{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Mesh Denoising\n",
    "==============\n",
    "\n",
    "*Important:* Please read the [installation page](http://gpeyre.github.io/numerical-tours/installation_python/) 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",
    "$\\newcommand{\\Om}{\\Omega}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "This tour explores denoising of 3-D meshes using linear filtering, heat\n",
    "diffusion and Sobolev regularization."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Loading required package: magic\n",
      "Loading required package: abind\n",
      "Loading required package: plyr\n",
      "Loading required package: magrittr\n",
      "\n",
      "Attaching package: 'imager'\n",
      "\n",
      "The following object is masked from 'package:magrittr':\n",
      "\n",
      "    add\n",
      "\n",
      "The following object is masked from 'package:plyr':\n",
      "\n",
      "    liply\n",
      "\n",
      "The following objects are masked from 'package:stats':\n",
      "\n",
      "    convolve, spectrum\n",
      "\n",
      "The following object is masked from 'package:graphics':\n",
      "\n",
      "    frame\n",
      "\n",
      "The following object is masked from 'package:base':\n",
      "\n",
      "    save.image\n",
      "\n",
      "\n",
      "Attaching package: 'akima'\n",
      "\n",
      "The following object is masked from 'package:imager':\n",
      "\n",
      "    interp\n",
      "\n",
      "network: Classes for Relational Data\n",
      "Version 1.13.0 created on 2015-08-31.\n",
      "copyright (c) 2005, Carter T. Butts, University of California-Irvine\n",
      "                    Mark S. Handcock, University of California -- Los Angeles\n",
      "                    David R. Hunter, Penn State University\n",
      "                    Martina Morris, University of Washington\n",
      "                    Skye Bender-deMoll, University of Washington\n",
      " For citation information, type citation(\"network\").\n",
      " Type help(\"network-package\") to get started.\n",
      "\n",
      "\n",
      "Attaching package: 'network'\n",
      "\n",
      "The following object is masked from 'package:plyr':\n",
      "\n",
      "    is.discrete\n",
      "\n",
      "\n",
      "Attaching package: 'tuneR'\n",
      "\n",
      "The following objects are masked from 'package:imager':\n",
      "\n",
      "    channel, play\n",
      "\n",
      "\n",
      "Attaching package: 'pracma'\n",
      "\n",
      "The following objects are masked _by_ '.GlobalEnv':\n",
      "\n",
      "    circshift, fftshift, grad, ifftshift\n",
      "\n",
      "The following objects are masked from 'package:magrittr':\n",
      "\n",
      "    and, mod, or\n",
      "\n",
      "The following objects are masked from 'package:geometry':\n",
      "\n",
      "    dot, polyarea\n",
      "\n",
      "The following object is masked from 'package:magic':\n",
      "\n",
      "    magic\n",
      "\n",
      "The following objects are masked from 'package:Matrix':\n",
      "\n",
      "    expm, lu, tril, triu\n",
      "\n"
     ]
    }
   ],
   "source": [
    "options(repr.plot.width=3.5, repr.plot.height=3.5)\n",
    "options(warn=-1) # turns off warnings, to turn on: \"options(warn=0)\"\n",
    "\n",
    "library(Matrix)\n",
    "library(geometry)\n",
    "library(imager)\n",
    "library(akima)\n",
    "library(plot3D)\n",
    "library(network)\n",
    "\n",
    "# Importing the libraries\n",
    "for (f in list.files(path=\"nt_toolbox/toolbox_general/\", pattern=\"*.R\")) {\n",
    "    source(paste(\"nt_toolbox/toolbox_general/\", f, sep=\"\"))\n",
    "}\n",
    "for (f in list.files(path=\"nt_toolbox/toolbox_signal/\", pattern=\"*.R\")) {\n",
    "    source(paste(\"nt_toolbox/toolbox_signal/\", f, sep=\"\"))\n",
    "}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "3-D Triangulated Meshes\n",
    "-----------------------\n",
    "The topology of a triangulation is defined via a set of indexes $\\Vv = \\{1,\\ldots,n\\}$\n",
    "that indexes the $n$ vertices, a set of edges $\\Ee \\subset \\Vv \\times \\Vv$\n",
    "and a set of $m$ faces $\\Ff \\subset \\Vv  \\times \\Vv \\times \\Vv$.\n",
    "\n",
    "\n",
    "We load a mesh. The set of faces $\\Ff$ is stored in a matrix $F \\in\n",
    "\\{1,\\ldots,n\\}^{3 \\times m}$.\n",
    "The positions $x_i \\in \\RR^3$, for $i \\in V$, of the $n$ vertices\n",
    "are stored in a matrix $X_0 = (x_{0,i})_{i=1}^n \\in \\RR^{3 \\times n}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "X0 = read_mesh(\"nt_toolbox/data/elephant-50kv.off\")$X0\n",
    "F = read_mesh(\"nt_toolbox/data/elephant-50kv.off\")$F0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Number $n$ of vertices and number $m$ of faces."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "n = dim(X0)[2]\n",
    "m = dim(F)[2]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Display the mesh in 3-D."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
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y7Ezj/JZPLu3bvSU0RZS0tLU1NTDl/o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\nTkSuQmCC",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "options(repr.plot.width=3.5, repr.plot.height=3.5)\n",
    "plot_mesh(X0, F, \"grey\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Noisy Mesh\n",
    "----------\n",
    "We generate artificially a noisy mesh by random normal displacement along the normal.\n",
    "We only perform normal displacements because tangencial displacements\n",
    "do not impact the geometry of the mesh.\n",
    "\n",
    "\n",
    "The parameter $\\rho>0$ controls the amount of noise."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "rho = 0.015"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "We compute the normals $N = (N_i)_{i=1}^n$ to the mesh.\n",
    "This is obtained by averaging the normal to the faces ajacent to each\n",
    "vertex."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "N = compute_normal(X0, F)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "We create a noisy mesh by displacement of the vertices along the\n",
    "normal direction\n",
    "$$ x_i = x_{0,i} + \\rho \\epsilon_i N_i \\in \\RR^3 $$\n",
    "where $\\epsilon_i \\sim \\Nn(0,1)$ is a realization of a Gaussian random\n",
    "variable,\n",
    "and where $N_i \\in \\RR^3$ is the normal of the mesh for each vertex index\n",
    "$i$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "X = X0 + t(cbind(rho*replicate(1, rnorm(n)),rho*replicate(1, rnorm(n)),rho*replicate(1, rnorm(n))))*N"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Display the noisy mesh."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_mesh(X, F, \"grey\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Adjacency Matrix\n",
    "----------------\n",
    "We define linear operators that compute local averages and differences on\n",
    "the mesh.\n",
    "\n",
    "\n",
    "First we compute the index of the edges that are in the mesh,\n",
    "by extracting pairs of index in the $F$ matrix."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "E = cbind(F[1:2,],F[2:3,],rbind(F[3,],F[1,]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Add the reversed edges. This defines the set of edges $\\Ee$\n",
    "that is stored in a matrix $E \\in \\{1,\\ldots,n\\}^{2 \\times p}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "E = unique_columns(cbind(E,rbind(E[2,],E[1,])))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "We keep only oriented pairs of index $(i,j)$ such that $i<j$,\n",
    "to avoid un-necessary computation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "E0 = E[,E[1,]<E[2,]]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "This defines a matrix\n",
    "$E \\in \\{1,\\ldots,n\\}^{2 \\times p_0}$ where $p_0=p/2$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "p0 = dim(E0)[2]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Display statistics of the mesh."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1] \"#vertices = 24955 , #faces = 49918 , #edges =  74877\"\n"
     ]
    }
   ],
   "source": [
    "print(paste(\"#vertices =\" , n , \", #faces =\" ,m , \", #edges = \" , p0))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "The weight matrix $W$ is the adjacency matrix\n",
    "defined by\n",
    "$$\n",
    "      W_{i,j} = \\choice{\n",
    "          1 \\qifq (i,j) \\in \\Ee, \\\\\n",
    "          0 \\quad \\text{otherwise.}\n",
    "      }\n",
    "$$\n",
    "Since most of the entries of $W$ are zero, we store it as a sparse\n",
    "matrix."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "W = sparseMatrix(i=E[1,]+1, j=E[2,]+1, x=1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Compute the connectivity weight vector $ d \\in \\NN^n $\n",
    "$$ d_i = \\sum_{j} W_{i,j} $$\n",
    "i.e. $d_i$ is the number of edges connected to $i$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "d = colSums(W)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Display the statistics of mesh connectivity."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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QUFFRUV3t7e9913X3BwcERExPDhw318\nfPTpBICBWBt2n3zyiYj07du33ni/fv1E5NNPP7VtW7ezbNmy+Ph47fmNtYqLi7Ozs7Ozszdt\n2jR37tzU1NSIiAh9+gFgFNbuxmonKDIzM+uNf/fdd9LA0xeNlpGRMXXq1IqKipEjR65du/bM\nmTOFhYWVlZXXr18/d+7ctm3bYmNjc3JyYmJiDh48qEM/AAzE2rAbMWKEiIwbN27p0qUXL14s\nLy/Py8vbuHGjNj5s2DA79vg/ycnJImI2m9evXz969OjOnTu3aNHC1dXV29s7KCgoJiZm69at\nCQkJ5eXl8+bN06EfAAZibdglJiaGhIQUFBRMmzatY8eOHh4eAQEBI0aMuHjxYvfu3RMTE+3a\npeb48eMicuelabNmzRKRb775Rod+ABiItWHn6+u7d+/edevWRUREtG7d2s3NzcfH56GHHnrz\nzTcPHTrUqlUru3apcXFxEZHy8vI7bOPq6ioiFRUVOvQDwEAacCMAk8k0atQoBz4Pu3fv3pmZ\nmUlJSUlJSbfbZvHixdqWOvYFwACMdFHx3LlzXV1dk5OTw8LC1q1bd/r06evXr1dXVxcXF58/\nf37jxo3R0dELFiwwmUyvv/66o5sF4FwaMLMrLCxMSUlJS0u7cuVKeXl5VVWV9hiKSZMm6XM2\ndtCgQZs3b548efK+ffv27dt3y218fX2XL18eHh6uQz8ADMTasCstLR00aNCJEyfqDp49e3by\n5MlZWVlr167VJ+9iYmLCw8PXr1//+eefHzt2LC8vr6ioyN3dPSAgoHv37mFhYePGjdPnACIA\nY7E27FJSUk6cONGzZ89NmzY9+OCD2uCXX345fvz49evXDxo0aPLkyXZr8v/x9vaeOHHixIkT\n9SkH4PfB2mN269atExGz2dytW7fawQEDBixZskREPvroI3s0BwC2Yu3MLjs7W0T69OlTb/zJ\nJ58UkVOnTtm2rTtgbSyARrA27Nzd3W/cuKFd6VaXtky1srLSxn3dBmtjATSOtbuxDz/8sIgc\nOHCg3nhGRoaIBAcH27atW2JtLIBGszbspk+fLiLx8fE//vijNnL16tUVK1Zoz42dMWOGnfqr\ni7WxABrN2rCLi4tLSEg4c+ZMz549tZG2bdu+8MIL165de/HFF/VZVsHaWACN1oAVFImJifv2\n7YuNjW3Tpo2bm9u9994bEhKycePGpUuX2q+/ulgbC6DRGvaQ7JCQkJCQEDu1clesjQXQaNbO\n7Dw9PU0mk2NnTKyNBdBo1s7sYmJiPvnkk6+//vqpp56ya0N3wNpYAI1mbditXLnS3d197Nix\n77///tNPP+3n52fXtm7H5mtjq6qqdu7ceePGjTtso13IUl1d3dTuATiOtWFXm24xMTG33EC3\nJyjadm1sVlZWdHS0NVvWXnMDwIgadoLi92fw4ME7duy468zObDb36NFDt64A2Jy1Yec8j762\n7dpYV1fXqKiou25mNptvXioHwEAMNrNjbSyAxrnTbMVkMulzS04rsTYWQKM1bNfMsfHH2lgA\njWak41CsjQXQaEY6Zsfa2JtZc52g/Tj6cEGNA3vw9PQcOnSo9nmDIRgp7Fgbe7Ndu3ZZeZ3g\n71GOiJjNZrPZ7JDy6enpkZGRDimNRjBS2M2dO/eLL75ITk4+fvz4hAkT+vbt2759ey8vr+vX\nr+fl5R05cmTNmjXp6elKrY0tLS0VEZGBIh0dUf+oyE+OqKupEhEHvfdfRL783w8fxmCksGNt\n7O11FNHjZtE3+cWhYadx1HuHwRgp7ITnxgJorLuH3c3Xmtzy6hODro0FoAgjXXoCAI12p7Cr\naQjdOq6rvLx8wYIF3bp18/Dw8Pf3j4yM3LNnj0M6AeDkjDSzq7d+o6KiIiIiYs6cOdnZ2eXl\n5fn5+RkZGUOGDNHnUWcAjMVIYVeP2WzOzMz08fFJTU21WCyXL19OTk729PT84IMP1qxZ4+ju\nADgXA4ddWlqaiCxatGjSpEn+/v7t2rWbNWtWYmKiiKxcudLBzQFwMgYOu3PnzolIXFxc3cHn\nnntORL7//nvH9ATAWRk47Fq0aCEizZo1qzuoXWRX74Z3AGC8sKtd5K9N4urd4OTQoUMi0qlT\nJ/0bA+DMjBd23t7eXbp0eeaZZ8rKykwmk3ZPJ82pU6defvllERk2bJjjGgTgjIy0XCw0NDQ7\nO/vSpUtnz549e/asNnj06NHaDR588EERCQwMjI+Pd0yLAJyVkcJu7969IlJaWnrmzJnTddRu\n4O/vP2TIkIULF2qH8wCglpHCTuPl5RUcHBwcfIsbXVgsFv37AWAIxjtmBwCNQNgBUAJhB0AJ\nhB0AJRB2AJRA2AFQAmEHQAmEHQAlEHYAlEDYAVACYQdACYQdACUQdgCUQNgBUAJhB0AJhB0A\nJRB2AJRA2AFQAmEHQAmEHQAlEHYAlEDYAVACYQdACYQdACUQdgCUQNgBUAJhB0AJhB0AJbg5\nuoEGu3jxYlpaWlZW1unTpwsKCioqKry9ve+7777g4OCIiIjhw4f7+Pg4ukcATsdgYbds2bL4\n+PiysrK6g8XFxdnZ2dnZ2Zs2bZo7d25qampERISjOgTgnIwUdhkZGVOnTnVxcRk5cmRUVNSj\njz7q7+/v5+dXVlaWm5t74sSJVatWbdu2LSYmZv/+/f3799enq6qqqp07d964cUOfcvUcPHjQ\nIXWVVyMO/eF7enoOHTrU1dXVUQ0YkZHCLjk5WUTMZvNLL71Ud9zb2zsoKCgoKCgmJmb27NlJ\nSUnz5s3btWuXPl3t2rUrOjpan1pwGjkiYjabzWazozpIT0+PjIx0VHUjMlLYHT9+XETGjx9/\nh21mzZqVlJT0zTffWPma1szLtD/g1dXVt/xuaWmpiIh0FGluZVGbuiJiEfnFEaVFxCIiSlbP\nFRHH/dKLRH7JzMz832dPb0adV9YYR4sWLUTEYrHcYZv8/HwR8fPzs/I19+7da+UP6u9///st\nXyE9Pd2Ovx7AKe3du7fB/4AdzUgzu969e2dmZiYlJSUlJd1um8WLF2tbWvmagwcP3rFjx51n\ndtXV1dnZ2QkJCbf8bkRExF1fwX6qq6t/+OGHnj17urg44CoiZasr+8Y1np6egwcPdkjppjDV\n1NQ4ugdr7d+/PzQ0tKqqKjQ0dMKECX379m3fvr2Xl9f169fz8vKOHDmyZs2a9PR0k8m0e/fu\n8PBwR/cLwIkYKexEZPv27ZMnT7ZYLLfbwNfXd/ny5WPGjNGzKwDOz2BhJyIlJSXr16///PPP\njx07lpeXV1RU5O7uHhAQ0L1797CwsHHjxrVq1crRPQJwOsYLOwBoBCOdoHBO5eXlixYt6tq1\nq6OOVZ88ebJXr15UV6S0M1TPzs5+9dVX3d3d9a/eFIRdUyUlJb3xxhuO7gLQlYuLy5w5cxzd\nRcMQdk3VpUsXEYmPj9dtgVpdBw8eNJvNVFentJNU1z72xkLYNZW2K9G/f/+4uDiHNGA2m6mu\nVGlnqO6oS/yawngdA0AjEHYAlEDYAVACYQdACYQdACUQdgCUQNgBUAJhB0AJhB0AJRB2TeXl\n5VX7X6orUl3ZN+7w6k3i2LvC/w5UVlbu2bOnsrKS6upUV/aNO7x6U3A/OwBKYDcWgBIIOwBK\nIOwAKIGwA6AEwg6AEgg7AEog7AAogbADoATCDoASCDsASiDsACiBsAOgBMIOgBIIOwBKIOwA\nKIGwA6AEwq4xLBbL5MmTO3To4OnpGRQUFBcXd/z4cUc18+uvv7Zs2dJkMulWMTQ01HQrujVw\n5cqVGTNmdO7c2cPDo3Xr1rGxsd9//70OdW/5rnV++//+97+ffvppX19fHx+fgQMH7t+/X5+6\nIlJRUfHOO+/06tXLy8srICAgLCxs9+7dulW3AUffKtl4rl271qlTp3o/RldX14yMDP2bqaqq\neuqpp3T+VQYGBjrws3T+/PkOHTrUK+3n53fu3Dl7l77DvyNPT097V6+pqTl8+LC7u3vdum5u\nbllZWTqUrq6ujo2NrfeuTSbT0qVLdahuE4Rdg7311lsi0qlTpwMHDty4ceM///lPaGioiPTp\n00f/ZubPn69z1pSUlJhMJjc3t/Lycn0q1qOF+5NPPvndd9+VlJQcPnz4oYceEpHRo0fr30xl\nZeXjjz8uIps3b9ahXFhYmIg8//zzFovl8uXLWvr069dPh9KrV68WEV9f32XLluXm5ubn569e\nvbply5aenp4XLlzQoYGmI+warFevXiKya9eu2pGcnBzd/rbXdfjwYTc3txYtWugZdtoOY+fO\nnfUpV8+BAwdEJDAw8Nq1a7WDP/74o4i0adNG/360PzZTp07Vp5y3t7eI5Obmal9euXJFRDw8\nPHQoPXDgQBGpN49bu3atiLz11ls6NNB0HLNrsPPnz4uI9vdcU1lZKSL+/v56tvHbb7+NHj26\nsrJy+fLletY9ffq0iPzhD3/Qs2itzz77TERmzJjh5+dXO9i9e/eamprc3Fydm/n555/nzZvX\noUOHpKQkfSr6+vqKSO3xQe2DFxAQoEPpb7/9VkSGDRtWdzA8PFxE9uzZo0MDTUfYNVhxcXFN\nTU2zZs20L3Nycl5++WURmThxop5tTJs27dy5c+PHjx8xYoSedbWwa9eu3dSpU9u2bevp6dmz\nZ8933nlH+4dnb1999ZWIhISEpKSkdO3a1cPD44EHHkhISCguLtahej0JCQnl5eXvvPOOj4+P\nPhVHjhwpItOmTcvNzb1y5cq0adNEZMyYMTqULi8vFxFPT8+6g9XV1SJy4cIFHRqwAUdPLY2t\n9sc4derUqqoq3epu2LBBRDp16qQlr56/ynHjxkmdyUWt8PBwHY7iaedGXnjhhXrVu3fvXlBQ\nYO/qdR06dEhEevfurWfR0tLSqVOn1n3jY8eOLSsr06F0165dRWTr1q11B7XdWH32o5uOsGuS\n2s+ci4vLpEmT9PnYXbhwoXnz5m5ubocPH67bhg6la2pqtP33Pn36ZGVllZSUXLly5f3339em\nNgsWLLB3dW0/rnnz5itWrLBYLL/99tunn36qHUCYOXOmvavX9eyzz4rIli1b9Cy6ffv2zp07\n1w27Vq1aLVu2TIfSc+fOFZF27dpt3bo1Pz8/Ly/vo48+uvfee0WkWbNmOjTQdIRdk1RXV1ss\nlh07dnTp0kVEXnvtNR2KPvnkkyIyf/782hGHT9JTU1NFpGvXrvYu5ObmJiKrVq2qO/jxxx+L\nvudMsrOzTSbTAw88oOd0fuvWrSaTqUOHDlu3bi0sLLRYLJ988km7du1EZPny5fauXlxc3LNn\nz3oT6ri4OG0Pw97VbYKws43Dhw+LSGBgoA61bndEwoGRZ7FYRMTd3d3ehbSpRF5eXt3B/Px8\nfarXmj17togsXLhQt4o1NTXaFTaZmZl1B/fu3atb3BQWFv7lL38JDAy85557goKC3n777fT0\ndBEJCQnRoXrTEXYNph2jtVgsdQevXbsmel194oRhd/nyZW2Xyt6FevfuLSJXrlypO6idh23R\nooW9q2uqq6vbt28vIjpfX+bh4SH/Oz9WSzszo2fQ16Wdmlu0aJFDqjcUZ2MbTDtSW2+ZjnaW\n8OaVFfZw82+x7ri9qwcEBJhMplOnTtUd1A5U9+vXz97VBwwYICIbN26sO7ht2zYRefTRR+1d\nXXP06NGcnJzevXt37NhRn4oa7dCkdmKk1smTJ0XkdmtabMjX19dkMl26dKl2xGKxrFq1SkQi\nIyPtXd029M3W34O3335bRNq3b79z586SkpK8vLy1a9e2bt1aHPcnTs9f5Z/+9CcRCQ4O3r9/\nf2lpaU5OznvvvafNdnfv3m3v6j/88IObm5u3t3dKSsrly5fz8/NXrVqlXXOXnp5u7+qav//9\n7yKSkJCgT7laM2bM0D54W7ZsycvLKyoqysjICAoKEpE5c+bYu/qQIUNEJDQ09KeffiorKzt4\n8ODDDz8sInFxcfYubSuEXYOVlJRoR0/qCQkJcdQKKj3D7uLFizcvTRWR6dOn69NASkrKzRe+\nTJkyRZ/qNTU1ERERIrJ9+3bdKmoKCgq0fKnnscceKykpsXf177777ubLCbt27Vq7nMP5EXaN\nUVhYOHPmzE6dOrm7u/v6+vbt29dsNldUVDiqH50n6QUFBXPnzu3Ro4eXl80AHJAAAALFSURB\nVFfz5s0HDBhQ7/Sove3bty8kJMTX19fb27tv376pqanV1dW6VW/btq2I5OTk6FaxVmlpaVJS\n0iOPPOLl5eXh4dGjR4+33npLh6TTnDx5Mi4uLiAgwN3dvVOnTjNnzqx35NrJmWrsf5QHAByO\nExQAlEDYAVACYQdACYQdACUQdgCUQNgBUAJhB0AJhB0AJRB2AJRA2AFQAmEHQAmEHQAlEHYA\nlEDYAVACYQdACYQdACUQdgCUQNgBUAJhB0AJhB0AJRB2AJRA2AFQAmEHQAmEHQAlEHYAlEDY\nAVACYQdACYQdACUQdgCUQNgBUAJhB0AJhB0AJRB2MDyTyWQymRzdBZwdYQdACYQdACUQdgCU\nQNjBYIqKiqZMmRIQENCiRYvnn3/+2rVrju4IxuDm6AaABqiuro6Kivrqq6+0L9esWVNSUuLY\nlmAUzOxgJBs2bPjqq69atmy5bdu2oqKiPXv2HDhwwNFNwRgIOxjJmjVrRGT+/PkxMTHNmjUL\nDw9ftGiRo5uCMZhqamoc3QNgrdatW1sslosXLwYGBmojly5d0v6fTzLujLCDkbi5uVVVVd24\nccPDw0MbKSsr8/T0FMIOd8NuLIzEx8dHRCwWS+1Ibm6u49qBkRB2MJIePXqISEZGRu1Ienq6\n49qBkRB2MJIRI0aIyOuvv56RkXHt2rVdu3bNnz/f0U3BGDhmByMpLy/v37//t99+Wzsye/bs\nxMRE4Zgd7oaZHYzE3d39X//619ixY/38/AICAl577bWFCxc6uikYAzM7AEpgZgdACYQdACUQ\ndgCUQNgBUAJhB0AJhB0AJRB2AJRA2AFQAmEHQAmEHQAlEHYAlEDYAVACYQdACYQdACUQdgCU\nQNgBUAJhB0AJhB0AJRB2AJRA2AFQAmEHQAmEHQAlEHYAlEDYAVACYQdACYQdACUQdgCUQNgB\nUML/AQ0fWfDV6qabAAAAAElFTkSuQmCC",
      "text/plain": [
       "Plot with title \"\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "h = hist(d, xlim=c(min(d),max(d)), col=\"DarkBlue\", main=\"\", breaks=5)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Store in sparse diagonal matices $D$ and $iD$\n",
    "respectively $D=\\text{diag}_i(d_i)$ and $D^{-1} = \\text{diag}_i(1/d_i)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "D = sparseMatrix(i=seq(from=1, to=n), j=seq(from=1, to=n), x=d)\n",
    "iD = sparseMatrix(i=seq(from=1, to=n), j=seq(from=1, to=n), x=1/d)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "The normalized weight matrix is defined as\n",
    "$$ \\tilde W_{i,j} = \\frac{1}{d_i} W_{i,j}, $$\n",
    "and hence $\\tilde W = D^{-1} W$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "tW = iD %*% W"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "It satisfies\n",
    "$$ \\forall i , \\quad \\sum_j \\tilde W_{i,j} = 1, $$\n",
    "i.e. $\\tilde W \\text{I} = \\text{I}$ where $\\text{I} \\in \\RR^n$ is the vector\n",
    "constant equal to one.\n",
    "\n",
    "\n",
    "The operator $\\tilde W \\in \\RR^{n \\times n} $, viewed as an operator\n",
    "$\\tilde W : \\RR^n \\rightarrow \\RR^n$, can be thought as a low pass\n",
    "filter.\n",
    "\n",
    "\n",
    "Laplacian and Gradient Operators\n",
    "--------------------------------\n",
    "The un-normalized Laplacian is on the contrary a symmetric high pass\n",
    "operator\n",
    "$$ L = D-W \\in \\RR^{n \\times n}. $$\n",
    "It satisfies $L \\text{I} = 0$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "L = D - W"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "The gradient operator compute directional derivative along edges.\n",
    "It can be used to factor the Laplacian operator, but in practice\n",
    "it is never computed explicitely since it is never needed in numerical\n",
    "computation.\n",
    "\n",
    "To represent the gradient, we index the set of (oriented) edges $ \\Ee_0 = (e_k)_{k=1}^{p_0} $\n",
    "where each edge is $e_k = (i,j) \\in \\{1,\\ldots,n\\}^2$ with $i<j$.\n",
    "\n",
    "\n",
    "The gradient operator is a matrix $G \\in \\RR^{p_0 \\times n}$ defined\n",
    "as, for all $e_k=(i,j)$ and all $\\ell \\notin \\{i,j\\}$,\n",
    "$$ G_{k,i}=1, \\quad G_{k,j}=-1, \\quad G_{k,\\ell}=0. $$\n",
    "\n",
    "\n",
    "It is stored as a sparse matrix, and can be thought as a derivative\n",
    "operator $G : \\RR^n \\rightarrow \\RR^{p_0} $ that maps signal defined\n",
    "on vertices to differences located along directed edges."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "G = sparseMatrix( i = cbind(seq(from=0,to=p0, by=p0/(p0-1)), seq(from=0,to=p0, by=p0/(p0-1)))+1,  \n",
    "                   j = cbind(E0[1,],E0[2,])+1,\n",
    "                   x = cbind(Matrix(1,nrow=1,ncol=p0),Matrix((-1),nrow=1,ncol=p0)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Display the non-zero entries of $G$ and $W$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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zUnpYWFjQkLG6NQ/DOJjaKIQkFwRaZuYlPYbd68mRCydevWpUuX1l8uEomcnJyc\nnJyCg4NXrVoVFha2fv16jO86kz//fPOLL64lJRX89Zc4O7skL0+s8S769bP48cdAJyeztn6w\nQbQh6XQWm8IuJSWFEDJ37twW2qxcuTIsLOzatWsM1QQ08/DYn55eRGsXQqFeYuL0oUO7NLlP\nCvVdufJowoSfystrRSK948eDJk1i0xxKNv3rqiZsSSSSFtrw+XxCiFSKc2GdQUhILN1J1727\n6ZMni154oRuSrlVVVdIRI46Ul9eqfn7ppZ8eP2bT+UA2/QMPGTKEEBIWFtZCmy1bttS1BLY7\ncyaPpjUbGemNGeOwd29ATs58U1N9mnrpZFauvEDIv2anTJv2i7aKaQc27cauWbPmwoULmzdv\nTklJmTdv3rBhw1RXn1RWVhYXF1+/fv3gwYMnT56kKGr16tXaLhY0gKZ7CG7fPm7pUvw5bLPk\n5AcNlmRl0Xj/Z41jU9iNHTs2KipqwYIF58+fP3/+fJNtjI2NIyIixo8fz3Bt0EFyuTIiIjU6\nOtvISG/hwkGlpTWrVyeXltZqvCM3N6uFCz00vloumD27/4cfXqi/ZPRoNt27lE1hRwgJDg4e\nP378kSNH4uPjb968WVxcLBaL9fX1bWxs3Nzc/P3958yZY2XV8BpU0H2bNv3+8cfJqp9jYu7R\n0UXfvpabNo2ePLkvHlzdPl5eXRssmT3bVSuVtA/Lwo4QIhKJQkNDQ0NDtV0IaEZVlTQm5t62\nbTROJnV2tnzvvaFvv40BXYe4uFgJBDy5XKFUEkIoHo8MGGCt7aLagH1hB51JcXG1p+eBvLwy\n+ro4dWraxIk96Vs/d3TpItq+fdzy5QlSqYLHIxs2jOrbl033KGVf2OXn5+/bty8xMTErK6uk\npEQqlYpEIjs7O3d394CAgOnTpxsZ4amd7KBUkgULztKadF5eXZOTH8yaFVteLpFK5Uol0dPj\nffLJiNWrh//yS84336RJpYrXX3eZN28g9m3VsXjxoODgPrdvF7u4WLLugWosC7vdu3evWLGi\ntvZfx61Vc2NV02PXrFmzZ8+egIAAbVUI6ps9Oy46Opu+9Y8d6+Dtbbdx49X6C6VSxX/+kywW\nSzZvvkZRFCHk/Pn7NTXyxYsH0VdJZ2JnZ1z/Xqoswqbr7FRzY6VS6YwZMw4dOpSdnV1aWiqT\nySorK+/duxcdHT158uSHDx8GBwdfuXJF28VCK/buvX3oEI1PivDxcYyPfw3sJgUAABnBSURB\nVO3QoaafGBsRkUoIUSiUCoWSoqj9++/QVwnoCDaN7DA3tnPIzX0eGBh9584z+roQCHgJCa8R\nQpq7UE91iP1/l8gqZTI8FLjzY9PITs25sYQQzI3VTRcvPhCJtjk5fUdr0hFCTE0NVD+EhLg1\n2WD2bDelUsnjURRFlEoSEsKmSyigfdg0ssPcWFYrLKwaN+6YXM7EGKrufp+bNo2uqJBERmZU\nVcnkciUhhM/nLV06eMsWnxEj7CMiUiUSxcyZru++O5iBqkC72BR2Q4YMSUhICAsLa2F6LObG\n6qb8fHGfPt8zk3QURUVG/nOGaudO3507fRs3CwlxxYCOU9gUdpgbyy5ice3XX6fcuVN88+bT\nnJxSJZ0POJw6tZ+pqd65c/e7dhVFRQX16NHm29KBmmpr5Xl5YkdHE5FIT9u1tA2bwg5zY1lE\nIpH7+BxNSSlkoK8dO8a9+y7G8kw4cSJn9uzYigqpgQE/PNzvzTcHaruiNmDTCQpCSHBwcF5e\n3nfffff66687OztbWlry+XyhUNi9e/eJEyd+9dVXeXl5ISEh2i4TyKlTucwkHSEkOfkhMx1x\nnFhc++qrv1RUSAkhtbXy+fPP5OZq/pbR9GHTyE4Fc2MZsGXLjS+/vE5R1H/+M1z9q23lcuWt\nW0/v3y//73+vavw5OM2hKFJZqevno/76S1xZKX30qOLPP0tGjLDLzX3ep4/F0KFdtF1X25w/\nf18m++dghFJJjh69+9FHL2ixpDZhX9gB3d55J37XrhTVz0uWnH/0qOKzz0a1+qmSkho/v2OM\njebqKJVkypS+DHeqvtpa+bRpMb/+2sR9XAYMsE5Pn8uiaWoNnq9G/r5ckTVYthtL8NxY+kVE\npNV/GRZ2XZ1PffbZFYaTzsCA37evxZdfjtXlI0dff32ryaQjhGRkFK9efZHhejpCJGo4NtLX\nZ1OAsGxkh7mxDGhwgYiaswvS0gpVF+gyQDXEuHBhxgsvdGOivw64ceNpC5slOfkRs+V0yMOH\nFQ2WPHtWo5VK2odNwYy5scxosGOl5n5WRYWUmaTT1+cPGGB99Gig7icdIaRPH/MWNouLC5tu\nkTRxYi/q39+GyZN19wBCY2wKu7q5sUeOHJk5c2afPn3Mzc35fL5qbmxwcPDPP//84YcfSiSS\n9evXa7tYFhMI+PVfGho2/JJIpQqF4l+/wRkZxWlpxbRXRoi3t929e/PT0uZMm+bMQHcd9957\nw5q76ZuZmcG2beMYrqcj7OyM168foRpWUxRZsmSwp2fDexfrMjbtxtLx3Fi5XB4XF1dT09Jo\nPC8vj9D28BcdpK/Pk0rldS/19P75klRUSBcsOBMVlc3nk7ff9ggIcHr8uMLb227SpJ9qa2W0\nVjV6tENCwnTWPfDQ0tLw9u25p07lFhVVXr36pKCgvH9/C7FY2qePxYcferLuf+fjj73nzRuY\nnl7k4mLZqxfLrtxmU9jRMTc2MTExKChInZa5ublqrpPtevQw/eOPfybq1x+YvPde4pEjdwkh\nUinZvv3W9u23CCEMHKoLDu4bHf0KvX3QxsCAHxzchxDy1lud4b7w9vbG9vasvJ8dm8KOjrmx\nPj4+MTExLY/swsPDk5KSevVi08PPO2LmTNe6Z98QQgIDe8vlSj6fIoRERjZx3ze6k87S0vCn\nn9T6gwTQAjaFHR1zY/l8fmBgYMtt4uLiyP/GlVywapVXaWnNvn135HKFTKZct+7S1q03wsP9\nzc0NamvlrX9ec3g8asaMft9+O6HxFV4AbcWmsMPcWGYIBLwvvxy7fv1IB4dvqqokhJCyMsnc\nuXEUs9e/BgX1OXRokrGxPpOdQifGprAjeG4sg+7ceVZa+vfevUKhlEgYvVjex8fxl1+CmewR\nOj2WhR3B3FimaPehKtOnu2ixd+iUuHIcCtrK3t54yRLt3L938uS+8+fr7gwwLpNI5BERaW++\neXrnzpSaGnovNtI49o3s8NxYxuzc6evn133ZssT8fBof7VrHyEgQH/+ahYWhszOb5hVwh1JJ\nRo368fr1x4SQvXszvvkmLTV1juo0PSuwLOwwN5ZuRUVVR49mPnlSOWNGv65djefMOVVW1tKF\njZpDnTgxmRUzwDjr2rVHqqRTycgoPns2LyCANZdksSnsVHNjeTzejBkzAgMDvby8rK2tTUxM\namtrnzx5kp6eHhkZGR0dHRwcnJSU5O3tre162Sc3V+zhsb+8XEII2bjxKkVRSkbmu/J41JAh\nXfz8ejDQF7RbfHxBgyXsCjs2HbPD3FhalZbWTJ58QpV0KnQnnZWVUHU1i6OjyfffT6C1L+g4\nPT1+gyXsuv6RTSM7OubGQp0pU2LS0ooY6EgoFLz77pBJk3qNGuXw6FFFSUmNm5uVnh6b/u5y\nk5WVYaMlIq1U0j5sCjs8N5Y+BQXlSUn5DHTE41H5+W9bWwtVLx0dTRwdTRjoFzouK6uk1SW6\njE1/TlUzXluYGEvw3Ni2qK2V79mT/uGHF7ZuvTFq1GFmOlUolCdO5DDTF2iWQNBwp9XAgE0B\nwqaRHZ4bq0EymcLX99ilS1p4Ltfu3am4jI6N3n136Oef/17vQC71wQdeWqynrdgUdpgbq0FJ\nSQVaSTpCSHU1yy5G5abS0pq1ay9duvRo+PBuGzaMtLISmprqW1gYlJT8feGXqal+165suqaV\nTWFHMDdWc54+rdJW11Onsule3twkkyn69fuhqKiKEJKS8vTo0btPniy+evVRXdIRQsrKapOS\nCl5+2Ul7ZbYNy8KOYG6shjB/0QCPR5mZGSxePGjt2hEMdw1tFRWVpUo6lZKSmgMH7jx82PDR\nfRcu5CPsQEfV1sp/++1BcvLDTz+9zGS/CxZ4fPONPyGkulpWWFhlZmbw8GG5tbWwulp6+fJj\nf/8eFhYNL2sALbpw4UHjJV5eDZ84YW2NS09Al1RWSjdsuHL2bJ6pqUFOTmnjB+JpFkURQqj+\n/a3u3CmuO5gdH59HCPnkk8sbN15t8tmMgwbZXrky09AQX0idEBzcJyIitf6SoKDePj7dly9P\nrPvn4/N5c+cO0EZ17cSmM8fQPosXn//ii2upqYUXLhTQn3SUr2+PX3+dPHKkfV3S8XiUjY3o\nzJm8Tz+93NxTaFNTC7/4ApeC6wpb24ZDNgcHE0tLw9jYKd26GQkEVJcuRidOvNKlC0Z2oDNk\nMsWxY3cJ/U+KIITw+dSpU1P9/XsSQnr2NDtw4E5V1d8nXleteiE5ueGeUQPx8fnr1uFwnk54\n/LjhH0XVn8nx43s+erRIGxVpAMKuk6MoiplnVxNCLC0NVUlHCHFzs7pzZ97evRk1NfKpU/t6\neXV7+rTh4e0G+vQxp71EaC+Gb8pPBzaFXZs2NzO369B9Fy4USCQMPSVHX/9fE8V79jT79NOR\ndS9nzXLbuTMlI6PpZ2kLBLyPPnqB3vpAbY6Opg2WdO/O+ll9bAq7+fPnx8TEFBYWarsQXSeR\nyM+fv19WJund22zz5muMxX5oaEvzIoyM9G7cmH38eGZ+fnlpaXVeXpmjo+nFiw8ePiz39raL\niPBvfJwItGXAAOvAwN4nT95TvZw4sdfgwbbaLanj2BR2e/bs2b59e0BAwMWLFzFwa05JSc2I\nEYczMxmaoa16QnbXrkaLFg36v/9rZWhmYMCfNcuNmcKgIyiK/PzzK5GRd9LTiwYMsJkzpz+7\n7ubUJDaFHSFEJBJ9+OGHFy9e1HYhumvz5uvMJJ25uQGfz7OxEX388fCQEFcGegQmCQS8N9/s\nVFOYWRZ2hJDhw4druwSdlpVVwswdhmNjp44YYUd3LwCawr6ws7Kywj5sk6RSxYkT2WlpRcxs\nn969cfIU2IR9YQdNqqqSenjsz8l5TndHPB6lUCgnTXJi1wWlAAi7zqCiQrpsWQKtSWdubrB0\n6ZD798tyc8WjRtmvWoXLRIBlEHas9+OPf86efaq5aVgasW2bz7JlQ+lbPwADEHbsJpUqQkJO\nKRQ0Jt2IEfZIOugEcCMAdjtxIpvWpCOEvP66C63rB2AGRnZsdenSwzfeiPvrLzHdHQ0b1vAu\nZgBshJEdK0kk8smTT+TlldGxckdHk7pZyPPmDRg+vBsdvQAwjOsjO7lcHhcXV1NT00KbvLw8\nQgjde4vqUyiUI0ceLiqq1uxqKYqytzcaM8bxyy/HFhVV37z5pG9fi5Ej7TXbC4C2cD3sEhMT\ng4KC1GmZm5tLdzHqkMuVc+eevnHjqcbXLBIJCgoWqn7u2tVo4EBrjXcBoEVcDzsfH5+YmJiW\nR3bh4eFJSUm9evVirKrmFBVVjR9/PDW1iI6V9+plRsdqAXQE18OOz+cHBga23CYuLo4QwuNp\n+fjmunVXPvvsskKhyalg+vp81d3uKIqqf+85gM6H62Gn+yorpamphXPmnLp3T8MTJCZNcvr+\n+wmRkXcqKqTBwX2GDOmi2fUD6BSEnY5SKsmyZQnHjmUWFlbRMbHf0FAQEeHftavRhx96aXzl\nADoIYaejXnzxSHLyQzrWbGlpOGlSr/XrRzk6sv5G2wDqQ9jpol27UmhKOiMjvbt337SxwQ1L\ngHNwUbHOuXbtybvvxmt2nWZmBg4OJkuWDLp/fwGSDrgJIzvdcvZs3ssvR2v2GF2/fpZpaXMM\nDPitNwXovBB2OuTOneJJk36WyzU2VaN/f2tv727r1o1A0gEg7HRCdbXs/feTIiJSNTWmMzTk\n797tP3fuAM2sDoD9EHbal5z8YOzYY5oa0FEU+eGHgLlz+2tkbQCdBsJOy3bsuLVsWYKm1mZo\nKLhxY1b//pjWCtAQwk4LlEpy+PCfMTE5WVmlqamFmlqttbUwI2MenoMD0CSEnRZs3nxt1aqL\nFEU0eNbVyEhQVLREY6sD6HRwnZ0WhIenUhSl2etLPv/8RU2uDqDTQdhpgVhcq9nprlZWwkWL\nBmlwhQCdD8JOCzQYdPr6/FGjHP74Y56eHv4pAVqCY3ZMe/q0Siyu7fh6evUyW7jQY+VKTx6P\nar01AOch7BhVWSm1s9vdwZUIhYIdO3znzx+okZIAOAL7PszZv/+Oicn2Dt5qeOPG0c+fv4uk\nA2grjOwYEhb2+6pVv3VkDRRFzp17zdfXUVMlAXAKwo52Eol8yJADd+4Ud2QlQqEgKioISQfQ\nbgg72g0derCDSRcS4nbw4CRN1QPATQg7en322ZWMjPY/+ZCiKEdHEyQdQMch7OiSlfV84MDv\nJJL2fLZuJhlFkS++wNQIAA1A2GmeUkl27Li5fHliWz8oEPACAnq98EK3KVP6JiTkP39eO2mS\n0+DBtnQUCcA1CDsNk0oVXbrsLi2tbtOnKIrs2TMhNPSfC0pcXa00XRoApyHsNKxbt4i2Jp2z\ns+WtW7ONjPRoKgkACMJOg2pqZCYmO2SyNtxwmKJIYuKMMWMc6KsKAFQQdprx5EmVnV14m2b4\nu7vbnj07DffaBGAGwq6jioqqnZy+raiQqv+Rt95y//bb8fSVBACNsS/s8vPz9+3bl5iYmJWV\nVVJSIpVKRSKRnZ2du7t7QEDA9OnTjYyMmKzHzm53m3ZdU1PneHjY0FcPADSJZWG3e/fuFStW\n1Nb+6xZJ5eXlmZmZmZmZx48fX7NmzZ49ewICApipR6FQqp90AgF5+vQdS0tDWksCgCax6a4n\nsbGxixcvlkqlM2bMOHToUHZ2dmlpqUwmq6ysvHfvXnR09OTJkx8+fBgcHHzlyhVmShIIvlKz\npYeHbVnZciQdgLawaWS3efNmQsjWrVuXLl1af7lIJHJycnJycgoODl61alVYWNj69etPnTpF\ndz1jxhxX54yEtbWoqGgx3cUAQMvYFHYpKSmEkLlz57bQZuXKlWFhYdeuXVNznXK5PC4urqam\npoU2eXl5hBCFouHuanLy/VbX/+OPL7/2mouaxQAAfdgUdjwejxAiaXG6KZ/PJ4RIpeqeG01M\nTAwKClKn5YMHDxosaXlYZ2DAr6lZoWYZAEA3NoXdkCFDEhISwsLCwsLCmmuzZcsWVUs11+nj\n4xMTE9PyyC42Nnb//v0zZ85ssNzUVF8sbjp5zc0NHj1apGYNAMAANoXdmjVrLly4sHnz5pSU\nlHnz5g0bNsze3l4oFFZWVhYXF1+/fv3gwYMnT56kKGr16tVqrpPP5wcGBrbc5tGjR/v379fT\nazid6/nzpXz+l/X3bo2N9e7cebN7d5O2/G8BABPYFHZjx46NiopasGDB+fPnz58/32QbY2Pj\niIiI8eMZumRXLl9JCCkqklpb61F4yBeADmNT2BFCgoODx48ff+TIkfj4+Js3bxYXF4vFYn19\nfRsbGzc3N39//zlz5lhZMX2/EBsbzOEH0HUsCztCiEgkCg0NDQ0N1XYhAMAm7As7bcnMzDQ0\nbOKSYKlUum/fvh49eqhOFusIhUKRk5PTp08fnaqK6GphqKpNFArF/fv3586d2/hAdmZmplZK\nUgfCrnWqf1GMJQHq++abb5p7q3EI6gKEXetCQkJkMll1ddO35ExPTz98+PCoUaN69OjBcGEt\nuH//fnJysq5VRXS1MFTVJqrCZs6c6e7u3vhdoVAYEhLCfFWtU0LHHDt2jBBy7NgxbRfyL7pZ\nlVJXC0NVbaKzhbVMt44FAADQBGEHAJyAsAMATkDYAQAnIOwAgBMQdgDACQg7AOAEhB0AcALC\nDgA4AWHXUUKhsO6/ukM3qyK6WhiqahOdLaxllFKdB2RB8+RyeXx8vK+vr+rxFzpCN6siuloY\nqmoTnS2sZQg7AOAE7MYCACcg7ACAExB2AMAJCDsA4ASEHQBwAsIOADgBYQcAnICwAwBOQNgB\nACcg7ACAExB2AMAJCDsA4ASEHQBwAsIOADgBYQcAnICwAwBOQNi1X1JSkp+fn6WlpYWFhY+P\nT2xsrGbX7+fnRzWlHWVoqo2ueeuttxpvEBVsvTrV1dWfffaZh4eHqampUCh0cXH54IMPSktL\n67fhxOZSQrucOHGi8T2pw8PDNdiFo6Njq/9k6pShqTa6prKy0tbWtrnvMLaeSmVl5bBhwxpv\nBxcXl9LS0rpmXNhcCLv2qPs1++CDDwoLC8Vi8bZt23g8noGBwf379zXSRVVVFUVRAoFAIpF0\npAxNtdEpz549i4uL8/HxafwLqYKtV2fTpk2EEBsbm6NHjz5//lwsFv/yyy8ODg6EkKVLl6ra\ncGRzIeza47vvviOE+Pn51V84Y8YMQsinn36qkS7S0tIIIX369OlgGZpqo1NaGH2oYOvVGTx4\nMCHk+PHj9RfGx8cTQuzs7FQvObK5cMyuPVTflXnz5tVfGBgYSAhJTEzUSBdZWVmEEGdn5w6W\noak2OqXu69tcA2y9Ovfu3SOE+Pn51V/o6elJCHn27JnqJUc2F8KuPW7evEkIaXAoxMvLi/zv\ne9NxqvV069Zt8eLFXbt2NTQ0HDBgwJdffimTydpUhqbasAu2Xh2xWKxUKs3NzesvTEhIIIS4\nuLioXnJlc9E6buysLC0tCSEVFRX1F6pObxkaGmqkizlz5hBCGp8OGz9+fN2BFXXK0FQb3dTc\ndxhbrwWJiYlWVlaEkIMHD6qWcGRzIezaQ3UuSS6X11+o+jPI5/M10sWIESMIIUOHDk1MTKyq\nqnr8+PGOHTuMjIwIIRs3blS/DE210U3NhR22XpOKi4tVF+tQFLVu3bq65RzZXAi79hAKhYSQ\nsrKy+gsrKysJISYmJvT1u2fPHkJIv3791C9DU210U5v2Tri89eRyeXh4uGpU1bdv3/j4+FY/\n0vk2F47ZtYeNjQ0hpLi4uP7CJ0+eEELs7Ozo6zc4OJgQkpubq34ZmmrTCXB26xUWFvr4+Cxe\nvFipVH711VcZGRnjxo1r9VOdb3Mh7NrDzc2NEHL9+vX6C9PT0wkh7u7u9PUrlUoJISYmJuqX\noak2nQA3t15NTY2/v//Fixdfeumlu3fvvvfee/r6+up8sPNtLoRde/j7+xNCDh06VH9hZGQk\nIWTixIka6cLGxoaiqLt379ZfqOpx+PDh6pehqTbsgq1X54cffkhPTx8zZsyJEydUV/M2xpXN\nRetOcmdVXFxsbGxMCPn4449LSkqKiopWrVpFCLG3t29wmqndZs2aRQhxd3dPSkqqrq5++PDh\n9u3bDQ0NCSGnT59WvwxNtdFNzX2HsfXqBAQEEEJu3rzZQhuObC6EXTsdOHCgwal6fX39uLg4\nTa0/Pz9fNaengXfeeaetZWiqjQ5qLuyw9eo0uR3qqNpwZHMh7NrvzJkzL774oomJiYWFxYQJ\nEy5duqTZ9ZeUlKxZs6Z///5CodDMzGz06NGRkZHtK0NTbXRNc2GnxNb7H9XZz5bDTsmNzUUp\nm59zAwDQaeAEBQBwAsIOADgBYQcAnICwAwBOQNgBACcg7ACAExB2AMAJCDsA4ASEHQBwAsIO\nADgBYQcAnICwAwBOQNgBACcg7ACAExB2AMAJCDsA4ASEHQBwAsIOADgBYQcAnICwAwBOQNgB\nACcg7ACAExB2AMAJCDsA4ASEHQBwAsIOADgBYQcAnICwAwBOQNgBACcg7ACAExB2AMAJCDsA\n4ASEHQBwAsIOADgBYQcAnICwAwBOQNgBACcg7ACAExB2AMAJCDsA4ASEHQBwAsIOADgBYQcA\nnICwAwBOQNgBACcg7ACAExB2AMAJCDsA4ASEHQBwAsIOADgBYQcAnICwAwBOQNgBACcg7ACA\nExB2AMAJCDsA4ASEHQBwAsIOADgBYQcAnICwAwBOQNgBACcg7ACAExB2AMAJCDsA4ASEHQBw\nAsIOADgBYQcAnICwAwBOQNgBACcg7ACAExB2AMAJCDsA4ASEHQBwAsIOADgBYQcAnICwAwBO\n+H/9L2xS1vVvTgAAAABJRU5ErkJggg==",
      "text/plain": [
       "Plot with title \"W\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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NJqQ5F0YHYshV3t3thdu3YFBwf36tWrbdu2UqlUtzc2KCjohx9+WLBgQU1NzZIl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\n2AEAFxB2AMAFhB0AcAFhBwBc+D99Ys/NcfV28gAAAABJRU5ErkJggg==",
      "text/plain": [
       "Plot with title \"G\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(E[1,]+1, E[2,]+1, col=\"DarkBlue\", xlim=c(0,max(E[1,]+1)), ylim=c(0,max(E[2,]+1)), xlab=\"\", ylab=\"\", type=\"p\", main=\"W\", pch=20, cex=0.5)\n",
    "\n",
    "\n",
    "plot(cbind(seq(from=0,to=p0, by=p0/(p0-1)), seq(from=0,to=p0, by=p0/(p0-1)))+1, cbind(E0[1,],E0[2,])+1, xlab=\"\", ylab=\"\", type=\"p\",cex=0.5,\n",
    "     col=\"Darkblue\", main=\"G\", pch=20)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "The Laplacian can be factored as follow\n",
    "$$ L = G^* G $$\n",
    "where $G^*$ is the transposed matrix (i.e. the adjoint operator, which\n",
    "can be thought as some kind of divergence).\n",
    "\n",
    "\n",
    "Check numerically that the factorization indeed hold."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1] \"Factorization error (should be 0) = 0\"\n"
     ]
    }
   ],
   "source": [
    "err = norm(t(G)%*%G - L)\n",
    "print(paste(\"Factorization error (should be 0) =\" , err))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Note that this factorization shows that $L$ is a positive semi-definite\n",
    "operator, i.e. it satisfies\n",
    "\n",
    "$$ \\dotp{L f}{f} = \\norm{G f}^2 \\geq 0. $$\n",
    "\n",
    "If the mesh is connected, then only constant signals $f \\in \\RR^n$ satisfies\n",
    "$Lf=0$.\n",
    "\n",
    "\n",
    "Note that this convention is the contrary to the usual convention of\n",
    "differential calculus, in which a Laplacian is a negative operator.\n",
    "\n",
    "\n",
    "\n",
    "Function Denoising with Filtering\n",
    "---------------------------------\n",
    "A signal defined on the mesh is a vector $f \\in \\RR^n$, where $f_i \\in \\RR$\n",
    "is the value at vertex $1 \\leq i \\leq n$.\n",
    "\n",
    "\n",
    "Load a texture image $I$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "M = load_image(\"nt_toolbox/data/lena.png\", n=256)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Compute spherical coordinates $ (\\theta_i,\\phi_i)$ for each vertex $x_{0,i}$\n",
    "on the mesh."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "v = X0 - Matrix(rowMeans(X0), nrow=dim(X0)[1], ncol=dim(X0)[2])\n",
    "theta = acos(v[1,]/sqrt(colSums(v**2)))/pi\n",
    "phi = (atan2(v[2,], v[3,])/pi+1)/2 "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Interpolate the texture on the mesh."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "x = seq(from=0, to=1, by=1/(dim(M)[1]-1))\n",
    "f = rescale(bicubic(x, x,  t(as.matrix(M)), theta, phi)$z)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Display the textured mesh."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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OTepk2bHnvsMavVWlVVpdfrFxYWqqurGYYJBoO1tbUnT54sLCxMJpM0TadSKZ/P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\nYII=",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "options(repr.plot.width=6, repr.plot.height=5)\n",
    "\n",
    "points3D(matrix(X0[1,]), matrix(X0[2,]), matrix(X0[3,]), axis=FALSE, grid=FALSE, box=FALSE, colkey=FALSE, type=\"p\", pch=20,cex=0.05,\n",
    "         col=as.color(f))\n",
    "par(new=TRUE)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "The operator $\\tilde W : \\RR^n \\rightarrow \\RR^n$ can be used to smooth\n",
    "a function $f$, simply by computing $\\tilde W f \\in \\RR^n$.\n",
    "\n",
    "\n",
    "To further smooth the mesh, it is possible to iterate this process, by\n",
    "defining $f^{(0)} = f$ and\n",
    "\n",
    "$$ f^{(\\ell+1)} = \\tilde W f^{(\\ell)}.$$\n",
    "\n",
    "Note that one has $ f^{(\\ell)} = \\tilde W^{\\ell} f, $\n",
    "but it is preferable to use the iterative algorithm to do the\n",
    "computations."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "__Exercise 1__\n",
    "\n",
    "Display the evolution of the image on the mesh as the number of\n",
    "iterations increases."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAtAAAAJYCAIAAADE1kZ+AAAABmJLR0QA/wD/AP+gvaeTAAAK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\nAElFTkSuQmCC",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAtAAAAJYCAIAAADE1kZ+AAAABmJLR0QA/wD/AP+gvaeTAAAK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\nAElFTkSuQmCC",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAtAAAAJYCAIAAADE1kZ+AAAABmJLR0QA/wD/AP+gvaeTAAAK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\nAElFTkSuQmCC",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAtAAAAJYCAIAAADE1kZ+AAAABmJLR0QA/wD/AP+gvaeTAAAK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\nAElFTkSuQmCC",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "source(\"nt_solutions/meshproc_3_denoising/exo1.R\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "## Insert your code here."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Mesh Denoising with Filtering\n",
    "-----------------------------\n",
    "The quality of a noisy mesh is improved by applying local averagings,\n",
    "that removes noise but also tends to smooth features.\n",
    "\n",
    "\n",
    "\n",
    "The operator $\\tilde W : \\RR^n \\rightarrow \\RR^n$ can be used to smooth\n",
    "a function, but it can also be applied to smooth the position $W \\in\n",
    "\\RR^{3 \\times n} $. Since they are stored as row of a matrix, one should\n",
    "applies $\\tilde W^*$ (transposed matrix) on the right side.\n",
    "$$ X^{(0)} = X \\qandq X^{(\\ell+1)} = X^{(\\ell)} W^* $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "niter = 5\n",
    "X1 = X\n",
    "for (i in (1:niter)){\n",
    "    X1 = t(tW %*% t(X1))\n",
    "}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "We can compute the errors in dB with respect to the clean mesh, using\n",
    "\n",
    "$$ \\text{SNR}(X,Y) = -20 \\log_{10} \\pa{ \\norm{X-Y}/\\norm{Y} }. $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1] \"Noisy = 26.8231946547605 dB Denoised =  39.971730546081  dB\"\n"
     ]
    }
   ],
   "source": [
    "pnoisy = snr(X0, X)\n",
    "pfilt  = snr(X0, X1)\n",
    "print(paste(\"Noisy =\", pnoisy, \"dB\", \"Denoised = \" , pfilt,\" dB\" ))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Display the results."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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p2STuFT7FxgLC0tPX36VDoFskE/8BheUGcwpboI3tyQn3IjzZ2s66bWBpRKMXts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\nAACgHIUDAAAo9/8A2QAK51/vBgMAAAAASUVORK5CYII=",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_mesh(data.matrix(X1), F, \"grey\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "__Exercise 2__\n",
    "\n",
    "Determine the optimal number of iterations to maximize the SNR.\n",
    "Record, for each number $i$ of iteration, the SNR in $err(i)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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PDI2tqadARkRil1/vx56RSWoHAAgEce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\ni8KB0PDw8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      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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9bQ7IC5NGlYtGo9IRPFVS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\nTkSuQmCC",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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2FhoXQQf+Gd5l+RSGRsbEw6hb/oPrdp2n5ubu7u7q7tm82eh7qC7K/eE6uLOk8p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\nrkJggg==",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "source(\"nt_solutions/meshproc_3_denoising/exo2.R\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "## Insert your code here."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Plot the error as a function of the number of iterations."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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//oo4/MDuuG4uO1ZYuefZad6AEAN8xKhWPNmjWSZs2a1ahRI0lxcXGSPvrooyef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\nAEA5AgcAAFDuL5df60lAhi3sAAAAAElFTkSuQmCC",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(err, type=\"l\", col=\"blue\", xlab = \"Iteration\", ylab = \"SNR\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Mesh Denoising with Linear Heat Diffusion\n",
    "-----------------------------------------\n",
    "Iterative filtering is closely related to the heat diffusion. The heat\n",
    "diffusion is a linear partial differential equation (PDE) that compute a continuous denoising result for\n",
    "arbitrary time $t$. It is thus more precise than simple iterative\n",
    "filterings.\n",
    "\n",
    "\n",
    "This PDE defines a function $f_t \\in \\RR^n$ parameterized by the time\n",
    "$t>0$ as\n",
    "$$ \\forall t>0, \\quad \\pd{f_t}{t} = -\\tilde L f_t\n",
    "      \\qandq f_0 = f, $$\n",
    "where $ \\tilde L $ is the symetric normaled Laplacian defined as\n",
    "$$ \\tilde L = D^{-1} L = \\text{Id}_n - \\tilde W. $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "tL = iD %*% L"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "This PDE is applied to the three components of a 3-D mesh to define a\n",
    "surface evolution\n",
    "$$ \\forall t>0, \\quad \\pd{X_t}{t} = -X_t \\tilde L^*\n",
    "      \\qandq f_0 = f. $$\n",
    "\n",
    "\n",
    "One can approximate the solution to this PDE using explicit finite\n",
    "difference in time (Euler explicit scheme)\n",
    "$$ X^{(\\ell+1)} = X^{(\\ell)} -  \\tau X^{(\\ell)} \\tilde L^*\n",
    "      = (1-\\tau) X^{(\\ell)} + \\tau  X^{(\\ell)} \\tilde W^* $$\n",
    "where $0 < \\tau < 1$ is a (small enough) time step and $f^{(\\ell)}$ is\n",
    "intended to be an approximation of $X_t$ at time $t=\\tau \\ell$.\n",
    "The smaller $\\tau$, the better the approximation.\n",
    "\n",
    "\n",
    "One can see that with $\\tau=1$, one recovers the iterative filtering\n",
    "method.\n",
    "\n",
    "\n",
    "Time step $\\tau$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "tau = .2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Maximum time of resolution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "Tmax = 40"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Number of iterations needed to reach this time."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "niter = as.integer(ceiling(Tmax/tau))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Initial solution at time $t=0$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "Xt = X"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "We use an explicit discretization in time of the PDE. Here is one\n",
    "iteration."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "Xt = Xt - tau*t(tL %*% t(Xt))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "__Exercise 3__\n",
    "\n",
    "Compute the linear heat diffusion.\n",
    "Monitor the denoising\n",
    "SNR $err(l)$ between $X_t$ and $X_0$ at iteration index $l$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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Vw3lwWDQekIXsUlFQAQ5tqVFHEGjJ3c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\niyd6SWVlZ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      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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fOndArAfczhAABdTE1N0TZ0FsxpK25h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\nmgdkWZZVV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      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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cIlVXx4WpGpG//mEAI5liTJ9u3b582bFzsICkfrOX78eKVSiZ2CtEtzD/tstoYH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\nENz/A9kEyOnEAqhyAAAAAElFTkSuQmCC",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "source(\"nt_solutions/meshproc_3_denoising/exo3.R\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "## Insert your code here."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Plot the error as a function of time."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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dzBgzA2Rrt2onOoPV1drFuHfv2wZQs+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\n",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "t = seq(0, Tmax, by=Tmax/(niter-1))\n",
    "\n",
    "plot(t, err, type=\"l\", col=\"blue\", xlab = \"Time\", ylab = \"SNR\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Mesh Denoising with Sobolev Regularization\n",
    "------------------------------------------\n",
    "Instead of solving an evolution PDE, it is possible to do denoising by\n",
    "solving a quadratic regularization.\n",
    "\n",
    "\n",
    "Denoting $G \\in \\RR^{p_0 \\times n}$ the gradient operator, the Soboleb\n",
    "norm of a signal $f \\in \\RR^n$ is defined as\n",
    "$$ J(f) = \\norm{G f}^2 = \\dotp{L f}{f}. $$\n",
    "It is extended to mesh poisition $X \\in \\RR^{3 \\times n}$ as\n",
    "$$ J(X) = \\norm{X G^*}^2 = \\dotp{X L}{X}, $$\n",
    "(remeber that $L$ is symmetric).\n",
    "\n",
    "\n",
    "\n",
    "Denoising of a noisy set of vertices $X$ is then defined as the solution of a quadratic minimization\n",
    "$$ X_\\mu = \\uargmin{Z \\in \\RR^{3 \\times n}} \\norm{Z-X}^2  + \\mu J(Z)^2. $$\n",
    "Here $\\mu \\geq 0$ controls the amount of denoising, and should be\n",
    "proportional to the noise level.\n",
    "\n",
    "\n",
    "The solution to this problem is obtained by solving the following\n",
    "symmetric linear system\n",
    "$$ X_\\mu^* = (\\text{Id}_n + \\mu L )^{-1} X^* $$\n",
    "(remember that the mesh vertex position are stored as rows, hence the transposed).\n",
    "\n",
    "\n",
    "We select a penalization weight $\\mu$. The larger, the smoother the result will\n",
    "be (more denoising)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "mu = 10"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "We set up the matrix of the system.\n",
    "It is important to use sparse matrix to have fast resolution scheme."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "A = Diagonal(x=1, n) + mu*L"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "We solve the system for each coordinate of the mesh.\n",
    "Since the matrix is highly sparse, it is very interesting\n",
    "to use an iterative method to solve the system, so here\n",
    "we use a conjugate gradient descent (function cg from sparse.linalg)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "# Computing Minv for the pcg\n",
    "dA = diag(A)\n",
    "A[which(dA == 0)] = 1e-04\n",
    "Minv = base::diag(1/dA, nrow = nrow(A))\n",
    "\n",
    "pcg = function (b, maxiter=1000) \n",
    "{\n",
    "    x = rep(0, length(b))\n",
    "    r = b - A %*% x\n",
    "    z = Minv %*% (r)\n",
    "    p = z\n",
    "    sumr2 = sum(r^2)\n",
    "    for(i in 1:maxiter)\n",
    "    {\n",
    "        Ap = A %*% p\n",
    "        a = as.numeric((t(r) %*% z)/(t(p) %*% Ap))\n",
    "        x = x + a * p\n",
    "        r1 = r - a * Ap\n",
    "        z1 = Minv %*% r1\n",
    "        bet = as.numeric((t(z1) %*% r1)/(t(z) %*% r))\n",
    "        p = z1 + bet * p\n",
    "        z = z1\n",
    "        r = r1\n",
    "        sumr2 = sum(r^2)\n",
    "    }\n",
    "    return(x)\n",
    "}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "max_iter = 50\n",
    "Xmu = X\n",
    "for (i in (1:3))\n",
    "{\n",
    "    b = X[i,]\n",
    "    Xmu[i,] = t(as.matrix(pcg(as.matrix(b), max_iter)))\n",
    "}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Display the result."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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/r16+PHj6VT4CBcG+xjstc6Wb3zP3LkSGtrK83DLna/55CNRCIxMDAgncIPpPbd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",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_mesh(Xmu, F, \"grey\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "__Exercise 4__\n",
    "\n",
    "Solve this problem for various $\\mu$ on a 3D mesh.\n",
    "Draw the evolution of the SNR denoising error as a function of $\\mu$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "source(\"nt_solutions/meshproc_3_denoising/exo4.R\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "## Insert your code here."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<script>\n",
    "  $(document).ready(function(){\n",
    "      $('div.prompt').hide();\n",
    "  });\n",
    "</script>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "<script>\n",
    "  $(document).ready(function(){\n",
    "      $('div.prompt').hide();\n",
    "  });\n",
    "</script>"
   ]
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "kernelspec": {
   "display_name": "R",
   "language": "R",
   "name": "ir"
  },
  "language_info": {
   "codemirror_mode": "r",
   "file_extension": ".r",
   "mimetype": "text/x-r-source",
   "name": "R",
   "pygments_lexer": "r",
   "version": "3.4.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}