Extremely fast 2D image convolution with a triangle filter. Convolves an image by a 2D triangle filter (the 1D triangle filter f is [1:r r+1 r:-1:1]/(r+1)^2, the 2D version is simply conv2(f,f')). 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 = [1:r r+1 r:-1:1]/(r+1)^2; 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 tri filter with radius r can be computed using [sigma=sqrt(r*(r+2)/6)]. For the first few values of r this translates to: r=1: sigma=1/sqrt(2), r=2: sigma=sqrt(4/3), r=3: sqrt(5/2), r=4: sigma=2. Given sigma, the equivalent value of r can be computed via [r=sqrt(6*sigma*sigma+1)-1]. For even finer grained control for very small amounts of smoothing, any value of r between 0 and 1 can be used (normally if r>=1 then r must be an integer). In this case a filter of the form fp=[1 p 1]/(2+p) is used, with p being determined automatically from r. The filter fp has a standard deviation of [sigma=sqrt(2/(p+2))]. Hence p can be determined from r by setting [sqrt(r*(r+2)/6)=sqrt(2/(p+2))], which gives [p=12/r/(r+2)-2]. Note that r=1 gives p=2, so fp=[1 2 1]/4 which is the same as the normal r=1 triangle filter. As r goes to 0, p goes to infinity, and fp becomes the delta function [0 1 0]. The computation for r<=1 is particularly fast. The related function convBox performs convolution with a box filter, which is slightly faster but has worse properties if used for smoothing. 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 = convTri( I, r, [s], [nomex] ) INPUTS I - [hxwxk] input k channel single image r - integer filter radius (or any value between 0 and 1) filter standard deviation is: sigma=sqrt(r*(r+2)/6) s - [1] integer downsampling amount after convolving nomex - [0] if true perform computation in matlab (for testing/timing) OUTPUTS J - [hxwxk] smoothed image EXAMPLE - matlab versus mex I = single(imResample(imread('cameraman.tif'),[480 640]))/255; r = 5; s = 2; % set parameters as desired tic, J1=convTri(I,r,s); toc % mex version (fast) tic, J2=convTri(I,r,s,1); toc % matlab version (slow) figure(1); im(J1); figure(2); im(abs(J2-J1)); EXAMPLE - triangle versus gaussian smoothing I = single(imResample(imread('cameraman.tif'),[480 640]))/255; sigma = 4; rg = ceil(3*sigma); f = filterGauss(2*rg+1,[],sigma^2); tic, J1=conv2(conv2(imPad(I,rg,'symmetric'),f,'valid'),f','valid'); toc r=sqrt(6*sigma*sigma+1)-1; tic, J2=convTri(I,r); toc figure(1); im(J1); figure(2); im(J2); figure(3); im(abs(J2-J1)); See also conv2, convBox, gaussSmooth 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]

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