This function is an implementation of N-dimensional convolution for the special case when the convolution kernel is separable.
A continuous function f(x1, x2, ... xN) is considered separable if there exist N functions f1, f2, ... fN such that f(x1, x2, ... xN) = f1(x1)f2(x2)f3(x3)...fN(xN).
In two dimensions, a discrete version of the separability condition is as follows: a 2D matrix is separable if it can be expressed as an outer product of two 1D vectors:
a=[-1 0 1]; b=[1 0 -1]; H=a'*b
In this case, supplying Matlab's conv2 function with the two vectors a & b (one to convolve the rows with, the second to convolve the columns with) results in a faster computation than supplying conv2 with the outer product H.
Convnsep.m extends this functionality to an arbitrary number of dimensions.
This may be useful for applications such as:
1. Smoothing a 3D image with separable filters
2. Bilateral filtering (requires 4D convolution for filtering a 3D image)
Notes and caveats:
- As the total size of the variable being convolved increases towards the largest size allowable by Matlab (platform/architecture dependent), performance significantly degrades due to memory limitations, and may become worse than that of convn. Because Matlab requires a contiguous block of memory to allocate space for a variable, the maximum size for which performance is reasonable may be significantly lower than the "theoretical" size limit (which is 1.5 GB on my platform).
- For convolution types other than 'full' (no discarding of any samples), only odd-sized kernels are currently working properly.
Igor Solovey (2021). Separable N-Dimensional Convolution (https://www.mathworks.com/matlabcentral/fileexchange/27957-separable-n-dimensional-convolution), MATLAB Central File Exchange. Retrieved .
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