{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
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   "source": [
    "Dijkstra and Fast Marching Algorithms\n",
    "=====================================\n",
    "$\\newcommand{\\dotp}[2]{\\langle #1, #2 \\rangle}$\n",
    "$\\newcommand{\\enscond}[2]{\\lbrace #1, #2 \\rbrace}$\n",
    "$\\newcommand{\\pd}[2]{ \\frac{ \\partial #1}{\\partial #2} }$\n",
    "$\\newcommand{\\umin}[1]{\\underset{#1}{\\min}\\;}$\n",
    "$\\newcommand{\\umax}[1]{\\underset{#1}{\\max}\\;}$\n",
    "$\\newcommand{\\umin}[1]{\\underset{#1}{\\min}\\;}$\n",
    "$\\newcommand{\\uargmin}[1]{\\underset{#1}{argmin}\\;}$\n",
    "$\\newcommand{\\norm}[1]{\\|#1\\|}$\n",
    "$\\newcommand{\\abs}[1]{\\left|#1\\right|}$\n",
    "$\\newcommand{\\choice}[1]{ \\left\\{  \\begin{array}{l} #1 \\end{array} \\right. }$\n",
    "$\\newcommand{\\pa}[1]{\\left(#1\\right)}$\n",
    "$\\newcommand{\\diag}[1]{{diag}\\left( #1 \\right)}$\n",
    "$\\newcommand{\\qandq}{\\quad\\text{and}\\quad}$\n",
    "$\\newcommand{\\qwhereq}{\\quad\\text{where}\\quad}$\n",
    "$\\newcommand{\\qifq}{ \\quad \\text{if} \\quad }$\n",
    "$\\newcommand{\\qarrq}{ \\quad \\Longrightarrow \\quad }$\n",
    "$\\newcommand{\\ZZ}{\\mathbb{Z}}$\n",
    "$\\newcommand{\\CC}{\\mathbb{C}}$\n",
    "$\\newcommand{\\RR}{\\mathbb{R}}$\n",
    "$\\newcommand{\\EE}{\\mathbb{E}}$\n",
    "$\\newcommand{\\Zz}{\\mathcal{Z}}$\n",
    "$\\newcommand{\\Ww}{\\mathcal{W}}$\n",
    "$\\newcommand{\\Vv}{\\mathcal{V}}$\n",
    "$\\newcommand{\\Nn}{\\mathcal{N}}$\n",
    "$\\newcommand{\\NN}{\\mathcal{N}}$\n",
    "$\\newcommand{\\Hh}{\\mathcal{H}}$\n",
    "$\\newcommand{\\Bb}{\\mathcal{B}}$\n",
    "$\\newcommand{\\Ee}{\\mathcal{E}}$\n",
    "$\\newcommand{\\Cc}{\\mathcal{C}}$\n",
    "$\\newcommand{\\Gg}{\\mathcal{G}}$\n",
    "$\\newcommand{\\Ss}{\\mathcal{S}}$\n",
    "$\\newcommand{\\Pp}{\\mathcal{P}}$\n",
    "$\\newcommand{\\Ff}{\\mathcal{F}}$\n",
    "$\\newcommand{\\Xx}{\\mathcal{X}}$\n",
    "$\\newcommand{\\Mm}{\\mathcal{M}}$\n",
    "$\\newcommand{\\Ii}{\\mathcal{I}}$\n",
    "$\\newcommand{\\Dd}{\\mathcal{D}}$\n",
    "$\\newcommand{\\Ll}{\\mathcal{L}}$\n",
    "$\\newcommand{\\Tt}{\\mathcal{T}}$\n",
    "$\\newcommand{\\si}{\\sigma}$\n",
    "$\\newcommand{\\al}{\\alpha}$\n",
    "$\\newcommand{\\la}{\\lambda}$\n",
    "$\\newcommand{\\ga}{\\gamma}$\n",
    "$\\newcommand{\\Ga}{\\Gamma}$\n",
    "$\\newcommand{\\La}{\\Lambda}$\n",
    "$\\newcommand{\\si}{\\sigma}$\n",
    "$\\newcommand{\\Si}{\\Sigma}$\n",
    "$\\newcommand{\\be}{\\beta}$\n",
    "$\\newcommand{\\de}{\\delta}$\n",
    "$\\newcommand{\\De}{\\Delta}$\n",
    "$\\newcommand{\\phi}{\\varphi}$\n",
    "$\\newcommand{\\th}{\\theta}$\n",
    "$\\newcommand{\\om}{\\omega}$\n",
    "$\\newcommand{\\Om}{\\Omega}$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "This numerical tours details the implementations\n",
    "of Dijkstra and Fast Marching algorithms in 2-D.\n",
    "\n",
    "\n",
    "The implementation are performed in Matlab, and are hence quite slow."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "options(warn=-1) # turns off warnings, to turn on: \"options(warn=0)\"\n",
    "\n",
    "\n",
    "library(imager)\n",
    "library(png)\n",
    "\n",
    "for (f in list.files(path=\"nt_toolbox/toolbox_general/\", pattern=\"*.R\")) {\n",
    "    source(paste(\"nt_toolbox/toolbox_general/\", f, sep=\"\"))\n",
    "}\n",
    "\n",
    "for (f in list.files(path=\"nt_toolbox/toolbox_signal/\", pattern=\"*.R\")) {\n",
    "    source(paste(\"nt_toolbox/toolbox_signal/\", f, sep=\"\"))\n",
    "}\n",
    "\n",
    "source(\"nt_toolbox/toolbox_wavelet_meshes/meshgrid.R\")\n",
    "\n",
    "source(\"nt_toolbox/toolbox_graph/perform_dijstra_fm.R\")\n",
    "\n",
    "options(repr.plot.width=3.5, repr.plot.height=3.5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Installation\n",
    "------------\n",
    "*Important:* Please read the [installation page](http://gpeyre.github.io/numerical-tours/installation_python/) for details about how to install the toolboxes."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Navigating on the Grid\n",
    "----------------------\n",
    "We use a cartesian grid of size $n \\times n$, and defines operators to navigate in the grid.\n",
    "\n",
    "\n",
    "We use a singe index $i \\in \\{1,\\ldots,n^2\\}$ to index a position on\n",
    "the 2-D grid.\n",
    "\n",
    "\n",
    "Size of the grid."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "n <- 40"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "The four displacement vector to go to the four neightbors."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "neigh <- array(c(1, 0, -1, 0, 0, 1, 0, -1), c(2,4))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "For simplicity of implementation, we use periodic boundary conditions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "boundary <- function(x){ mod(x,n) }"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "For a given grid index |k|, and a given neighboring index k in ${1,2,3,4}$,\n",
    "|Neigh(k,i)| gives the corresponding grid neighboring index."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "ind2sub1 <- function(k){ c(as.integer( (k-1-mod(k-1,n))/n + 1 ), mod(k-1,n) + 1) }\n",
    "sub2ind1 <- function(u){ as.integer( (u[1]-1)*n + (u[2]-1) + 1 ) }\n",
    "Neigh <- function(k,i){ sub2ind1(boundary(ind2sub1(k) + neigh[,i])) }"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Check that these functions are indeed bijections."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1] 13 27\n",
      "[1] 134\n"
     ]
    }
   ],
   "source": [
    "print( ind2sub1( sub2ind1(c(13, 27)) ) )\n",
    "print( sub2ind1( ind2sub1(134) ) )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Dikstra Algorithm\n",
    "-----------------\n",
    "The Dijkstra algorithm compute the geodesic distance on a graph.\n",
    "We use here a graph whose nodes are the pixels, and whose edges defines the\n",
    "usual 4-connectity relationship.\n",
    "\n",
    "\n",
    "In the following, we use the notation $i \\sim j$ to indicate that an\n",
    "index $j$ is a neighbor of $i$ on the graph defined by the discrete\n",
    "grid.\n",
    "\n",
    "\n",
    "The metric $W(x)$. We use here a constant metric."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "W <- array(1, c(n,n))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Set $\\Ss = \\{x_0\\}$ of initial points."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "x0 <- c(n/2 + 1 , n/2 + 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Initialize the stack of available indexes."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "I <- c(sub2ind1(x0))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Initialize the distance to $+\\infty$, excepted for the boundary conditions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "D <- array(1, c(n,n)) + Inf\n",
    "u <- ind2sub1(I)\n",
    "D[u[1],u[2]] <- 0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Initialize the state to 0 (unexplored), excepted for the boundary point $\\Ss$ (indexed by |I|)\n",
    "to $1$ (front)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "S <- array(0, c(n,n))\n",
    "S[u[1],u[2]] <- 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Define a callbabk to use a 1-D indexing on a 2-D array."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "extract   <- function(x,I){ x[I] }\n",
    "extract1d <- function(x,I){ extract(as.vector(t(x)),I) }"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "The first step of each iteration of the method is\n",
    "to pop the from stack the element $i$ with smallest current distance $D_i$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "j <- order( extract1d(D,I)  )\n",
    "j <- j[1]\n",
    "i <- I[j]       \n",
    "a <-  I[j]\n",
    "I <- I[-c(j)]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "We update its state $S$ to be dead (-1)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "u <- ind2sub1(i)\n",
    "S[u[1],u[2]] <- -1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Retrieve the list of the neighbors that are not dead and add to I those that are not yet in it."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "J <- c()\n",
    "for (k in 1:4){\n",
    "    j <- Neigh(i,k)\n",
    "    if ( extract1d(S,j)!=-1 ){\n",
    "        # add to the list of point to update\n",
    "        J <- c(J, j) \n",
    "        if ( extract1d(S,j)==0 ){\n",
    "            # add to the front\n",
    "            u <- ind2sub1(j)\n",
    "            S[u[1],u[2]] <- 1\n",
    "            I <- c(I, j)\n",
    "        }\n",
    "    }\n",
    "}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
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      "text/plain": [
       "Plot with title \"\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "imageplot(S)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Update neighbor values.\n",
    "For each neightbo $j$ of $i$, perform the update,\n",
    "assuming the length of the edge between $j$ and $k$ is $W_j$.\n",
    "$$ D_j \\leftarrow \\umin{k \\sim j} D_k + W_j. $$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "DNeigh <- function(D,k){ extract1d(D,Neigh(j,k)) }\n",
    "for (j in J){\n",
    "    dx <- min(DNeigh(D,1), DNeigh(D,2))\n",
    "    dy <- min(DNeigh(D,3), DNeigh(D,4))\n",
    "    u <- ind2sub1(j)\n",
    "    w <- extract1d(W,j);\n",
    "    D[u[1],u[2]] <- min(dx + w, dy + w)\n",
    "}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "__Exercise 1__\n",
    "\n",
    "Implement the Dijkstra algorithm by iterating these step while the\n",
    "stack |I| is non empty.\n",
    "Display from time to time the front that propagates."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAlgAAAJYCAIAAAAxBA+LAAAABmJLR0QA/wD/AP+gvaeTAAAU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\nAAAAAElFTkSuQmCC",
      "text/plain": [
       "Plot with title \"\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "options(repr.plot.width=5, repr.plot.height=5)\n",
    "\n",
    "source(\"nt_solutions/fastmarching_0_implementing/exo1.R\")\n",
    "D <- exo1(x0,W)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "## Insert your code here."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Display the geodesic distance map using a cosine modulation to make the\n",
    "level set appears more clearly."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAaQAAAGkCAIAAADxLsZiAAAABmJLR0QA/wD/AP+gvaeTAAAK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",
      "text/plain": [
       "Plot with title \"\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "displ <- function(D){ cos(2*pi*5*D/max(D) ) }\n",
    "\n",
    "# color map function\n",
    "cmap_jet <- function(v){ return( rgb(v, (sin(v*2*pi)+1)/2, (cos(v*2*pi)+1)/2) ) }\n",
    "\n",
    "options(repr.plot.width=3.5, repr.plot.height=3.5)\n",
    "\n",
    "plot(as.cimg(displ(D)), colourscale=cmap_jet, interpolate = FALSE, axes = FALSE)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "source": [
    "Fast Marching\n",
    "-------------\n",
    "The Dijstra algorithm suffers from a strong metrization problem, and it\n",
    "actually computes the $\\ell^1$ distance on the grid.\n",
    "\n",
    "\n",
    "The Fast Marching algorithm replace the graph update by a local\n",
    "resolution of the Eikonal equation. This reduces significantly the grid\n",
    "bias, and can be shown to converge to the underlying geodesic distance\n",
    "when the grid step size tends to zero.\n",
    "\n",
    "\n",
    "Over a continuous domain, the distance map $D(x)$ to a set of seed\n",
    "points $ \\Ss $ is the unique solution in the viscosity sense\n",
    "$$ \\forall x \\notin \\Ss, \\quad \\norm{\\nabla D(x)} = W(x)\n",
    "      \\qandq\n",
    "    \\forall y \\in \\Ss, \\quad D(y) = 0. $$\n",
    "\n",
    "\n",
    "The equation is then discretized on a grid of $n \\times n$ pixel, and a\n",
    "solution $ (D_{k,\\ell})_{k,\\ell=1}^n \\in \\RR^{n \\times n} $ is found by using\n",
    "an upwind finite difference approximation, that is faithful to the\n",
    "viscosity solution\n",
    "$$ \\forall (k,\\ell) \\notin \\tilde \\Ss,  \\quad\n",
    "     \\norm{ (\\nabla D)_{k,\\ell} } = W_{k,\\ell}$\n",
    "  \\qandq\n",
    "  \\forall (k,\\ell) \\notin \\tilde \\Ss, \\quad D_{k,\\ell}=0,\n",
    "$$\n",
    "where $\\tilde \\Ss$ is the set of discrete starting points (defined here\n",
    "by |x0|).\n",
    "\n",
    "\n",
    "To be consisten with the viscosity solution, one needs to use a\n",
    "non-linear upwind gradient derivative. This corresponds to computing\n",
    "the norm of the gradient as\n",
    "$$ \\norm{ (\\nabla D)_{k,\\ell} }^2 =\n",
    "      \\max( D_{k+1,\\ell}-D_{k,\\ell}, D_{k-1,\\ell}-D_{k,\\ell}, 0 )^2 +\n",
    "      \\max( D_{k,\\ell+1}-D_{k,\\ell}, D_{k,\\ell-1}-D_{k,\\ell}, 0 )^2.\n",
    "$$\n",
    "\n",
    "\n",
    "A each step of the FM propagation, one update $ D_{k,\\ell} \\leftarrow d $\n",
    "by solving the eikonal equation with respect to $D_{k,\\ell}$ alone.\n",
    "This is equivalent to solving the quadratic equation\n",
    "$$ (d-d_x)^2 + (d-d_y)^2 = w^2 \\qwhereq w=W_{k,\\ell}. $$\n",
    "and where\n",
    "$$ d_x = \\min(D_{k+1,\\ell},D_{k-1,\\ell}) \\qandq\n",
    "   d_y = \\min(D_{k,\\ell+1},D_{k,\\ell-1}). $$\n",
    "\n",
    "\n",
    "The update is thus defined as\n",
    "$$\n",
    "      d = \\choice{\n",
    "          \\frac{d_x+d_y+ \\sqrt{\\De}}{2} \\quad\\text{when}\\quad \\De \\geq 0, \\\\\n",
    "          \\min(d_x,d_y)+w \\quad \\text{otherwise.}$\n",
    "      }$\n",
    "      \\qwhereq\n",
    "      \\De = 2 w^2 - (d_x-d_y)^2.\n",
    "$$\n",
    "\n",
    "\n",
    "Note that in the case where $\\De<0$, one has to use the Dijkstra\n",
    "update.\n",
    "\n",
    "\n",
    "Once the Dijstra algorithm is implemented, the implementation of the Fast\n",
    "Marching is trivial. It just corresponds to replacing the graph udpate"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "D[u[1],u[2]] <- min(dx + w, dy + w)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "by the eikonal update."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "Delta <- 2*w - (dx-dy)**2\n",
    "if (Delta>=0){\n",
    "    D[u[1],u[2]] <- (dx + dy + sqrt(Delta))/ 2 }  \n",
    "if (Delta<0){\n",
    "    D[u[1],u[2]] <- min(dx + w, dy + w) }"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "__Exercise 2__\n",
    "\n",
    "Implement the Fast Marching algorithm.\n",
    "Display from time to time the front that propagates."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
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IRGIAQAhEZq9N2rpkrUpHbzV5orKT+TJs6nMtE2KTq9owK2+UMvK/1AE94XH3ja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\nrkJggg==",
      "text/plain": [
       "Plot with title \"\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "options(repr.plot.width=5, repr.plot.height=5)\n",
    "\n",
    "source(\"nt_solutions/fastmarching_0_implementing/exo2.R\")\n",
    "D <- exo2(x0,W)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "## Insert your code here."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Display the geodesic distance map using a cosine modulation to make the\n",
    "level set appears more clearly."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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8bTpfL5LFt7yMbyHbtsjGuRKk91aYKN0fVlY6ayNTJ0omDWuzYGk62DORLBrk3B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\ngg==",
      "text/plain": [
       "Plot with title \"\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "options(repr.plot.width=3.5, repr.plot.height=3.5)\n",
    "\n",
    "plot(as.cimg(displ(D)), colourscale=cmap_jet, interpolate = FALSE, axes = FALSE)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Computation of Geodesic Paths\n",
    "-----------------------------\n",
    "We use a more complicated, non-constant metric, with a bump in the\n",
    "middle."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "n <- 100\n",
    "x <- seq(-1, 1, length=n)\n",
    "grid <- meshgrid_2d(x, x)\n",
    "Y <- grid$X ; X <- grid$Y\n",
    "sigma <- 0.2\n",
    "W <- 1 + 8 * exp(-(X**2 + Y**2)/ (2*sigma**2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Display it."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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IAIwg6IIOyACMIOiCDsgAjCDogg7IAI\nwg6IIOyACMIOiCDsgAjCDogg7IAIwg6IIOyACMIOiCDsgAjCDogg7IAIwg6IIOyACMIOiCDs\ngAjCDogg7IAIwg6IIOyACMIOiCDsgAjCDogg7IAIwg6IIOyACMIOiCDsgAjCDogg7IAIwg6I\nIOyACMIOiCDsgAjCDogg7IAIwg6IIOyACMIOiCDsgAjCDogg7IAIwg6IIOyACMIOiCDsgAjC\nDogg7IAIwg6IIOyACMIOiCDsgAjCDogg7IAIwg6IIOyACMIOiCDsgAjCDogg7IAIwg6IIOyA\nCMIOiCDsgAjCDogg7IAIwg6IIOyACMIOiCDsgAjCDogg7IAIwg6IIOyACMIOiCDsgAjCDogg\n7IAIwg6IIOyACMIOiCDsgAjCDogg7IAIwg6IIOyACMIOiCDsgAjCDogg7IAIwg6IIOyACMIO\niCDsgAjCDogg7IAIwg6IIOyACMIOiCDsgAj/A1sBIsD+DM9bAAAAAElFTkSuQmCC",
      "text/plain": [
       "Plot with title \"\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "imageplot(W)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Starting points."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "x0 <- c(round(.1*n)+1, round(.1*n)+1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "__Exercise 3:__ \n",
    "\n",
    "Compute the distance map to these starting point using the FM algorithm. _Important:_ use symetric boundary conditions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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WChfiDcL9vbsZrbNKh7DTUS4T5dOHsN+3Qh0x1Z/IuYOfIjGH/iKIoLB6mnudLD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y9mYPrdVwSr6F2tId7Gx8hK3eu9iYeWdQ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\ng10URSMhBrsoikbC/wlK68ldOsvq2gAAAABJRU5ErkJggg==",
      "text/plain": [
       "Plot with title \"\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "options(repr.plot.width=3.5, repr.plot.height=3.5)\n",
    "\n",
    "source(\"nt_solutions/fastmarching_0_implementing/exo3.R\")\n",
    "D <- exo3(x0,W)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "## Insert your code here."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Once the geodesic distance map to $\\Ss$\n",
    "has been computed, the geodesic curve between any point $x_1$ and $\\Ss$\n",
    "extracted through gradient descent\n",
    "$$ \\ga'(t) = - \\eta_t \\nabla D(\\ga(t)) \\qandq \\ga(0)=x_1 $$\n",
    "where $\\eta_t>0$ controls the parameterization speed of the resulting\n",
    "curve.\n",
    "\n",
    "\n",
    "To obtain unit speed parameterization, one can use $\\eta_t =\n",
    "\\norm{\\nabla D(\\ga(t))}^{-1}$ (one need to be careful when\n",
    "$\\ga$ approaches $\\Ss$ since $D$ is not smooth on $\\Ss$).\n",
    "\n",
    "\n",
    "Compute the gradient $G_0(x) = \\nabla D(x) \\in \\RR^2$ of the distance map.\n",
    "Use centered differences."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "G0 <- grad(D)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Normalize the gradient to obtained $G(x) = G_0(x)/\\norm{G_0(x)}$, in order to have unit speed geodesic curve (parameterized\n",
    "by arc length)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "d <- sqrt(apply(G0**2, c(1,2), sum))\n",
    "U <- array(0, c(n,n,2))\n",
    "U[,,1] <- d\n",
    "U[,,2] <- d\n",
    "G <- G0 / U"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "The geodesic is then numerically computed using a discretized gradient\n",
    "descent, which defines a discret curve $ (\\ga_k)_k $ using\n",
    "$$ \\ga_{k+1} = \\ga_k - \\tau G(\\ga_k) $$\n",
    "where $\\ga_k \\in \\RR^2$ is an approximation of $\\ga(t)$ at time\n",
    "$t=k\\tau$, and the step size $\\tau>0$ should be small enough.\n",
    "\n",
    "\n",
    "Step size $\\tau$ for the gradient descent."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "tau <- .8"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Initialize the path with the ending point."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "x1 <- round(.9*n) + 1i*round(.88*n)\n",
    "gamma <- c(x1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Define a shortcut to interpolate $G$ at a 2-D points.\n",
    "_Warning:_ the |interp2| switches the role of the axis ..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "Geval <- function(G,x){ bilinear_interpolate(G[,,1], Im(x), Re(x) ) + 1i * bilinear_interpolate(G[,,2],Im(x), Re(x)) }"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Compute the gradient at the last point in the path, using interpolation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "g <- Geval(G, gamma[length(gamma)])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Perform the descent and add the new point to the path."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "gamma <- c(gamma, gamma[length(gamma)] - tau*g)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "__Exercise 4__\n",
    "\n",
    "Perform the full geodesic path extraction by iterating the gradient\n",
    "descent. You must be very careful when the path become close to\n",
    "$x_0$, because the distance function is not differentiable at this\n",
    "point. You must stop the iteration when the path is close to $x_0$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": true,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "source(\"nt_solutions/fastmarching_0_implementing/exo4.R\")\n",
    "gamma <- exo4(tau,x0,x1,G)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [],
   "source": [
    "## Insert your code here."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "deletable": true,
    "editable": true
   },
   "source": [
    "Display the geodesic curve."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "collapsed": false,
    "deletable": true,
    "editable": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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jBhaHaQJOc+7v3r51e9P7gst+b5NcGW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\nAAAASUVORK5CYII=",
      "text/plain": [
       "Plot with title \"\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "imageplot(W)\n",
    "lines(Im(gamma), Re(gamma), col=\"blue\", lwd=3)\n",
    "points(x0[2], x0[1], col=\"red\", pch=19, lwd=4)\n",
    "points(Im(x1), Re(x1), col=\"green\", pch=19, lwd=4)"
   ]
  },
  {
   "cell_type": "raw",
   "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": {
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