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  <title>Chapter 0 - OKAI</title>

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          <p>Chapter 0</p>
          <h2>A Taste of Artificial Intelligence (AI)</h2>
          <p>Are you wondering what AI can do? Below is an interactive AI module
              that can recognize any digit that you write! Reading the following
              chapters will enable you to understand how this module works
              behind the scenes.</p>
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          <h2>Handwritten Digit Recognition</h1>
          <p>Draw a single digit (0-9) in the white box below and click the "recognize" button.
              The digit you write can be recognized with a success rate of 98.08%.

            <br>
            <small id="credit" data-toggle="tooltip" title="This module is based on Hubert Eichner's
            JavaScript implementation of a neural net for handwritten digit recognition, optimized for mobile.
            The source code is available in OKAI's github">credit</small>
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               <button type="button" id="recognize" onclick="recognize()" class="btn btn-light mt-3">Recognize</button>
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    <!-- BEGIN WEIGHTS -->
    <script type="text/javascript">
      // some of the code below was taken from a stackoverflow flag I cannot find anymore, and adapted to my needs.
      // Thanks a ton to the original author!
      var canvas;
      var ctx;

      var prevX = 0;
      var currX = 0;
      var prevY = 0;
      var currY = 0;
      var paths = []; // recording paths
      var paintFlag = false;
      var color = "black";
      var lineWidth = 20;

      var clearBeforeDraw = false; // controls whether canvas will be cleared on next mousedown event. Set to true after digit recognition

      // the neural network's weights (unit-unit weights, and unit biases)
      // training was done in Matlab with the MNIST dataset.
      // this data is for a 784-200-10 unit, with logistic non-linearity in the hidden and softmax in the output layer.
      // The input is a [-1;1] gray level image, background == 1, 28x28 pixels linearized in column order (i.e. column1(:); column2(:); ...)
      // i-th output being the maximum means the network thinks the input encodes (i-1)
      // the weights below showed a 1.92% error rate on the test data set (9808/10000 digits recognized correctly).
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          0.97618, -0.428155, -0.825335, 0.0663525, -0.252102, 1.42023, -0.211406, -0.549629, 0.288938, -0.367544, 0.981119, -0.861998, -0.0978796, -2.13574, 1.67448, -0.486979, -0.655064, 0.0876539, -0.920079, 0.838986, -0.385184, -0.768065,
          0.0278169, 0.764948, -0.279529, -0.762522, 0.563328, -0.513939, -0.0184082, 0.895262, -0.857057, -0.219892, 0.309356, -0.485254, -0.106315, 0.601012, 0.5126, -0.45509, 1.07242, 0.745894, -0.662426, -1.51229, 0.0423359, 0.617365, 0.163451,
          -0.553203, -0.161091, -0.615686, -0.760901, -0.864786, -0.540975, 0.200036, 0.540387, 0.0481334, -0.75039, -0.254785, -0.333893, 0.161339, 0.734239, 0.391471, -0.476712, 0.598803
        ],
        [-0.823418, 0.380799, 0.441713, 0.428736, 2.65236, 0.905039, -0.25199, 0.0723711, 0.392547, 0.706137, 2.35118, 1.28268, -0.197731, 0.472847, 2.32248, 0.275431, 0.269705, 1.52667, -0.0931101, -0.987194, 1.41352, -0.848697, -0.569718,
          2.11607, 0.295744, -0.262944, -0.712598, 1.6751, -0.531171, -1.00579, -0.201727, -0.786184, -0.296217, 0.543134, 0.601636, -0.394478, 0.619219, 1.0598, -0.463994, -0.868514, -0.185332, 1.70221, -1.77063, 0.424553, -0.540194, 1.01335,
          0.67229, -0.078009, -0.975557, 0.0275916, 1.49203, 0.66986, -0.602254, -1.57721, 0.0540897, 0.180679, -0.73641, 0.858529, -0.228447, -0.674176, -0.823261, -0.0124315, 0.166844, 0.947693, 0.793836, -1.44588, 1.98137, 1.00171, 1.45505,
          0.431462, -2.16347, 0.480645, -0.378093, 0.494897, -0.3219, -0.183139, -0.363703, -0.963229, -0.681948, 0.96694, -2.04823, 0.0700168, 0.650468, -0.404429, 0.375098, 0.053157, -1.37709, 0.202017, -0.0302393, 0.543329, 0.570758, 0.608443,
          -1.21033, -0.478777, 0.166424, 1.10919, -1.44998, 1.9565, -0.629106, -0.692402, 1.0042, 1.40405, 0.0770993, 1.39343, -0.869654, -0.944405, -0.529465, 0.561616, 0.948276, -0.381202, -0.0625773, 0.265479, 0.927656, 0.183621, 0.469049, -
          1.59988, 1.69697, 0.591354, 0.126989, -1.25269, 0.352193, 0.742284, -0.329177, -0.437253, -0.0491613, -0.544682, -0.358365, -0.292597, 0.741614, -1.51014, -0.273552, 0.681179, 1.05271, 0.516688, -0.342525, 1.18099, -2.05598, -0.545017,
          0.100126, 0.856157, 0.439645, 0.435219, -0.163375, 0.569578, 0.203825, -0.204001, 0.402037, 0.380089, -0.965807, 0.736224, -1.56064, 0.887663, -1.94346, 0.390256, -0.14638, -0.528429, -1.30915, -1.83072, -1.05213, 0.432365, 0.611385, -
          1.27376, 0.436293, -0.426238, 0.371424, -0.122245, -0.892942, 0.784511, 0.87657, -0.843596, 0.821356, -0.723744, -0.188603, -0.770648, 0.216358, 0.147172, 0.608664, -0.344758, -0.0988283, 0.134224, 0.0870816, -0.782166, -0.519631, -
          1.6077, -0.229582, 0.664242, 0.645405, -0.269772, -0.539242, 0.218015, 1.09658, 1.1972, -0.272101, 0.809838, -0.497889, -0.672209, -1.75552, -1.43563, 1.19184, 0.681575
        ],
        [-0.598903, -0.550104, -0.52355, -0.188229, 0.757833, 0.122506, 1.44765, -1.51711, -0.299931, -0.968524, -0.749438, -0.157857, 1.04299, -0.164433, 0.860262, -0.357564, -0.922937, 1.0352, -0.341289, 0.173465, 1.40978, 0.806245, -0.0201346,
          0.155425, 0.367049, -0.468195, 0.26194, -0.237561, -0.514518, -0.105776, 0.420625, 0.909893, 1.06559, -0.357176, 0.638831, -0.549053, -0.415928, 1.08195, -0.0815873, -0.517886, 0.296801, -0.0327398, 0.824502, -0.525542, -0.56462,
          0.360483, -1.28647, -0.636515, 0.768898, -0.402619, -0.181267, -1.17698, -0.675118, 0.20146, -0.605803, -0.392849, -0.411201, 0.1126, -0.452895, -1.14425, -0.0706288, -0.281412, -0.31817, -0.884616, -0.33598, 0.222194, -0.861537, -
          0.935402, 0.374003, 1.54709, -1.06125, 0.127386, -0.153661, 0.293125, -0.648531, 1.05186, 0.00292419, 1.11123, -0.760627, -2.05218, -0.165579, -0.359261, -0.287854, -1.38551, -0.192489, -0.0142184, -1.25271, -0.843711, -0.985648, -
          2.14909, -0.0186729, 0.493857, 0.775097, 0.0621715, 0.538066, -0.741633, 0.360059, -1.20396, 0.67766, -0.0761122, 0.645486, -0.153375, 0.930736, 0.45499, 0.763076, -0.136051, -0.770157, -0.806827, -0.441221, -0.546789, -0.976304, -
          0.129629, -0.912183, -0.00837599, -0.0856566, 0.0697607, 1.35127, -0.0672679, 0.0487817, 0.190426, 0.486384, -0.593641, -0.626597, 0.570286, 1.31472, 0.727246, -1.52312, 0.531825, 0.449111, 0.436264, -0.577608, 1.21207, -0.0324695, -
          0.658513, 0.0238966, -0.495386, 0.521196, -0.719584, 0.528344, 1.49786, -0.143551, -0.301187, -0.53113, 0.00854805, -0.219704, 0.245421, 0.0791451, -0.584985, 0.227921, -0.957152, -0.53843, -1.32728, -0.767072, -0.662043, 0.232316, -
          0.0224666, 0.745536, -1.20467, -0.212089, 0.951174, 1.31489, -0.152599, -0.0401533, 0.997693, -0.0702789, 0.110847, 0.623449, -1.20259, -0.229778, -0.708925, -0.0090411, 1.05347, -0.553273, -0.23238, -0.831694, -0.74302, -1.64773,
          0.678227, 1.25874, -1.08337, -0.206111, -0.341693, -0.0548461, 0.021161, 0.492893, 1.81201, -0.632714, 1.22094, 0.511431, 0.139088, -1.1215, 0.814548, -0.4984, 0.724605, 1.36276, 0.740758, 1.1258, 0.687195, -0.80727, -0.210761
        ],
        [0.136472, 0.0676158, -0.840637, 2.00275, 0.310164, 0.703495, 0.0530234, 0.642718, -0.272215, -0.698208, -0.724258, -0.795768, 0.0919211, -0.809739, 1.05545, 1.01084, 0.0934803, -0.39034, 1.19008, 0.204798, -0.312871, -0.89373, 0.372958,
          1.37577, -0.111703, 1.42117, 0.111701, -0.251531, -0.636292, 1.43268, -0.238121, 1.67919, -0.265951, 0.414543, -1.13401, 0.480393, -2.05768, -0.0825599, -0.228999, -0.178089, 0.628501, 0.997306, 0.306325, -0.737603, -0.329901, 0.927823,
          0.0195058, 0.456329, 2.0744, 0.942688, -0.0017025, 0.741748, 0.751472, -0.301618, -0.054383, 0.768873, 2.09881, -0.349266, -0.532971, -0.507109, -0.683053, -0.458929, -0.610649, -0.499344, -0.563861, 1.32329, -0.541804, 1.81619, 0.421569,
          -0.110533, 0.173826, -1.2713, -0.248694, -0.353467, 1.28057, -1.89039, 0.0628244, 1.06295, 0.476042, -0.186707, 0.121963, 0.786043, 0.00588527, 0.491155, -0.443082, 0.0581695, 0.333261, 0.498635, 0.0321047, -0.52187, 0.0203444, 0.384157,
          0.907186, 0.759188, -0.743464, 0.801651, -0.386188, -0.716903, 0.336286, 1.86066, -0.0452195, 0.535105, 0.152163, -0.397392, -1.01934, -0.440914, 1.39284, 0.618149, -0.361474, -1.34618, -0.0977204, 0.703735, 1.46512, -0.990949, 0.284948,
          -0.209649, 0.367366, 0.810345, -1.59966, -0.0882599, -0.309574, -0.158926, 1.3708, -1.20885, -1.71313, -1.06225, 0.68935, 0.304657, -0.0273226, 0.632294, -0.0936451, -0.76886, 1.47763, 0.385299, 0.471012, -1.30475, 0.148354, 1.31088,
          0.697501, 0.328678, 0.485987, -0.226797, 1.55139, 0.168321, 0.529315, 0.509285, -0.382232, -0.965476, 1.30988, 0.146991, 0.526487, 1.4614, 0.876483, -0.0493464, -1.35586, 0.463778, -0.558769, 0.174925, 1.40458, -1.33121, 0.0203772, -
          0.615263, 0.467958, -0.361831, -1.03102, 1.79953, 0.634536, -0.0437456, -0.670317, 0.892622, -0.310001, -0.52676, 1.07314, -1.28013, 0.588799, -0.844059, 0.818576, -0.643092, 0.856823, 0.182973, 0.876918, 1.28605, 0.235656, 0.207455, -
          0.573538, -0.290554, 2.35392, -0.406035, 1.27002, 0.111846, -0.589686, 0.0954991, -0.562721, -0.427945, -0.31688, -1.00782, 0.0796161, 1.30002, -0.0054727, -0.331399
        ],
        [-0.809901, 0.0569591, -0.492214, -0.802176, -0.67174, -1.85251, -0.149186, -0.604944, 0.958413, 0.495526, -1.59882, -0.96047, 0.341518, -1.48144, -0.914552, -0.504466, -0.508587, -0.31523, 0.401154, -0.000257909, 0.855992, 0.247601, -
          1.05374, -0.484457, -0.997409, -0.234968, -0.683218, -0.890052, 0.130158, -0.390313, -0.233486, -0.449485, 0.229263, -0.13047, -0.913744, -0.389641, 1.43609, -0.0844235, -0.48974, 0.457275, 0.202622, -1.01636, -1.07737, -0.00506024,
          0.260488, -0.062374, -0.422665, 0.166343, -0.417602, -0.561984, -0.525589, -0.404152, -0.175043, 0.579527, 1.9171, 0.46353, -1.27779, 0.498285, 0.942672, -0.699087, 0.564096, 0.388429, -1.31016, 0.823139, 0.0647525, -1.02034, -2.71481, -
          0.884012, -2.27181, -1.33846, 0.575995, -0.907516, -0.369825, 0.278956, -1.2406, -0.395051, 0.368301, 0.691069, -0.746027, -0.230036, -0.682997, -0.326476, 0.136648, -0.171453, 0.991118, -0.044616, -1.27587, 1.55318, 0.328649, 1.12438, -
          1.52436, -0.1066, -0.263798, -1.52143, -0.214258, -0.962471, 0.0534518, -0.633848, -1.34541, -1.29834, 0.877802, 0.503422, 0.907654, 0.562595, -0.844768, 1.65498, -2.27302, 0.00878335, -0.82218, 2.26561, -1.43774, -0.688249, -0.715462,
          0.296556, -0.295884, 1.26701, 0.0528664, 0.631008, 1.28768, 0.303625, 0.679467, 0.426767, -1.49539, 0.533715, -0.912027, -0.329454, -0.333994, 0.505127, 1.09778, -0.520088, -0.153515, -0.662371, -1.23342, -0.542458, -0.121921, 1.02143, -
          0.261008, -0.918588, 1.47385, -1.14802, -0.70963, -0.273309, 1.36197, -1.57803, -0.298641, -1.26737, 0.522564, 0.819519, -1.94213, -1.24474, 0.763026, 1.79811, -0.255928, -0.441871, -1.35628, -0.0375556, 0.049629, -0.141569, 1.53467,
          0.176309, -1.38526, 0.303159, 0.508887, 0.0132563, 0.925258, 0.100075, -0.218879, -0.720119, -0.101902, -0.13577, 1.73847, 1.39844, 0.0973974, 0.364443, -0.304954, 2.32384, 0.851702, 1.21847, -0.476916, 0.104079, -2.22563, 0.614057, -
          1.89411, -0.590563, 0.299338, -0.16394, -0.970441, -1.4343, -0.591618, 0.0314549, 0.97616, -1.15865, 0.598869, 0.643538, -1.22864, -0.635935, -0.887539, 0.480097, -0.390531, 0.464681
        ],
        [0.556977, -1.91918, -0.133978, -1.13368, 0.517309, -1.34561, -0.601899, -0.554714, -0.138805, -0.869225, -0.626074, -1.50575, -0.960188, 0.14965, 0.458502, -1.90492, 0.887959, 0.242981, -0.701652, -0.531657, -0.75281, 0.893896, -0.464351,
          -1.73633, -0.626516, -1.22222, -0.797585, -0.117463, 0.237913, 0.752908, -0.118893, -1.01523, 0.545527, -0.0673889, 0.264197, 0.21078, 1.979, -0.390557, 1.39546, -1.77115, -0.239506, 0.644214, 0.496313, 0.983665, 1.01098, 0.837308, -
          0.268252, 0.150553, -1.27665, 0.418294, 0.834484, 0.0765808, 0.740093, 0.658202, -0.723915, 0.421127, -1.91303, -0.485427, 0.238296, 0.869054, 1.39391, -1.48023, -0.241704, 0.132516, 0.508713, 0.360598, 1.76791, -0.224022, -0.215825, -
          0.623694, 0.414653, -1.24123, -1.50825, 1.14963, 1.00851, -0.339593, -0.129326, -1.3616, -0.153948, 0.381251, -0.689582, -0.018699, 0.065132, -0.0985039, -0.728613, -0.0119607, -0.00243543, 1.47538, -0.311773, -0.0320127, 1.89922, -
          1.64606, -1.58737, 1.18614, -0.277943, -1.0503, 0.371198, 1.04802, 1.04481, -1.79989, -1.67937, 0.279675, -0.936314, -1.48916, 0.569204, 0.820942, 1.28958, -0.104987, -0.0670011, 0.925491, 1.80079, 0.628888, -1.99493, 0.985972, -1.74251,
          -0.63889, 0.421421, 0.47508, -0.48704, -0.761577, -1.35395, -0.556659, 1.04058, 1.15504, 0.07308, 0.208226, 1.89169, 0.38174, -0.176511, 0.841196, -1.3791, -1.44921, -0.937534, 0.511366, 0.336981, 1.25253, 0.999645, 0.510472, 0.19874,
          1.15297, 0.627525, 0.212395, -1.87288, -1.44533, -1.96709, -0.543088, -0.67488, 0.363721, 1.39207, -0.494536, 0.723111, 1.14041, -0.542509, 0.169748, 0.763965, 0.341389, 1.49063, 0.659432, -2.39621, -0.508731, -1.10062, 1.09885, -
          0.199547, 0.142887, 0.749443, -0.0950408, -0.181612, 0.651813, -0.310566, 1.2885, 0.622912, -0.927178, -0.208322, 1.07954, -0.19662, 0.00771559, 0.147359, 1.51327, -0.781276, 0.147958, 0.579119, -0.241808, 0.0331761, 0.151339, 0.0397303,
          -0.947662, -1.34192, -0.62616, 0.0720814, -0.669598, -0.684889, 1.36728, 0.599631, -0.271235, -0.582245, 0.188591, -1.21257, -0.929924, 0.629918, -1.02826
        ]
      ];
      var bias3 = [-0.0972961, 0.0300511, -0.0173903, -0.00445253, 0.0983954, -0.211495, -0.157007, 0.113646, 0.259432, -0.0774153];

      //END WEIGHTS
      //neural net with one hidden layer; sigmoid for hidden, softmax for output
      function nn(data, w12, bias2, w23, bias3) {
        // just some incomplete sanity checks to find the most obvious errors
        if (!Array.isArray(data) || data.length == 0 ||
          !Array.isArray(w12) || w12.length == 0 || !Array.isArray(w12[0]) || data.length != w12[0].length || !Array.isArray(bias2) || bias2.length != w12.length ||
          !Array.isArray(w23) || w23.length == 0 || !Array.isArray(w23[0]) || w12.length != w23[0].length || !Array.isArray(bias3) || bias3.length != w23.length) {
          console.error('nn(): invalid parameters');
          console.log('d: ' + data.length + ', w12: ' + w12.length + ', w12[0]: ' + w12[0].length + ', bias2: ' + bias2.length +
            ', w23: ' + w23.length + ', w23[0]: ' + w23[0].length + ', bias3: ' + bias3.length);
          return undefined;
        }
        var t1 = new Date();

        // compute layer2 output
        var out2 = [];
        for (var i = 0; i < w12.length; i++) {
          out2[i] = bias2[i];
          for (var j = 0; j < w12[i].length; j++) {
            out2[i] += data[j] * w12[i][j];
          }
          out2[i] = 1 / (1 + Math.exp(-out2[i]));
        }
        //compute layer3 activation
        var out3 = [];
        for (var i = 0; i < w23.length; i++) {
          out3[i] = bias3[i];
          for (var j = 0; j < w23[i].length; j++) {
            out3[i] += out2[j] * w23[i][j];
          }
        }
        // compute layer3 output (softmax)
        var max3 = out3.reduce(function(p, c) {
          return p > c ? p : c;
        });
        var nominators = out3.map(function(e) {
          return Math.exp(e - max3);
        });
        var denominator = nominators.reduce(function(p, c) {
          return p + c;
        });
        var output = nominators.map(function(e) {
          return e / denominator;
        });

        // timing measurement
        var dt = new Date() - t1;
        console.log('NN time: ' + dt + 'ms');
        return output;
      }

      // computes center of mass of digit, for centering
      // note 1 stands for black (0 white) so we have to invert.
      function centerImage(img) {
        var meanX = 0;
        var meanY = 0;
        var rows = img.length;
        var columns = img[0].length;
        var sumPixels = 0;
        for (var y = 0; y < rows; y++) {
          for (var x = 0; x < columns; x++) {
            var pixel = (1 - img[y][x]);
            sumPixels += pixel;
            meanY += y * pixel;
            meanX += x * pixel;
          }
        }
        meanX /= sumPixels;
        meanY /= sumPixels;

        var dY = Math.round(rows / 2 - meanY);
        var dX = Math.round(columns / 2 - meanX);
        return {
          transX: dX,
          transY: dY
        };
      }

      // given grayscale image, find bounding rectangle of digit defined
      // by above-threshold surrounding
      function getBoundingRectangle(img, threshold) {
        var rows = img.length;
        var columns = img[0].length;
        var minX = columns;
        var minY = rows;
        var maxX = -1;
        var maxY = -1;
        for (var y = 0; y < rows; y++) {
          for (var x = 0; x < columns; x++) {
            if (img[y][x] < threshold) {
              if (minX > x) minX = x;
              if (maxX < x) maxX = x;
              if (minY > y) minY = y;
              if (maxY < y) maxY = y;
            }
          }
        }
        return {
          minY: minY,
          minX: minX,
          maxY: maxY,
          maxX: maxX
        };
      }

      // take canvas image and convert to grayscale. Mainly because my
      // own functions operate easier on grayscale, but some stuff like
      // resizing and translating is better done with the canvas functions
      function imageDataToGrayscale(imgData) {
        var grayscaleImg = [];
        for (var y = 0; y < imgData.height; y++) {
          grayscaleImg[y] = [];
          for (var x = 0; x < imgData.width; x++) {
            var offset = y * 4 * imgData.width + 4 * x;
            var alpha = imgData.data[offset + 3];
            // weird: when painting with stroke, alpha == 0 means white;
            // alpha > 0 is a grayscale value; in that case I simply take the R value
            if (alpha == 0) {
              imgData.data[offset] = 255;
              imgData.data[offset + 1] = 255;
              imgData.data[offset + 2] = 255;
            }
            imgData.data[offset + 3] = 255;
            // simply take red channel value. Not correct, but works for
            // black or white images.
            grayscaleImg[y][x] = imgData.data[y * 4 * imgData.width + x * 4 + 0] / 255;
          }
        }
        return grayscaleImg;
      }

      // takes the image in the canvas, centers & resizes it, then puts into 10x10 px bins
      // to give a 28x28 grayscale image; then, computes class probability via neural network
      function recognize() {
        var t1 = new Date();

        // convert RGBA image to a grayscale array, then compute bounding rectangle and center of mass
        var imgData = ctx.getImageData(0, 0, 280, 280);
        grayscaleImg = imageDataToGrayscale(imgData);
        var boundingRectangle = getBoundingRectangle(grayscaleImg, 0.01);
        var trans = centerImage(grayscaleImg); // [dX, dY] to center of mass

        // copy image to hidden canvas, translate to center-of-mass, then
        // scale to fit into a 200x200 box (see MNIST calibration notes on
        // Yann LeCun's website)
        var canvasCopy = document.createElement("canvas");
        canvasCopy.width = imgData.width;
        canvasCopy.height = imgData.height;
        var copyCtx = canvasCopy.getContext("2d");
        var brW = boundingRectangle.maxX + 1 - boundingRectangle.minX;
        var brH = boundingRectangle.maxY + 1 - boundingRectangle.minY;
        var scaling = 190 / (brW > brH ? brW : brH);
        // scale
        copyCtx.translate(canvas.width / 2, canvas.height / 2);
        copyCtx.scale(scaling, scaling);
        copyCtx.translate(-canvas.width / 2, -canvas.height / 2);
        // translate to center of mass
        copyCtx.translate(trans.transX, trans.transY);

        // if (document.getElementById('scaleStrokeWidth').checked == true) {
        //   // redraw the image with a scaled lineWidth first.
        //   // not this is a bit buggy; the bounding box we computed above (which contributed to "scaling") is not valid anymore because
        //   // the line width has changed. This is mostly a problem for extreme cases (very small digits) where the rescaled digit will
        //   // be smaller than the bounding box. I could change this but it'd screw up the code.
        for (var p = 0; p < paths.length; p++) {
          for (var i = 0; i < paths[p][0].length - 1; i++) {
            var x1 = paths[p][0][i];
            var y1 = paths[p][1][i];
            var x2 = paths[p][0][i + 1];
            var y2 = paths[p][1][i + 1];
            draw(copyCtx, color, lineWidth / scaling, x1, y1, x2, y2);
          }
        }
        // } else {
        // default take image from original canvas
        // copyCtx.drawImage(ctx.canvas, 0, 0);
        // }


        // now bin image into 10x10 blocks (giving a 28x28 image)
        imgData = copyCtx.getImageData(0, 0, 280, 280);
        grayscaleImg = imageDataToGrayscale(imgData);
        var nnInput = new Array(784);
        for (var y = 0; y < 28; y++) {
          for (var x = 0; x < 28; x++) {
            var mean = 0;
            for (var v = 0; v < 10; v++) {
              for (var h = 0; h < 10; h++) {
                mean += grayscaleImg[y * 10 + v][x * 10 + h];
              }
            }
            mean = (1 - mean / 100); // average and invert
            nnInput[x * 28 + y] = (mean - .5) / .5;
          }
        }

        // for visualization/debugging: paint the input to the neural net.
        // if (document.getElementById('preprocessing').checked == true) {
        ctx.clearRect(0, 0, canvas.width, canvas.height);
        ctx.drawImage(copyCtx.canvas, 0, 0);
        for (var y = 0; y < 28; y++) {
          for (var x = 0; x < 28; x++) {
            var block = ctx.getImageData(x * 10, y * 10, 10, 10);
            var newVal = 255 * (0.5 - nnInput[x * 28 + y] / 2);
            for (var i = 0; i < 4 * 10 * 10; i += 4) {
              block.data[i] = newVal;
              block.data[i + 1] = newVal;
              block.data[i + 2] = newVal;
              block.data[i + 3] = 255;
            }
            ctx.putImageData(block, x * 10, y * 10);
          }
        }
        // }

        //for copy & pasting the digit into matlab
        //document.getElementById('nnInput').innerHTML=JSON.stringify(nnInput)+';<br><br><br><br>';
        var maxIndex = 0;
        var nnOutput = nn(nnInput, w12, bias2, w23, bias3);
        console.log(nnOutput);
        nnOutput.reduce(function(p, c, i) {
          if (p < c) {
            maxIndex = i;
            return c;
          } else return p;
        });
        console.log('maxIndex: ' + maxIndex);
        document.getElementById('nnOut').innerHTML = "<div>" + maxIndex + "</div>";
        clearBeforeDraw = true;
        var dt = new Date() - t1;
        console.log('recognize time: ' + dt + 'ms');
      }

      function init() {

        canvas = document.getElementById('can');
        ctx = canvas.getContext("2d");

        canvas.addEventListener("mousemove", function(e) {
          findxy('move', e)
        }, false);
        canvas.addEventListener("mousedown", function(e) {
          findxy('down', e)
        }, false);
        canvas.addEventListener("mouseup", function(e) {
          findxy('up', e)
        }, false);
        canvas.addEventListener("mouseout", function(e) {
          findxy('out', e)
        }, false);

        // Set up touch events for mobile, etc
        canvas.addEventListener("touchstart", function (e) {
          findxy('start', e)
          // mousePos = getTouchPos(canvas, e);
          // var touch = e.touches[0];
          // var mouseEvent = new MouseEvent("mousedown", {
          //   clientX: touch.clientX,
          //   clientY: touch.clientY
          // });
          // canvas.dispatchEvent(mouseEvent);

        }, false);

        canvas.addEventListener("touchend", function (e) {
          findxy('end', e)
          // var mouseEvent = new MouseEvent("mouseup", {});
          // canvas.dispatchEvent(mouseEvent);
        }, false);

        canvas.addEventListener("touchmove", function (e) {
          findxy('touchMove', e)
          // var touch = e.touches[0];
          // var mouseEvent = new MouseEvent("mousemove", {
          //   clientX: touch.clientX,
          //   clientY: touch.clientY
          // });
          // canvas.dispatchEvent(mouseEvent);
        }, false);
      }

      // Get the position of a touch relative to the canvas
      function getTouchPos(canvasDom, touchEvent) {
        var rect = canvasDom.getBoundingClientRect();
        return {
          x: touchEvent.touches[0].clientX - rect.left,
          y: touchEvent.touches[0].clientY - rect.top
        };
      }

      // draws a line from (x1, y1) to (x2, y2) with nice rounded caps
      function draw(ctx, color, lineWidth, x1, y1, x2, y2) {
        ctx.beginPath();
        ctx.strokeStyle = color;
        ctx.lineWidth = lineWidth;
        ctx.lineCap = 'round';
        ctx.lineJoin = 'round';
        ctx.moveTo(x1, y1);
        ctx.lineTo(x2, y2);
        ctx.stroke();
        ctx.closePath();
      }

      function erase() {
        ctx.clearRect(0, 0, canvas.width, canvas.height);
        document.getElementById('nnInput').innerHTML = '';
        document.getElementById('nnOut').innerHTML = '';
        paths = [];
      }

      function findxy(res, e) {
        //for mouse events
        if (res == 'down') {
          if (clearBeforeDraw == true) {
            ctx.clearRect(0, 0, canvas.width, canvas.height);
            document.getElementById('nnInput').innerHTML = '';
            document.getElementById('nnOut').innerHTML = '';
            paths = [];
            clearBeforeDraw = false;
          }

          let bound = canvas.getBoundingClientRect();
          currX = e.clientX - bound.left - canvas.clientLeft;
          currY = e.clientY - bound.top - canvas.clientTop;

          //draw a circle
          ctx.beginPath();
          ctx.lineWidth = 1;
          ctx.arc(currX, currY, lineWidth / 2, 0, 2 * Math.PI);
          ctx.stroke();
          ctx.closePath();
          ctx.fill();

          paths.push([
            [currX],
            [currY]
          ]);
          paintFlag = true;
        }

        if (res == 'up' || res == "out") {
          paintFlag = false;
          //console.log(paths);
        }

        if (res == 'move') {
          if (paintFlag) {
            // draw a line to previous point
            prevX = currX;
            prevY = currY;

            let bound = canvas.getBoundingClientRect();

            currX = e.clientX - bound.left - canvas.clientLeft;
            currY = e.clientY - bound.top - canvas.clientTop;

            currPath = paths[paths.length - 1];
            currPath[0].push(currX);
            currPath[1].push(currY);
            paths[paths.length - 1] = currPath;
            draw(ctx, color, lineWidth, prevX, prevY, currX, currY);
          }
        }

        let adjust_val = 1;
        //for touch events
        if (res == 'start') {
          if (clearBeforeDraw == true) {
            ctx.clearRect(0, 0, canvas.width, canvas.height);
            document.getElementById('nnInput').innerHTML = '';
            document.getElementById('nnOut').innerHTML = '';
            paths = [];
            clearBeforeDraw = false;
          }

          var touch = e.touches[0];
          let bound = canvas.getBoundingClientRect();
          currX = touch.pageX - bound.left - canvas.clientLeft + adjust_val*touch.target.offsetLeft;
          currY = touch.clientY - bound.top - canvas.clientTop + adjust_val*touch.target.offsetTop;

          //draw a circle
          ctx.beginPath();
          ctx.lineWidth = 1;
          ctx.arc(currX, currY, lineWidth / 2, 0, 2 * Math.PI);
          ctx.stroke();
          ctx.closePath();
          ctx.fill();

          paths.push([
            [currX],
            [currY]
          ]);
          paintFlag = true;
          e.preventDefault();
        }

        if (res == 'end') {
          paintFlag = false;
          //console.log(paths);
          e.preventDefault();
        }

        if (res == 'touchMove') {
          if (paintFlag) {
            // draw a line to previous point
            prevX = currX;
            prevY = currY;

            var touch = e.touches[0];
            let bound = canvas.getBoundingClientRect();
            currX = touch.clientX - bound.left - canvas.clientLeft + adjust_val*touch.target.offsetLeft;
            currY = touch.clientY - bound.top - canvas.clientTop + adjust_val*touch.target.offsetTop;

            currPath = paths[paths.length - 1];
            currPath[0].push(currX);
            currPath[1].push(currY);
            paths[paths.length - 1] = currPath;
            draw(ctx, color, lineWidth, prevX, prevY, currX, currY);
            e.preventDefault();
          }
        }

      }
      init();
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