MATLAB亚像素边缘提取：基于Zernike矩的边缘检测算法

以下是MATLAB中使用Zernike矩进行边缘检测，并进行亚像素边缘提取的代码：

% 计算Zernike矩7 × 7模板
M = 7; % 模板大小
n = (M+1)/2; % 中心位置
[x, y] = meshgrid(1:M, 1:M);
rho = sqrt((x-n).^2 + (y-n).^2) / n;
theta = atan2((y-n), (x-n));

Z = zeros(M, M, 5);
Z(:,:,1) = 1; % M00
Z(:,:,2) = rho .* cos(theta); % M11
Z(:,:,3) = (2*rho.^2 - 1); % M20
Z(:,:,4) = rho .* cos(theta) .* (3*rho.^2 - 2); % M31
Z(:,:,5) = (6*rho.^4 - 6*rho.^2 + 1); % M40

% 读取图像
image = imread('image.jpg');
image = double(rgb2gray(image));

% 初始化边缘点坐标
edge_points = [];

% 遍历图像的每个像素点
for i = n+1:size(image, 1)-n
    for j = n+1:size(image, 2)-n
        % 计算Zernike矩
        Z_prime = zeros(1, 5);
        for k = 1:5
            Z_prime(k) = sum(sum(Z(:,:,k) .* image(i-n:i+n, j-n:j+n)));
        end
        
        % 计算边缘角度
        phi = atan2(imag(Z_prime(4)), real(Z_prime(4)));
        
        % 计算l1和l2
        l1 = sqrt((5*real(Z_prime(5)) + 3*real(Z_prime(3))) / (8*real(Z_prime(3))));
        l2 = sqrt((5*real(Z_prime(4)) + real(Z_prime(2))) / (6*real(Z_prime(2))));
        
        % 计算l
        l = (l1 + l2) / 2;
        
        % 计算k和h
        k = 3 * real(Z_prime(2)) / (2 * (1 - l2^2)^(3/2));
        h = (real(Z_prime(1)) - (k*pi)/2 + k * asin(l2) + k * l2 * sqrt(1 - l2^2)) / pi;
        
        % 判断是否为边缘点
        kt = 0.1; % k的阈值
        lt = 0.1; % l的阈值
        if k >= kt && abs(l2 - l1) <= lt
            edge_points = [edge_points; j, i, phi, h, k];
        end
    end
end

% 边缘检测
edge_image = zeros(size(image));
for i = 1:size(edge_points, 1)
    x = edge_points(i, 1);
    y = edge_points(i, 2);
    edge_image(y, x) = 255;
end

% 显示结果
figure;
subplot(1, 2, 1);
imshow(uint8(image));
title('原始图像');
subplot(1, 2, 2);
imshow(uint8(edge_image));
title('边缘检测结果');

请注意，此代码中的图像路径应根据实际情况进行修改。此外，代码中的阈值kt和lt也需要根据具体需求进行调整。