Extremely fast 2D image convolution with a triangle filter.


function J = convTri( I, r, s, nomex )


 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:

  J = convTri( I, r, [s], [nomex] )

  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)

  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.  []
 Licensed under the Simplified BSD License [see external/bsd.txt]

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