Convex_Hull.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Convex Hull\n",
    "### Dr. Tirthajyoti Sarkar, Fremont CA 94536\n",
    "The convex hull of a binary image is the set of pixels included in the smallest convex polygon that surround all white pixels in the input.\n",
    "\n",
    "A good overview of the algorithm is given on [this blog](http://blogs.mathworks.com/steve/2011/10/04/binary-image-convex-hull-algorithm-notes/)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "\n",
    "from skimage.morphology import convex_hull_image\n",
    "from skimage import data, img_as_float\n",
    "from skimage.util import invert"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The original image is inverted as the object must be white"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "image = invert(data.horse())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Create a convex hull"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "chull = convex_hull_image(image)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Show the result"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 576x288 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, axes = plt.subplots(1, 2, figsize=(8, 4))\n",
    "ax = axes.ravel()\n",
    "\n",
    "ax[0].set_title('Original picture')\n",
    "ax[0].imshow(image, cmap=plt.cm.gray, interpolation='nearest')\n",
    "ax[0].set_axis_off()\n",
    "\n",
    "ax[1].set_title('Transformed picture - convex hull')\n",
    "ax[1].imshow(chull, cmap=plt.cm.gray, interpolation='nearest')\n",
    "ax[1].set_axis_off()\n",
    "\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Showing the difference by fixing the pixel intensity on the convex hull"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "chull_diff = img_as_float(chull.copy())\n",
    "chull_diff[image] = 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()\n",
    "ax.imshow(chull_diff, cmap=plt.cm.gray, interpolation='nearest')\n",
    "ax.set_title('Difference')\n",
    "plt.show()"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.2"
  },
  "latex_envs": {
   "LaTeX_envs_menu_present": true,
   "autoclose": false,
   "autocomplete": true,
   "bibliofile": "biblio.bib",
   "cite_by": "apalike",
   "current_citInitial": 1,
   "eqLabelWithNumbers": true,
   "eqNumInitial": 1,
   "hotkeys": {
    "equation": "Ctrl-E",
    "itemize": "Ctrl-I"
   },
   "labels_anchors": false,
   "latex_user_defs": false,
   "report_style_numbering": false,
   "user_envs_cfg": false
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}