Mercurial > ift6266
annotate writeup/techreport.tex @ 568:ae6ba0309bf9
nouveaux graphes
| author | Yoshua Bengio <bengioy@iro.umontreal.ca> |
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| date | Thu, 03 Jun 2010 13:19:16 -0400 |
| parents | 8aad1c6ec39a |
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| rev | line source |
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1 \documentclass[12pt,letterpaper]{article} |
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2 \usepackage[utf8]{inputenc} |
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3 \usepackage{graphicx} |
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4 \usepackage{times} |
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5 \usepackage{mlapa} |
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6 \usepackage{subfigure} |
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7 |
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8 \begin{document} |
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9 \title{Generating and Exploiting Perturbed and Multi-Task Handwritten Training Data for Deep Architectures} |
| 381 | 10 \author{The IFT6266 Gang} |
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11 \date{April 2010, Technical Report, Dept. IRO, U. Montreal} |
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12 |
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13 \maketitle |
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14 |
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15 \begin{abstract} |
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16 Recent theoretical and empirical work in statistical machine learning has |
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17 demonstrated the importance of learning algorithms for deep |
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18 architectures, i.e., function classes obtained by composing multiple |
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19 non-linear transformations. In the area of handwriting recognition, |
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20 deep learning algorithms |
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21 had been evaluated on rather small datasets with a few tens of thousands |
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22 of examples. Here we propose a powerful generator of variations |
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23 of examples for character images based on a pipeline of stochastic |
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24 transformations that include not only the usual affine transformations |
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25 but also the addition of slant, local elastic deformations, changes |
| 541 | 26 in thickness, background images, grey level, contrast, occlusion, and |
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27 various types of pixel and spatially correlated noise. |
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28 We evaluate a deep learning algorithm (Stacked Denoising Autoencoders) |
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29 on the task of learning to classify digits and letters transformed |
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30 with this pipeline, using the hundreds of millions of generated examples |
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31 and testing on the full 62-class NIST test set. |
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32 We find that the SDA outperforms its |
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33 shallow counterpart, an ordinary Multi-Layer Perceptron, |
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34 and that it is better able to take advantage of the additional |
| 438 | 35 generated data, as well as better able to take advantage of |
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36 the multi-task setting, i.e., |
| 438 | 37 training from more classes than those of interest in the end. |
| 38 In fact, we find that the SDA reaches human performance as | |
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39 estimated by the Amazon Mechanical Turk on the 62-class NIST test characters. |
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40 \end{abstract} |
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41 |
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42 \section{Introduction} |
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43 |
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44 Deep Learning has emerged as a promising new area of research in |
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45 statistical machine learning (see~\emcite{Bengio-2009} for a review). |
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46 Learning algorithms for deep architectures are centered on the learning |
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47 of useful representations of data, which are better suited to the task at hand. |
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48 This is in great part inspired by observations of the mammalian visual cortex, |
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49 which consists of a chain of processing elements, each of which is associated with a |
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50 different representation. In fact, |
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51 it was found recently that the features learnt in deep architectures resemble |
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52 those observed in the first two of these stages (in areas V1 and V2 |
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53 of visual cortex)~\cite{HonglakL2008}. |
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54 Processing images typically involves transforming the raw pixel data into |
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55 new {\bf representations} that can be used for analysis or classification. |
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56 For example, a principal component analysis representation linearly projects |
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57 the input image into a lower-dimensional feature space. |
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58 Why learn a representation? Current practice in the computer vision |
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59 literature converts the raw pixels into a hand-crafted representation |
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60 (e.g.\ SIFT features~\cite{Lowe04}), but deep learning algorithms |
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61 tend to discover similar features in their first few |
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62 levels~\cite{HonglakL2008,ranzato-08,Koray-08,VincentPLarochelleH2008-very-small}. |
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63 Learning increases the |
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64 ease and practicality of developing representations that are at once |
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65 tailored to specific tasks, yet are able to borrow statistical strength |
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66 from other related tasks (e.g., modeling different kinds of objects). Finally, learning the |
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67 feature representation can lead to higher-level (more abstract, more |
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68 general) features that are more robust to unanticipated sources of |
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69 variance extant in real data. |
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70 |
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71 Whereas a deep architecture can in principle be more powerful than a |
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72 shallow one in terms of representation, depth appears to render the |
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73 training problem more difficult in terms of optimization and local minima. |
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74 It is also only recently that successful algorithms were proposed to |
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75 overcome some of these difficulties. All are based on unsupervised |
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76 learning, often in an greedy layer-wise ``unsupervised pre-training'' |
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77 stage~\cite{Bengio-2009}. One of these layer initialization techniques, |
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78 applied here, is the Denoising |
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79 Auto-Encoder~(DEA)~\cite{VincentPLarochelleH2008-very-small}, which |
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80 performed similarly or better than previously proposed Restricted Boltzmann |
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81 Machines in terms of unsupervised extraction of a hierarchy of features |
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82 useful for classification. The principle is that each layer starting from |
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83 the bottom is trained to encode their input (the output of the previous |
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84 layer) and try to reconstruct it from a corrupted version of it. After this |
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85 unsupervised initialization, the stack of denoising auto-encoders can be |
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86 converted into a deep supervised feedforward neural network and trained by |
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87 stochastic gradient descent. |
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88 |
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89 |
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90 \section{Perturbation and Transformation of Character Images} |
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91 |
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92 This section describes the different transformations we used to generate data, in their order. |
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93 The code for these transformations (mostly python) is available at |
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94 {\tt http://anonymous.url.net}. All the modules in the pipeline share |
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95 a global control parameter ($0 \le complexity \le 1$) that allows one to modulate the |
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96 amount of deformation or noise introduced. |
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97 |
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98 We can differentiate two important parts in the pipeline. The first one, |
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99 from slant to pinch, performs transformations of the character. The second |
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100 part, from blur to contrast, adds noise to the image. |
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101 |
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102 \subsection{Slant} |
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103 |
| 541 | 104 We mimic slant by shifting each row of the image |
| 105 proportionally to its height: $shift = round(slant \times height)$. | |
| 106 The $slant$ coefficient can be negative or positive with equal probability | |
| 107 and its value is randomly sampled according to the complexity level: | |
| 108 $slant \sim U[0,complexity]$, so the | |
| 109 maximum displacement for the lowest or highest pixel line is of | |
| 110 $round(complexity \times 32)$. | |
| 111 | |
| 112 --- | |
| 113 | |
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114 In order to mimic a slant effect, we simply shift each row of the image |
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115 proportionnaly to its height: $shift = round(slant \times height)$. We |
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116 round the shift in order to have a discret displacement. We do not use a |
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117 filter to smooth the result in order to save computing time and also |
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118 because latter transformations have similar effects. |
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119 |
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120 The $slant$ coefficient can be negative or positive with equal probability |
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121 and its value is randomly sampled according to the complexity level. In |
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122 our case we take uniformly a number in the range $[0,complexity]$, so the |
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123 maximum displacement for the lowest or highest pixel line is of |
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124 $round(complexity \times 32)$. |
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125 |
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126 |
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127 \subsection{Thickness} |
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128 |
| 541 | 129 Morphological operators of dilation and erosion~\citep{Haralick87,Serra82} |
| 130 are applied. The neighborhood of each pixel is multiplied | |
| 131 element-wise with a {\em structuring element} matrix. | |
| 132 The pixel value is replaced by the maximum or the minimum of the resulting | |
| 133 matrix, respectively for dilation or erosion. Ten different structural elements with | |
| 134 increasing dimensions (largest is $5\times5$) were used. For each image, | |
| 135 randomly sample the operator type (dilation or erosion) with equal probability and one structural | |
| 136 element from a subset of the $n$ smallest structuring elements where $n$ is | |
| 137 $round(10 \times complexity)$ for dilation and $round(6 \times complexity)$ | |
| 138 for erosion. A neutral element is always present in the set, and if it is | |
| 139 chosen no transformation is applied. Erosion allows only the six | |
| 140 smallest structural elements because when the character is too thin it may | |
| 141 be completely erased. | |
| 142 | |
| 143 --- | |
| 144 | |
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145 To change the thickness of the characters we used morpholigical operators: |
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146 dilation and erosion~\cite{Haralick87,Serra82}. |
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147 |
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148 The basic idea of such transform is, for each pixel, to multiply in the |
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149 element-wise manner its neighbourhood with a matrix called the structuring |
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150 element. Then for dilation we remplace the pixel value by the maximum of |
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151 the result, or the minimum for erosion. This will dilate or erode objects |
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152 in the image and the strength of the transform only depends on the |
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153 structuring element. |
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154 |
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155 We used ten different structural elements with increasing dimensions (the |
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156 biggest is $5\times5$). for each image, we radomly sample the operator |
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157 type (dilation or erosion) with equal probability and one structural |
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158 element from a subset of the $n$ smallest structuring elements where $n$ is |
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159 $round(10 \times complexity)$ for dilation and $round(6 \times complexity)$ |
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160 for erosion. A neutral element is always present in the set, if it is |
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161 chosen the transformation is not applied. Erosion allows only the six |
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162 smallest structural elements because when the character is too thin it may |
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163 erase it completly. |
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164 |
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165 \subsection{Affine Transformations} |
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166 |
| 541 | 167 A $2 \times 3$ affine transform matrix (with |
| 168 6 parameters $(a,b,c,d,e,f)$) is sampled according to the $complexity$ level. | |
| 169 Each pixel $(x,y)$ of the output image takes the value of the pixel | |
| 170 nearest to $(ax+by+c,dx+ey+f)$ in the input image. This | |
| 171 produces scaling, translation, rotation and shearing. | |
| 172 The marginal distributions of $(a,b,c,d,e,f)$ have been tuned by hand to | |
| 173 forbid important rotations (not to confuse classes) but to give good | |
| 174 variability of the transformation: $a$ and $d$ $\sim U[1-3 \times | |
| 175 complexity,1+3 \times complexity]$, $b$ and $e$ $\sim[-3 \times complexity,3 | |
| 176 \times complexity]$ and $c$ and $f$ $\sim U[-4 \times complexity, 4 \times | |
| 177 complexity]$. | |
| 178 | |
| 179 ---- | |
| 180 | |
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181 We generate an affine transform matrix according to the complexity level, |
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182 then we apply it directly to the image. The matrix is of size $2 \times |
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183 3$, so we can represent it by six parameters $(a,b,c,d,e,f)$. Formally, |
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184 for each pixel $(x,y)$ of the output image, we give the value of the pixel |
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185 nearest to : $(ax+by+c,dx+ey+f)$, in the input image. This allows to |
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186 produce scaling, translation, rotation and shearing variances. |
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187 |
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188 The sampling of the parameters $(a,b,c,d,e,f)$ have been tuned by hand to |
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189 forbid important rotations (not to confuse classes) but to give good |
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190 variability of the transformation. For each image we sample uniformly the |
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191 parameters in the following ranges: $a$ and $d$ in $[1-3 \times |
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192 complexity,1+3 \times complexity]$, $b$ and $e$ in $[-3 \times complexity,3 |
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193 \times complexity]$ and $c$ and $f$ in $[-4 \times complexity, 4 \times |
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194 complexity]$. |
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195 |
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196 |
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197 \subsection{Local Elastic Deformations} |
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198 |
| 541 | 199 This filter induces a ``wiggly'' effect in the image, following~\citet{SimardSP03-short}, |
| 200 which provides more details. | |
| 201 Two ``displacements'' fields are generated and applied, for horizontal | |
| 202 and vertical displacements of pixels. | |
| 203 To generate a pixel in either field, first a value between -1 and 1 is | |
| 204 chosen from a uniform distribution. Then all the pixels, in both fields, are | |
| 205 multiplied by a constant $\alpha$ which controls the intensity of the | |
| 206 displacements (larger $\alpha$ translates into larger wiggles). | |
| 207 Each field is convolved with a Gaussian 2D kernel of | |
| 208 standard deviation $\sigma$. Visually, this results in a blur. | |
| 209 $\alpha = \sqrt[3]{complexity} \times 10.0$ and $\sigma = 10 - 7 \times | |
| 210 \sqrt[3]{complexity}$. | |
| 211 | |
| 212 ---- | |
| 213 | |
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214 This filter induces a "wiggly" effect in the image. The description here |
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215 will be brief, as the algorithm follows precisely what is described in |
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216 \cite{SimardSP03}. |
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217 |
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218 The general idea is to generate two "displacements" fields, for horizontal |
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219 and vertical displacements of pixels. Each of these fields has the same |
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220 size as the original image. |
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221 |
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222 When generating the transformed image, we'll loop over the x and y |
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223 positions in the fields and select, as a value, the value of the pixel in |
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224 the original image at the (relative) position given by the displacement |
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225 fields for this x and y. If the position we'd retrieve is outside the |
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226 borders of the image, we use a 0 value instead. |
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227 |
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228 To generate a pixel in either field, first a value between -1 and 1 is |
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229 chosen from a uniform distribution. Then all the pixels, in both fields, is |
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230 multiplied by a constant $\alpha$ which controls the intensity of the |
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231 displacements (bigger $\alpha$ translates into larger wiggles). |
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232 |
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233 As a final step, each field is convoluted with a Gaussian 2D kernel of |
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234 standard deviation $\sigma$. Visually, this results in a "blur" |
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235 filter. This has the effect of making values next to each other in the |
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236 displacement fields similar. In effect, this makes the wiggles more |
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237 coherent, less noisy. |
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238 |
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239 As displacement fields were long to compute, 50 pairs of fields were |
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240 generated per complexity in increments of 0.1 (50 pairs for 0.1, 50 pairs |
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241 for 0.2, etc.), and afterwards, given a complexity, we selected randomly |
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242 among the 50 corresponding pairs. |
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243 |
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244 $\sigma$ and $\alpha$ were linked to complexity through the formulas |
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245 $\alpha = \sqrt[3]{complexity} \times 10.0$ and $\sigma = 10 - 7 \times |
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246 \sqrt[3]{complexity}$. |
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247 |
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248 |
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249 \subsection{Pinch} |
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250 |
| 541 | 251 This is a GIMP filter called ``Whirl and |
| 252 pinch'', but whirl was set to 0. A pinch is ``similar to projecting the image onto an elastic | |
| 253 surface and pressing or pulling on the center of the surface'' (GIMP documentation manual). | |
| 254 For a square input image, this is akin to drawing a circle of | |
| 255 radius $r$ around a center point $C$. Any point (pixel) $P$ belonging to | |
| 256 that disk (region inside circle) will have its value recalculated by taking | |
| 257 the value of another ``source'' pixel in the original image. The position of | |
| 258 that source pixel is found on the line that goes through $C$ and $P$, but | |
| 259 at some other distance $d_2$. Define $d_1$ to be the distance between $P$ | |
| 260 and $C$. $d_2$ is given by $d_2 = sin(\frac{\pi{}d_1}{2r})^{-pinch} \times | |
| 261 d_1$, where $pinch$ is a parameter to the filter. | |
| 262 The actual value is given by bilinear interpolation considering the pixels | |
| 263 around the (non-integer) source position thus found. | |
| 264 Here $pinch \sim U[-complexity, 0.7 \times complexity]$. | |
| 265 | |
| 266 --- | |
| 267 | |
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268 This is another GIMP filter we used. The filter is in fact named "Whirl and |
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269 pinch", but we don't use the "whirl" part (whirl is set to 0). As described |
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270 in GIMP, a pinch is "similar to projecting the image onto an elastic |
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271 surface and pressing or pulling on the center of the surface". |
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272 |
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273 Mathematically, for a square input image, think of drawing a circle of |
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274 radius $r$ around a center point $C$. Any point (pixel) $P$ belonging to |
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275 that disk (region inside circle) will have its value recalculated by taking |
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276 the value of another "source" pixel in the original image. The position of |
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277 that source pixel is found on the line thats goes through $C$ and $P$, but |
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278 at some other distance $d_2$. Define $d_1$ to be the distance between $P$ |
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279 and $C$. $d_2$ is given by $d_2 = sin(\frac{\pi{}d_1}{2r})^{-pinch} \times |
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280 d_1$, where $pinch$ is a parameter to the filter. |
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281 |
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282 If the region considered is not square then, before computing $d_2$, the |
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283 smallest dimension (x or y) is stretched such that we may consider the |
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284 region as if it was square. Then, after $d_2$ has been computed and |
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285 corresponding components $d_2\_x$ and $d_2\_y$ have been found, the |
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286 component corresponding to the stretched dimension is compressed back by an |
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287 inverse ratio. |
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288 |
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289 The actual value is given by bilinear interpolation considering the pixels |
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290 around the (non-integer) source position thus found. |
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291 |
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292 The value for $pinch$ in our case was given by sampling from an uniform |
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293 distribution over the range $[-complexity, 0.7 \times complexity]$. |
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294 |
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295 \subsection{Motion Blur} |
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296 |
| 541 | 297 This is a ``linear motion blur'' in GIMP |
| 298 terminology, with two parameters, $length$ and $angle$. The value of | |
| 299 a pixel in the final image is approximately the mean value of the $length$ first pixels | |
| 300 found by moving in the $angle$ direction. | |
| 301 Here $angle \sim U[0,360]$ degrees, and $length \sim {\rm Normal}(0,(3 \times complexity)^2)$. | |
| 302 | |
| 303 ---- | |
| 304 | |
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305 This is a GIMP filter we applied, a "linear motion blur" in GIMP |
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306 terminology. The description will be brief as it is a well-known filter. |
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307 |
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308 This algorithm has two input parameters, $length$ and $angle$. The value of |
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309 a pixel in the final image is the mean value of the $length$ first pixels |
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310 found by moving in the $angle$ direction. An approximation of this idea is |
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311 used, as we won't fall onto precise pixels by following that |
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312 direction. This is done using the Bresenham line algorithm. |
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313 |
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314 The angle, in our case, is chosen from a uniform distribution over |
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315 $[0,360]$ degrees. The length, though, depends on the complexity; it's |
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316 sampled from a Gaussian distribution of mean 0 and standard deviation |
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317 $\sigma = 3 \times complexity$. |
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318 |
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319 \subsection{Occlusion} |
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320 |
| 541 | 321 Selects a random rectangle from an {\em occluder} character |
| 322 images and places it over the original {\em occluded} character | |
| 323 image. Pixels are combined by taking the max(occluder,occluded), | |
| 324 closer to black. The rectangle corners | |
| 325 are sampled so that larger complexity gives larger rectangles. | |
| 326 The destination position in the occluded image are also sampled | |
| 327 according to a normal distribution (see more details in~\citet{ift6266-tr-anonymous}). | |
| 328 This filter has a probability of 60\% of not being applied. | |
| 329 | |
| 330 --- | |
| 331 | |
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332 This filter selects random parts of other (hereafter "occlusive") letter |
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333 images and places them over the original letter (hereafter "occluded") |
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334 image. To be more precise, having selected a subregion of the occlusive |
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335 image and a desination position in the occluded image, to determine the |
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336 final value for a given overlapping pixel, it selects whichever pixel is |
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337 the lightest. As a reminder, the background value is 0, black, so the value |
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338 nearest to 1 is selected. |
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339 |
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340 To select a subpart of the occlusive image, four numbers are generated. For |
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341 compability with the code, we'll call them "haut", "bas", "gauche" and |
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342 "droite" (respectively meaning top, bottom, left and right). Each of these |
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343 numbers is selected according to a Gaussian distribution of mean $8 \times |
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344 complexity$ and standard deviation $2$. This means the largest the |
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345 complexity is, the biggest the occlusion will be. The absolute value is |
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346 taken, as the numbers must be positive, and the maximum value is capped at |
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347 15. |
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348 |
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349 These four sizes collectively define a window centered on the middle pixel |
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350 of the occlusive image. This is the part that will be extracted as the |
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351 occlusion. |
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352 |
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353 The next step is to select a destination position in the occluded |
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354 image. Vertical and horizontal displacements $y\_arrivee$ and $x\_arrivee$ |
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355 are selected according to Gaussian distributions of mean 0 and of standard |
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356 deviations of, respectively, 3 and 2. Then an horizontal placement mode, |
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357 $place$, is selected to be of three values meaning |
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358 left, middle or right. |
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359 |
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360 If $place$ is "middle", the occlusion will be horizontally centered |
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361 around the horizontal middle of the occluded image, then shifted according |
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362 to $x\_arrivee$. If $place$ is "left", it will be placed on the left of |
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363 the occluded image, then displaced right according to $x\_arrivee$. The |
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364 contrary happens if $place$ is $right$. |
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365 |
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366 In both the horizontal and vertical positionning, the maximum position in |
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367 either direction is such that the selected occlusion won't go beyond the |
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368 borders of the occluded image. |
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369 |
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370 This filter has a probability of not being applied, at all, of 60\%. |
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371 |
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372 |
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373 \subsection{Pixel Permutation} |
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374 |
| 541 | 375 This filter permutes neighbouring pixels. It selects first |
| 376 $\frac{complexity}{3}$ pixels randomly in the image. Each of them are then | |
| 377 sequentially exchanged with one other pixel in its $V4$ neighbourhood. The number | |
| 378 of exchanges to the left, right, top, bottom is equal or does not differ | |
| 379 from more than 1 if the number of selected pixels is not a multiple of 4. | |
| 380 % TODO: The previous sentence is hard to parse | |
| 381 This filter has a probability of 80\% of not being applied. | |
| 382 | |
| 383 --- | |
| 384 | |
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385 This filter permuts neighbouring pixels. It selects first |
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386 $\frac{complexity}{3}$ pixels randomly in the image. Each of them are then |
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387 sequentially exchanged to one other pixel in its $V4$ neighbourhood. Number |
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388 of exchanges to the left, right, top, bottom are equal or does not differ |
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389 from more than 1 if the number of selected pixels is not a multiple of 4. |
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390 |
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391 It has has a probability of not being applied, at all, of 80\%. |
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392 |
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393 |
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394 \subsection{Gaussian Noise} |
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395 |
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396 This filter simply adds, to each pixel of the image independently, a |
| 541 | 397 noise $\sim Normal(0(\frac{complexity}{10})^2)$. |
| 398 It has a probability of 70\% of not being applied. | |
| 399 | |
| 400 --- | |
| 401 | |
| 402 This filter simply adds, to each pixel of the image independently, a | |
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403 Gaussian noise of mean $0$ and standard deviation $\frac{complexity}{10}$. |
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404 |
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405 It has has a probability of not being applied, at all, of 70\%. |
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406 |
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407 |
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408 \subsection{Background Images} |
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409 |
| 541 | 410 Following~\citet{Larochelle-jmlr-2009}, this transformation adds a random |
| 411 background behind the letter. The background is chosen by first selecting, | |
| 412 at random, an image from a set of images. Then a 32$\times$32 sub-region | |
| 413 of that image is chosen as the background image (by sampling position | |
| 414 uniformly while making sure not to cross image borders). | |
| 415 To combine the original letter image and the background image, contrast | |
| 416 adjustments are made. We first get the maximal values (i.e. maximal | |
| 417 intensity) for both the original image and the background image, $maximage$ | |
| 418 and $maxbg$. We also have a parameter $contrast \sim U[complexity, 1]$. | |
| 419 Each background pixel value is multiplied by $\frac{max(maximage - | |
| 420 contrast, 0)}{maxbg}$ (higher contrast yield darker | |
| 421 background). The output image pixels are max(background,original). | |
| 422 | |
| 423 --- | |
| 424 | |
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425 Following~\cite{Larochelle-jmlr-2009}, this transformation adds a random |
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426 background behind the letter. The background is chosen by first selecting, |
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427 at random, an image from a set of images. Then we choose a 32x32 subregion |
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428 of that image as the background image (by sampling x and y positions |
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429 uniformly while making sure not to cross image borders). |
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430 |
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431 To combine the original letter image and the background image, contrast |
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432 adjustments are made. We first get the maximal values (i.e. maximal |
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433 intensity) for both the original image and the background image, $maximage$ |
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434 and $maxbg$. We also have a parameter, $contrast$, given by sampling from a |
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435 uniform distribution over $[complexity, 1]$. |
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436 |
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437 Once we have all these numbers, we first adjust the values for the |
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438 background image. Each pixel value is multiplied by $\frac{max(maximage - |
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439 contrast, 0)}{maxbg}$. Therefore the higher the contrast, the darkest the |
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440 background will be. |
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441 |
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442 The final image is found by taking the brightest (i.e. value nearest to 1) |
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443 pixel from either the background image or the corresponding pixel in the |
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444 original image. |
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445 |
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446 \subsection{Salt and Pepper Noise} |
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447 |
| 541 | 448 This filter adds noise $\sim U[0,1]$ to random subsets of pixels. |
| 449 The number of selected pixels is $0.2 \times complexity$. | |
| 450 This filter has a probability of not being applied at all of 75\%. | |
| 451 | |
| 452 --- | |
| 453 | |
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454 This filter adds noise to the image by randomly selecting a certain number |
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455 of them and, for those selected pixels, assign a random value according to |
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456 a uniform distribution over the $[0,1]$ ranges. This last distribution does |
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457 not change according to complexity. Instead, the number of selected pixels |
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458 does: the proportion of changed pixels corresponds to $complexity / 5$, |
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459 which means, as a maximum, 20\% of the pixels will be randomized. On the |
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460 lowest extreme, no pixel is changed. |
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461 |
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462 This filter also has a probability of not being applied, at all, of 75\%. |
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463 |
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464 \subsection{Spatially Gaussian Noise} |
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465 |
| 541 | 466 Different regions of the image are spatially smoothed. |
| 467 The image is convolved with a symmetric Gaussian kernel of | |
| 468 size and variance chosen uniformly in the ranges $[12,12 + 20 \times | |
| 469 complexity]$ and $[2,2 + 6 \times complexity]$. The result is normalized | |
| 470 between $0$ and $1$. We also create a symmetric averaging window, of the | |
| 471 kernel size, with maximum value at the center. For each image we sample | |
| 472 uniformly from $3$ to $3 + 10 \times complexity$ pixels that will be | |
| 473 averaging centers between the original image and the filtered one. We | |
| 474 initialize to zero a mask matrix of the image size. For each selected pixel | |
| 475 we add to the mask the averaging window centered to it. The final image is | |
| 476 computed from the following element-wise operation: $\frac{image + filtered | |
| 477 image \times mask}{mask+1}$. | |
| 478 This filter has a probability of not being applied at all of 75\%. | |
| 479 | |
| 480 ---- | |
| 481 | |
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482 The aim of this transformation is to filter, with a gaussian kernel, |
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483 different regions of the image. In order to save computing time we decided |
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484 to convolve the whole image only once with a symmetric gaussian kernel of |
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485 size and variance choosen uniformly in the ranges: $[12,12 + 20 \times |
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486 complexity]$ and $[2,2 + 6 \times complexity]$. The result is normalized |
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487 between $0$ and $1$. We also create a symmetric averaging window, of the |
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488 kernel size, with maximum value at the center. For each image we sample |
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489 uniformly from $3$ to $3 + 10 \times complexity$ pixels that will be |
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490 averaging centers between the original image and the filtered one. We |
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491 initialize to zero a mask matrix of the image size. For each selected pixel |
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492 we add to the mask the averaging window centered to it. The final image is |
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493 computed from the following element-wise operation: $\frac{image + filtered |
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494 image \times mask}{mask+1}$. |
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495 |
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496 This filter has a probability of not being applied, at all, of 75\%. |
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497 |
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498 \subsection{Scratches} |
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499 |
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500 The scratches module places line-like white patches on the image. The |
| 541 | 501 lines are heavily transformed images of the digit ``1'' (one), chosen |
| 502 at random among five thousands such 1 images. The 1 image is | |
| 503 randomly cropped and rotated by an angle $\sim Normal(0,(100 \times | |
| 504 complexity)^2$, using bi-cubic interpolation, | |
| 505 Two passes of a grey-scale morphological erosion filter | |
| 506 are applied, reducing the width of the line | |
| 507 by an amount controlled by $complexity$. | |
| 508 This filter is only applied only 15\% of the time. When it is applied, 50\% | |
| 509 of the time, only one patch image is generated and applied. In 30\% of | |
| 510 cases, two patches are generated, and otherwise three patches are | |
| 511 generated. The patch is applied by taking the maximal value on any given | |
| 512 patch or the original image, for each of the 32x32 pixel locations. | |
| 513 | |
| 514 --- | |
| 515 | |
| 516 The scratches module places line-like white patches on the image. The | |
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517 lines are in fact heavily transformed images of the digit "1" (one), chosen |
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518 at random among five thousands such start images of this digit. |
| 428 | 519 |
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520 Once the image is selected, the transformation begins by finding the first |
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521 $top$, $bottom$, $right$ and $left$ non-zero pixels in the image. It is |
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522 then cropped to the region thus delimited, then this cropped version is |
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523 expanded to $32\times32$ again. It is then rotated by a random angle having a |
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524 Gaussian distribution of mean 90 and standard deviation $100 \times |
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525 complexity$ (in degrees). The rotation is done with bicubic interpolation. |
| 428 | 526 |
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527 The rotated image is then resized to $50\times50$, with anti-aliasing. In |
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528 that image, we crop the image again by selecting a region delimited |
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529 horizontally to $left$ to $left+32$ and vertically by $top$ to $top+32$. |
| 428 | 530 |
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531 Once this is done, two passes of a greyscale morphological erosion filter |
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532 are applied. Put briefly, this erosion filter reduces the width of the line |
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533 by a certain $smoothing$ amount. For small complexities (< 0.5), |
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534 $smoothing$ is 6, so the line is very small. For complexities ranging from |
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535 0.25 to 0.5, $smoothing$ is 5. It is 4 for complexities 0.5 to 0.75, and 3 |
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536 for higher complexities. |
| 428 | 537 |
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538 To compensate for border effects, the image is then cropped to 28x28 by |
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539 removing two pixels everywhere on the borders, then expanded to 32x32 |
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540 again. The pixel values are then linearly expanded such that the minimum |
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541 value is 0 and the maximal one is 1. Then, 50\% of the time, the image is |
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542 vertically flipped. |
| 428 | 543 |
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544 This filter is only applied only 15\% of the time. When it is applied, 50\% |
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545 of the time, only one patch image is generated and applied. In 30\% of |
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546 cases, two patches are generated, and otherwise three patches are |
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547 generated. The patch is applied by taking the maximal value on any given |
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548 patch or the original image, for each of the 32x32 pixel locations. |
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549 |
| 541 | 550 \subsection{Grey Level and Contrast Changes} |
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551 |
| 541 | 552 This filter changes the contrast and may invert the image polarity (white |
| 553 on black to black on white). The contrast $C$ is defined here as the | |
| 554 difference between the maximum and the minimum pixel value of the image. | |
| 555 Contrast $\sim U[1-0.85 \times complexity,1]$ (so contrast $\geq 0.15$). | |
| 556 The image is normalized into $[\frac{1-C}{2},1-\frac{1-C}{2}]$. The | |
| 557 polarity is inverted with $0.5$ probability. | |
| 558 | |
| 559 --- | |
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560 This filter changes the constrast and may invert the image polarity (white |
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561 on black to black on white). The contrast $C$ is defined here as the |
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562 difference between the maximum and the minimum pixel value of the image. A |
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563 contrast value is sampled uniformly between $1$ and $1-0.85 \times |
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564 complexity$ (this insure a minimum constrast of $0.15$). We then simply |
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565 normalize the image to the range $[\frac{1-C}{2},1-\frac{1-C}{2}]$. The |
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566 polarity is inverted with $0.5$ probability. |
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567 |
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568 |
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569 \begin{figure}[h] |
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570 \resizebox{.99\textwidth}{!}{\includegraphics{images/example_t.png}}\\ |
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571 \caption{Illustration of the pipeline of stochastic |
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572 transformations applied to the image of a lower-case t |
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573 (the upper left image). Each image in the pipeline (going from |
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574 left to right, first top line, then bottom line) shows the result |
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575 of applying one of the modules in the pipeline. The last image |
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576 (bottom right) is used as training example.} |
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577 \label{fig:pipeline} |
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578 \end{figure} |
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579 |
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580 |
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581 \begin{figure}[h] |
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582 \resizebox{.99\textwidth}{!}{\includegraphics{images/transfo.png}}\\ |
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583 \caption{Illustration of each transformation applied to the same image |
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584 of the upper-case h (upper-left image). first row (from left to rigth) : original image, slant, |
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585 thickness, affine transformation, local elastic deformation; second row (from left to rigth) : |
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586 pinch, motion blur, occlusion, pixel permutation, gaussian noise; third row (from left to rigth) : |
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587 background image, salt and pepper noise, spatially gaussian noise, scratches, |
| 541 | 588 grey level and contrast changes.} |
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589 \label{fig:transfo} |
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590 \end{figure} |
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591 |
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592 |
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593 \section{Experimental Setup} |
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594 |
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595 \subsection{Training Datasets} |
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596 |
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597 \subsubsection{Data Sources} |
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598 |
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599 \begin{itemize} |
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600 \item {\bf NIST} |
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601 The NIST Special Database 19 (NIST19) is a very widely used dataset for training and testing OCR systems. |
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602 The dataset is composed with over 800 000 digits and characters (upper and lower cases), with hand checked classifications, |
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603 extracted from handwritten sample forms of 3600 writers. The characters are labelled by one of the 62 classes |
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604 corresponding to "0"-"9","A"-"Z" and "a"-"z". The dataset contains 8 series of different complexity. |
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605 The fourth series, $hsf_4$, experimentally recognized to be the most difficult one for classification task is recommended |
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606 by NIST as testing set and is used in our work for that purpose. |
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607 The performances reported by previous work on that dataset mostly use only the digits. |
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608 Here we use the whole classes both in the training and testing phase. |
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609 |
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610 |
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611 \item {\bf Fonts} |
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612 In order to have a good variety of sources we downloaded an important number of free fonts from: {\tt http://anonymous.url.net} |
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613 %real adress {\tt http://cg.scs.carleton.ca/~luc/freefonts.html} |
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614 in addition to Windows 7's, this adds up to a total of $9817$ different fonts that we can choose uniformly. |
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615 The ttf file is either used as input of the Captcha generator (see next item) or, by producing a corresponding image, |
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616 directly as input to our models. |
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617 %Guillaume are there other details I forgot on the font selection? |
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618 |
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619 \item {\bf Captchas} |
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620 The Captcha data source is an adaptation of the \emph{pycaptcha} library (a python based captcha generator library) for |
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621 generating characters of the same format as the NIST dataset. The core of this data source is composed with a random character |
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622 generator and various kinds of tranformations similar to those described in the previous sections. |
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623 In order to increase the variability of the data generated, different fonts are used for generating the characters. |
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624 Transformations (slant, distorsions, rotation, translation) are applied to each randomly generated character with a complexity |
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625 depending on the value of the complexity parameter provided by the user of the data source. Two levels of complexity are |
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626 allowed and can be controlled via an easy to use facade class. |
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627 \item {\bf OCR data} |
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628 \end{itemize} |
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629 |
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630 \subsubsection{Data Sets} |
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631 \begin{itemize} |
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632 \item {\bf P07} |
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633 The dataset P07 is sampled with our transformation pipeline with a complexity parameter of $0.7$. |
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634 For each new exemple to generate, we choose one source with the following probability: $0.1$ for the fonts, |
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635 $0.25$ for the captchas, $0.25$ for OCR data and $0.4$ for NIST. We apply all the transformations in their order |
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636 and for each of them we sample uniformly a complexity in the range $[0,0.7]$. |
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637 \item {\bf NISTP} {\em ne pas utiliser PNIST mais NISTP, pour rester politically correct...} |
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638 NISTP is equivalent to P07 (complexity parameter of $0.7$ with the same sources proportion) except that we only apply transformations from slant to pinch. Therefore, the character is transformed |
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639 but no additionnal noise is added to the image, this gives images closer to the NIST dataset. |
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640 \end{itemize} |
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641 |
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642 We noticed that the distribution of the training sets and the test sets differ. |
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643 Since our validation sets are sampled from the training set, they have approximately the same distribution, but the test set has a completely different distribution as illustrated in figure \ref {setsdata}. |
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644 |
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645 \begin{figure} |
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646 \subfigure[NIST training]{\includegraphics[width=0.5\textwidth]{images/nisttrainstats}} |
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647 \subfigure[NIST validation]{\includegraphics[width=0.5\textwidth]{images/nistvalidstats}} |
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648 \subfigure[NIST test]{\includegraphics[width=0.5\textwidth]{images/nistteststats}} |
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649 \subfigure[NISTP validation]{\includegraphics[width=0.5\textwidth]{images/nistpvalidstats}} |
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650 \caption{Proportion of each class in some of the data sets} |
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651 \label{setsdata} |
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652 \end{figure} |
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653 |
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654 \subsection{Models and their Hyperparameters} |
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655 |
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656 \subsubsection{Multi-Layer Perceptrons (MLP)} |
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657 |
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658 An MLP is a family of functions that are described by stacking layers of of a function similar to |
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659 $$g(x) = \tanh(b+Wx)$$ |
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660 The input, $x$, is a $d$-dimension vector. |
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661 The output, $g(x)$, is a $m$-dimension vector. |
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662 The parameter $W$ is a $m\times d$ matrix and is called the weight matrix. |
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663 The parameter $b$ is a $m$-vector and is called the bias vector. |
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664 The non-linearity (here $\tanh$) is applied element-wise to the output vector. |
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665 Usually the input is referred to a input layer and similarly for the output. |
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666 You can of course chain several such functions to obtain a more complex one. |
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667 Here is a common example |
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668 $$f(x) = c + V\tanh(b+Wx)$$ |
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669 In this case the intermediate layer corresponding to $\tanh(b+Wx)$ is called a hidden layer. |
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670 Here the output layer does not have the same non-linearity as the hidden layer. |
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671 This is a common case where some specialized non-linearity is applied to the output layer only depending on the task at hand. |
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672 |
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673 If you put 3 or more hidden layers in such a network you obtain what is called a deep MLP. |
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674 The parameters to adapt are the weight matrix and the bias vector for each layer. |
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675 |
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676 \subsubsection{Stacked Denoising Auto-Encoders (SDAE)} |
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677 \label{SdA} |
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678 |
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679 Auto-encoders are essentially a way to initialize the weights of the network to enable better generalization. |
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680 This is essentially unsupervised training where the layer is made to reconstruct its input through and encoding and decoding phase. |
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681 Denoising auto-encoders are a variant where the input is corrupted with random noise but the target is the uncorrupted input. |
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682 The principle behind these initialization methods is that the network will learn the inherent relation between portions of the data and be able to represent them thus helping with whatever task we want to perform. |
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683 |
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684 An auto-encoder unit is formed of two MLP layers with the bottom one called the encoding layer and the top one the decoding layer. |
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685 Usually the top and bottom weight matrices are the transpose of each other and are fixed this way. |
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686 The network is trained as such and, when sufficiently trained, the MLP layer is initialized with the parameters of the encoding layer. |
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687 The other parameters are discarded. |
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688 |
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689 The stacked version is an adaptation to deep MLPs where you initialize each layer with a denoising auto-encoder starting from the bottom. |
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690 During the initialization, which is usually called pre-training, the bottom layer is treated as if it were an isolated auto-encoder. |
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691 The second and following layers receive the same treatment except that they take as input the encoded version of the data that has gone through the layers before it. |
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692 For additional details see \cite{vincent:icml08}. |
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693 |
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694 \section{Experimental Results} |
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695 |
| 438 | 696 \subsection{SDA vs MLP vs Humans} |
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697 |
| 438 | 698 We compare here the best MLP (according to validation set error) that we found against |
| 699 the best SDA (again according to validation set error), along with a precise estimate | |
| 700 of human performance obtained via Amazon's Mechanical Turk (AMT) | |
| 701 service\footnote{http://mturk.com}. AMT users are paid small amounts | |
| 702 of money to perform tasks for which human intelligence is required. | |
| 703 Mechanical Turk has been used extensively in natural language | |
| 704 processing \cite{SnowEtAl2008} and vision | |
| 705 \cite{SorokinAndForsyth2008,whitehill09}. AMT users where presented | |
| 706 with 10 character images and asked to type 10 corresponding ascii | |
| 707 characters. Hence they were forced to make a hard choice among the | |
| 708 62 character classes. Three users classified each image, allowing | |
| 709 to estimate inter-human variability (shown as +/- in parenthesis below). | |
| 710 | |
| 711 \begin{table} | |
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712 \caption{Overall comparison of error rates ($\pm$ std.err.) on 62 character classes (10 digits + |
| 438 | 713 26 lower + 26 upper), except for last columns -- digits only, between deep architecture with pre-training |
| 714 (SDA=Stacked Denoising Autoencoder) and ordinary shallow architecture | |
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715 (MLP=Multi-Layer Perceptron). The models shown are all trained using perturbed data (NISTP or P07) |
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716 and using a validation set to select hyper-parameters and other training choices. |
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717 \{SDA,MLP\}0 are trained on NIST, |
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718 \{SDA,MLP\}1 are trained on NISTP, and \{SDA,MLP\}2 are trained on P07. |
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719 The human error rate on digits is a lower bound because it does not count digits that were |
| 461 | 720 recognized as letters. For comparison, the results found in the literature |
| 721 on NIST digits classification using the same test set are included.} | |
| 438 | 722 \label{tab:sda-vs-mlp-vs-humans} |
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723 \begin{center} |
| 438 | 724 \begin{tabular}{|l|r|r|r|r|} \hline |
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725 & NIST test & NISTP test & P07 test & NIST test digits \\ \hline |
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726 Humans& 18.2\% $\pm$.1\% & 39.4\%$\pm$.1\% & 46.9\%$\pm$.1\% & $>1.1\%$ \\ \hline |
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727 SDA0 & 23.7\% $\pm$.14\% & 65.2\%$\pm$.34\% & 97.45\%$\pm$.06\% & 2.7\% $\pm$.14\%\\ \hline |
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728 SDA1 & 17.1\% $\pm$.13\% & 29.7\%$\pm$.3\% & 29.7\%$\pm$.3\% & 1.4\% $\pm$.1\%\\ \hline |
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729 SDA2 & 18.7\% $\pm$.13\% & 33.6\%$\pm$.3\% & 39.9\%$\pm$.17\% & 1.7\% $\pm$.1\%\\ \hline |
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730 MLP0 & 24.2\% $\pm$.15\% & 68.8\%$\pm$.33\% & 78.70\%$\pm$.14\% & 3.45\% $\pm$.15\% \\ \hline |
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731 MLP1 & 23.0\% $\pm$.15\% & 41.8\%$\pm$.35\% & 90.4\%$\pm$.1\% & 3.85\% $\pm$.16\% \\ \hline |
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732 MLP2 & 24.3\% $\pm$.15\% & 46.0\%$\pm$.35\% & 54.7\%$\pm$.17\% & 4.85\% $\pm$.18\% \\ \hline |
| 461 | 733 [5] & & & & 4.95\% $\pm$.18\% \\ \hline |
| 734 [2] & & & & 3.71\% $\pm$.16\% \\ \hline | |
| 735 [3] & & & & 2.4\% $\pm$.13\% \\ \hline | |
| 736 [4] & & & & 2.1\% $\pm$.12\% \\ \hline | |
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737 \end{tabular} |
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738 \end{center} |
| 438 | 739 \end{table} |
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740 |
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741 \subsection{Perturbed Training Data More Helpful for SDAE} |
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742 |
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743 \begin{table} |
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744 \caption{Relative change in error rates due to the use of perturbed training data, |
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745 either using NISTP, for the MLP1/SDA1 models, or using P07, for the MLP2/SDA2 models. |
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746 A positive value indicates that training on the perturbed data helped for the |
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747 given test set (the first 3 columns on the 62-class tasks and the last one is |
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748 on the clean 10-class digits). Clearly, the deep learning models did benefit more |
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749 from perturbed training data, even when testing on clean data, whereas the MLP |
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750 trained on perturbed data performed worse on the clean digits and about the same |
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751 on the clean characters. } |
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752 \label{tab:sda-vs-mlp-vs-humans} |
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753 \begin{center} |
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754 \begin{tabular}{|l|r|r|r|r|} \hline |
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755 & NIST test & NISTP test & P07 test & NIST test digits \\ \hline |
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756 SDA0/SDA1-1 & 38\% & 84\% & 228\% & 93\% \\ \hline |
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757 SDA0/SDA2-1 & 27\% & 94\% & 144\% & 59\% \\ \hline |
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758 MLP0/MLP1-1 & 5.2\% & 65\% & -13\% & -10\% \\ \hline |
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759 MLP0/MLP2-1 & -0.4\% & 49\% & 44\% & -29\% \\ \hline |
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760 \end{tabular} |
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761 \end{center} |
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762 \end{table} |
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763 |
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764 |
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765 \subsection{Multi-Task Learning Effects} |
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766 |
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767 As previously seen, the SDA is better able to benefit from the |
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768 transformations applied to the data than the MLP. In this experiment we |
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769 define three tasks: recognizing digits (knowing that the input is a digit), |
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770 recognizing upper case characters (knowing that the input is one), and |
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771 recognizing lower case characters (knowing that the input is one). We |
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772 consider the digit classification task as the target task and we want to |
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773 evaluate whether training with the other tasks can help or hurt, and |
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774 whether the effect is different for MLPs versus SDAs. The goal is to find |
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775 out if deep learning can benefit more (or less) from multiple related tasks |
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776 (i.e. the multi-task setting) compared to a corresponding purely supervised |
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777 shallow learner. |
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778 |
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779 We use a single hidden layer MLP with 1000 hidden units, and a SDA |
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780 with 3 hidden layers (1000 hidden units per layer), pre-trained and |
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781 fine-tuned on NIST. |
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782 |
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783 Our results show that the MLP benefits marginally from the multi-task setting |
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784 in the case of digits (5\% relative improvement) but is actually hurt in the case |
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785 of characters (respectively 3\% and 4\% worse for lower and upper class characters). |
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786 On the other hand the SDA benefitted from the multi-task setting, with relative |
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787 error rate improvements of 27\%, 15\% and 13\% respectively for digits, |
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788 lower and upper case characters, as shown in Table~\ref{tab:multi-task}. |
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789 |
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790 \begin{table} |
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791 \caption{Test error rates and relative change in error rates due to the use of |
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792 a multi-task setting, i.e., training on each task in isolation vs training |
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793 for all three tasks together, for MLPs vs SDAs. The SDA benefits much |
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794 more from the multi-task setting. All experiments on only on the |
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795 unperturbed NIST data, using validation error for model selection. |
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796 Relative improvement is 1 - single-task error / multi-task error.} |
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797 \label{tab:multi-task} |
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798 \begin{center} |
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799 \begin{tabular}{|l|r|r|r|} \hline |
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800 & single-task & multi-task & relative \\ |
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801 & setting & setting & improvement \\ \hline |
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802 MLP-digits & 3.77\% & 3.99\% & 5.6\% \\ \hline |
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803 MLP-lower & 17.4\% & 16.8\% & -4.1\% \\ \hline |
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804 MLP-upper & 7.84\% & 7.54\% & -3.6\% \\ \hline |
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805 SDA-digits & 2.6\% & 3.56\% & 27\% \\ \hline |
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806 SDA-lower & 12.3\% & 14.4\% & 15\% \\ \hline |
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parents:
458
diff
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807 SDA-upper & 5.93\% & 6.78\% & 13\% \\ \hline |
|
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Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
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changeset
|
808 \end{tabular} |
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parents:
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diff
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|
809 \end{center} |
|
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parents:
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diff
changeset
|
810 \end{table} |
|
437
479f2f518fc9
added Training with More Classes than Necessary
Guillaume Sicard <guitch21@gmail.com>
parents:
434
diff
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811 |
|
379
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added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff
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|
812 \section{Conclusions} |
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added writeup skeleton
Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff
changeset
|
813 |
|
407
fe2e2964e7a3
description des transformations en cours ajout d un fichier special.bib pour des references specifiques
Xavier Glorot <glorotxa@iro.umontreal.ca>
parents:
393
diff
changeset
|
814 \bibliography{strings,ml,aigaion,specials} |
|
379
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parents:
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|
815 \bibliographystyle{mlapa} |
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Yoshua Bengio <bengioy@iro.umontreal.ca>
parents:
diff
changeset
|
816 |
|
407
fe2e2964e7a3
description des transformations en cours ajout d un fichier special.bib pour des references specifiques
Xavier Glorot <glorotxa@iro.umontreal.ca>
parents:
393
diff
changeset
|
817 \end{document} |