notebooks/Example.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%load_ext autoreload\n",
    "%autoreload 2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "from tda_digits import *"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Extraction of topological features from a handwritten digits"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Load the image:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "image = get_image(n=17, show=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Produce the binary image B by thresholding the input image:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "binary_image = get_binary_image(image, show=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Reduce the binary image to 1 pixel width to expose its topology using the Zhang-Suen Thinning algorithm:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "skeleton = get_skeleton(binary_image, show=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Transform the pixels of skeleton to points in the plane:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 504x504 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "points = get_points(skeleton, sweep_direction='top', show=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Construct an embedded graph G in the plane where we treat the points as vertices and add edges between adjacent points similar to [1] and then remove all cycles of length 3. Intuitively, we connect the points while trying not to create new topological features."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 504x504 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "point_list = PointList(points)\n",
    "emb_graph = point_list.get_emb_graph(show=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Construct a simplex stream for computing the persistent homology using the filtration on the vertices of the graph G. Filtration is the following. We are adding the vertices and edges of the embedded graph G as we sweep across the plane in sweep_direction. In this example, we sweep across the plane from $y = -\\inf$ to $y = \\inf$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "simplices = get_simplices(emb_graph, show=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Compute the persistent homology given the simplex stream to get the so-called Betti barcodes.\n",
    "\n",
    "*Betti $k$ barcode* is a finite set of intervals. Each interval represents the first filtration level where the topological feature of dimension $k$ appears and the filtration level where it disappears. This are called birth and death times of the topological feature. \n",
    "\n",
    "A connected component (connected cluster of points) is 0 dimensional and a cycle (loop) is 1 dimensional feature. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Betti 0\n",
      "(4.0, inf)\n",
      "\n",
      "Betti 1\n",
      "(12.0, inf)\n",
      "(20.0, inf)\n",
      "\n"
     ]
    }
   ],
   "source": [
    "intervals = get_betti_barcodes(simplices, show=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "draw_betti_barcodes(intervals, image_size, show=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We note that the Betti 0 barcode (the upper picture) clearly shows the single connected component with the birth time of 4.\n",
    "\n",
    "The Betti 1 barcode shows two lines with birth times 12 and 20 corresponding to the births of 2 cycles (the value of y when the loop closes) in the drawing of number 8 when we sweep in $y$ direction across the plane from $y = -\\inf$ to $y = \\inf$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Extract 4 features:\n",
    "$$\n",
    "        \t\\sum_i { x_i (y_i - x_i) } \\\\\n",
    "        \t\\sum_i { (y_{\\text{max}} - y_i) (y_i - x_i) } \\\\\n",
    "        \t\\sum_i { x_i^2 \\cdot (y_i - x_i)^4 } \\\\\n",
    "        \t\\sum_i { (y_{\\text{max}} - y_i)^2 (y_i - x_i)^4 }\n",
    "$$\n",
    "From each Betti barcode intervals:\n",
    "$$\n",
    "        (x1, y1), (x2, y2), ..., (x_n, y_n)\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[96.0, 0.0, 5308416.0, 0.0]"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "extract_features(intervals[0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[352.0, 0.0, 11075584.0, 0.0]"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "extract_features(intervals[1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There are 4 sweep directions: top, bottom, right, left. We compute 2 barcodes for 0 and 1 dimension with 4 features each. All together 32 features per image of a handwritten digit:\n",
    "$$\n",
    "4 \\text{ sweeps } \\cdot 2 \\text{ barcodes } \\cdot 4 \\text{ features } = 32 \\text{ features }\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "32\n"
     ]
    }
   ],
   "source": [
    "all_features = extract_all_features(n=17)\n",
    "print(len(all_features))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[184.0,\n",
       " 28.0,\n",
       " 10242304.0,\n",
       " 3136.0,\n",
       " 367.0,\n",
       " 0.0,\n",
       " 9898057.0,\n",
       " 0.0,\n",
       " 184.0,\n",
       " 28.0,\n",
       " 10242304.0,\n",
       " 3136.0,\n",
       " 367.0,\n",
       " 0.0,\n",
       " 9898057.0,\n",
       " 0.0,\n",
       " 96.0,\n",
       " 0.0,\n",
       " 5308416.0,\n",
       " 0.0,\n",
       " 352.0,\n",
       " 0.0,\n",
       " 11075584.0,\n",
       " 0.0,\n",
       " 129.0,\n",
       " 38.0,\n",
       " 6996809.0,\n",
       " 5776.0,\n",
       " 311.0,\n",
       " 0.0,\n",
       " 7860161.0,\n",
       " 0.0]"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "all_features"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "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": "3.8.10"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}