MATLAB 亚像素边缘提取算法：利用Zernike矩和卷积运算

以下是使用 MATLAB 编写的代码，用于计算 Zernike 矩和边缘提取：

% 计算Zernike矩
function [Z00, Z11, Z20, Z31, Z40] = calculateZernikeMoments(image)
    % 图像大小
    [m, n] = size(image);
    
    % 初始化Zernike矩
    Z00 = 0;
    Z11 = 0;
    Z20 = 0;
    Z31 = 0;
    Z40 = 0;
    
    % 计算Zernike矩
    for i = 1:m
        for j = 1:n
            rho = sqrt((2*i-m-1)^2 + (2*j-n-1)^2) / min(m, n);
            theta = atan2((m-1-2*i+n-1), (2*j-n-1));
            
            R = zernikeRadialPoly(0, 0, rho);
            Z00 = Z00 + image(i, j) * R * exp(-1i * 0 * theta);
            
            R = zernikeRadialPoly(1, 1, rho);
            Z11 = Z11 + image(i, j) * R * exp(-1i * 1 * theta);
            
            R = zernikeRadialPoly(2, 0, rho);
            Z20 = Z20 + image(i, j) * R * exp(-1i * 2 * theta);
            
            R = zernikeRadialPoly(3, 1, rho);
            Z31 = Z31 + image(i, j) * R * exp(-1i * 3 * theta);
            
            R = zernikeRadialPoly(4, 0, rho);
            Z40 = Z40 + image(i, j) * R * exp(-1i * 4 * theta);
        end
    end
end

% Zernike径向多项式
function R = zernikeRadialPoly(n, m, rho)
    R = zeros(size(rho));
    
    for s = 0:(n-abs(m))/2
        c = (-1)^s * factorial(n-s) / (factorial(s) * factorial((n+abs(m))/2-s) * factorial((n-abs(m))/2-s));
        R = R + c * rho.^(n-2*s);
    end
    
    R = R .* sqrt(n+1);
end

% 边缘提取
function [edges_x, edges_y] = extractEdges(image, kt, lt)
    % 图像大小
    [m, n] = size(image);
    
    % 初始化边缘点坐标
    edges_x = [];
    edges_y = [];
    
    for i = 2:(m-1)
        for j = 2:(n-1)
            % 计算Zernike矩
            [Z00, Z11, Z20, Z31, Z40] = calculateZernikeMoments(image(i-1:i+1, j-1:j+1));
            
            % 计算边缘角度
            phi = atan(imag(Z31) / real(Z31));
            
            % 计算边缘角度方向垂直直线边缘
            if abs(phi - pi/2) < 0.1
                % 计算l1和l2
                l1 = sqrt((5*real(Z40)+3*real(Z20)) / (8*real(Z20)));
                l2 = sqrt((5*real(Z31)+real(Z11)) / (6*real(Z11)));
                
                % 计算l
                l = (l1 + l2) / 2;
                
                % 计算k和h
                k = 3*real(Z11) / (2*(1-l2^2)^(3/2));
                h = (real(Z00) - (k*pi)/2 + k*asin(l2) + k*l2*sqrt(1-l2^2)) / pi;
                
                % 判断是否为边缘点
                if k >= kt && abs(l2 - l1) <= lt
                    % 计算亚像素边缘点坐标
                    x = i + l * cos(phi);
                    y = j + l * sin(phi);
                    
                    edges_x = [edges_x, x];
                    edges_y = [edges_y, y];
                end
            end
        end
    end
end

使用示例：

% 读取图像
image = imread('image.png');

% 灰度化
image_gray = rgb2gray(image);

% 提取边缘
kt = 0.1; % k的阈值
lt = 0.1; % l2和l1之差的阈值
[edges_x, edges_y] = extractEdges(image_gray, kt, lt);

% 绘制边缘
imshow(image);
hold on;
plot(edges_y, edges_x, 'r.');
hold off;

请注意，这只是一个简单的示例代码，具体的实现方式可能因不同的需求而有所不同。