% edgeNL: Edge detection function based on nonlinear derivatives
% (elementary demo):
% This function differientiates the image to estimate the gradient.
% The principle is based on polarized derivatives to automatically select
% the best edge localization.
% The main benefits are
% - univocal edge localization for synthetic and real images
% - noise reduction with no regularization: the noise level is weaker than
% the noise level in the original image
% - better direction estimation of the gradient
% - can product a confident edge reference map for synthetic images
% - extremely efficient on salt noise OR pepper noise (this last case needs
% a change in the nonlinear derivatives)
% - still noise reduction with regularized schemes (Canny, Demigny, ...,
% can also be adapted to the asymetrical filters (Prewitt, Sobel, ...)
% Drawback
% - no detection of vertical and horizontal "white" thin (1 pixel) lines
% Rk: this demo only performs edge detection and does not include edge
% extraction (local maxima) and other steps to obtain a binary
% edge map.
% Written by O. Laligant - 2009, University of Burgundy
% Ref: A Nonlinear Derivative Scheme Applied to Edge Detection,
% Olivier Laligant, Frederic Truchetet, IEEE Transactions on Pattern
% Analysis and Machine Intelligence - PAMI , vol. 32, no. 2,
% pp. 242-257, 2010
function edgeNL()
% ------------- Edge detection on a simple synthetic image -------------
Im = example();
figure(1), imagesc(Im);
title ('Original image');
% edge detection (nonlinear gradient estimate)
% better localization and direction estimation than the linear scheme
gI = algoNL(Im);
% For a synthetic object, the edges are localized inside the object shape
figure(2), imagesc(gI);
title('Edge detection');
% ------------- Additive gaussian white noise ------------------------
Imn = imnoise(Im, 'gaussian', 0, 0.01);
figure(3), imagesc(Imn);
title ('Noisy image');
gIn = algoNL(Imn);
figure(4), imagesc(gIn);
title('Edge detection on the noisy image');
end
% ----------------------------------------------------------------------------
%
% --------------------------- functions -------------------------------------
%
% ------------- computation of the nonlinear derivatives -------------
% polarized derivatives
% lead to an univocal localization of edges
function [gm, gh, gv] = algoNL(I);
% for classical regularization schemes, I can be replaced by a regularized
% version
% for asymetrical schemes (Prewitt, Sobel, etc), I must be replaced by two
% regularized versions (the regularization is different on columns and rows)
dph = thresh0(conv2(I, [0 1 -1], 'same'));
dnh = -thresh0(-conv2(I, [1 -1 0], 'same'));
gh = dph+dnh;
dpv = thresh0(conv2(I, [0; 1; -1], 'same'));
dnv = -thresh0(-conv2(I, [1; -1; 0], 'same'));
gv = dpv + dnv;
gm = sqrt(gh.*gh + gv.*gv);
end
% ------------- threshold -------------
function st = thresh0(s)
st = s.*(sign(s)+1)/2;
end%function
% ------------- test image -------------
% an example of image to illustrate the localization of the method
function Im = example()
Im =[
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 255 255 255 0 0 0 0 0 0 0 255 0 0
0 0 255 255 255 0 0 0 0 0 0 255 255 0 0
0 0 255 255 255 0 0 0 0 0 255 255 255 0 0
0 0 0 0 0 0 0 0 0 255 255 255 255 0 0
0 0 0 0 0 0 0 0 255 255 255 255 255 0 0
0 0 0 0 0 0 0 255 255 255 255 0 0 0 0
0 0 0 0 0 0 255 255 255 255 0 0 0 0 0
0 0 0 0 0 255 255 255 255 255 0 0 0 0 0
0 0 0 0 255 255 255 0 255 255 0 0 0 0 0
0 0 0 255 255 255 0 0 0 255 255 0 0 0 0
0 0 255 255 255 255 255 0 255 255 255 255 255 0 0
0 0 0 255 255 255 255 255 255 255 255 255 0 0 0
0 0 0 0 255 255 255 255 255 255 255 255 255 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
];
end
