好的,以下是一个MATLAB程序,用于实现所述的法向量加权最小二乘法。需要注意的是,在使用这段代码之前,必须首先定义离散点坐标和法向量,以及多项式的阶数。
- % 定义离散点坐标和法向量
- x = [1,2,3,4,5];
- y = [1,2,3,4,5];
- z = [1,2,3,4,5];
- nx = [0.2,-0.4,0.6,-0.8,1];
- ny = [1.2,-1.4,1.6,-1.8,2];
- nz = [2.2,-2.4,2.6,-2.8,3];
- % 多项式阶数
- order = 2;
- % 构造矩阵A和向量b
- n = length(x);
- m = (order+1)*(order+2)/2; % 计算系数数量
- A = zeros(n,m);
- b = zeros(n,1);
- for i = 1:n
- % 构造A矩阵
- row = 1;
- for j = 0:order
- for k = 0:(order-j)
- if ((j+k)<=order)
- A(i,row) = x(i)^j*y(i)^k;
- row = row+1;
- end
- end
- end
- % 构造b向量
- b(i) = z(i)-nx(i)*A(i,1)-(ny(i)*A(i,2))-(nz(i)*A(i,3));
- end
- % 加权最小二乘法求解系数向量
- W = diag([nx',ny',nz']);
- C = A'*W*A;
- D = A'*W*b;
- c = C\D;
- % 构造自由曲面
- [X,Y] = meshgrid(1:0.1:5,1:0.1:5);
- Z = zeros(size(X));
- for i = 1:(order+1)
- for j = 1:(order+1-i)
- k = (order+1-i-j+1);
- coef = c((i-1)*j+k);
- Z = Z + coef*X.^(i-1).*Y.^(j-1);
- end
- end
- % 绘制拟合曲面和离散点
- figure;
- scatter3(x,y,z,'filled','b');
- hold on;
- surf(X,Y,Z,'EdgeColor','none','FaceAlpha',0.5);
- xlabel('X');
- ylabel('Y');
- zlabel('Z');
- title('Polynomial Freeform Surface Fitting');
复制代码
上述代码中的过程包括:
- 构造矩阵A和向量b
- 利用加权最小二乘法求解系数向量c
- 构造拟合的自由曲面
- 绘制自由曲面和离散点的图形。
可以根据需要自行调整参数,例如多项式的阶数,以得到更好的拟合效果。
--- 光学专家Gpt |