--- a/writeup/nips2010_submission.tex	Fri May 28 17:39:22 2010 -0600
+++ b/writeup/nips2010_submission.tex	Sat May 29 16:50:03 2010 -0400
@@ -108,116 +108,71 @@
 
 \section{Perturbation and Transformation of Character Images}
 
-This section describes the different transformations we used to generate data, in their order.
+This section describes the different transformations we used to stochastically
+transform source images in order to obtain data. More details can
+be found in this technical report~\cite{ift6266-tr-anonymous}.
 The code for these transformations (mostly python) is available at 
 {\tt http://anonymous.url.net}. All the modules in the pipeline share
 a global control parameter ($0 \le complexity \le 1$) that allows one to modulate the
-amount of deformation or noise introduced.
+amount of deformation or noise introduced. 
 
-We can differentiate two important parts in the pipeline. The first one,
-from slant to pinch, performs transformations of the character. The second
-part, from blur to contrast, adds noise to the image.
-
-\subsection{Slant}
+There are two main parts in the pipeline. The first one,
+from slant to pinch below, performs transformations. The second
+part, from blur to contrast, adds different kinds of noise.
 
-In order to mimic a slant effect, we simply shift each row of the image
-proportionnaly to its height: $shift = round(slant \times height)$.  We
-round the shift in order to have a discret displacement. We do not use a
-filter to smooth the result in order to save computing time and also
-because latter transformations have similar effects.
-
+{\large\bf Transformations}\\
+{\bf Slant}\\
+We mimic slant by shifting each row of the image
+proportionnaly to its height: $shift = round(slant \times height)$.  
 The $slant$ coefficient can be negative or positive with equal probability
-and its value is randomly sampled according to the complexity level.  In
-our case we take uniformly a number in the range $[0,complexity]$, so the
+and its value is randomly sampled according to the complexity level:
+e $slant \sim U[0,complexity]$, so the
 maximum displacement for the lowest or highest pixel line is of
-$round(complexity \times 32)$.
-
-
-\subsection{Thickness}
-
-To change the thickness of the characters we used morpholigical operators:
-dilation and erosion~\cite{Haralick87,Serra82}.
-
-The basic idea of such transform is, for each pixel, to multiply in the
-element-wise manner its neighbourhood with a matrix called the structuring
-element.  Then for dilation we remplace the pixel value by the maximum of
-the result, or the minimum for erosion.  This will dilate or erode objects
-in the image and the strength of the transform only depends on the
-structuring element.
-
-We used ten different structural elements with increasing dimensions (the
-biggest is $5\times5$).  for each image, we radomly sample the operator
-type (dilation or erosion) with equal probability and one structural
+$round(complexity \times 32)$.\\
+{\bf Thickness}\\
+Morpholigical operators of dilation and erosion~\cite{Haralick87,Serra82}
+are applied. The neighborhood of each pixel is multiplied
+element-wise with a {\em structuring element} matrix.
+The pixel value is replaced by the maximum or the minimum of the resulting
+matrix, respectively for dilation or erosion. Ten different structural elements with 
+increasing dimensions (largest is $5\times5$) were used.  For each image, 
+randomly sample the operator type (dilation or erosion) with equal probability and one structural
 element from a subset of the $n$ smallest structuring elements where $n$ is
 $round(10 \times complexity)$ for dilation and $round(6 \times complexity)$
-for erosion.  A neutral element is always present in the set, if it is
-chosen the transformation is not applied.  Erosion allows only the six
+for erosion.  A neutral element is always present in the set, and if it is
+chosen no transformation is applied.  Erosion allows only the six
 smallest structural elements because when the character is too thin it may
-erase it completly.
-
-\subsection{Affine Transformations}
-
-We generate an affine transform matrix according to the complexity level,
-then we apply it directly to the image.  The matrix is of size $2 \times
-3$, so we can represent it by six parameters $(a,b,c,d,e,f)$.  Formally,
-for each pixel $(x,y)$ of the output image, we give the value of the pixel
-nearest to : $(ax+by+c,dx+ey+f)$, in the input image.  This allows to
-produce scaling, translation, rotation and shearing variances.
-
-The sampling of the parameters $(a,b,c,d,e,f)$ have been tuned by hand to
+be completely erased.\\
+{\bf Affine Transformations}\\
+A $2 \times 3$ affine transform matrix (with
+6 parameters $(a,b,c,d,e,f)$) is sampled according to the $complexity$ level.
+Each pixel $(x,y)$ of the output image takes the value of the pixel
+nearest to $(ax+by+c,dx+ey+f)$ in the input image.  This 
+produces scaling, translation, rotation and shearing.
+The marginal distributions of $(a,b,c,d,e,f)$ have been tuned by hand to
 forbid important rotations (not to confuse classes) but to give good
-variability of the transformation. For each image we sample uniformly the
-parameters in the following ranges: $a$ and $d$ in $[1-3 \times
-complexity,1+3 \times complexity]$, $b$ and $e$ in $[-3 \times complexity,3
-\times complexity]$ and $c$ and $f$ in $[-4 \times complexity, 4 \times
-complexity]$.
-
-
-\subsection{Local Elastic Deformations}
-
-This filter induces a "wiggly" effect in the image. The description here
-will be brief, as the algorithm follows precisely what is described in
-\cite{SimardSP03}.
-
-The general idea is to generate two "displacements" fields, for horizontal
-and vertical displacements of pixels. Each of these fields has the same
-size as the original image.
-
-When generating the transformed image, we'll loop over the x and y
-positions in the fields and select, as a value, the value of the pixel in
-the original image at the (relative) position given by the displacement
-fields for this x and y. If the position we'd retrieve is outside the
-borders of the image, we use a 0 value instead.
-
+variability of the transformation: $a$ and $d$ $\sim U[1-3 \times
+complexity,1+3 \times complexity]$, $b$ and $e$ $\sim[-3 \times complexity,3
+\times complexity]$ and $c$ and $f$ $\sim U[-4 \times complexity, 4 \times
+complexity]$.\\
+{\bf Local Elastic Deformations}\\
+This filter induces a "wiggly" effect in the image, following~\cite{SimardSP03},
+which provides more details. 
+Two "displacements" fields are generated and applied, for horizontal
+and vertical displacements of pixels. 
 To generate a pixel in either field, first a value between -1 and 1 is
-chosen from a uniform distribution. Then all the pixels, in both fields, is
+chosen from a uniform distribution. Then all the pixels, in both fields, are
 multiplied by a constant $\alpha$ which controls the intensity of the
-displacements (bigger $\alpha$ translates into larger wiggles).
-
-As a final step, each field is convoluted with a Gaussian 2D kernel of
-standard deviation $\sigma$. Visually, this results in a "blur"
-filter. This has the effect of making values next to each other in the
-displacement fields similar. In effect, this makes the wiggles more
-coherent, less noisy.
-
-As displacement fields were long to compute, 50 pairs of fields were
-generated per complexity in increments of 0.1 (50 pairs for 0.1, 50 pairs
-for 0.2, etc.), and afterwards, given a complexity, we selected randomly
-among the 50 corresponding pairs.
-
-$\sigma$ and $\alpha$ were linked to complexity through the formulas
+displacements (larger $\alpha$ translates into larger wiggles).
+Each field is convoluted with a Gaussian 2D kernel of
+standard deviation $\sigma$. Visually, this results in a blur.
 $\alpha = \sqrt[3]{complexity} \times 10.0$ and $\sigma = 10 - 7 \times
-\sqrt[3]{complexity}$.
-
-
-\subsection{Pinch}
-
-This is another GIMP filter we used. The filter is in fact named "Whirl and
-pinch", but we don't use the "whirl" part (whirl is set to 0). As described
-in GIMP, a pinch is "similar to projecting the image onto an elastic
-surface and pressing or pulling on the center of the surface".
-
-Mathematically, for a square input image, think of drawing a circle of
+\sqrt[3]{complexity}$.\\
+{\bf Pinch}\\
+This GIMP filter is named "Whirl and
+pinch", but whirl was set to 0. A pinch is ``similar to projecting the image onto an elastic
+surface and pressing or pulling on the center of the surface''~\cite{GIMP-manual}.
+For a square input image, think of drawing a circle of
 radius $r$ around a center point $C$. Any point (pixel) $P$ belonging to
 that disk (region inside circle) will have its value recalculated by taking
 the value of another "source" pixel in the original image. The position of
@@ -225,139 +180,58 @@
 at some other distance $d_2$. Define $d_1$ to be the distance between $P$
 and $C$. $d_2$ is given by $d_2 = sin(\frac{\pi{}d_1}{2r})^{-pinch} \times
 d_1$, where $pinch$ is a parameter to the filter.
-
-If the region considered is not square then, before computing $d_2$, the
-smallest dimension (x or y) is stretched such that we may consider the
-region as if it was square. Then, after $d_2$ has been computed and
-corresponding components $d_2\_x$ and $d_2\_y$ have been found, the
-component corresponding to the stretched dimension is compressed back by an
-inverse ratio.
-
 The actual value is given by bilinear interpolation considering the pixels
 around the (non-integer) source position thus found.
-
-The value for $pinch$ in our case was given by sampling from an uniform
-distribution over the range $[-complexity, 0.7 \times complexity]$.
-
-\subsection{Motion Blur}
-
-This is a GIMP filter we applied, a "linear motion blur" in GIMP
-terminology. The description will be brief as it is a well-known filter.
-
-This algorithm has two input parameters, $length$ and $angle$. The value of
-a pixel in the final image is the mean value of the $length$ first pixels
-found by moving in the $angle$ direction. An approximation of this idea is
-used, as we won't fall onto precise pixels by following that
-direction. This is done using the Bresenham line algorithm.
-
-The angle, in our case, is chosen from a uniform distribution over
-$[0,360]$ degrees. The length, though, depends on the complexity; it's
-sampled from a Gaussian distribution of mean 0 and standard deviation
-$\sigma = 3 \times complexity$.
-
-\subsection{Occlusion}
-
-This filter selects random parts of other (hereafter "occlusive") letter
-images and places them over the original letter (hereafter "occluded")
-image. To be more precise, having selected a subregion of the occlusive
-image and a desination position in the occluded image, to determine the
-final value for a given overlapping pixel, it selects whichever pixel is
-the lightest. As a reminder, the background value is 0, black, so the value
-nearest to 1 is selected.
+Here $pinch \sim U[-complexity, 0.7 \times complexity]$.\\
 
-To select a subpart of the occlusive image, four numbers are generated. For
-compability with the code, we'll call them "haut", "bas", "gauche" and
-"droite" (respectively meaning top, bottom, left and right). Each of these
-numbers is selected according to a Gaussian distribution of mean $8 \times
-complexity$ and standard deviation $2$. This means the largest the
-complexity is, the biggest the occlusion will be. The absolute value is
-taken, as the numbers must be positive, and the maximum value is capped at
-15.
-
-These four sizes collectively define a window centered on the middle pixel
-of the occlusive image. This is the part that will be extracted as the
-occlusion.
-
-The next step is to select a destination position in the occluded
-image. Vertical and horizontal displacements $y\_arrivee$ and $x\_arrivee$
-are selected according to Gaussian distributions of mean 0 and of standard
-deviations of, respectively, 3 and 2. Then an horizontal placement mode,
-$place$, is selected to be of three values meaning
-left, middle or right.
-
-If $place$ is "middle", the occlusion will be horizontally centered
-around the horizontal middle of the occluded image, then shifted according
-to $x\_arrivee$. If $place$ is "left", it will be placed on the left of
-the occluded image, then displaced right according to $x\_arrivee$. The
-contrary happens if $place$ is $right$.
-
-In both the horizontal and vertical positionning, the maximum position in
-either direction is such that the selected occlusion won't go beyond the
-borders of the occluded image.
-
-This filter has a probability of not being applied, at all, of 60\%.
-
-
-\subsection{Pixel Permutation}
-
-This filter permuts neighbouring pixels. It selects first
+{\large\bf Injecting Noise}\\
+{\bf Motion Blur}\\
+This GIMP filter is a ``linear motion blur'' in GIMP
+terminology, with two parameters, $length$ and $angle$. The value of
+a pixel in the final image is the approximately mean value of the $length$ first pixels
+found by moving in the $angle$ direction. 
+Here $angle \sim U[0,360]$ degrees, and $length \sim {\rm Normal}(0,(3 \times complexity)^2)$.\\
+{\bf Occlusion}\\
+This filter selects a random rectangle from an {\em occluder} character
+images and places it over the original {\em occluded} character
+image. Pixels are combined by taking the max(occluder,occluded),
+closer to black. The corners of the occluder  The rectangle corners
+are sampled so that larger complexity gives larger rectangles.
+The destination position in the occluded image are also sampled
+according to a normal distribution (see more details in~\cite{ift6266-tr-anonymous}.
+It has has a probability of not being applied at all of 60\%.\\
+{\bf Pixel Permutation}\\
+This filter permutes neighbouring pixels. It selects first
 $\frac{complexity}{3}$ pixels randomly in the image. Each of them are then
 sequentially exchanged to one other pixel in its $V4$ neighbourhood. Number
 of exchanges to the left, right, top, bottom are equal or does not differ
 from more than 1 if the number of selected pixels is not a multiple of 4.
-
-It has has a probability of not being applied, at all, of 80\%.
-
-
-\subsection{Gaussian Noise}
-
+It has has a probability of not being applied at all of 80\%.\\
+{\bf Gaussian Noise}\\
 This filter simply adds, to each pixel of the image independently, a
-Gaussian noise of mean $0$ and standard deviation $\frac{complexity}{10}$.
-
-It has has a probability of not being applied, at all, of 70\%.
-
-
-\subsection{Background Images}
-
+noise $\sim Normal(0(\frac{complexity}{10})^2)$.
+It has has a probability of not being applied at all of 70\%.\\
+{\bf Background Images}\\
 Following~\cite{Larochelle-jmlr-2009}, this transformation adds a random
 background behind the letter. The background is chosen by first selecting,
-at random, an image from a set of images. Then we choose a 32x32 subregion
-of that image as the background image (by sampling x and y positions
+at random, an image from a set of images. Then a 32$\times$32 subregion
+of that image is chosen as the background image (by sampling position
 uniformly while making sure not to cross image borders).
-
 To combine the original letter image and the background image, contrast
 adjustments are made. We first get the maximal values (i.e. maximal
 intensity) for both the original image and the background image, $maximage$
-and $maxbg$. We also have a parameter, $contrast$, given by sampling from a
-uniform distribution over $[complexity, 1]$.
-
-Once we have all these numbers, we first adjust the values for the
-background image. Each pixel value is multiplied by $\frac{max(maximage -
-  contrast, 0)}{maxbg}$. Therefore the higher the contrast, the darkest the
-background will be.
-
-The final image is found by taking the brightest (i.e. value nearest to 1)
-pixel from either the background image or the corresponding pixel in the
-original image.
-
-\subsection{Salt and Pepper Noise}
-
-This filter adds noise to the image by randomly selecting a certain number
-of them and, for those selected pixels, assign a random value according to
-a uniform distribution over the $[0,1]$ ranges. This last distribution does
-not change according to complexity. Instead, the number of selected pixels
-does: the proportion of changed pixels corresponds to $complexity / 5$,
-which means, as a maximum, 20\% of the pixels will be randomized. On the
-lowest extreme, no pixel is changed.
-
-This filter also has a probability of not being applied, at all, of 75\%.
-
-\subsection{Spatially Gaussian Noise}
-
-The aim of this transformation is to filter, with a gaussian kernel,
-different regions of the image. In order to save computing time we decided
-to convolve the whole image only once with a symmetric gaussian kernel of
-size and variance choosen uniformly in the ranges: $[12,12 + 20 \times
+and $maxbg$. We also have a parameter $contrast \sim U[complexity, 1]$.
+Each background pixel value is multiplied by $\frac{max(maximage -
+  contrast, 0)}{maxbg}$ (higher contrast yield darker
+background). The output image pixels are max(background,original).\\
+{\bf Salt and Pepper Noise}\\
+This filter adds noise $\sim U[0,1]$ to random subsets of pixels.
+The number of selected pixels is $0.2 \times complexity$.
+This filter has a probability of not being applied at all of 75\%.\\
+{\bf Spatially Gaussian Noise}\\
+Different regions of the image are spatially smoothed.
+The image is convolved with a symmetric Gaussian kernel of
+size and variance choosen uniformly in the ranges $[12,12 + 20 \times
 complexity]$ and $[2,2 + 6 \times complexity]$. The result is normalized
 between $0$ and $1$.  We also create a symmetric averaging window, of the
 kernel size, with maximum value at the center.  For each image we sample
@@ -367,53 +241,27 @@
 we add to the mask the averaging window centered to it.  The final image is
 computed from the following element-wise operation: $\frac{image + filtered
   image \times mask}{mask+1}$.
-
-This filter has a probability of not being applied, at all, of 75\%.
-
-\subsection{Scratches}
-
+This filter has a probability of not being applied at all of 75\%.\\
+{\bf Scratches}\\
 The scratches module places line-like white patches on the image.  The
-lines are in fact heavily transformed images of the digit "1" (one), chosen
-at random among five thousands such start images of this digit.
-
-Once the image is selected, the transformation begins by finding the first
-$top$, $bottom$, $right$ and $left$ non-zero pixels in the image. It is
-then cropped to the region thus delimited, then this cropped version is
-expanded to $32\times32$ again. It is then rotated by a random angle having a
-Gaussian distribution of mean 90 and standard deviation $100 \times
-complexity$ (in degrees). The rotation is done with bicubic interpolation.
-
-The rotated image is then resized to $50\times50$, with anti-aliasing. In
-that image, we crop the image again by selecting a region delimited
-horizontally to $left$ to $left+32$ and vertically by $top$ to $top+32$.
-
-Once this is done, two passes of a greyscale morphological erosion filter
-are applied. Put briefly, this erosion filter reduces the width of the line
-by a certain $smoothing$ amount. For small complexities (< 0.5),
-$smoothing$ is 6, so the line is very small. For complexities ranging from
-0.25 to 0.5, $smoothing$ is 5. It is 4 for complexities 0.5 to 0.75, and 3
-for higher complexities.
-
-To compensate for border effects, the image is then cropped to 28x28 by
-removing two pixels everywhere on the borders, then expanded to 32x32
-again. The pixel values are then linearly expanded such that the minimum
-value is 0 and the maximal one is 1. Then, 50\% of the time, the image is
-vertically flipped.
-
+lines are heavily transformed images of the digit "1" (one), chosen
+at random among five thousands such 1 images. The 1 image is
+randomly cropped and rotated by an angle $\sim Normal(0,(100 \times
+complexity)^2$, using bicubic interpolation,
+Two passes of a greyscale morphological erosion filter
+are applied, reducing the width of the line
+by an amount controlled by $complexity$.
 This filter is only applied only 15\% of the time. When it is applied, 50\%
 of the time, only one patch image is generated and applied. In 30\% of
 cases, two patches are generated, and otherwise three patches are
 generated. The patch is applied by taking the maximal value on any given
-patch or the original image, for each of the 32x32 pixel locations.
-
-\subsection{Color and Contrast Changes}
-
+patch or the original image, for each of the 32x32 pixel locations.\\
+{\bf Color and Contrast Changes}\\
 This filter changes the constrast and may invert the image polarity (white
 on black to black on white). The contrast $C$ is defined here as the
-difference between the maximum and the minimum pixel value of the image. A
-contrast value is sampled uniformly between $1$ and $1-0.85 \times
-complexity$ (this insure a minimum constrast of $0.15$). We then simply
-normalize the image to the range $[\frac{1-C}{2},1-\frac{1-C}{2}]$. The
+difference between the maximum and the minimum pixel value of the image. 
+Contrast $\sim U[1-0.85 \times complexity,1]$ (so constrast $\geq 0.15$). 
+The image is normalized into $[\frac{1-C}{2},1-\frac{1-C}{2}]$. The
 polarity is inverted with $0.5$ probability.
 
 
@@ -560,10 +408,10 @@
 MLP0   &  24.2\% $\pm$.15\%  & 68.8\%$\pm$.33\%  & 78.70\%$\pm$.14\%  & 3.45\% $\pm$.15\% \\ \hline 
 MLP1   &  23.0\% $\pm$.15\%  &  41.8\%$\pm$.35\%  & 90.4\%$\pm$.1\%  & 3.85\% $\pm$.16\% \\ \hline 
 MLP2   &  24.3\% $\pm$.15\%  &  46.0\%$\pm$.35\%  & 54.7\%$\pm$.17\%  & 4.85\% $\pm$.18\% \\ \hline 
-[5]    &                     &                    &                   & 4.95\% $\pm$.18\% \\ \hline
-[2]    &                     &                    &                   & 3.71\% $\pm$.16\% \\ \hline
-[3]    &                     &                    &                   & 2.4\% $\pm$.13\% \\ \hline
-[4]    &                     &                    &                   & 2.1\% $\pm$.12\% \\ \hline
+\cite{Granger+al-2007} &     &                    &                   & 4.95\% $\pm$.18\% \\ \hline
+\cite{Cortes+al-2000} &      &                    &                   & 3.71\% $\pm$.16\% \\ \hline
+\cite{Oliveira+al-2002} &    &                    &                   & 2.4\% $\pm$.13\% \\ \hline
+\cite{Migram+al-2005} &      &                    &                   & 2.1\% $\pm$.12\% \\ \hline
 \end{tabular}
 \end{center}
 \end{table}