{"nbformat_minor": 0, "worksheets": [{"cells": [{"source": ["Dijkstra and Fast Marching Algorithms\n", "=====================================\n", "\n*Important:* Please read the [installation page](http://gpeyre.github.io/numerical-tours/installation_matlab/) for details about how to install the toolboxes.\n", "$\\newcommand{\\dotp}[2]{\\langle #1, #2 \\rangle}$\n", "$\\newcommand{\\enscond}[2]{\\lbrace #1, #2 \\rbrace}$\n", "$\\newcommand{\\pd}[2]{ \\frac{ \\partial #1}{\\partial #2} }$\n", "$\\newcommand{\\umin}[1]{\\underset{#1}{\\min}\\;}$\n", "$\\newcommand{\\umax}[1]{\\underset{#1}{\\max}\\;}$\n", "$\\newcommand{\\umin}[1]{\\underset{#1}{\\min}\\;}$\n", "$\\newcommand{\\uargmin}[1]{\\underset{#1}{argmin}\\;}$\n", "$\\newcommand{\\norm}[1]{\\|#1\\|}$\n", "$\\newcommand{\\abs}[1]{\\left|#1\\right|}$\n", "$\\newcommand{\\choice}[1]{ \\left\\{ \\begin{array}{l} #1 \\end{array} \\right. }$\n", "$\\newcommand{\\pa}[1]{\\left(#1\\right)}$\n", "$\\newcommand{\\diag}[1]{{diag}\\left( #1 \\right)}$\n", "$\\newcommand{\\qandq}{\\quad\\text{and}\\quad}$\n", "$\\newcommand{\\qwhereq}{\\quad\\text{where}\\quad}$\n", "$\\newcommand{\\qifq}{ \\quad \\text{if} \\quad }$\n", "$\\newcommand{\\qarrq}{ \\quad \\Longrightarrow \\quad }$\n", "$\\newcommand{\\ZZ}{\\mathbb{Z}}$\n", "$\\newcommand{\\CC}{\\mathbb{C}}$\n", "$\\newcommand{\\RR}{\\mathbb{R}}$\n", "$\\newcommand{\\EE}{\\mathbb{E}}$\n", "$\\newcommand{\\Zz}{\\mathcal{Z}}$\n", "$\\newcommand{\\Ww}{\\mathcal{W}}$\n", "$\\newcommand{\\Vv}{\\mathcal{V}}$\n", "$\\newcommand{\\Nn}{\\mathcal{N}}$\n", "$\\newcommand{\\NN}{\\mathcal{N}}$\n", "$\\newcommand{\\Hh}{\\mathcal{H}}$\n", "$\\newcommand{\\Bb}{\\mathcal{B}}$\n", "$\\newcommand{\\Ee}{\\mathcal{E}}$\n", "$\\newcommand{\\Cc}{\\mathcal{C}}$\n", "$\\newcommand{\\Gg}{\\mathcal{G}}$\n", "$\\newcommand{\\Ss}{\\mathcal{S}}$\n", "$\\newcommand{\\Pp}{\\mathcal{P}}$\n", "$\\newcommand{\\Ff}{\\mathcal{F}}$\n", "$\\newcommand{\\Xx}{\\mathcal{X}}$\n", "$\\newcommand{\\Mm}{\\mathcal{M}}$\n", "$\\newcommand{\\Ii}{\\mathcal{I}}$\n", "$\\newcommand{\\Dd}{\\mathcal{D}}$\n", "$\\newcommand{\\Ll}{\\mathcal{L}}$\n", "$\\newcommand{\\Tt}{\\mathcal{T}}$\n", "$\\newcommand{\\si}{\\sigma}$\n", "$\\newcommand{\\al}{\\alpha}$\n", "$\\newcommand{\\la}{\\lambda}$\n", "$\\newcommand{\\ga}{\\gamma}$\n", "$\\newcommand{\\Ga}{\\Gamma}$\n", "$\\newcommand{\\La}{\\Lambda}$\n", "$\\newcommand{\\si}{\\sigma}$\n", "$\\newcommand{\\Si}{\\Sigma}$\n", "$\\newcommand{\\be}{\\beta}$\n", "$\\newcommand{\\de}{\\delta}$\n", "$\\newcommand{\\De}{\\Delta}$\n", "$\\newcommand{\\phi}{\\varphi}$\n", "$\\newcommand{\\th}{\\theta}$\n", "$\\newcommand{\\om}{\\omega}$\n", "$\\newcommand{\\Om}{\\Omega}$\n"], "metadata": {}, "cell_type": "markdown"}, {"source": ["This numerical 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."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 2, "cell_type": "code", "language": "python", "metadata": {}, "input": ["warning off\n", "addpath('toolbox_signal')\n", "addpath('toolbox_general')\n", "addpath('toolbox_graph')\n", "addpath('solutions/fastmarching_0_implementing')\n", "warning on"]}, {"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."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 3, "cell_type": "code", "language": "python", "metadata": {}, "input": ["n = 40;"]}, {"source": ["The four displacement vector to go to the four neightbors."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 4, "cell_type": "code", "language": "python", "metadata": {}, "input": ["neigh = [[1;0] [-1;0] [0;1] [0;-1]];"]}, {"source": ["For simplicity of implementation, we use periodic boundary conditions."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 5, "cell_type": "code", "language": "python", "metadata": {}, "input": ["boundary = @(x)mod(x-1,n)+1;"]}, {"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."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 6, "cell_type": "code", "language": "python", "metadata": {}, "input": ["ind2sub1 = @(k)[rem(k-1, n)+1; (k - rem(k-1, n) - 1)/n + 1]; \n", "sub2ind1 = @(u)(u(2)-1)*n+u(1);\n", "Neigh = @(k,i)sub2ind1( boundary(ind2sub1(k)+neigh(:,i)) );"]}, {"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."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 7, "cell_type": "code", "language": "python", "metadata": {}, "input": ["W = ones(n);"]}, {"source": ["Set $\\Ss = \\{x_0\\}$ of initial points."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 8, "cell_type": "code", "language": "python", "metadata": {}, "input": ["x0 = [n/2;n/2];"]}, {"source": ["Initialize the stack of available indexes."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 9, "cell_type": "code", "language": "python", "metadata": {}, "input": ["I = sub2ind1(x0);"]}, {"source": ["Initialize the distance to $+\\infty$, excepted for the boundary conditions."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 10, "cell_type": "code", "language": "python", "metadata": {}, "input": ["D = zeros(n)+Inf; \n", "D(I) = 0;"]}, {"source": ["Initialize the state to 0 (unexplored), excepted for the boundary point $\\Ss$ (indexed by |I|)\n", "to $1$ (front)."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 11, "cell_type": "code", "language": "python", "metadata": {}, "input": ["S = zeros(n);\n", "S(I) = 1;"]}, {"source": ["The first step of each iteration of the method is\n", "to pop the from stack the element $i$ with smallest current distance $D_i$."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 12, "cell_type": "code", "language": "python", "metadata": {}, "input": ["[tmp,j] = sort(D(I)); j = j(1);\n", "i = I(j); I(j) = [];"]}, {"source": ["We update its state $S$ to be dead (-1)."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 13, "cell_type": "code", "language": "python", "metadata": {}, "input": ["S(i) = -1;"]}, {"source": ["Retrieve the list of the four neighbors."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 14, "cell_type": "code", "language": "python", "metadata": {}, "input": ["J = [Neigh(i,1); Neigh(i,2); Neigh(i,3); Neigh(i,4)];"]}, {"source": ["Remove those that are dead (no need to consider them anymore)."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 15, "cell_type": "code", "language": "python", "metadata": {}, "input": ["J(S(J)==-1) = [];"]}, {"source": ["Add those that are not yet considered (state 0) to the front stack $I$ (state 1)."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 16, "cell_type": "code", "language": "python", "metadata": {}, "input": ["J1 = J(S(J)==0);\n", "I = [I; J1];\n", "S(J1) = 1;"]}, {"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", "$$D_j \\leftarrow \\umin{k \\sim j} D_k + W_j.$$"], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 17, "cell_type": "code", "language": "python", "metadata": {}, "input": ["for j=J'\n", " dx = min( D([Neigh(j,1) Neigh(j,2)]) );\n", " dy = min( D([Neigh(j,3) Neigh(j,4)]) );\n", " D(j) = min(dx+W(j), dy+W(j));\n", "end"]}, {"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."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [{"metadata": {}, "png": 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"output_type": "display_data"}], "prompt_number": 18, "cell_type": "code", "language": "python", "metadata": {}, "input": ["exo1()"]}, {"collapsed": false, "outputs": [], "prompt_number": 19, "cell_type": "code", "language": "python", "metadata": {}, "input": ["%% Insert your code here."]}, {"source": ["Display the geodesic distance map using a cosine modulation to make the\n", "level set appears more clearly."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [{"metadata": {}, "png": 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"output_type": "display_data"}], "prompt_number": 20, "cell_type": "code", "language": "python", "metadata": {}, "input": ["displ = @(D)cos(2*pi*5*D/max(D(:)));\n", "clf;\n", "imageplot(displ(D));\n", "colormap jet(256);"]}, {"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"], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 21, "cell_type": "code", "language": "python", "metadata": {}, "input": ["D(j) = min(dx+W(j), dy+W(j));"]}, {"source": ["by the eikonal update."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 22, "cell_type": "code", "language": "python", "metadata": {}, "input": ["Delta = 2*W(j) - (dx-dy)^2;\n", "if Delta>=0\n", " D(j) = (dx+dy+sqrt(Delta))/2;\n", "else\n", " D(j) = min(dx+W(j), dy+W(j));\n", "end"]}, {"source": ["__Exercise 2__\n", "\n", "Implement the Fast Marching algorithm.\n", "Display from time to time the front that propagates."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [{"metadata": {}, "png": 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J5r/92udz4Py7nvttWsIpJa7AAABBkcAAACGRwAAAIdEDO2e3vFZBknbPZCaP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\nAABJRU5ErkJggg==\n", "output_type": "display_data"}], "prompt_number": 23, "cell_type": "code", "language": "python", "metadata": {}, "input": ["exo2()"]}, {"collapsed": false, "outputs": [], "prompt_number": 24, "cell_type": "code", "language": "python", "metadata": {}, "input": ["%% Insert your code here."]}, {"source": ["Display the geodesic distance map using a cosine modulation to make the\n", "level set appears more clearly."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [{"metadata": {}, "png": 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At3JEfHSop7o+z98m+lwjYu8ujFq+7HE9zLPePsC3UoW/wCRJKVnAJEkpWcAk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\nkySlZAGTJKX0fwEapR7xKnMniwAAAABJRU5ErkJggg==\n", "output_type": "display_data"}], "prompt_number": 25, "cell_type": "code", "language": "python", "metadata": {}, "input": ["clf;\n", "imageplot(displ(D));\n", "colormap jet(256);"]}, {"source": ["Computation of Geodesic Paths\n", "-----------------------------\n", "We use a more complicated, non-constant metric, with a bump in the\n", "middle."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 26, "cell_type": "code", "language": "python", "metadata": {}, "input": ["n = 100;\n", "x = linspace(-1,1,n);\n", "[Y,X] = meshgrid(x,x);\n", "sigma = .2;\n", "W = 1 + 8 * exp(-(X.^2+Y.^2)/(2*sigma^2));"]}, {"source": ["Display it."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [{"metadata": {}, "png": 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DIJIAAyCSHhinONgkG7em+vfsWzJvHeyB9W++NzWqHa/PrypH4K/Pl5outnZXeeblyt79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\nSAIMgEj/BfjMoJRe8D4cAAAAAElFTkSuQmCC\n", "output_type": "display_data"}], "prompt_number": 27, "cell_type": "code", "language": "python", "metadata": {}, "input": ["clf;\n", "imageplot(W);"]}, {"source": ["Starting points."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 28, "cell_type": "code", "language": "python", "metadata": {}, "input": ["x0 = round([.1;.1]*n); % [.8;.8]]*n);"]}, {"source": ["__Exercise 3__\n", "\n", "Compute the distance map to these starting point using the FM algorithm."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [{"metadata": {}, "png": 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wPg3gNleA40L40cNzRJXse82fJX/0DfPTeDls9PFaue259VeRrlIyofK/5jle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\ni3fJYHx/fixH5+jjWArgYdTequn0ox73L9PwH8RfL8Pft/h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"output_type": "display_data"}], "prompt_number": 29, "cell_type": "code", "language": "python", "metadata": {}, "input": ["exo3()"]}, {"collapsed": false, "outputs": [], "prompt_number": 30, "cell_type": "code", "language": "python", "metadata": {}, "input": ["%% Insert your code here."]}, {"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."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 31, "cell_type": "code", "language": "python", "metadata": {}, "input": ["options.order = 2;\n", "G0 = grad(D, options);"]}, {"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)."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 32, "cell_type": "code", "language": "python", "metadata": {}, "input": ["G = G0 ./ repmat( sqrt( sum(G0.^2, 3) ), [1 1 2]);"]}, {"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."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 33, "cell_type": "code", "language": "python", "metadata": {}, "input": ["tau = .8;"]}, {"source": ["Initialize the path with the ending point."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 34, "cell_type": "code", "language": "python", "metadata": {}, "input": ["x1 = round([.9;.88]*n);\n", "gamma = x1;"]}, {"source": ["Define a shortcut to interpolate $G$ at a 2-D points.\n", "_Warning:_ the |interp2| switches the role of the axis ..."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 35, "cell_type": "code", "language": "python", "metadata": {}, "input": ["Geval = @(G,x)[interp2(1:n,1:n,G(:,:,1),x(2),x(1)); ...\n", " interp2(1:n,1:n,G(:,:,2),x(2),x(1)) ];"]}, {"source": ["Compute the gradient at the last point in the path, using interpolation."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 36, "cell_type": "code", "language": "python", "metadata": {}, "input": ["g = Geval(G, gamma(:,end));"]}, {"source": ["Perform the descent and add the new point to the path."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 37, "cell_type": "code", "language": "python", "metadata": {}, "input": ["gamma(:,end+1) = gamma(:,end) - tau*g;"]}, {"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 $x_0$."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [], "prompt_number": 38, "cell_type": "code", "language": "python", "metadata": {}, "input": ["exo4()"]}, {"collapsed": false, "outputs": [], "prompt_number": 39, "cell_type": "code", "language": "python", "metadata": {}, "input": ["%% Insert your code here."]}, {"source": ["Display the geodesic curve."], "metadata": {}, "cell_type": "markdown"}, {"collapsed": false, "outputs": [{"metadata": {}, "png": 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"output_type": "display_data"}], "prompt_number": 40, "cell_type": "code", "language": "python", "metadata": {}, "input": ["clf; hold on;\n", "imageplot(W); colormap gray(256);\n", "h = plot(gamma(2,:),gamma(1,:), '.b'); set(h, 'LineWidth', 2);\n", "h = plot(x0(2),x0(1), '.r'); set(h, 'MarkerSize', 25);\n", "h = plot(x1(2),x1(1), '.b'); set(h, 'MarkerSize', 25);\n", "axis ij;"]}], "metadata": {}}], "nbformat": 3, "metadata": {"kernelspec": {"name": "matlab_kernel", "language": "matlab", "display_name": "Matlab"}, "language_info": {"mimetype": "text/x-matlab", "name": "matlab", "file_extension": ".m", "help_links": [{"url": "https://github.com/calysto/metakernel/blob/master/metakernel/magics/README.md", "text": "MetaKernel Magics"}]}}}