% DERIVATIVE5 - 5-Tap 1st and 2nd discrete derivatives
%
% This function computes 1st and 2nd derivatives of an image using the 5-tap
% coefficients given by Farid and Simoncelli. The results are significantly
% more accurate than MATLAB's GRADIENT function on edges that are at angles
% other than vertical or horizontal. This in turn improves gradient orientation
% estimation enormously. If you are after extreme accuracy try using DERIVATIVE7.
%
% Usage: [gx, gy, gxx, gyy, gxy] = derivative5(im, derivative specifiers)
%
% Arguments:
% im - Image to compute derivatives from.
% derivative specifiers - A comma separated list of character strings
% that can be any of 'x', 'y', 'xx', 'yy' or 'xy'
% These can be in any order, the order of the
% computed output arguments will match the order
% of the derivative specifier strings.
% Returns:
% Function returns requested derivatives which can be:
% gx, gy - 1st derivative in x and y
% gxx, gyy - 2nd derivative in x and y
% gxy - 1st derivative in y of 1st derivative in x
%
% Examples:
% Just compute 1st derivatives in x and y
% [gx, gy] = derivative5(im, 'x', 'y');
%
% Compute 2nd derivative in x, 1st derivative in y and 2nd derivative in y
% [gxx, gy, gyy] = derivative5(im, 'xx', 'y', 'yy')
%
% See also: DERIVATIVE7
% Reference: Hany Farid and Eero Simoncelli "Differentiation of Discrete
% Multi-Dimensional Signals" IEEE Trans. Image Processing. 13(4): 496-508 (2004)
% Copyright (c) 2010 Peter Kovesi
% Centre for Exploration Targeting
% The University of Western Australia
% peterkovesi.com
%
% Permission is hereby granted, free of charge, to any person obtaining a copy
% of this software and associated documentation files (the "Software"), to deal
% in the Software without restriction, subject to the following conditions:
%
% The above copyright notice and this permission notice shall be included in
% all copies or substantial portions of the Software.
%
% The Software is provided "as is", without warranty of any kind.
%
% April 2010
function varargout = derivative5(im, varargin)
varargin = varargin(:);
varargout = cell(size(varargin));
% Check if we are just computing 1st derivatives. If so use the
% interpolant and derivative filters optimized for 1st derivatives, else
% use 2nd derivative filters and interpolant coefficients.
% Detection is done by seeing if any of the derivative specifier
% arguments is longer than 1 char, this implies 2nd derivative needed.
secondDeriv = false;
for n = 1:length(varargin)
if length(varargin{n}) > 1
secondDeriv = true;
break
end
end
if ~secondDeriv
% 5 tap 1st derivative cofficients. These are optimal if you are just
% seeking the 1st deriavtives
p = [0.037659 0.249153 0.426375 0.249153 0.037659];
d1 =[0.109604 0.276691 0.000000 -0.276691 -0.109604];
else
% 5-tap 2nd derivative coefficients. The associated 1st derivative
% coefficients are not quite as optimal as the ones above but are
% consistent with the 2nd derivative interpolator p and thus are
% appropriate to use if you are after both 1st and 2nd derivatives.
p = [0.030320 0.249724 0.439911 0.249724 0.030320];
d1 = [0.104550 0.292315 0.000000 -0.292315 -0.104550];
d2 = [0.232905 0.002668 -0.471147 0.002668 0.232905];
end
% Compute derivatives. Note that in the 1st call below MATLAB's conv2
% function performs a 1D convolution down the columns using p then a 1D
% convolution along the rows using d1. etc etc.
gx = false;
for n = 1:length(varargin)
if strcmpi('x', varargin{n})
varargout{n} = conv2(p, d1, im, 'same');
gx = true; % Record that gx is available for gxy if needed
gxn = n;
elseif strcmpi('y', varargin{n})
varargout{n} = conv2(d1, p, im, 'same');
elseif strcmpi('xx', varargin{n})
varargout{n} = conv2(p, d2, im, 'same');
elseif strcmpi('yy', varargin{n})
varargout{n} = conv2(d2, p, im, 'same');
elseif strcmpi('xy', varargin{n}) || strcmpi('yx', varargin{n})
if gx
varargout{n} = conv2(d1, p, varargout{gxn}, 'same');
else
gx = conv2(p, d1, im, 'same');
varargout{n} = conv2(d1, p, gx, 'same');
end
else
error('''%s'' is an unrecognized derivative option',varargin{n});
end
end