Fast way to implement 2D convolution in C

Question

I am trying to implement a vision algorithm, which includes a prefiltering stage with a 9x9 Laplacian-of-Gaussian filter. Can you point to a document which explains fast filter implementations briefly? I think I should make use of FFT for most efficient filtering.

Answer 1

Are you sure you want to use FFT? That will be a whole-array transform, which will be expensive. If you've already decided on a 9x9 convolution filter, you don't need any FFT.

Generally, the cheapest way to do convolution in C is to set up a loop that moves a pointer over the array, summing the convolved values at each point and writing the data to a new array. This loop can then be parallelised using your favourite method (compiler vectorisation, MPI libraries, OpenMP, etc).

Regarding the boundaries:

• If you assume the values to be 0 outside the boundaries, then add a 4 element border of 0 to your 2d array of points. This will avoid the need for `if` statements to handle the boundaries, which are expensive.
• If your data wraps at the boundaries (ie it is periodic), then use a modulo or add a 4 element border which copies the opposite side of the grid (abcdefg -> fgabcdefgab for 2 points). **Note: this is what you are implicitly assuming with any kind of Fourier transform, including FFT**. If that is not the case, you would need to account for it before any FFT is done.

The 4 points are because the maximum boundary overlap of a 9x9 kernel is 4 points outside the main grid. Thus, n points of border needed for a 2n+1 x 2n+1 kernel.

If you need this convolution to be really fast, and/or your grid is large, consider partitioning it into smaller pieces that can be held in the processor's cache, and thus calculated far more quickly. This also goes for any GPU-offloading you might want to do (they are ideal for this type of floating-point calculation).

Answer 2

Here is a theory link http://hebb.mit.edu/courses/9.29/2002/readings/c13-1.pdf

And here is a link to fftw, which is a pretty good FFT library that I've used in the past (check licenses to make sure it is suitable) http://www.fftw.org/

All you do is FFT your image and kernel (the 9x9 matrix). Multiply together, then back transform.

However, with a 9x9 matrix you may still be better doing it in real coordinates (just with a double loop over the image pixels and the matrix). Try both ways!

Answer 3

Actually you don't need to use a FFT size large enough to hold the entire image. You can do a lot of smaller overlapping 2d ffts. You can search for "fast convolution" "overlap save" "overlap add".

However, for a 9x9 kernel. You may not see much advantage speedwise.