{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Data Compression and Shannon Theory"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This tour reproduces the figures for my introduction article \n",
    "on Shannon theory of data compression."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First add the toolbox additional functions into the path."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {},
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "warning off; addpath('toolbox'); warning on;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#Image loading and quantizing"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Set parameters values."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {},
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "name = 'hibiscus'; % image name\n",
    "q = 4; % quantization step\n",
    "qmax = q;\n",
    "n = 256; % image size"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Useful helper functions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {},
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "normalize = @(x)x/sum(x(:));\n",
    "myhist = @(x,y)normalize(hist(x,y));\n",
    "Quant = @(x,q)min(floor( rescale(x)*q  ), q-1);\n",
    "Entropy = @(h)-sum(h.*log2(max(h,1e-20)));\n",
    "myaxis = @(x,h)axis([min(x)-1/2 max(x)+1/2 0 max(h)*1.02]);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Load the initial image $x_0$ and quantize it to $q$ values to obtain $x$ with pixel values in $\\{0,\\ldots,q-1\\}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {},
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x0 = load_image(name, n);\n",
    "x0 = rescale(sum(x0,3));\n",
    "x = Quant(x0,q);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Display the images $x_0$ and $x$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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       "EoEESQghRCKQIAkhhEgEEiQhhBCJQIIkhBAiEUiQhBBCJAIJkhBCiEQgQRJCCJEIJEhCCCESgQRJ\n",
       "CCFEIpAgCSGESAQSJCGEEIlAgiSEECIRSJCEEEIkAgmSEEKIRCBBEkIIkQgkSEIIIRKBBEkIIUQi\n",
       "kCAJIYRIBBIkIYQQiUCCJIQQIhFIkIQQQiQCCZIQQohEIEESQgiRCCRIQgghEoEESQghRCKQIAkh\n",
       "hEgEEiQhhBCJQIIkhBAiEUiQhBBCJAIJkhBCiEQgQRJCCJEIJEhCCCESgQRJCCFEIpAgCSGESAQS\n",
       "JCGEEIlAgiSEECIRSJCEEEIkAgmSEEKIRCBBEkIIkQgkSEIIIRKBBEkIIUQikCAJIYRIBBIkIYQQ\n",
       "iUCCJIQQIhFIkIQQQiQCCZIQQohEqGsr8AAAADVJREFUIEESQgiRCCRIQgghEoEESQghRCKQIAkh\n",
       "hEgEEiQhhBCJQIIkhBAiEUiQhBBCJIL/B4GjUkaNhZ8YAAAAAElFTkSuQmCC\n"
      ],
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {},
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "imageplot({x0,x}, {'Original', 'Quantized'});"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Huffman coding of pixel values"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Compute the empirical distributions (histogram) of the pixels."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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       "7+KHsRO0ZcuWsLCwzz///N133+UqQUFBr7322rp164RtbPrCA/qEgaffAjwYnE7ntWvXuru7FQpF\n",
       "YmLi6B8awsQhkAAAgApY1AAAAFRAIAEAABUQSAAAQAUEEgAAUAGBBAAAVEAgAQAAFRBIAABABQQS\n",
       "AABQAYEEAABUQCABAAAVEEgAAEAFBBIAAFABgQQAAFRAIAEAABUQSAAAQAUEEgAAUAGBBAAAVEAg\n",
       "AQAAFRBIAABAhf8Dhc8aWidphzkAAAAASUVORK5CYII=\n"
      ],
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {},
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "hx = myhist(x(:),0:q-1);\n",
    "clf; \n",
    "bar(0:q-1, hx); axis tight;\n",
    "myaxis(0:qmax-1, hx);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Run Huffman's algorithm to compute the tree storing in its leaves\n",
    "the codewords of an optimal coding scheme."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {},
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Tx = compute_hufftree(hx);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Display this tree, with the codewords in the leaves."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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       "hJAnnnji1KlT/fv3p49ms5lLI6VS+fLLL+fk5Dh7GhFCCgoKWJcAAGBnQvuEROn1+ry8vNzc3Nzc\n",
       "3Obm5piYmJiYmMTERDqGBwAAPCTMQAIAAKcjtJYdAAA4KQQSAADwAgIJAAB4AYEEAAC8gEACAABe\n",
       "QCABAAAvIJAAAIAXEEgAAMALCCQAAOAFBBIAAPACAgkAAHjh/wNow39pl34m4gAAAABJRU5ErkJg\n",
       "gg==\n"
      ],
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {},
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "clf; plot_hufftree(Tx, -1); "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Generate the list of all the codewords."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {},
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[Cx,Lx] = huffman_gencode(Tx);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Perform the actual coding of the values in $x$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {},
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bx = perform_huffcoding(x(:)+1,Tx,+1);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Compare the length of the stream with the entropy bound, to check its tightness. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Entropy=1.541, Huffman=1.681, log2(q)=2.000"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Ex = Entropy(hx);\n",
    "Ex1 = sum(hx(:).*Lx);\n",
    "Ex2 = length(bx)/length(x(:)); % should be equal to Ex1\n",
    "fprintf('Entropy=%.3f, Huffman=%.3f, log2(q)=%.3f\\n', Ex, Ex1, log2(q));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This shows that the entropy is a bit smaller than the actual average number \n",
    "of bits per symbol. This is because the optimal codeword \n",
    "length predicted by Shannon's \n",
    "theory is not an integer, whereas Hufman coding can only generate \n",
    "codewords with integer length.\n",
    "This can be alleviated using arithmetic coding, which is a more advancesd\n",
    "coding technic. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#Coding of pixel differences"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Compute difference of consecutive pixels, $y_i=x_{i}-x_{i-1}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {},
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y = diff(x(:))+q-1;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Compute the empirical frequency (histogram) of these differences."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {},
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "hy = myhist(y(:),0:2*q-2);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Display this histogram."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false,
    "scrolled": false
   },
   "outputs": [
    {
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       "DiGE11YlmzfeeCM4OPiJJ564DDd0jfGaY40F6LYd0PsTX+AI6eJ85zvfWbRo0c6dO3/xi1/k5ubG\n",
       "x8fX1NS8/fbbJ0+enDNnjswnCgYHB+vr6ydPnvzcc88N/21mZuavf/1r7bu6VOXl5VVUVFit1qGh\n",
       "IfdrjsPCwoqLi/3d2iUocLftwN2f+AiBdNEefvjh0NDQP/7xj0899ZRaCQ4OvvPOO++9917/Nja6\n",
       "jz76aGBgYGBg4L333hv+W5lvHgxEkZGRW7duzc/PLy8vLy8vF1+/5jgpKcnfrV2CAnrbDtD9iY/w\n",
       "gr5vqKOj48iRI21tbQaD4corrxx+0wkgeM0xLgz7ExWBBACQApMaAABSIJAAAFIgkAAAUiCQAABS\n",
       "IJAAAFIgkAAAUiCQAABSIJAAAFIgkAAAUiCQAABSIJAAAFIgkAAAUiCQAABSIJAAAFIgkAAAUiCQ\n",
       "AABSIJAAAFIgkAAAUiCQAABS+H9IFDnHHfVbRQAAAABJRU5ErkJggg==\n"
      ],
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {},
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "clf; \n",
    "bar(-(q-1):q-1, hy); axis tight;\n",
    "myaxis(-(qmax-1):qmax-1, hy);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Compute the Huffman tree."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {},
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Ty = compute_hufftree(hy);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Display this Huffman tree."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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      ],
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {},
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "clf; plot_hufftree(Ty, -q); "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Generate the list of codewords associated to this tree."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {},
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "[Cy,Ly] = huffman_gencode(Ty);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Perform the codding of the list of differences to obtain a binary stream."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {},
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "by = perform_huffcoding(y(:)+1,Ty,+1);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Compare the length of the stream with the entropy bound, to check its tightness. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Entropy=0.612, Huffman=1.164, log2(q)=2.000"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Ey = Entropy(hy);\n",
    "Ey1 = sum(hy(:).*Ly);\n",
    "Ey2 = length(by)/length(y); % should be equal to Ey1\n",
    "fprintf('Entropy=%.3f, Huffman=%.3f, log2(q)=%.3f\\n', Ey, Ey1, log2(q));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#Conclusion"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This simple experiment shows that by a simple re-transformation of the data, \n",
    "one can obtain a set of values with a lower entropy, thus leading to \n",
    "a higher compression rate. This is possible because the initial data\n",
    "(here a quantized image) violates the independency assumption of Shannon's \n",
    "theory. In this example, pixels that are neighbors tend to have similar \n",
    "values  (for instance in constant areas), and computtting the differences\n",
    "leads to lots of 0's, thus to a high entropy. \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Matlab",
   "language": "matlab",
   "name": "matlab_kernel"
  },
  "language_info": {
   "file_extension": ".m",
   "help_links": [
    {
     "text": "MetaKernel Magics",
     "url": "https://github.com/calysto/metakernel/blob/master/metakernel/magics/README.md"
    }
   ],
   "mimetype": "text/x-matlab",
   "name": "matlab"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}