--- a/writeup/nips2010_submission.tex	Tue Jun 01 11:04:09 2010 -0700
+++ b/writeup/nips2010_submission.tex	Tue Jun 01 11:14:48 2010 -0700
@@ -157,7 +157,7 @@
 proportionally 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:
-e $slant \sim U[0,complexity]$, so the
+$slant \sim U[0,complexity]$, so the
 maximum displacement for the lowest or highest pixel line is of
 $round(complexity \times 32)$.\\
 {\bf Thickness.}
@@ -187,9 +187,9 @@
 \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~\citet{SimardSP03-short},
+This filter induces a ``wiggly'' effect in the image, following~\citet{SimardSP03-short},
 which provides more details. 
-Two "displacements" fields are generated and applied, for horizontal
+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, are
@@ -200,13 +200,13 @@
 $\alpha = \sqrt[3]{complexity} \times 10.0$ and $\sigma = 10 - 7 \times
 \sqrt[3]{complexity}$.\\
 {\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
+This is a GIMP filter called ``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''~\citep{GIMP-manual}.
-For a square input image, think of drawing a circle of
+For a square input image, this is akin to 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
+the value of another ``source'' pixel in the original image. The position of
 that source pixel is found on the line that goes through $C$ and $P$, but
 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
@@ -222,31 +222,32 @@
 \vspace*{1mm}
 
 {\bf Motion Blur.}
-This GIMP filter is a ``linear motion blur'' in GIMP
+This 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
+a pixel in the final image is approximately the  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
+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
+closer to black. 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~\citet{ift6266-tr-anonymous}).
-It has has a probability of not being applied at all of 60\%.\\
+This filter has a probability of 60\% of not being applied.\\
 {\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
+sequentially exchanged with one other pixel in its $V4$ neighbourhood. The number
+of exchanges to the left, right, top, bottom is 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\%.\\
+% TODO: The previous sentence is hard to parse
+This filter has a probability of 80\% of not being applied.\\
 {\bf Gaussian Noise.}
 This filter simply adds, to each pixel of the image independently, a
 noise $\sim Normal(0(\frac{complexity}{10})^2)$.
-It has has a probability of not being applied at all of 70\%.\\
+It has a probability of 70\% of not being applied.\\
 {\bf Background Images.}
 Following~\citet{Larochelle-jmlr-2009}, this transformation adds a random
 background behind the letter. The background is chosen by first selecting,
@@ -280,7 +281,7 @@
 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 heavily transformed images of the digit "1" (one), chosen
+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 bi-cubic interpolation,