Reversing digital filters

[WIP]

Motivation: we are trying to record raw video with existing HDMI recorders (that don't know anything about recording raw).

Unfortunately, it seems that some of these recorders apply some processing on the image, like sharpening or blurring. Therefore, it may be a good idea to attempt to undo some of these filters applied to the image without our permission :P

General idea: feed some test images to the HDMI, compare with the output image from the recorder, and attempt to undo the transformations in order to recover the original image.

We will start by experimenting with simple linear filters on grayscale images, as they are easiest to work with.

1 Linear filters on grayscale images

Input:

source image
altered image (with an unknown filter)

To find a digital linear filter that would undo the alteration on our test image, we may solve a linear system: each pixel can be expressed as a linear combination of its neighbouring pixels. For a MxN image and a PxP filter, we will have P*P unknowns and M*N equations.

To simplify things, we'll consider filters with odd diameters, so filter size would be PxP = (2*n+1) x (2*n+1).

If we assume our filter is horizontally and vertically symmetrical, the number of unknowns decreases to (n+1) * (n+1).

If we assume our filter is also diagonally symmetrical, the number of unknowns becomes n * (n+1) / 2.

Let's try some examples.

We will use a training data set (a sample image used to compute the filter), and a validation data set (a different image, to check how well the filter does when the input data doesn't match). This is a simple strategy to avoid overfitting [1][2][3]. Maybe not the best one [4], but for a quick experiment, it should do the trick.

Blur:

f1 = @(x) imfilter(x, fspecial('disk', 1));
  0.025079   0.145344   0.025079
  0.145344   0.318310   0.145344
  0.025079   0.145344   0.025079

3x3 averaging blur:

f2 = @(x) imfilter(x, fspecial('average', 3));
  0.11111   0.11111   0.11111
  0.11111   0.11111   0.11111
  0.11111   0.11111   0.11111

Sharpen:

f3 = @(x) imfilter(x, fspecial('unsharp'));
 -0.16667  -0.66667  -0.16667
 -0.66667   4.33333  -0.66667
 -0.16667  -0.66667  -0.16667

Blur followed by sharpen:

f4 = @(x) imfilter(imfilter(x, fspecial('disk', 1)), fspecial('unsharp'));
 -0.0041798  -0.0409430  -0.1052555  -0.0409430  -0.0041798
 -0.0409430  -0.1381697   0.3357311  -0.1381697  -0.0409430
 -0.1052555   0.3357311   0.9750399   0.3357311  -0.1052555
 -0.0409430  -0.1381697   0.3357311  -0.1381697  -0.0409430
 -0.0041798  -0.0409430  -0.1052555  -0.0409430  -0.0041798

Laplacian of Gaussian (edge detector):

f5 = @(x) imfilter(x, fspecial('log'));

Laplacian of Gaussian plus the original image:

f6 = @(x) imfilter(x, fspecial('log')) + x;

2 Different filters on odd/even columns

Average odd/even columns (1 with 2, 3 with 4, similar to a YUV422 subsampling)

g1 = @(x) imresize((x(:,1:2:end) + x(:,2:2:end))/2, size(x), 'nearest');
 0 0.5 0.5 on columns 1:2:N
 0.5 0.5 0 on columns 2:2:N

Blur on odd columns, sharpen on even columns

function y = g2aux(x,f1,f2)
   y = x;
   y(:,1:2:end) = imfilter(x, f1(x))(:,1:2:end);
   y(:,2:2:end) = imfilter(x, f2(x))(:,2:2:end);
end
g2 = @(x) g2aux(x,f1,f2)

Green on odd columns, red on even columns, attempt to recover green (similar to the debayering problem)

function y = g3aux(g,r)
   y = g;
   y(:,2:2:end) = r(:,2:2:end);
end
g3 = @(g) g2aux(g,r)

3 Nonlinear filters

Median filter:

h1 = @(x) medfilt2(x, [3 3])

4 Added noise

Gaussian:

n1 = @(x) x + randn(size(x)) * 50;

Row noise:

n2 = @(x) x + ones(size(x,1),1) * randn(1,size(x,2)) * 10;