# Computationally Efficient Ways of Determining If Pixels Are Clumped

I'm working on an application where I get a video frame that's already segmented into good and bad pixels (as in -- the hardware either coughs up a pixel value or a flag that says "bad"). I want to come up with a figure of merit for the frame that's based both on the number of bad pixels and the amount that they are clumped, with one big clump being worse than a bunch of little ones.

Here's two pictures to illustrate: the left picture has 17 dots 10 pixels in diameter -- it's better, on the theory that the dots are artifacts that can be "seen around". The right picture has about the same number of bad pixels, but they're all in one dot 41 pixels in diameter -- it's worse, on the theory that having so many bad pixels in a clump may occlude something interesting.

Things I can think to do:

• Walk the perimeter of each clump (really inefficient)
• Low-pass filter (originally I was thinking to use an FFT and look for high frequency content (not terribly efficient, and not terribly accurate, either, I think))
• draw a box around the bad pixels by finding the largest vertical and horizontal extents, and compare the area of that box to the bad pixel count (this could have false negatives if there's a large blob and a few little dots)

Are there known good ways to do this?

• Could you share some of those frames so we could build algorithm on real samples? – Royi Jan 30 at 20:50
• Unfortunately no. But the pixels are distinctly segmented. So any images can be treated as 2-valued pixels with white denoting "bad" and black denoting "good". A pretty good stand-in would be an image with about 2% of the bad pixels all concentrated into one circle, and another one with about 5% into around 20 same-sized circles. – TimWescott Jan 30 at 22:30
• So why not use analysis functions like bwlabel()? It seems I don't get what you want to get. – Royi Jan 31 at 5:59
• Can you point me to the algorithm that bwlabel() implements? How computationally efficient is it? Perhaps it's my answer -- I'm all over 1D processing, but this 2D stuff isn't my forte. – TimWescott Jan 31 at 16:00
• Could you share an image? It doesn't have to be real world image you use which might be with IP. Generate something on your own and share with us. We'll try to assist you. – Royi Jan 31 at 16:17

I've not tried this, but I'm wondering if something like a simple low pass filtering, followed by a column sum will do.

No time to explain code right now, but this seems to do something like what you want. Will return in a few hours and explain what it's doing.  # R Code Below

library(imager)

N <- 128

noise <- array(runif(N*N*1*1),c(N,N,1,1)) #5x5 pixels, 1 frames, 1 colours. All noise
small_clumps <- as.cimg(noise)
blurry <- isoblur(small_clumps,5)

layout(matrix(c(1,1,2,3), 2, 2, byrow = TRUE))
plot(colSums(blurry-mean(blurry)))
plot(small_clumps)
plot(blurry)

large_clumps <- small_clumps
large_clumps[65:75, 65:75] <- 1
large_clumps[15:25, 35:45] <- 1
blurry_large <- isoblur(large_clumps,5)
# par(mfrow=c(3,1))
layout(matrix(c(1,1,2,3), 2, 2, byrow = TRUE))
plot(colSums(blurry_large-mean(blurry_large)))
plot(large_clumps)
plot(blurry_large)

• It looks like it's doing the same thing that I was musing about with doing an FFT -- i.e., emphasize the low-frequency effects that's kinda synonymous with "clumping". It needs a computationally efficient LPF, though. – TimWescott Jan 30 at 18:05
• @TimWescott Right: you should be able to code it pretty efficiently. Let me think about it and will update the post later today. – Peter K. Jan 30 at 20:23

Median filter? Use it to «thin out» bad pixels that appear locally sparse (for some definition of «local»). Then count the remaining bad pixels.

Or convolve with a large-ish 2d kernel (eg flat rectangular window) and decimate to get a number for «how many bad pixels are there inside each eg 16x16 window». Then accumulate the score for each block in a nonlinear fashion (so as to punish really dense blocks).

Im = randi([0 1], 640, 480);

Im_lp = conv2(ones(16), Im);

Im_lp_dec = Im_lp(8:16:end, 8:16:end);

score = sum(Im_lp_dec(:).^2);

Erode the image using a structuring element which is the size/shape of the maximum allowable "bad region". Then dilate using the same structuring element. This will remove the bad-but-good-enough regions. From there you can work on characterizing/measuring what's left. Example given below using Matlab.

% Create a blank image.
Mimg = 1000;
Nimg = 1000;
img = zeros( Mimg, Nimg );

% Make up some coordinates.
m = 0 : Mimg - 1;
n = 0 : Nimg  - 1;
[ MM, NN ] = ndgrid( m(:), n(:).' );

Ncirc = 50;
dmax = 40;                              % Max possible radius
r = randi( dmax, [ Ncirc, 1 ] );        % Radius of circles
x0 = randi( Mimg, [ Ncirc, 1 ] ) - 1;   % x-coorindate of center
y0 = randi( Mimg, [ Ncirc, 1 ] ) - 1;   % y-coordinate of center
for ii = 1 : Ncirc
t = sqrt( ( x0(ii) - MM ).^2 + ( y0(ii) - NN ).^2 );
img( t <= r(ii) ) = 1;
end

d = 20;

% Create the structuring element.
s = sqrt( bsxfun( @plus, ( (1:d*2)' - d ).^2, ( (1:d*2) - d ).^2 ) ) <= d;

% Remove the bad pixel regions that are too small to care about.
img_bad = imdilate( imerode( img, s ), s );

figure();
set( gcf(), 'color', 'w' );
subplot( 1, 2, 1 );
imagesc( img );
colormap( gray );
subplot( 1, 2, 2 );
title( 'Too Big To Ignore Mask' );
colormap( gray ); The approach I took was using MATLAB's functions, either regionprops() or bwconncomp() and bwdist().

The idea is to give a grade for each pixel which is a part of bad pixels object.
The grade is the radius of the circle bounding the object the pixel resides in.

One way to calculate the radius of the bounding circle is the MajorAxisLength property in the output of regionprops().
Another nice trick is using the Binary Image Distance Transform. If you apply the distance transform to the image where each bad pixels is black and the rest of the pixels is white, then for each object the maximum value represents the radius of the bounding image and the coordinate represent the center of the bounding circle.

So for pixel I gave a grade and then summed the value of all pixels.
The result is as expected:

The score above completely ignores the content of the frame and only takes into account the map of the bad pixels.
In real world application I'd do one extra step. I would take Saliency Map into account. Then for pixels which are in an important location I'd add more to their score.

The full MATLAB code is available on my StackExchange Signal Processing Q63549 GitHub Repository.

• @TimWescott, Could you point me what's missing in my answer? – Royi Feb 6 at 22:46