A faster approach to Laplacian of Gaussian

Question

I am currently in the process of optimizing my code to make image processing more efficient. my first problem was due to the vision.VideoFileReader and step where it took a long time to open each frame. I speed up my code by compressing my grayscale image into 3 frames in 1 RGB frame. This way I could load 1 RGB frame using vid.step() and have 3 frames imported ready for processing.

Now my code is running slow on the Laplacian of Gaussian (LoG) filtering. I read that using the function imfilter can be used to perform a LoG but it appears to be the next rate limiting step.

Upon further reading, it appears that imfilter is not the best option for speed. Apperently MATLAB introduced a LoG function but it was introduced in R2016b and I'm unfortunately using R2016a.

Is there a way to speed up imfilter or is there a better function to use to perform a LoG filtering?

Should I call python to speed up the process?

Code:

Hei = gh.Video.reader.info.VideoSize(2);
Wid = gh.Video.reader.info.VideoSize(1);

Log_filter = fspecial('log', filterdot, thresh); % fspecial creat predefined filter.Return a filter.
    % 25X25 Gaussian filter with SD =25 is created.

tic
ii = 1;

bkgd = zeros(Hei,Wid,3);
bkgd(:,:,1) = gh.Bkgd;
bkgd(:,:,2) = gh.Bkgd;
bkgd(:,:,3) = gh.Bkgd;

bkgdmod = reshape(bkgd,720,[]);

while ~isDone(gh.Video.reader)
    frame = gh.readFrame();
    img_temp = double(frame);

    img_temp2 = reshape(img_temp,720,[]);
    subbk = img_temp2 - bkgdmod;

    img_LOG = imfilter(subbk, Log_filter, 'symmetric', 'conv');

    img_LOG =  imbinarize(img_LOG,.002);
    [~, centroids, ~] = gh.Video.blobAnalyser.step(img_LOG);

    toc
end

Answer 1

The Laplace of Gaussian is not directly separable into two 1D kernels. Therefore, imfilter will do a full convolution, which is quite expensive. But we can manually separate it out into simpler filters.

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The Laplace of Gaussian is defined as the sum of two second-order-derivatives of the Gaussian:

LoG = d²/dx² G + d²/dy² G

The Gaussian itself, and its derivatives, are separable. Therefore, the above can be computed using 4 1D convolutions, which is much cheaper than a single 2D convolution unless the kernel is very small (e.g. if the kernel is 7x7, we need 49 multiplications and additions per pixel for the 2D kernel, or 4*7=28 multiplications and additions per pixel for the 4 1D kernels; this difference grows as the kernel gets larger). The computations would be:

sigma = 3;
cutoff = ceil(3*sigma);
G = fspecial('gaussian',[1,2*cutoff+1],sigma);
d2G = G .* ((-cutoff:cutoff).^2 - sigma^2)/ (sigma^4);
dxx = conv2(d2G,G,img,'same');
dyy = conv2(G,d2G,img,'same');
LoG = dxx + dyy;

If you are really strapped for time, and don't care about precision, you could set cutoff to 2*sigma (for a smaller kernel), but this is not ideal.

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An alternative, less precise, is to separate the operation differently:

LoG * f = ( d²/dx² G + d²/dy² G ) * f
        = ( d²/dx² 𝛿 * G + d²/dy² 𝛿 * G ) * f
        = ( d²/dx^2 𝛿 + d²/dy² 𝛿 ) * G * f

(with * there representing convolution, and 𝛿 the Dirac delta, convolution's equivalent to multiplying by 1). The d²/dx² 𝛿 + d²/dy² 𝛿 operator doesn't really exist in the discrete world, but you can take the finite difference approximation, which leads to the famous 3x3 Laplace kernel:

[ 1  1  1             [ 0  1  0
  1 -8  1       or:     1 -4  1
  1  1  1 ]             0  1  0 ]

Now we get a rougher approximation, but it's faster to compute (2 1D convolutions, and a convolution with a trivial 3x3 kernel):

sigma = 3;
cutoff = ceil(3*sigma);
G = fspecial('gaussian',[1,2*cutoff+1],sigma);
tmp = conv2(G,G,img,'same');
h = fspecial('laplacian',0);
LoG = conv2(tmp,h,'same'); % or use imfilter