Extremely fast 2D image convolution with a box filter.
Convolves an image by a F=ones(2*r+1,2*r+1)/(2*r+1)^2 filter. The
convolution can be performed in constant time per-pixel, independent of
the radius r. In fact the implementation is nearly optimal, with the
convolution taking only slightly more time than creating a copy of the
input array. Boundary effects are handled as if the image were padded
symmetrically prior to performing the convolution. An optional integer
downsampling parameter "s" can be specified, in which case the output is
downsampled by s (the implementation is efficient with downsampling
occurring simultaneously with smoothing, saving additional time).
The output is exactly equivalent to the following Matlab operations:
f = ones(1,2*r+1); f=f/sum(f);
J = padarray(I,[r r],'symmetric','both');
J = convn(convn(J,f,'valid'),f','valid');
if(s>1), t=floor(s/2)+1; J=J(t:s:end-s+t,t:s:end-s+t,:); end
The computation, however, is an order of magnitude faster than the above.
When used as a smoothing filter, the standard deviation (sigma) of a box
filter with radius r can be computed using [sigma=sqrt(r*(r+1)/3)]. For
the first few values of r this translates to: r=1: sigma=sqrt(2/3), r=2:
sigma=sqrt(2), r=3: sigma=2. Given sigma, the equivalent value of r can
be computed via [r=sqrt(12*sigma*sigma+1)/2-.5].
The related function convTri performs convolution with a triangle filter,
which has nicer properties if used for smoothing, but is slightly slower.
This code requires SSE2 to compile and run (most modern Intel and AMD
processors support SSE2). Please see: http://en.wikipedia.org/wiki/SSE2.
USAGE
J = convBox( I, r, [s], [nomex] )
INPUTS
I - [hxwxk] input k channel single image
r - integer filter radius
s - [1] integer downsampling amount after convolving
nomex - [0] if true perform computation in matlab (for testing/timing)
OUTPUTS
J - [hxwxk] smoothed image
EXAMPLE
I = single(imResample(imread('cameraman.tif'),[480 640]))/255;
r = 5; s = 2; % set parameters as desired
tic, J1=convBox(I,r,s); toc % mex version (fast)
tic, J2=convBox(I,r,s,1); toc % matlab version (slow)
figure(1); im(J1); figure(2); im(abs(J2-J1));
See also conv2, convTri
Piotr's Computer Vision Matlab Toolbox Version 3.02
Copyright 2014 Piotr Dollar & Ron Appel. [pdollar-at-gmail.com]
Licensed under the Simplified BSD License [see external/bsd.txt]