matlab zernike矩

从一篇论文里面看到了zernike矩，一直不懂矩是个什么东西，在网上找了一些资料，记录一下
在网上看到的一篇博客里面，理论部分归纳得很好
https://www.cnblogs.com/ronny/p/3985810.html

zernike矩的matlab 代码来自https://www.mathworks.com/matlabcentral/fileexchange/38900-zernike-moments
具体代码如下：
Zernikmoment(p,n,m)的代码

function [Z, A, Phi] = Zernikmoment(p,n,m)
% -------------------------------------------------------------------------
% Copyright C 2014 Amir Tahmasbi
% Texas A&M University
% amir.tahmasbi@tamu.edu
% http://people.tamu.edu/~amir.tahmasbi/index.html
%
% License Agreement: To acknowledge the use of the code please cite the
%                    following papers:
%
% [1] A. Tahmasbi, F. Saki, S. B. Shokouhi,
%     Classification of Benign and Malignant Masses Based on Zernike Moments,
%     Comput. Biol. Med., vol. 41, no. 8, pp. 726-735, 2011.
%
% [2] F. Saki, A. Tahmasbi, H. Soltanian-Zadeh, S. B. Shokouhi,
%     Fast opposite weight learning rules with application in breast cancer
%     diagnosis, Comput. Biol. Med., vol. 43, no. 1, pp. 32-41, 2013.
%
% -------------------------------------------------------------------------
% Function to find the Zernike moments for an N x N binary ROI
%
% [Z, A, Phi] = Zernikmoment(p,n,m)
% where
%   p = input image N x N matrix (N should be an even number)
%   n = The order of Zernike moment (scalar)
%   m = The repetition number of Zernike moment (scalar)
% and
%   Z = Complex Zernike moment
%   A = Amplitude of the moment
%   Phi = phase (angle) of the mement (in degrees)
%
% Example:
%   1- calculate the Zernike moment (n,m) for an oval shape,
%   2- rotate the oval shape around its centeroid,
%   3- calculate the Zernike moment (n,m) again,
%   4- the amplitude of the moment (A) should be the same for both images
%   5- the phase (Phi) should be equal to the angle of rotation
N = size(p,1);
x = 1:N; y = x;
[X,Y] = meshgrid(x,y);
R = sqrt((2.*X-N-1).^2+(2.*Y-N-1).^2)/N;
Theta = atan2((N-1-2.*Y+2),(2.*X-N+1-2));
R = (R<=1).*R;
Rad = radialpoly(R,n,m);    % get the radial polynomial
Product = p(x,y).*Rad.*exp(-1i*m*Theta);
Z = sum(Product(:));        % calculate the moments
cnt = nnz(R)+1;             % count the number of pixels inside the unit circle
Z = (n+1)*Z/cnt;            % normalize the amplitude of moments
A = abs(Z);                 % calculate the amplitude of the moment
Phi = angle(Z)*180/pi;      % calculate the phase of the mement (in degrees)

其中需要调用radialpoly(r,n,m)，代码如下
radialpoly(r,n,m)

function rad = radialpoly(r,n,m)
% -------------------------------------------------------------------------
% Copyright C 2014 Amir Tahmasbi
% Texas A&M University
% amir.tahmasbi@tamu.edu
% http://people.tamu.edu/~amir.tahmasbi/index.html
%
% License Agreement: To acknowledge the use of the code please cite the
%                    following papers:
%
% [1] A. Tahmasbi, F. Saki, S. B. Shokouhi,
%     Classification of Benign and Malignant Masses Based on Zernike Moments,
%     Comput. Biol. Med., vol. 41, no. 8, pp. 726-735, 2011.
%
% [2] F. Saki, A. Tahmasbi, H. Soltanian-Zadeh, S. B. Shokouhi,
%     Fast opposite weight learning rules with application in breast cancer
%     diagnosis, Comput. Biol. Med., vol. 43, no. 1, pp. 32-41, 2013.
%
% -------------------------------------------------------------------------
% Function to compute Zernike Polynomials:
%
% f = radialpoly(r,n,m)
% where
%   r = radius
%   n = the order of Zernike polynomial
%   m = the repetition of Zernike moment
rad = zeros(size(r));                     % Initilization
for s = 0:(n-abs(m))/2c = (-1)^s*factorial(n-s)/(factorial(s)*factorial((n+abs(m))/2-s)*...factorial((n-abs(m))/2-s));rad = rad + c*r.^(n-2*s);
end

测试zernike矩的代码如下
Zernike_main.m

% -------------------------------------------------------------------------
% Copyright C 2014 Amir Tahmasbi
% Texas A&M University
% amir.tahmasbi@tamu.edu
% http://people.tamu.edu/~amir.tahmasbi/index.html
%
% License Agreement: To acknowledge the use of the code please cite the
%                    following papers:
%
% [1] A. Tahmasbi, F. Saki, S. B. Shokouhi,
%     Classification of Benign and Malignant Masses Based on Zernike Moments,
%     Comput. Biol. Med., vol. 41, no. 8, pp. 726-735, 2011.
%
% [2] F. Saki, A. Tahmasbi, H. Soltanian-Zadeh, S. B. Shokouhi,
%     Fast opposite weight learning rules with application in breast cancer
%     diagnosis, Comput. Biol. Med., vol. 43, no. 1, pp. 32-41, 2013.
%
% -------------------------------------------------------------------------
% A demo of how to use the Zernike moment function.
%
% Example:
%   1- calculate the Zernike moment (n,m) for an oval shape,
%   2- rotate the oval shape around its centeroid,
%   3- calculate the Zernike moment (n,m) again,
%   4- the amplitude of the moment (A) should be the same for both images
%   5- the phase (Phi) should be equal to the angle of rotation
clc; clear all; close all;
n = 4; m = 2;           % Define the order and the repetition of the moment
disp('------------------------------------------------');
disp(['Calculating Zernike moments ..., n = ' num2str(n) ', m = ' num2str(m)]);
%--------------------------------------------------------------------------
% row 1
p = rgb2gray(imread('Oval_H.png'));
figure(1);subplot(2,3,1);imshow(p);
title('Horizontal oval');
p = logical(not(p));
tic
[~, AOH, PhiOH] = Zernikmoment(p,n,m);      % Call Zernikemoment fuction
Elapsed_time = toc;
xlabel({['A = ' num2str(AOH)]; ['\phi = ' num2str(PhiOH)]});
p = rgb2gray(imread('Oval_45.png'));
figure(1);subplot(2,3,2);imshow(p);
title('-45 degree oval');
p = logical(not(p));
[~, AOH, PhiOH] = Zernikmoment(p,n,m);      % Call Zernikemoment fuction
xlabel({['A = ' num2str(AOH)]; ['\phi = ' num2str(PhiOH)]});
p = rgb2gray(imread('Oval_V.png'));
figure(1);subplot(2,3,3);imshow(p);
title('Vertical oval');
p = logical(not(p));
[~, AOH, PhiOH] = Zernikmoment(p,n,m);      % Call Zernikemoment fuction
xlabel({['A = ' num2str(AOH)]; ['\phi = ' num2str(PhiOH)]});
%--------------------------------------------------------------------------
% row 2
p = rgb2gray(imread('shape_0.png'));
figure(1);subplot(2,3,4);imshow(p);
title('Horizontal shape');
p = logical(not(p));
[~, AOH, PhiOH] = Zernikmoment(p,n,m);      % Call Zernikemoment fuction
xlabel({['A = ' num2str(AOH)]; ['\phi = ' num2str(PhiOH)]});
p = rgb2gray(imread('shape_90.png'));
figure(1);subplot(2,3,5);imshow(p);
title('Vertical shape');
p = logical(not(p));
[~, AOV, PhiOV] = Zernikmoment(p,n,m);      % Call Zernikemoment fuction
xlabel({['A = ' num2str(AOV)]; ['\phi = ' num2str(PhiOV)]});
p = rgb2gray(imread('Rectangular_H.png'));
figure(1);subplot(2,3,6);imshow(p);
title('Horizontal Rectangle');
p = logical(not(p));
[~, AOH, PhiOH] = Zernikmoment(p,n,m);      % Call Zernikemoment fuction
xlabel({['A = ' num2str(AOH)]; ['\phi = ' num2str(PhiOH)]});
%--------------------------------------------------------------------------
% show the elapsed time
disp('Calculation is complete.');
disp(['The elapsed time per image is ' num2str(Elapsed_time) ' seconds']);

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