}, {"source": ["The operator is compressed by computing a sparse matrix $ S_\\tau $\n", "that approximate $S$. Since most of the entries of $S$ are small, one\n", "achieve this compression using a non-linear hard thresholding\n", "$$ (S_\\tau)_{i,j} = \\choice{\n", "      S_{i,j} \\qifq \\abs{S_{i,j}}>\\tau, \\\\\n", "      0 \\quad \\text{otherwise}.\n", "  }$\n", "$$\n", "One thus defined $S_\\tau = \\Gamma_\\tau(S)$ where\n", "$\\Gamma_\\tau$ is the hard thresholding operator."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 23, "cell_type": "code", "language": "python", "metadata": {}, "input": ["Gamma = @(S,tau)S .* ( abs(S)>tau );"]}, {"source": ["The threshold $\\tau$ can be adapted to ensure that\n", "$S_\\tau$ has only $m$ non-zero entries.\n", "This defines $\\tau=\\tau(m)$."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 24, "cell_type": "code", "language": "python", "metadata": {}, "input": ["subs = @(u,m)u(m);\n", "tau = @(m)subs( sort(abs(S(:)), 'descend'), m);"]}, {"source": ["Set the targeted number of coefficients."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 25, "cell_type": "code", "language": "python", "metadata": {}, "input": ["m = 4*N;"]}, {"source": ["In order to save computation time when applying $S_\\tau$, it should be\n", "stored as a sparse matrix $S_1$."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 26, "cell_type": "code", "language": "python", "metadata": {}, "input": ["S1 = sparse(Gamma(S,tau(m)));"]}, {"source": ["Display the support of the $m$ largest coefficients that are used for\n", "the approximation."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [{"metadata": {}, "png": 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"output_type": "display_data"}], "prompt_number": 27, "cell_type": "code", "language": "python", "metadata": {}, "input": ["clf; \n", "plot_wavelet(zeros(N), Jmin, options); hold on;\n", "spy(S1);"]}, {"source": ["One can use this compressed operator to approximate\n", "$$ g =  \\Psi^* S \\Psi f   $$\n", "by\n", "$$ g_\\tau = \\Psi^* S_\\tau \\Psi f. $$\n", "When $\\tau \\rightarrow 0 $,  $g_\\tau \\rightarrow g$.\n", "When $\\tau$ is large, since $S_\\tau$ is a sparse matrix, computing\n", "$g_\\gau$ is faster than computing $g$."], "metadata": {}, "cell_type": "markdown"}, {"source": ["__Exercise 1__\n", "\n", "Display the result $g_{\\tau(m)}$ for different values of $m$."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [{"metadata": {}, "png": 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"output_type": "display_data"}], "prompt_number": 28, "cell_type": "code", "language": "python", "metadata": {}, "input": ["exo1()"]}, {"collapsed": false, "outputs": [], "prompt_number": 29, "cell_type": "code", "language": "python", "metadata": {}, "input": ["%% Insert your code here."]}, {"source": ["__Exercise 2__\n", "\n", "Study the speed gain as a function of $m$ of using the sparse multiplication with respect\n", "to the direct computation of $T f$.\n", "o correction for this exercise."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 30, "cell_type": "code", "language": "python", "metadata": {}, "input": ["exo2()"]}, {"collapsed": false, "outputs": [], "prompt_number": 31, "cell_type": "code", "language": "python", "metadata": {}, "input": ["%% Insert your code here."]}, {"source": ["Operator compression is not only useful to compute efficiently\n", "approximation of product $T f$. It can also be used for instance to\n", "compute approximation of the inverse $T^{-1}$, see for instance [BeyCoifRokh91](#biblio).\n", "\n", "\n", "Bibliography\n", "------------\n", "<html><a name=\"biblio\"></a></html>\n", "\n", "\n", "* [ChaMalYap00] E-C. Chang, S. Mallat, C. Yap, [Wavelet Foveation][1], Applied and Computational Harmonic Analysis, Volume 9, Issue 3, Pages 312?335, 2000\n", "* [Burt88] P.J. Burt, [Smart Sensing with a Pyramid Vision Machine][2], PIEEE(76), No. 8, pp. 1006-1015, 1988.\n", "* [BramPascXu90] J.H. Bramble, J. Pasciak e J. Xu, [Parallel multilevel preconditioners][3], Math. Comput., 55, 1-22, 1990.\n", "* [BeyCoifRokh91] G. Beylkin, R. Coifman, V. Rokhlin, [Fast wavelet transforms and numerical algorithms][4], Comm. Pure Appl. Math., 44, 141-183, 1991.\n", "* [Beylkin92] G. Beylkin, [On the Representation of Operators in Bases of Compactly Supported Wavelets][5] SIAM Journal on Numerical Analysis, 29:6, 1716-1, 1992.\n", "* [DahmKun92] W. Dahmen and A. Kunoth, _Multilevel preconditioning_, Numer. Math., 63, pp. 315? 344, 1992.\n", "\n", "[1]:http://dx.doi.org/10.1006/acha.2000.0324\n", "[2]:http://dx.doi.org/10.1109/5.5971\n", "[3]:http://www.jstor.org/stable/2008789\n", "[4]:http://dx.doi.org/10.1002/cpa.3160440202\n", "[5]:http://dx.doi.org/10.1137/0729097"], "metadata": {}, "cell_type": "markdown"}], "metadata": {}}], "nbformat": 3, "metadata": {"kernelspec": {"name": "matlab_kernel", "language": "matlab", "display_name": "Matlab"}, "language_info": {"mimetype": "text/x-matlab", "name": "matlab", "file_extension": ".m", "help_links": [{"url": "https://github.com/calysto/metakernel/blob/master/metakernel/magics/README.md", "text": "MetaKernel Magics"}]}}}