function y = convolve2(x, m, shape, tol)%CONVOLVE2 Two dimensional convolution.%   Y = CONVOLVE2(X, M) performs the 2-D convolution of matrices X and%   M. If [mx,nx] = size(X) and [mm,nm] = size(M), then size(Y) =%   [mx+mm-1,nx+nm-1]. Values near the boundaries of the output array are%   calculated as if X was surrounded by a border of zero values.%%   Y = CONVOLVE2(X, M, SHAPE) where SHAPE is a string returns a%   subsection of the 2-D convolution with size specified by SHAPE:%%       'full'    - (default) returns the full 2-D convolution,%       'same'    - returns the central part of the convolution%                   that is the same size as A (using zero padding),%       'valid'   - returns only those parts of the convolution%                   that are computed without the zero-padded%                   edges, size(Y) = [mx-mm+1,nx-nm+1] when%                   size(X) > size(M),%       'wrap'    - as for 'same' except that instead of using%                   zero-padding the input A is taken to wrap round as%                   on a toroid.%       'reflect' - as for 'same' except that instead of using%                   zero-padding the input A is taken to be reflected%                   at its boundaries.%%   CONVOLVE2 is fastest when mx > mm and nx > nm - i.e. the first%   argument is the input and the second is the mask.%%   If the rank of the mask M is low, CONVOLVE2 will decompose it into a%   sum of outer product masks, each of which is applied efficiently as%   convolution with a row vector and a column vector, by calling CONV2.%   The function will often be faster than CONV2 or FILTER2 (in some%   cases much faster) and will produce the same results as CONV2 to%   within a small tolerance.%%   Y = CONVOLVE2(... , TOL) where TOL is a number in the range 0.0 to%   1.0 computes the convolution using a reduced-rank approximation to%   M, provided this will speed up the computation. TOL limits the%   relative sum-squared error in the effective mask; that is, if the%   effective mask is E, the error is controlled such that%%       sum(sum( (M-E) .* (M-E) ))%       --------------------------    <=  TOL%            sum(sum( M .* M ))%%   See also CONV2, FILTER2.% David Young, Department of Informatics, University of Sussex, February 2002,%   revised January 2005.% Deal with optional argumentserror(nargchk(2,4,nargin));if nargin < 3    shape = 'full';    % shape default as for CONV2    tol = 0;elseif nargin < 4    if isnumeric(shape)        tol = shape;        shape = 'full';    else        tol = 0;    endend;% Set up to do the wrap & reflect operations, not handled by conv2if strcmp(shape, 'wrap')    x = wraparound(x, m);    shape = 'valid';elseif strcmp(shape, 'reflect')    x = reflectborders(x, m);    shape = 'valid';end% do the convolution itselfy = doconv(x, m, shape, tol);%-----------------------------------------------------------------------function y = doconv(x, m, shape, tol);% Carry out convolution[mx, nx] = size(x);[mm, nm] = size(m);% If the mask is bigger than the input, or it is 1-D already,% just let CONV2 handle it.if mm > mx | nm > nx | mm == 1 | nm == 1    y = conv2(x, m, shape);else    % Get svd of mask    if mm < nm; m = m'; end        % svd(..,0) wants m > n    [u,s,v] = svd(m, 0);    s = diag(s);    rank = trank(m, s, tol);    if rank*(mm+nm) < mm*nm         % take advantage of low rank        if mm < nm;  t = u; u = v; v = t; end  % reverse earlier transpose        vp = v';        % For some reason, CONV2(H,C,X) is very slow, so use the normal call        y = conv2(conv2(x, u(:,1)*s(1), shape), vp(1,:), shape);        for r = 2:rank            y = y + conv2(conv2(x, u(:,r)*s(r), shape), vp(r,:), shape);        end    else        if mm < nm; m = m'; end     % reverse earlier transpose        y = conv2(x, m, shape);    endend%-----------------------------------------------------------------------function r = trank(m, s, tol)% Approximate rank function - returns rank of matrix that fits given% matrix to within given relative rms error. Expects original matrix% and vector of singular values.if tol < 0 | tol > 1    error('Tolerance must be in range 0 to 1');endif tol == 0             % return estimate of actual rank    tol = length(m) * max(s) * eps;    r = sum(s > tol);else    ss = s .* s;    t = (1 - tol) * sum(ss);    r = 0;    sm = 0;    while sm < t        r = r + 1;        sm = sm + ss(r);    endend%-----------------------------------------------------------------------function y = wraparound(x, m)% Extend x so as to wrap around on both axes, sufficient to allow a% "valid" convolution with m to return the cyclical convolution.% We assume mask origin near centre of mask for compatibility with% "same" option.[mx, nx] = size(x);[mm, nm] = size(m);if mm > mx | nm > nx    error('Mask does not fit inside array')endmo = floor((1+mm)/2); no = floor((1+nm)/2);  % reflected mask originml = mo-1;            nl = no-1;             % mask left/above originmr = mm-mo;           nr = nm-no;            % mask right/below originme = mx-ml+1;         ne = nx-nl+1;          % reflected margin in inputmt = mx+ml;           nt = nx+nl;            % top of image in outputmy = mx+mm-1;         ny = nx+nm-1;          % output sizey = zeros(my, ny);y(mo:mt, no:nt) = x;      % central regionif ml > 0    y(1:ml, no:nt) = x(me:mx, :);                   % top side    if nl > 0        y(1:ml, 1:nl) = x(me:mx, ne:nx);            % top left corner    end    if nr > 0        y(1:ml, nt+1:ny) = x(me:mx, 1:nr);          % top right corner    endendif mr > 0    y(mt+1:my, no:nt) = x(1:mr, :);                 % bottom side    if nl > 0        y(mt+1:my, 1:nl) = x(1:mr, ne:nx);          % bottom left corner    end    if nr > 0        y(mt+1:my, nt+1:ny) = x(1:mr, 1:nr);        % bottom right corner    endendif nl > 0    y(mo:mt, 1:nl) = x(:, ne:nx);                   % left sideendif nr > 0    y(mo:mt, nt+1:ny) = x(:, 1:nr);                 % right sideend%-----------------------------------------------------------------------function y = reflectborders(x, m)% Extend x so as to reflect at each boundary, sufficient to allow a% "valid" convolution with m to return a matrix the same size as% the orginal.% We assume mask origin near centre of mask for compatibility with% "same" option.[mx, nx] = size(x);[mm, nm] = size(m);if mm > mx | nm > nx    error('Mask does not fit inside array')endmo = floor((1+mm)/2); no = floor((1+nm)/2);  % reflected mask originml = mo-1;            nl = no-1;             % mask left/above originmr = mm-mo;           nr = nm-no;            % mask right/below originme = mx-mr+1;         ne = nx-nr+1;          % translated margin in inputmt = mx+ml;           nt = nx+nl;            % top/right of image in outputmy = mx+mm-1;         ny = nx+nm-1;          % output sizey = zeros(my, ny);y(mo:mt, no:nt) = x;      % central regionif ml > 0    y(1:ml, no:nt) = x(ml:-1:1, :);                   % top side    if nl > 0        y(1:ml, 1:nl) = x(ml:-1:1, nl:-1:1);          % top left corner    end    if nr > 0        y(1:ml, nt+1:ny) = x(ml:-1:1, nx:-1:ne);      % top right corner    endendif mr > 0    y(mt+1:my, no:nt) = x(mx:-1:me, :);               % bottom side    if nl > 0        y(mt+1:my, 1:nl) = x(mx:-1:me, nl:-1:1);      % bottom left corner    end    if nr > 0        y(mt+1:my, nt+1:ny) = x(mx:-1:me, nx:-1:ne);  % bottom right corner    endendif nl > 0    y(mo:mt, 1:nl) = x(:, nl:-1:1);                   % left sideendif nr > 0    y(mo:mt, nt+1:ny) = x(:, nx:-1:ne);               % right sideend