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What makes a convolution kernel separable? How would I be able to tell what those separable parts were in order to do two 1D convolutions instead of a 2D convolution>
Thanks
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A two-dimensional filter kernel is separable if it can be expressed as the outer product of two vectors.
A Separable Convolution is a process in which a single convolution can be divided into two or more convolutions to produce the same output. A single process is divided into two or more sub-processes to achieve the same effect.
A separable kernel gives separate control of the frequency-smoothing and time-smoothing of the WVD which is an improvement over the spectrogram which does not have flexibility to independently adjust smoothing along the time and/or the frequency axis [62].
Separable filters are one of the most useful tools in image processing and they can turn algorithms from “theoretical and too expensive” to practical under the same computational constraints.
If the 2D filter kernel has a rank of 1 then it is separable. You can test this in e.g. Matlab or Octave:
octave-3.2.3:1> sobel = [-1 0 1 ; -2 0 2 ; -1 0 1];
octave-3.2.3:2> rank(sobel)
ans = 1
octave-3.2.3:3>
See also: http://blogs.mathworks.com/steve/2006/11/28/separable-convolution-part-2/ - this covers using SVD (Singular Value Decomposition) to extract the two 1D kernels from a separable 2D kernel.
See also this question on DSP.stackexchange.com: Fast/efficient way to decompose separable integer 2D filter coefficients
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you can also split the matrix into symmetric and skew parts and separate each part, which can be effective for larger 2d convolutions.
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