Description: This function does the decomposition of a separable nD kernel into its 1D components, such that a convolution with each of these components yields the same result as a convolution with the full nD kernel, at a drastic reduction in computational cost.
SYNTAX: ======= [K1,KN,ERR] = DECOMPOSE_KERNEL(H) computes a set of 1D kernels K1{1}, K1{2}, ... K1{N} such that the convolution of an image with all of these in turn yields the same result as a convolution with the Ndimensional input kernel H: RES1 = CONVN(IMG,H); RES2 = IMG; FOR II=1:LENGTH(K1) RES2 = CONVN(RES2,K1{II}); END
KN is the reconstruction of the original kernel H from the 1D kernels K1, and ERR is the sum of absolute differences between H and KN. The syntax mimics DirkJan Kroon's submission to the FileExchange (see below).
EXPLANATION: ============
In general, for a 2D kernel H, the convolution with 2D image F: G = F * H is identical to the convolution of the image with column vector H1 and convolution of the result with row vector H2: G = ( F * H1 ) * H2 . In MATLAB speak, this means that > CONV2(F,H) == CONV2(CONV2(F,H1),H2)
Because of the properties of the convolution, ( F * H1 ) * H2 = F * ( H1 * H2 ) , meaning that the convolution of the two 1D filters with each other results in the original filter H. And because H1 is a column vector and H2 a row vector, H = H1 * H2 = H1 H2 . Thus, we need to find two vectors whose product yields the matrix H. In MATLAB speak we need to solve the equation > H1*H2 == H
The function in the standard MATLAB toolbox, FILTER2, does just this, and it does it using singular value decomposition: U S V' = H , H1(i) = U(i,1) S(1,1)^0.5 , H2(i) = V(i,1)* S(1,1)^0.5 . (the * here is the conjugate!)
Note that, if the kernel H is separable, all values of S are zero except S(1,1). Also note that this is an underdetermined problem, in the sense that H = H1 H2 = ( a H1 ) ( 1/a H2 ) ; that is, it is possible to multiply one 1D kernel with any value and compensate by dividing the other kernel with the same value. Our solution will, in effect, just choose one of the infinite number of (equivalent) solutions.
To extend this concept to nD, what we need to understand is that it is possible to collapse all dimensions except one, obtaining a 2D matrix, and solve the above equation. This results in a 1D kernel and an (n1)D kernel. Repeat the process until all you have is a set of 1D kernels and you're done!
This function is inspired by a solution to this problem that DirkJan Kroon posted on the File Exchange recently: http://www.mathworks.com/matlabcentral/fil...lin1dkernels His solution does the whole decomposition in one go, by setting up one big set of equations. He noted a problem with negative values, which produce complex 1D kernels. The magnitude of the result is correct, but the sign is lost. He needs to resort to some heuristic to determine the sign of each element. What he didn't notice (or didn't mention) is the problem that his solution has with 0 values. The SVD solution doesn't have this problem, although it sometimes does produce a slightly worse solution. For example, in the first example below, DirkJan Kroon's solution is exact, whereas this one produces a very small error. Where DirkJan Kroon's solution cannot find the exact solution, this algorithm generally does better.
EXAMPLES: =========
Simplest 5D example:
H = ones(5,7,4,1,5);
[K1,~,err] = SeparateKernel(H); % D.Kroon's submission to FileEx. err
[k1,~,err] = decompose_kernel(H); err
2D example taken from DirkJan Kroon's submission:
a = permute(rand(4,1),[1 2 3])0.5; b = permute(rand(4,1),[2 1 3])0.5; H = repmat(a,[1 4]).*repmat(b,[4 1]);
[K1,~,err] = SeparateKernel(H); err
[k1,~,err] = decompose_kernel(H); err
2D example for which DirkJan Kroon's solution has problems:
H = [1,2,3,2,1]'*[1,1,3,0,3,1,1]; [K1,~,err] = SeparateKernel(H); err [k1,~,err] = decompose_kernel(H); err
3D example that's not separable:
H = rand(5,5,3);
[K1,~,err] = SeparateKernel(H); err
[k1,~,err] = decompose_kernel(H);
Example to apply a convolution using the decomposed kernel:
img = randn(50,50,50); h = ones(7,7,7); tic; res1 = convn(img,h); toc k1 = decompose_kernel(h); tic; res2 = img; for ii=1:length(k1) res2 = convn(res2,k1{ii}); end toc rms_diff = sqrt(mean((res1(:)res2(:)).^2))
License: Shareware Related: correct, Resort, heuristic, magnitude, Complex, Negative, produce, determine O/S:BSD, Linux, Solaris, Mac OS X File Size: 10.0 KB Downloads: 0

