dawehner/image-size-model.ipynb

## image-size-model.ipynb
{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Problem"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Today I stumpled upon a pretty interesting plot. Let's see whether we can figure out the x scaling, as it is pretty obviously not linear."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAtgAAAEeCAYAAACwimIxAAAQsUlEQVR42u3deW8UZ4LA4fke81FW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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from IPython.display import Image\n",
    "Image(filename='data.png')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As we see, this is obviously not trivial. It is certainly not linear, but maybe logarithmic or a quite high polynomial might work. First though let's try to extract the size of the bars using scikit-image."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Read in the data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from skimage import data, io, filters, measure"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "data = io.imread('data.png', as_grey=True)\n",
    "# Strip the top out\n",
    "data = data[20:,]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(266, 728)"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "contours = measure.find_contours(data, 0.8)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "from scipy.optimize import curve_fit"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Data extraction"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Split up the images up manually. We could use some fancy algorithm, but this seems easier. After then look at an individual line through the dataset."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXcAAAA9CAYAAABWdClAAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nswpycTczqyAXdzOzCvofbaX0cx4at6oAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1146d5950>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAW0AAACdCAYAAAB/5CmPAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x117385cd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAVcAAAD+CAYAAACOVUaKAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nVZIaGFdJamBcJamBcZWkBsZVkhr8Lxl7pm5bfs3xAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1168ec5d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "d1 = data[0:50,0:600]\n",
    "plt.imshow(d1)\n",
    "plt.show()\n",
    "\n",
    "d2 = data[90:150,0:160]\n",
    "plt.imshow(d2)\n",
    "plt.show()\n",
    "\n",
    "\n",
    "d3 = data[170:220,0:70]\n",
    "plt.imshow(d3)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here we print out an individual line of the dataset, this gives us an idea  how to extract an exact line of the data."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x116a28cd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(d1[25,])\n",
    "plt.plot(d2[30,])\n",
    "plt.plot(d3[30,])\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In order to extract a single line lenght, we just check how long a single line is over 0.5 and then sums up the boolean values. This gives us the effective X length of the individual bar."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "560 130 24\n"
     ]
    }
   ],
   "source": [
    "len1 = np.sum(d1[25,] > 0.5)\n",
    "len2 = np.sum(d2[30,] > 0.5)\n",
    "len3 = np.sum(d3[30,] > 0.5)\n",
    "\n",
    "points = np.array([[len1, 65.2], [len2, 1.5], [len3, 1.1]])\n",
    "print len1, len2, len3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Finding the right model/scaling"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ -2.18158824e-12   1.00000000e+00] [[ inf  inf]\n",
      " [ inf  inf]]\n",
      "[ -1.89337435e-12   1.00000000e+00  -2.02977291e-12] [[ inf  inf  inf]\n",
      " [ inf  inf  inf]\n",
      " [ inf  inf  inf]]\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/Users/dawehner/anaconda/lib/python2.7/site-packages/scipy/optimize/minpack.py:690: OptimizeWarning: Covariance of the parameters could not be estimated\n",
      "  category=OptimizeWarning)\n"
     ]
    }
   ],
   "source": [
    "linear = lambda x, a, b: a + b * x\n",
    "square = lambda x, a, b, c: a + b * x + c * x **2\n",
    "\n",
    "popt, pcov = curve_fit(linear, points[:,0], points[:,0])\n",
    "print popt, pcov\n",
    "\n",
    "popt, pcov = curve_fit(square, points[:,0], points[:,0])\n",
    "print popt, pcov"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "mh, this didn't out worked so well, so let's try logarithmic"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-0.33248244  0.00794913] [[  1.00585771e-01  -2.16923734e-04]\n",
      " [ -2.16923734e-04   9.11444246e-07]]\n"
     ]
    }
   ],
   "source": [
    "points2 = np.log(points[:,1])\n",
    "popt, pcov = curve_fit(linear, points[:,0], points2)\n",
    "print popt, pcov"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Lovely!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\begin{equation}\n",
    "a + b x = \\log(y) <=> \\\\\n",
    "\\exp(a + bx) = y <=>\\\\\n",
    "\\exp(a) \\exp(bx) = y => \\\\\n",
    "0.71 \\exp(0.0079 \\; x) = y\n",
    "\\end{equation}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Summary"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This plot was fun to read. The resulting xscale is:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\begin{equation}\n",
    "x' = 0.71 \\; e^{\\; 0.0079 x}\n",
    "\\end{equation}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.11"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}