Beziars from Iterated Function Systems
authorSam Moore <matches@ucc.asn.au>
Thu, 16 Jan 2014 05:36:01 +0000 (13:36 +0800)
committerSam Moore <matches@ucc.asn.au>
Thu, 16 Jan 2014 05:36:01 +0000 (13:36 +0800)
Using Goldman's Iterated Function System to generate a Bezier is horribly inefficient.
I guess the point is that it proves that a Bezier is a fractal.

The subdivision still looks like it gives a slightly different curve to
just sampling P(t), but maybe I aren't using enough points.

ipython_notebooks/de_Casteljau.ipynb
ipython_notebooks/fractals_basic.ipynb

index 976a6b6..c7a2805 100644 (file)
      "source": [
       "Described in Goldman, used as argument that Beziers are fractals because they are fixed points of an iterated function system.\n",
       "\n",
      "source": [
       "Described in Goldman, used as argument that Beziers are fractals because they are fixed points of an iterated function system.\n",
       "\n",
-      "The de Casteljau algorithm splits a Bezier curve into two Bezier segments.\n",
+      "I think the algorithm developed by Goldman might be worse than just applying the Bezier definition...\n",
       "\n"
      ]
     },
       "\n"
      ]
     },
-    {
-     "cell_type": "heading",
-     "level": 3,
-     "metadata": {},
-     "source": [
-      "Bezier Curve Definition"
-     ]
-    },
-    {
-     "cell_type": "markdown",
-     "metadata": {},
-     "source": [
-      "\n",
-      "Bezier curve $P(t)$ of degree $n$\n",
-      "\n",
-      "$ P(t) = \\sum_{j=0}^{n} B_j^n (t) P_j  \\quad \\quad \\quad 0 \\leq t \\leq 1$\n",
-      "\n",
-      "For control points $P_0 ... P_n$, with Bernstein basis functions $B_j^n(t) = \\left(^n_j\\right)t^j(1 - t)^{n-j} \\quad \\quad j=0,...n$\n",
-      "\n",
-      "Binomial Coefficients: $\\left(^n_j\\right) = \\frac{n!}{n!(n-j)!}$"
-     ]
-    },
     {
      "cell_type": "heading",
      "level": 2,
      "metadata": {},
      "source": [
     {
      "cell_type": "heading",
      "level": 2,
      "metadata": {},
      "source": [
-      "Helpers"
+      "Binomial Coefficients"
      ]
     },
     {
      ]
     },
     {
@@ -78,7 +56,7 @@
      "language": "python",
      "metadata": {},
      "outputs": [],
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 216
+     "prompt_number": 2
     },
     {
      "cell_type": "code",
     },
     {
      "cell_type": "code",
        "output_type": "stream",
        "stream": "stdout",
        "text": [
        "output_type": "stream",
        "stream": "stdout",
        "text": [
-        "0.000294923782349\n",
-        "0.000556945800781\n"
+        "0.00108003616333\n",
+        "0.000107049942017\n"
        ]
       }
      ],
        ]
       }
      ],
-     "prompt_number": 215
+     "prompt_number": 3
     },
     {
      "cell_type": "heading",
     },
     {
      "cell_type": "heading",
       "Bezier from Definition"
      ]
     },
       "Bezier from Definition"
      ]
     },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "\n",
+      "Bezier curve $P(t)$ of degree $n$\n",
+      "\n",
+      "$ P(t) = \\sum_{j=0}^{n} B_j^n (t) P_j  \\quad \\quad \\quad 0 \\leq t \\leq 1$\n",
+      "\n",
+      "For control points $\\{P_0 ... P_n\\}$, with Bernstein basis functions $B_j^n(t) = \\left(^n_j\\right)t^j(1 - t)^{n-j} \\quad \\quad j=0,...n$\n",
+      "\n",
+      "Binomial Coefficients: $\\left(^n_j\\right) = \\frac{n!}{n!(n-j)!} = \\text{nCj}$"
+     ]
+    },
     {
      "cell_type": "code",
      "collapsed": false,
     {
      "cell_type": "code",
      "collapsed": false,
       "    \"\"\" Apply Bezier definition to produce a point on the Bezier curve with control points P \"\"\"\n",
       "    n = len(P)-1\n",
       "    return [sum([B(n, j, t) * P[j][i] for j in xrange(n+1)]) for i in xrange(len(P[0]))]\n",
       "    \"\"\" Apply Bezier definition to produce a point on the Bezier curve with control points P \"\"\"\n",
       "    n = len(P)-1\n",
       "    return [sum([B(n, j, t) * P[j][i] for j in xrange(n+1)]) for i in xrange(len(P[0]))]\n",
+      "\n",
+      "def BezierCurve(P, nPoints):\n",
+      "    return map(lambda t : Bezier(P,t), linspace(0,1,nPoints))\n",
       "        \n",
       "        \n",
-      "def PlotBezier(P,nPoints=50,style='o-'):\n",
+      "def PlotBezier(P,nPoints=50,style='-'):\n",
       "    \"\"\" Plot a Bezier \"\"\"\n",
       "    points = map(lambda t : Bezier(P,t), linspace(0,1,nPoints))\n",
       "    x = [p[0] for p in points]\n",
       "    \"\"\" Plot a Bezier \"\"\"\n",
       "    points = map(lambda t : Bezier(P,t), linspace(0,1,nPoints))\n",
       "    x = [p[0] for p in points]\n",
      "language": "python",
      "metadata": {},
      "outputs": [],
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 266
+     "prompt_number": 4
     },
     {
      "cell_type": "code",
     },
     {
      "cell_type": "code",
      "input": [
       "title(\"Bezier Curves of Degree 2\")\n",
       "PlotBezier([(0,0),(0.4,0.8),(1,1)],nPoints=100,style='-')\n",
      "input": [
       "title(\"Bezier Curves of Degree 2\")\n",
       "PlotBezier([(0,0),(0.4,0.8),(1,1)],nPoints=100,style='-')\n",
-      "PlotBezier([(0,0),(0.4,-0.8),(1,1)],nPoints=100,style='-')\n",
-      "legend()"
+      "PlotBezier([(0,0),(0.4,-0.8),(1,1)],nPoints=100,style='-')"
      ],
      "language": "python",
      "metadata": {},
      "outputs": [
      ],
      "language": "python",
      "metadata": {},
      "outputs": [
+      {
+       "output_type": "stream",
+       "stream": "stderr",
+       "text": [
+        "/usr/lib/pymodules/python2.7/matplotlib/axes.py:4486: UserWarning: No labeled objects found. Use label='...' kwarg on individual plots.\n",
+        "  warnings.warn(\"No labeled objects found. \"\n"
+       ]
+      },
       {
        "metadata": {},
        "output_type": "display_data",
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c6BiKpo75erx16hRA1u3boWrqyukUimmT58OW1tbeHt7AwA8PT1hY2ODAQMGoF27dtDU1ISHhwda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\n8tDFtAu6mnZFV7Ou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        "text": [
       {
        "metadata": {},
        "output_type": "display_data",
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c6BiKpo75erx16hRA1u3boWrqyukUimmT58OW1tbeHt7AwA8PT1hY2ODAQMGoF27dtDU1ISHhwda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\ngg==\n",
        "text": [
-        "<matplotlib.figure.Figure at 0x3c4ee50>"
+        "<matplotlib.figure.Figure at 0x2bb2bd0>"
        ]
       }
      ],
        ]
       }
      ],
-     "prompt_number": 267
+     "prompt_number": 5
     },
     {
      "cell_type": "heading",
      "level": 2,
      "metadata": {},
      "source": [
     },
     {
      "cell_type": "heading",
      "level": 2,
      "metadata": {},
      "source": [
-      "Fractal Beziers"
+      "De Casteljau Algorithm"
      ]
     },
     {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
      ]
     },
     {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "$ L P = Q$ and $ M P = R$"
+      "Subdivide the original $n$ control points $\\{P_0 ... P_n\\}$ into $\\{Q_0 ... Q_n\\}$ and $\\{R_0 ... R_n\\}$ (ie: Makes $2n$ points)\n",
+      "\n",
+      "Defining $(n+1)\\times(n+1)$ matrices $L$ and $M$ such that $ L P = Q$ and $ M P = R$\n",
+      "\n",
+      "Then it can be shown (subdividing at $t = 1/2$)\n",
+      "\n",
+      "$L_{i,j} = \\left(\\frac{\\left(^j_k\\right)}{2^j}\\right)$ and $M_{i,j} =  \\left(\\frac{\\left(^{n-j}_{n-k}\\right)}{2^{n-j}}\\right)$"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "**Confusion:** Do $Q$ and $R$ converge to the same curve as applying the definition to produce points along $P(t)$?\n",
+      "\n",
+      "It is implied in Goldman, but it doesn't look like it, unless I made a mistake. See below."
      ]
     },
     {
      ]
     },
     {
      "collapsed": false,
      "input": [
       "def L(n):\n",
      "collapsed": false,
      "input": [
       "def L(n):\n",
+      "    \"\"\" Gives L of size (n+1)*(n+1) \"\"\"\n",
       "    try:\n",
       "        return L.dynamic[n]\n",
       "    except KeyError:\n",
       "    try:\n",
       "        return L.dynamic[n]\n",
       "    except KeyError:\n",
       "L.dynamic = {}                        \n",
       "\n",
       "def M(n):\n",
       "L.dynamic = {}                        \n",
       "\n",
       "def M(n):\n",
+      "    \"\"\" Gives M of size (n+1)*(n+1) \"\"\"\n",
       "    try:\n",
       "        return M.dynamic[n]\n",
       "    except KeyError:\n",
       "    try:\n",
       "        return M.dynamic[n]\n",
       "    except KeyError:\n",
      "language": "python",
      "metadata": {},
      "outputs": [],
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 296
+     "prompt_number": 6
     },
     {
      "cell_type": "code",
      "collapsed": false,
      "input": [
     },
     {
      "cell_type": "code",
      "collapsed": false,
      "input": [
-      "points = [(0,0), (0.5,1), (1,0)]"
+      "L(2)"
      ],
      "language": "python",
      "metadata": {},
      ],
      "language": "python",
      "metadata": {},
-     "outputs": [],
-     "prompt_number": 298
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "pyout",
+       "prompt_number": 7,
+       "text": [
+        "array([[ 1.  ,  0.  ,  0.  ],\n",
+        "       [ 0.5 ,  0.5 ,  0.  ],\n",
+        "       [ 0.25,  0.5 ,  0.25]])"
+       ]
+      }
+     ],
+     "prompt_number": 7
     },
     {
      "cell_type": "code",
      "collapsed": false,
      "input": [
       "def Casteljau(P, nPoints):\n",
     },
     {
      "cell_type": "code",
      "collapsed": false,
      "input": [
       "def Casteljau(P, nPoints):\n",
+      "    \"\"\" Approximates the Bezier curve with control points P using the de Casteljau subdivision algorithm \"\"\"\n",
       "    while len(P) < nPoints:\n",
       "        Q = dot(L(len(P)-1), P)\n",
       "        R = dot(M(len(P)-1), P)\n",
       "    while len(P) < nPoints:\n",
       "        Q = dot(L(len(P)-1), P)\n",
       "        R = dot(M(len(P)-1), P)\n",
-      "        P = list(Q) + list(R)[1:]\n",
+      "        P = list(Q) + list(R)\n",
       "    return P\n",
       "    "
      ],
      "language": "python",
      "metadata": {},
      "outputs": [],
       "    return P\n",
       "    "
      ],
      "language": "python",
      "metadata": {},
      "outputs": [],
-     "prompt_number": 329
+     "prompt_number": 8
     },
     {
      "cell_type": "code",
      "collapsed": false,
      "input": [
     },
     {
      "cell_type": "code",
      "collapsed": false,
      "input": [
-      "points = [(0,0), (0.5,1), (1,0)]\n",
+      "points = [(0,0), (0.5,1),(1,0),(1.4,-2)]\n",
       "c = Casteljau(points,50)\n",
       "PlotBezier(points)\n",
       "c = Casteljau(points,50)\n",
       "PlotBezier(points)\n",
-      "plot([p[0] for p in c], [p[1] for p in c], 'go-')"
+      "plot([p[0] for p in c], [p[1] for p in c], 'g-')"
      ],
      "language": "python",
      "metadata": {},
      ],
      "language": "python",
      "metadata": {},
       {
        "metadata": {},
        "output_type": "pyout",
       {
        "metadata": {},
        "output_type": "pyout",
-       "prompt_number": 332,
+       "prompt_number": 9,
        "text": [
        "text": [
-        "[<matplotlib.lines.Line2D at 0x5a3a6d0>]"
+        "[<matplotlib.lines.Line2D at 0x2af9b10>]"
        ]
       },
       {
        "metadata": {},
        "output_type": "display_data",
        ]
       },
       {
        "metadata": {},
        "output_type": "display_data",
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1SOEKCHVbKuxrv86urh3ocWCFuy+uDvX6y++qNZUCQlRT/yKskfnO/4bN27Qr18/Ll68iKurK6tW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\nh1Ugnfv4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        "text": [
        "text": [
-        "<matplotlib.figure.Figure at 0x5942790>"
+        "<matplotlib.figure.Figure at 0x2af9950>"
        ]
       }
      ],
        ]
       }
      ],
-     "prompt_number": 332
+     "prompt_number": 9
     },
     {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
     },
     {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "So, it sort of works but not giving the same curve as the vanilla method"
+      "So, it sort of works but not giving the same curve? Or maybe I just need *lots* more points and they will converge."
+     ]
+    },
+    {
+     "cell_type": "heading",
+     "level": 2,
+     "metadata": {},
+     "source": [
+      "Iterated Function Systems"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Defining $L_p = P^{-1} L P$ and $M_p = P^{-1} M P$, then $P L_p = L P = Q$ and $P M_p = M P = R$\n",
+      "\n",
+      "\n",
+      "$\\{L_p, M_p\\}$ is an iterated function system, $\\implies$ Beziers are fractals.\n",
+      "\n",
+      "Problem: $P$ needs to be invertable; introduce homogeneous coordinates $\\langle$insert hand waving here$\\rangle$ to \"lift\" it and then it all magically works"
      ]
     },
     {
      "cell_type": "code",
      "collapsed": false,
      ]
     },
     {
      "cell_type": "code",
      "collapsed": false,
-     "input": [],
+     "input": [
+      "def lift(P):\n",
+      "    \"\"\" Lift P to make it square by introducing homogeneous coordinates\"\"\"\n",
+      "    n = len(P)\n",
+      "    result = []\n",
+      "    for i in xrange(len(P)):\n",
+      "        ones = min(n-len(P[i]), max(1, i-1))\n",
+      "        zeroes = max(n-len(P[i])-ones, 0)\n",
+      "        result += [asarray(list(P[i]) + [1. for _ in xrange(ones)] + [0. for _ in xrange(zeroes)])]\n",
+      "    return asarray(result)\n",
+      "\n",
+      "def lower(P):\n",
+      "    \"\"\" Lower square P back to list of 2D points \"\"\"\n",
+      "    return [row[0:2] for row in P]\n",
+      "\n",
+      "def Lp(P):\n",
+      "    return dot(inv(P), dot(L(len(P)-1), P))\n",
+      "\n",
+      "def Mp(P):\n",
+      "    return dot(inv(P), dot(M(len(P)-1), P)) # Typo in Goldman? Reads \"P^-1 * R * P\" instead of \"P^-1 * M * P\""
+     ],
      "language": "python",
      "metadata": {},
      "language": "python",
      "metadata": {},
-     "outputs": []
+     "outputs": [],
+     "prompt_number": 10
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "So, Goldman's method is:\n",
+      "\n",
+      "<ol>\n",
+      " <li> lift the control points\n",
+      "P from the ambient affine space of dimension $d$ to the\n",
+      "ambient affine space of dimension $n$;\n",
+      " </li><li> generate the corresponding higher dimensional Bezier curve using the\n",
+      "iterated function system $\\{L_p , M_p \\}$ ; \n",
+      "</li><li> project the\n",
+      "resulting $n$-dimensional Bezier curve orthogonally back\n",
+      "down to the original dimension $d$.\n",
+      "</ol>\n"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "def Goldman(P, nPoints):\n",
+      "    while len(P) < nPoints:\n",
+      "        Pl = lift(P)\n",
+      "        LP = Lp(Pl)\n",
+      "        MP = Mp(Pl)\n",
+      "        Q = dot(Pl, LP)\n",
+      "        R = dot(Pl, MP)\n",
+      "        P = lower(Q) + lower(R)\n",
+      "    return P\n",
+      "        "
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 11
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "points = [(0,0), (0.5,1),(1,0),(1.4,-2)]\n",
+      "c = Casteljau(points,50)\n",
+      "c2 = Goldman(points,50)\n",
+      "PlotBezier(points)\n",
+      "plot([p[0] for p in c], [p[1] for p in c], 'g-')\n",
+      "plot([p[0] for p in c2], [p[1] for p in c2], 'r-')"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "pyout",
+       "prompt_number": 12,
+       "text": [
+        "[<matplotlib.lines.Line2D at 0x30a8750>]"
+       ]
+      },
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": "iVBORw0KGgoAAAANSUhEUgAAAX4AAAD9CAYAAAC7iRw+AAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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RBk4e/hy+YeZvLHvAyZP3Aaenvx/CS/zZHDiX7BgAYGBgVhbW5OQkFBsOy8vL4KCgmjd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\nIaoYSfxCCFHF/A+4kFooABS92QAAAABJRU5ErkJggg==\n",
+       "text": [
+        "<matplotlib.figure.Figure at 0x2bfe7d0>"
+       ]
+      }
+     ],
+     "prompt_number": 12
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "The algorithm works, I guess, but is it useful? It will be a lot slower than de Casteljau, because it does a lot more matrix manipulation stuff, but it gives no more accuracy on each iteration."
+     ]
+    },
+    {
+     "cell_type": "heading",
+     "level": 2,
+     "metadata": {},
+     "source": [
+      "Time Performance of Casteljau, Goldman, and Basic Bezier creation"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "from time import time\n",
+      "points = [(0,0), (0.5,1),(1,0),(1.4,-2)]\n",
+      "costs = {BezierCurve : [], Casteljau : [], Goldman : []}\n",
+      "stddev = {}\n",
+      "nAverages = 10\n",
+      "depth=8\n",
+      "for method in costs:\n",
+      "\n",
+      "    stddev.update({method : []})\n",
+      "    for n in xrange(1,depth):\n",
+      "        #display(\"depth %d for %s\" % (n, str(method)))\n",
+      "        c = []\n",
+      "        for _ in xrange(nAverages):\n",
+      "            t0 = time()\n",
+      "            p = method(points, nPoints=len(points) * 2.**n)\n",
+      "            c += [time()-t0]\n",
+      "        costs[method] += [mean(c)]\n",
+      "        stddev[method] += [var(c)**0.5]"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 17
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "title(\"Time to produce $n$-dimensional Bezier Curve\")\n",
+      "xlabel(\"$n$\")\n",
+      "ylabel(\"$t$ (s)\")\n",
+      "x = [len(points) * 2.**n for n in xrange(1,depth)]\n",
+      "for method in costs:\n",
+      "    errorbar(x,costs[method],yerr=stddev[method],label =str(method).split(\" \")[1])\n",
+      "legend(loc=\"upper left\")"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "pyout",
+       "prompt_number": 18,
+       "text": [
+        "<matplotlib.legend.Legend at 0x2f7f790>"
+       ]
+      },
+      {
+       "metadata": {},
+       "output_type": "display_data",
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3btZuXIl3t7euLm5GabrBfjxxx8JCgrCwcEBb29v5syZY1iWkpKCpaUlq1atwsfHBxcXF8Osi0LU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\nSAAATigSAIATigQA4OQDGQo6bPj82VkAAAAASUVORK5CYII=\n",
+       "text": [
+        "<matplotlib.figure.Figure at 0x3a1be50>"
+       ]
+      }
+     ],
+     "prompt_number": 18
     }
    ],
    "metadata": {}
     }
    ],
    "metadata": {}
index 38b7eee..7ce2dea 100644 (file)
       "    normal = asarray([-s[1], s[0]])\n",
       "    normal = normal / sum(normal*normal)**0.5\n",
       "    # Calculate the three points \n",
       "    normal = asarray([-s[1], s[0]])\n",
       "    normal = normal / sum(normal*normal)**0.5\n",
       "    # Calculate the three points \n",
-      "    # Are these are the maps that form the iterated function system?\n",
+      "    # Are these are the maps that form the iterated function system? I don't think so.\n",
       "    # \"Notice that in each case the entire fractal can be recovered from a single point.\" Goldman\n",
       "    # But for this I need *two* points :S\n",
       "    \n",
       "    # \"Notice that in each case the entire fractal can be recovered from a single point.\" Goldman\n",
       "    # But for this I need *two* points :S\n",
       "    \n",

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