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  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Active Contours using Parameteric Curves"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This tour explores image segmentation using parametric active contours.\n",
      "$\\newcommand{\\dotp}[2]{\\langle #1, #2 \\rangle}\n",
      "\\newcommand{\\qandq}{\\quad\\text{and}\\quad}\n",
      "\\newcommand{\\qwhereq}{\\quad\\text{where}\\quad}\n",
      "\\newcommand{\\qifq}{ \\quad \\text{if} \\quad }\n",
      "\\newcommand{\\ZZ}{\\mathbb{Z}}\n",
      "\\newcommand{\\RR}{\\mathbb{R}}\n",
      "\\newcommand{\\CC}{\\mathbb{C}}\n",
      "\\newcommand{\\pa}[1]{\\left(#1\\right)}\n",
      "\\newcommand{\\si}{\\sigma}\n",
      "\\newcommand{\\Nn}{\\mathcal{N}}\n",
      "\\newcommand{\\Bb}{\\mathcal{B}}\n",
      "\\newcommand{\\EE}{\\mathbb{E}}\n",
      "\\newcommand{\\norm}[1]{\\|#1\\|}\n",
      "\\newcommand{\\abs}[1]{\\left|#1\\right|}\n",
      "\\newcommand{\\choice}[1]{ \\left\\{  \\begin{array}{l} #1 \\end{array} \\right. }\n",
      "\\newcommand{\\al}{\\alpha}\n",
      "\\newcommand{\\la}{\\lambda}\n",
      "\\newcommand{\\ga}{\\gamma}\n",
      "\\newcommand{\\Ga}{\\Gamma}\n",
      "\\newcommand{\\La}{\\Lambda}\n",
      "\\newcommand{\\si}{\\sigma}\n",
      "\\newcommand{\\Si}{\\Sigma}\n",
      "\\newcommand{\\be}{\\beta}\n",
      "\\newcommand{\\de}{\\delta}\n",
      "\\newcommand{\\De}{\\Delta}\n",
      "\\renewcommand{\\phi}{\\varphi}\n",
      "\\renewcommand{\\th}{\\theta}\n",
      "\\newcommand{\\om}{\\omega}\n",
      "\\newcommand{\\Om}{\\Omega}\n",
      "$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "*Important:* You need to download the file `nt_toolbox.py` from the \n",
      "root of the github repository. "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from nt_toolbox import *\n",
      "%load_ext autoreload\n",
      "%autoreload 2"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The autoreload extension is already loaded. To reload it, use:\n",
        "  %reload_ext autoreload\n"
       ]
      }
     ],
     "prompt_number": 196
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Parameteric Curves\n",
      "-------------------\n",
      "\n",
      "In this tours, the active contours are represented using parametric\n",
      "curve $ \\ga : [0,1] \\rightarrow \\RR^2 $. \n",
      "\n",
      "\n",
      "This curve is discretized using a piewise linear curve with \n",
      "$p$ segments, and is stored as a complex vector of points in the plane\n",
      "$\\ga \\in \\CC^p$.\n",
      "\n",
      "Initial polygon."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "gamma0 = array([.78, .14, .42, .18, .32, .16, .75, .83, .57, .68, .46, .40, .72, .79, .91, .90]) + 1j*array([.87, .82, .75, .63, .34, .17, .08, .46, .50, .25, .27, .57, .73, .57, .75, .79])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Display the initial curve."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def periodize(gamma): return concatenate((gamma, [gamma[0]]));\n",
      "def cplot(gamma,s='b',lw=1): plot(real(periodize(gamma)), imag(periodize(gamma)), s, linewidth=lw); axis('equal'); axis('off');\n",
      "cplot(gamma0,'b.-'); "
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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r5WjId0qn0NK4kWfZy17AhcD7oZryo4Ij6fOOzEZ1DXAp8JrSSeZwA4/Ub9m12ajzztJJ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\nRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x108022410>"
       ]
      }
     ],
     "prompt_number": 2
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Number of points of the discrete curve."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "p = 256;"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Shortcut to re-sample a curve according to arc length."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def interpc(x,xf,yf): return interp(x,xf,real(yf)) + 1j * interp(x,xf,imag(yf));\n",
      "def curvabs(gamma): return concatenate( ([0], cumsum( 1e-5 + abs(gamma[:-1:]-gamma[1::]) ) ) );\n",
      "def resample1(gamma,d): return interpc(arange(0,p)/float(p),  d/d[-1],gamma);\n",
      "def resample(gamma): return resample1( periodize(gamma), curvabs(periodize(gamma)) );"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 4
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Initial curve $ \\ga_1(t)$."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "gamma1 = resample(gamma0);"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 5
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Display the initial curve."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "cplot(gamma1, 'k');"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x1084d0f90>"
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Shortcut for forward and backward finite differences."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def shiftR(c): return concatenate( ([c[-1]],c[:-1:]) );\n",
      "def shiftL(c): return concatenate( (c[1::],[c[0]]) );\n",
      "def BwdDiff(c): return c - shiftR(c);\n",
      "def FwdDiff(c): return shiftL(c) - c;\n",
      "def dotp(c1,c2): return dot(real(c1),real(c2)) + dot(imag(c1),imag(c2));"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 7
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The tangent to the curve is computed as\n",
      "$$ t_\\ga(s) = \\frac{\\ga'(t)}{\\norm{\\ga'(t)}} $$\n",
      "and the normal is $ n_\\ga(t) = t_\\ga(t)^\\bot. $\n",
      "\n",
      "Shortcut to compute the tangent and the normal to a curve."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def normalize(v): return v/maximum(abs(v),1e-10)\n",
      "def tangent(gamma): return normalize( FwdDiff(gamma) )\n",
      "def normal(gamma): return -1j*tangent(gamma)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 8
    },
    {
     "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",
     "collapsed": false,
     "input": [
      "delta = .03;\n",
      "gamma2 = gamma1 + delta * normal(gamma1);\n",
      "gamma3 = gamma1 - delta * normal(gamma1);"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 9
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Display the curves."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "cplot(gamma1, 'k');\n",
      "cplot(gamma2, 'r--');\n",
      "cplot(gamma3, 'b--');\n",
      "axis('tight'); axis('off');"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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6mOFeLsO1kfYQ7C14uAO/31jgeywDIGWGgZqypY9pK9BqZTQ0dYgcaAzTtFkm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\nSvCYdYgc0BXczykvmAnz+X6BFe585r8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       "text": [
        "<matplotlib.figure.Figure at 0x108521350>"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Evolution by Mean Curvature"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "A curve evolution is a series of curves $ s \\mapsto \\ga_s $ indexed by\n",
      "an evolution parameter $s \\geq 0$. The intial curve $\\ga_0$ for\n",
      "$s=0$ is evolved, usually by minizing some energy $E(\\ga)$ in a gradient descent\n",
      "$$ \\frac{\\partial \\ga_s}{\\partial s} = \\nabla E(\\ga_s). $$\n",
      "\n",
      "\n",
      "Note that the gradient of an energy is defined with respect to the\n",
      "curve-dependent inner product\n",
      "$$ \\dotp{a}{b} = \\int_0^1 \\dotp{a(t)}{b(t)} \\norm{\\ga'(t)} d t.  $$\n",
      "The set of curves can thus be thought as being a Riemannian surface.\n",
      "\n",
      "\n",
      "The simplest evolution is the mean curvature evolution.\n",
      "It corresponds to minimization of the curve length\n",
      "$$ E(\\ga) = \\int_0^1 \\norm{\\ga'(t)} d t $$\n",
      "\n",
      "\n",
      "The gradient of the length is \n",
      "$$ \\nabla E(\\ga)(t) = -\\kappa_\\ga(t) n_\\ga(t)  $$\n",
      "where $ \\kappa_\\ga $ is the curvature, defined as\n",
      "$$ \\kappa_\\ga(t) = \\frac{1}{\\norm{\\ga'(t)}} \\dotp{ t_\\ga'(t) }{ n_\\ga(t) }\u00a0. $$\n",
      "\n",
      "\n",
      "\n",
      "Shortcut for normal times curvature $ \\kappa_\\ga(t) n_\\ga(t) $."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def normalC(gamma): return BwdDiff(tangent(gamma)) / abs( FwdDiff(gamma) )"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 11
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Time step for the evolution.\n",
      "It should be very small because we use an explicit time stepping and the\n",
      "curve has strong curvature."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "dt = 0.001 / 100"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 12
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Number of iterations."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Tmax = 3.0 / 100\n",
      "niter = round(Tmax/dt)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 13
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Initialize the curve for $s=0$."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "gamma = gamma1;"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 14
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Evolution of the curve."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "gamma = gamma + dt * normalC(gamma);"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 15
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "To stabilize the evolution, it is important to re-sample the curve so\n",
      "that it is unit-speed parametrized. You do not need to do it every time\n",
      "step though (to speed up)."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "gamma = resample(gamma);"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 16
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Perform the curve evolution.\n",
      "You need to resample it a few times."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "gamma = gamma1\n",
      "displist = around(linspace(0,niter,10))\n",
      "k = 0;\n",
      "for i in arange(0,niter+1):\n",
      "    gamma = resample( gamma + dt * normalC(gamma) );\n",
      "    if i==displist[k]:\n",
      "        lw = 1;\n",
      "        if i==0 or i==niter:\n",
      "            lw = 4;\n",
      "        cplot(gamma, 'r', lw);\n",
      "        k = k+1;\n",
      "        axis('tight');  axis('off');"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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yPz5ggn9H1oDqobTSqymD8n5V9rU3RaGVTbeYcP2limN/QmDPEm2WpeiIl1me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       "text": [
        "<matplotlib.figure.Figure at 0x1085444d0>"
       ]
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Geodesic Active Contours"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Geodesic active contours minimize a weighted length\n",
      "$$ E(\\ga) = \\int_0^1 W(\\ga(t)) \\norm{\\ga'(t)} d t, $$\n",
      "where $W(x)>0$ is the geodesic metric, that should be small in areas\n",
      "where the image should be segmented."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Create a synthetic weight $W(x)$."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "n = 200;\n",
      "nbumps = 40;\n",
      "theta = random.rand(nbumps,1)*2*pi;\n",
      "r = .6*n/2; \n",
      "a = array([.62*n,.6*n]);\n",
      "x = around( a[0] + r*cos(theta) );\n",
      "y = around( a[1] + r*sin(theta) );\n",
      "W = zeros([n,n]); \n",
      "for i in arange(0,nbumps):\n",
      "    W[int(x[i]),int(y[i])] = 1;\n",
      "W = gaussian_blur(W,6.0);\n",
      "W = rescale( -minimum(W,.05), .3,1);"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 154
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Display the metric $W$."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "imageplot(W);"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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5AUC/RB3JwKtOqVXoxO7tqiIBP0Y/dypCBJ9AlRpSQgD+WQTv7++YzWa9RkA3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\nxwAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10a714dd0>"
       ]
      }
     ],
     "prompt_number": 156
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Pre-compute the gradient $\\nabla W(x)$ of the metric."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "G = grad(W);\n",
      "G = G[:,:,0] + 1j*G[:,:,1];"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 180
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Display the image of the magnitude $\\norm{\\nabla W(x)}$ of the gradient."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "imageplot(abs(G))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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2OKeNR9VS1CIi3YaWgAabNetQKKzWvVSxGWMPmxZBiWDhEyDPNVAiYIzAZg00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\nQCAQCAQCgUAgENhH/D8qPxw2wSvs1wAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10a6e5b50>"
       ]
      }
     ],
     "prompt_number": 182
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "def EvalG(gamma): interp(1:n,1:n, G, imag(gamma), real(gamma));\n",
      "def EvalW(gamma): interp(1:n,1:n, W, imag(gamma), real(gamma));"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x = linspace(0,10,5)\n",
      "y = linspace(0,6,5)\n",
      "x = [0.5, 2]\n",
      "y =  [0.5, 1]\n",
      "interp2([],[],f,x,y)\n",
      "print x,y\n",
      "print [x,y]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "ename": "RuntimeError",
       "evalue": "input and output rank must be > 0",
       "output_type": "pyerr",
       "traceback": [
        "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m\n\u001b[0;31mRuntimeError\u001b[0m                              Traceback (most recent call last)",
        "\u001b[0;32m<ipython-input-201-b8baedde3359>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[1;32m      3\u001b[0m \u001b[0mx\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;36m0.5\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m2\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m      4\u001b[0m \u001b[0my\u001b[0m \u001b[0;34m=\u001b[0m  \u001b[0;34m[\u001b[0m\u001b[0;36m0.5\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;36m1\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 5\u001b[0;31m \u001b[0minterp2\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0my\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m      6\u001b[0m \u001b[0;32mprint\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0my\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m      7\u001b[0m \u001b[0;32mprint\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0my\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
        "\u001b[0;32m/Users/gpeyre/Dropbox/github/numerical-tours/python/nt_toolbox.py\u001b[0m in \u001b[0;36minterp2\u001b[0;34m(x, y, f, xi, yi, k)\u001b[0m\n\u001b[1;32m    275\u001b[0m                 \u001b[0minterp2\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0mn\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0mn\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mG\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mimag\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mgamma\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mreal\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mgamma\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m;\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    276\u001b[0m \t\"\"\"\n\u001b[0;32m--> 277\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mndimage\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmap_coordinates\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0my\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0morder\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mk\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m;\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    278\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    279\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0mgrad\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
        "\u001b[0;32m/Users/gpeyre/anaconda/lib/python2.7/site-packages/scipy/ndimage/interpolation.pyc\u001b[0m in \u001b[0;36mmap_coordinates\u001b[0;34m(input, coordinates, output, order, mode, cval, prefilter)\u001b[0m\n\u001b[1;32m    295\u001b[0m     \u001b[0moutput_shape\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcoordinates\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mshape\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    296\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0minput\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mndim\u001b[0m \u001b[0;34m<\u001b[0m \u001b[0;36m1\u001b[0m \u001b[0;32mor\u001b[0m \u001b[0mlen\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0moutput_shape\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m<\u001b[0m \u001b[0;36m1\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 297\u001b[0;31m         \u001b[0;32mraise\u001b[0m \u001b[0mRuntimeError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'input and output rank must be > 0'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    298\u001b[0m     \u001b[0;32mif\u001b[0m \u001b[0mcoordinates\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mshape\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m!=\u001b[0m \u001b[0minput\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mndim\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    299\u001b[0m         \u001b[0;32mraise\u001b[0m \u001b[0mRuntimeError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'invalid shape for coordinate array'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
        "\u001b[0;31mRuntimeError\u001b[0m: input and output rank must be > 0"
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