{"worksheets": [{"cells": [{"cell_type": "markdown", "metadata": {}, "source": ["Active Contours using Parameteric Curves", "=========================================", "", "*Important:* Please read the [installation page](http://gpeyre.github.io/numerical-tours/installation_python/) for details about how to install the toolboxes.", "$\\newcommand{\\dotp}[2]{\\langle #1, #2 \\rangle}$", "$\\newcommand{\\enscond}[2]{\\lbrace #1, #2 \\rbrace}$", "$\\newcommand{\\pd}[2]{ \\frac{ \\partial #1}{\\partial #2} }$", "$\\newcommand{\\umin}[1]{\\underset{#1}{\\min}\\;}$", "$\\newcommand{\\umax}[1]{\\underset{#1}{\\max}\\;}$", "$\\newcommand{\\umin}[1]{\\underset{#1}{\\min}\\;}$", "$\\newcommand{\\uargmin}[1]{\\underset{#1}{argmin}\\;}$", "$\\newcommand{\\norm}[1]{\\|#1\\|}$", "$\\newcommand{\\abs}[1]{\\left|#1\\right|}$", "$\\newcommand{\\choice}[1]{ \\left\\{  \\begin{array}{l} #1 \\end{array} \\right. }$", "$\\newcommand{\\pa}[1]{\\left(#1\\right)}$", "$\\newcommand{\\diag}[1]{{diag}\\left( #1 \\right)}$", "$\\newcommand{\\qandq}{\\quad\\text{and}\\quad}$", "$\\newcommand{\\qwhereq}{\\quad\\text{where}\\quad}$", "$\\newcommand{\\qifq}{ \\quad \\text{if} \\quad }$", "$\\newcommand{\\qarrq}{ \\quad \\Longrightarrow \\quad }$", "$\\newcommand{\\ZZ}{\\mathbb{Z}}$", "$\\newcommand{\\CC}{\\mathbb{C}}$", "$\\newcommand{\\RR}{\\mathbb{R}}$", "$\\newcommand{\\EE}{\\mathbb{E}}$", "$\\newcommand{\\Zz}{\\mathcal{Z}}$", "$\\newcommand{\\Ww}{\\mathcal{W}}$", "$\\newcommand{\\Vv}{\\mathcal{V}}$", "$\\newcommand{\\Nn}{\\mathcal{N}}$", "$\\newcommand{\\NN}{\\mathcal{N}}$", "$\\newcommand{\\Hh}{\\mathcal{H}}$", "$\\newcommand{\\Bb}{\\mathcal{B}}$", "$\\newcommand{\\Ee}{\\mathcal{E}}$", "$\\newcommand{\\Cc}{\\mathcal{C}}$", "$\\newcommand{\\Gg}{\\mathcal{G}}$", "$\\newcommand{\\Ss}{\\mathcal{S}}$", "$\\newcommand{\\Pp}{\\mathcal{P}}$", "$\\newcommand{\\Ff}{\\mathcal{F}}$", "$\\newcommand{\\Xx}{\\mathcal{X}}$", "$\\newcommand{\\Mm}{\\mathcal{M}}$", "$\\newcommand{\\Ii}{\\mathcal{I}}$", "$\\newcommand{\\Dd}{\\mathcal{D}}$", "$\\newcommand{\\Ll}{\\mathcal{L}}$", "$\\newcommand{\\Tt}{\\mathcal{T}}$", "$\\newcommand{\\si}{\\sigma}$", "$\\newcommand{\\al}{\\alpha}$", "$\\newcommand{\\la}{\\lambda}$", "$\\newcommand{\\ga}{\\gamma}$", "$\\newcommand{\\Ga}{\\Gamma}$", "$\\newcommand{\\La}{\\Lambda}$", "$\\newcommand{\\si}{\\sigma}$", "$\\newcommand{\\Si}{\\Sigma}$", "$\\newcommand{\\be}{\\beta}$", "$\\newcommand{\\de}{\\delta}$", "$\\newcommand{\\De}{\\Delta}$", "$\\newcommand{\\phi}{\\varphi}$", "$\\newcommand{\\th}{\\theta}$", "$\\newcommand{\\om}{\\omega}$", "$\\newcommand{\\Om}{\\Omega}$"]}, {"cell_type": "markdown", "metadata": {}, "source": ["This tour explores image segmentation using parametric active contours."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["from __future__ import division", "import nt_toolbox as nt", "from nt_solutions import segmentation_2_snakes_param as solutions", "%matplotlib inline", "%load_ext autoreload", "%autoreload 2"]}, {"cell_type": "markdown", "metadata": {}, "source": ["Parameteric Curves", "------------------", "In this tours, the active contours are represented using parametric", "curve $ \\ga : [0,1] \\rightarrow \\RR^2 $.", "", "", "This curve is discretized using a piewise linear curve with", "$p$ segments, and is stored as a complex vector of points in the plane", "$\\ga \\in \\CC^p$."]}, {"cell_type": "markdown", "metadata": {}, "source": ["Initial polygon."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["gamma0 = [0.78 0.14 0.42 0.18 0.32 0.16 0.75 0.83 0.57 0.68 0.46 0.40 0.72 0.79 0.91 0.90]' + ...", "     1i* [0.87 0.82 0.75 0.63 0.34 0.17 0.08 0.46 0.50 0.25 0.27 0.57 0.73 0.57 0.75 0.79]'"]}, {"cell_type": "markdown", "metadata": {}, "source": ["Number of points of the discrete curve."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["p = 256"]}, {"cell_type": "markdown", "metadata": {}, "source": ["Shortcut to re-sample a curve according to arc length."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["curvabs = lambda gamma: [0; cumsum(1e-5 + abs(gamma(1: end-1)-gamma(2: end)))]", "resample1 = lambda gamma, d: interp1(d/ d(end), gamma, (0: p-1)'/ p, 'linear')", "resample = lambda gamma: resample1([gamma; gamma(1)], curvabs([gamma; gamma(1)]))"]}, {"cell_type": "markdown", "metadata": {}, "source": ["Initial curve $ \\ga_1(t) $."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["gamma1 = resample(gamma0)"]}, {"cell_type": "markdown", "metadata": {}, "source": ["Display the initial curve."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["h = plot(gamma1([1: end 1]), 'k')", "set(h, 'LineWidth', 2); axis('tight'); axis('off')"]}, {"cell_type": "markdown", "metadata": {}, "source": ["Shortcut for forward and backward finite differences."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["BwdDiff = lambda c: c - c([end 1: end-1])", "FwdDiff = lambda c: c([2: end 1]) - c", "dotp = lambda c1, c2: real(c1.*conj(c2))"]}, {"cell_type": "markdown", "metadata": {}, "source": ["The tangent to the curve is computed as", "$$ t_\\ga(s) = \\frac{\\ga'(t)}{\\norm{\\ga'(t)}} $$", "and the normal is $ n_\\ga(t) = t_\\ga(t)^\\bot. $", "", "", "Shortcut to compute the tangent and the normal to a curve."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["normalize = lambda v: v./ max(abs(v), eps)", "tangent = lambda gamma: normalize(FwdDiff(gamma))", "normal = lambda gamma: -1i*tangent(gamma)"]}, {"cell_type": "markdown", "metadata": {}, "source": ["Move the curve in the normal direction, by computing $ \\ga_1(t) \\pm \\delta n_{\\ga_1}(t) $."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["delta = .03", "gamma2 = gamma1 + delta * normal(gamma1)", "gamma3 = gamma1 - delta * normal(gamma1)"]}, {"cell_type": "markdown", "metadata": {}, "source": ["Display the curves."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["hold on", "h = plot(gamma1([1: end 1]), 'k'); set(h, 'LineWidth', 2)", "h = plot(gamma2([1: end 1]), 'r--'); set(h, 'LineWidth', 2)", "h = plot(gamma3([1: end 1]), 'b--'); set(h, 'LineWidth', 2)", "axis('tight'); axis('off')"]}, {"cell_type": "markdown", "metadata": {}, "source": ["Evolution by Mean Curvature", "---------------------------", "A curve evolution is a series of curves $ s \\mapsto \\ga_s $ indexed by", "an evolution parameter $s \\geq 0$. The intial curve $\\ga_0$ for", "$s=0$ is evolved, usually by minizing some energy $E(\\ga)$ in a gradient descent", "$$ \\frac{\\partial \\ga_s}{\\partial s} = \\nabla E(\\ga_s). $$", "", "", "Note that the gradient of an energy is defined with respect to the", "curve-dependent inner product", "$$ \\dotp{a}{b} = \\int_0^1 \\dotp{a(t)}{b(t)} \\norm{\\ga'(t)} d t.  $$", "The set of curves can thus be thought as being a Riemannian surface.", "", "", "The simplest evolution is the mean curvature evolution.", "It corresponds to minimization of the curve length", "$$ E(\\ga) = \\int_0^1 \\norm{\\ga'(t)} d t $$", "", "", "The gradient of the length is", "$$ \\nabla E(\\ga)(t) = -\\kappa_\\ga(t) n_\\ga(t)  $$", "where $ \\kappa_\\ga $ is the curvature, defined as", "$$ \\kappa_\\ga(t) = \\frac{1}{\\norm{\\ga'(t)}} \\dotp{ t_\\ga'(t) }{ n_\\ga(t) } . $$", "", "", "", "Shortcut for normal times curvature $ \\kappa_\\ga(t) n_\\ga(t) $."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["normalC = lambda gamma: BwdDiff(tangent(gamma)) ./ abs(FwdDiff(gamma))"]}, {"cell_type": "markdown", "metadata": {}, "source": ["Time step for the evolution.", "It should be very small because we use an explicit time stepping and the", "curve has strong curvature."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["dt = 0.001 / 100"]}, {"cell_type": "markdown", "metadata": {}, "source": ["Number of iterations."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["Tmax = 3 / 100", "niter = round(Tmax/ dt)"]}, {"cell_type": "markdown", "metadata": {}, "source": ["Initialize the curve for $s=0$."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["gamma = gamma1"]}, {"cell_type": "markdown", "metadata": {}, "source": ["Evolution of the curve."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["gamma = gamma + dt * normalC(gamma)"]}, {"cell_type": "markdown", "metadata": {}, "source": ["To stabilize the evolution, it is important to re-sample the curve so", "that it is unit-speed parametrized. You do not need to do it every time", "step though (to speed up)."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["gamma = resample(gamma)"]}, {"cell_type": "markdown", "metadata": {}, "source": ["__Exercise 1__", "", "Perform the curve evolution.", "You need to resample it a few times."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["solutions.exo1()"]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["## Insert your code here."]}, {"cell_type": "markdown", "metadata": {}, "source": ["Geodesic Active Contours", "------------------------", "Geodesic active contours minimize a weighted length", "$$ E(\\ga) = \\int_0^1 W(\\ga(t)) \\norm{\\ga'(t)} d t, $$", "where $W(x)>0$ is the geodesic metric, that should be small in areas", "where the image should be segmented.", "", "", "Create a synthetic weight $W(x)$."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["n = 200", "nbumps = 40", "theta = rand(nbumps, 1)*2*pi", "r = .6*n/ 2; a = [.62*n .6*n]", "x = round(a(1) + r*cos(theta))", "y = round(a(2) + r*sin(theta))", "W = zeros(n); W(x + (y-1)*n) = 1", "W = perform_blurring(W, 10)", "W = rescale(-min(W, .05), .3, 1)"]}, {"cell_type": "markdown", "metadata": {}, "source": ["Display the metric."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["imageplot(W)"]}, {"cell_type": "markdown", "metadata": {}, "source": ["Pre-compute the gradient $\\nabla W(x)$ of the metric."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["options.order = 2", "G = grad(W, options)", "G = G(: , : , 1) + 1i*G(: , : , 2)"]}, {"cell_type": "markdown", "metadata": {}, "source": ["Shortcut to evaluate the gradient and the potential along a curve."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["EvalG = lambda gamma: interp2(1: n, 1: n, G, imag(gamma), real(gamma))", "EvalW = lambda gamma: interp2(1: n, 1: n, W, imag(gamma), real(gamma))"]}, {"cell_type": "markdown", "metadata": {}, "source": ["Create a circular curve $\\ga_0$."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["r = .98*n/ 2", "p = 128; % number of points on the curve", "theta = linspace(0, 2*pi, p + 1)'; theta(end) = []", "gamma0 = n/ 2*(1 + 1i) +  r*(cos(theta) + 1i*sin(theta))"]}, {"cell_type": "markdown", "metadata": {}, "source": ["Initialize the curve at time $t=0$ with a circle."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["gamma = gamma0"]}, {"cell_type": "markdown", "metadata": {}, "source": ["For this experiment, the time step should be larger, because the", "curve is in $[1,n] \\times [1,n]$."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["dt = 1"]}, {"cell_type": "markdown", "metadata": {}, "source": ["Number of iterations."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["Tmax = 5000", "niter = round(Tmax/ dt)"]}, {"cell_type": "markdown", "metadata": {}, "source": ["Display the curve on the back ground;"]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["lw = 2", "clf; hold on", "imageplot(W)", "h = plot(imag(gamma([1: end 1])), real(gamma([1: end 1])), 'r')", "set(h, 'LineWidth', lw)", "axis('ij')"]}, {"cell_type": "markdown", "metadata": {}, "source": ["The gradient of the energy is", "$$ \\nabla E(\\ga) = -W(\\ga(t)) \\kappa_\\ga(t) n_\\ga(t) + \\dotp{\\nabla W(\\ga(t))}{ n_\\ga(t) } n_\\ga(t). $$", "", "", "Evolution of the curve according to this gradient."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["N = normal(gamma)", "g = - EvalW(gamma).*normalC(gamma) + dotp(EvalG(gamma), N) .* N", "gamma = gamma - dt*g"]}, {"cell_type": "markdown", "metadata": {}, "source": ["To avoid the curve from being poorly sampled, it is important to", "re-sample it evenly."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["gamma = resample(gamma)"]}, {"cell_type": "markdown", "metadata": {}, "source": ["__Exercise 2__", "", "Perform the curve evolution."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["solutions.exo2()"]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["## Insert your code here."]}, {"cell_type": "markdown", "metadata": {}, "source": ["Medical Image Segmentation", "--------------------------", "One can use a gradient-based metric to perform edge detection in medical", "images.", "", "", "Load an image $f$."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["n = 256", "f = rescale(sum(load_image('cortex', n), 3))"]}, {"cell_type": "markdown", "metadata": {}, "source": ["Display."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["imageplot(f)"]}, {"cell_type": "markdown", "metadata": {}, "source": ["An edge detector metric can be defined as a decreasing function of the", "gradient magnitude.", "$$ W(x) = \\psi( d \\star h_a(x) )", "      \\qwhereq d(x) = \\norm{\\nabla f(x)}. $$", "where $h_a$ is a blurring kernel of width $a>0$.", "", "", "Compute the magnitude of the gradient."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["options.order = 2", "G = grad(f, options)", "d = sqrt(sum(G.^2, 3))"]}, {"cell_type": "markdown", "metadata": {}, "source": ["Blur it by $h_a$."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["a = 3", "d = perform_blurring(d, a)"]}, {"cell_type": "markdown", "metadata": {}, "source": ["Compute a decreasing function of the gradient to define $W$."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["d = min(d, .4)", "W = rescale(-d, .8, 1)"]}, {"cell_type": "markdown", "metadata": {}, "source": ["Display it."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["imageplot(W)"]}, {"cell_type": "markdown", "metadata": {}, "source": ["Number of points."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["p = 128"]}, {"cell_type": "markdown", "metadata": {}, "source": ["__Exercise 3__", "", "Create an initial circle $\\gamma_0$ of $p$ points."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["solutions.exo3()"]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["## Insert your code here."]}, {"cell_type": "markdown", "metadata": {}, "source": ["Step size."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["dt = 2"]}, {"cell_type": "markdown", "metadata": {}, "source": ["Number of iterations."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["Tmax = 9000", "niter = round(Tmax/ dt)"]}, {"cell_type": "markdown", "metadata": {}, "source": ["__Exercise 4__", "", "Perform the curve evolution."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["solutions.exo4()"]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["## Insert your code here."]}, {"cell_type": "markdown", "metadata": {}, "source": ["Evolution of a Non-closed Curve", "-------------------------------", "It is possible to perform the evolution of a non-closed curve by adding", "boundary constraint", "$$ \\ga(0)=x_0 \\qandq \\ga(1)=x_1. $$", "", "", "In this case, the algorithm find a local minimizer of the geodesic", "distance between the two points.", "", "", "Note that a much more efficient way to solve this problem is to use the", "Fast Marching algorithm to find the global minimizer of the geodesic", "length.", "", "", "Load an image $f$."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["n = 256", "f = rescale(sum(load_image('cortex', n), 3))", "f = f(46: 105, 61: 120)", "n = size(f, 1)"]}, {"cell_type": "markdown", "metadata": {}, "source": ["Display."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["imageplot(f)"]}, {"cell_type": "markdown", "metadata": {}, "source": ["__Exercise 5__", "", "Compute an edge attracting criterion $W(x)>0$, that is small in area of strong", "gradient."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["solutions.exo5()"]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["## Insert your code here."]}, {"cell_type": "markdown", "metadata": {}, "source": ["Start and end points $x_0$ and $x_1$."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["x0 = 4 + 55i", "x1 = 53 + 4i"]}, {"cell_type": "markdown", "metadata": {}, "source": ["Initial curve $\\ga_0$."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["p = 128", "t = linspace(0, 1, p)'", "gamma0 = t*x1 + (1-t)*x0"]}, {"cell_type": "markdown", "metadata": {}, "source": ["Initialize the evolution."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["gamma = gamma0"]}, {"cell_type": "markdown", "metadata": {}, "source": ["Display."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["clf; hold on", "imageplot(W)", "h = plot(imag(gamma([1: end])), real(gamma([1: end])), 'r'); set(h, 'LineWidth', 2)", "h = plot(imag(gamma([1 end])), real(gamma([1 end])), 'b.'); set(h, 'MarkerSize', 30)", "axis('ij')"]}, {"cell_type": "markdown", "metadata": {}, "source": ["Re-sampling for non-periodic curves."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["curvabs = lambda gamma: [0; cumsum(1e-5 + abs(gamma(1: end-1)-gamma(2: end)))]", "resample1 = lambda gamma, d: interp1(d/ d(end), gamma, (0: p-1)'/ (p-1), 'linear')", "resample = lambda gamma: resample1(gamma, curvabs(gamma))"]}, {"cell_type": "markdown", "metadata": {}, "source": ["Time step."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["dt = 1/ 10"]}, {"cell_type": "markdown", "metadata": {}, "source": ["Number of iterations."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["Tmax = 2000*4/ 7", "niter = round(Tmax/ dt)"]}, {"cell_type": "markdown", "metadata": {}, "source": ["__Exercise 6__", "", "Perform the curve evolution.", "Be careful to impose the boundary conditions at each step."]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["solutions.exo6()"]}, {"cell_type": "code", "language": "python", "metadata": {}, "outputs": [], "collapsed": false, "input": ["## Insert your code here."]}]}], "nbformat": 3, "metadata": {"name": ""}, "nbformat_minor": 0}