泽尼克多项式是一个正交多项式,分为奇偶两类。
奇多项式:
偶多项式:
其中:
这里fai为方位角,范围[0-2pi];p为径向距离,范围[0,1];n-m大于等于0;
如果n-m=0,则R=0。
根据不同的m和n值,可以得到不同的多项式,用j表示不同的多项式,通常称为Noll序列:
n,m | 0,0 | 1,1 | 1,−1 | 2,0 | 2,−2 |
---|---|---|---|---|---|
j | 1 | 2 | 3 | 4 | 5 |
n,m | 2,2 | 3,−1 | 3,1 | 3,−3 | 3,3 |
j | 6 | 7 | 8 | 9 | 10 |
n,m | 4,0 | 4,2 | 4,−2 | 4,4 | 4,−4 |
j | 11 | 12 | 13 | 14 | 15 |
n,m | 5,1 | 5,−1 | 5,3 | 5,−3 | 5,5 |
j | 16 | 17 | 18 | 19 | 20 |
Noll序列前15项泽尼克多项式为:
1 | Piston | |
2 | Tip (lateral position) (X-Tilt) | |
3 | Tilt (lateral position) (Y-Tilt) | |
4 | Defocus (longitudinal position) | |
5 | Astigmatism | |
6 | Astigmatism | |
7 | Coma | |
8 | Coma | |
9 | Trefoil | |
10 | Trefoil | |
11 | Third-order spherical | |
12 | — | |
13 | — | |
14 | — | |
15 | — |
下面用15阶泽尼克多项式拟合了一个高斯函数。
matlab代码如下:
clear all;close all;clc; [x,y]=meshgrid(-100:100); sigma=50; target_img = 10000*(1/(2*pi*sigma^2))*exp(-(x.^2+y.^2)/(2*sigma^2)); B = target_img(:); srcA = zeros(length(B),15); for i=1:15 [x,y] = meshgrid(-1:0.01:1); p = sqrt(x.^2+y.^2); theta = atan2(y,x); img = zernike_polynomials(i,p,theta); img(p>1) = nan; subplot(3,5,i); imagesc(img); srcA(:,i) = img(:); ax=gca; ax.XAxis.Visible='off'; ax.YAxis.Visible='off'; end A = srcA; B(isnan(srcA(:,1))) = []; A(isnan(srcA(:,1)),:) = []; x = inv(A'*A)*A'*B; fit_img = zeros(size(target_img)); for i=1:15 fit_img = fit_img + x(i)*reshape(srcA(:,i),size(target_img)); end figure; mesh(target_img); hold on; surf(fit_img); function re = zernike_polynomials(n,p,fai) switch n case 1 re = ones(size(p)); case 2 re = 2*p.*sin(fai); case 3 re = 2*p.*cos(fai); case 4 re = sqrt(6)*p.^2.*sin(2*fai); case 5 re = sqrt(3)*(2*p.^2-1); case 6 re = sqrt(6)*p.^2.*cos(2*fai); case 7 re = sqrt(8)*p.^3.*sin(3*fai); case 8 re = sqrt(8)*(3*p.^3-2*p).*sin(fai); case 9 re = sqrt(8)*(3*p.^3-2*p).*cos(fai); case 10 re = sqrt(8)*p.^3.*cos(3*fai); case 11 re = sqrt(10)*p.^4.*sin(4*fai); case 12 re = sqrt(10)*(4*p.^4-3*p.^2).*sin(2*fai); case 13 re = sqrt(5)*(6*p.^4-6*p.^2+1); case 14 re = sqrt(10)*(4*p.^4-3*p.^2).*cos(2*fai); case 15 re = sqrt(10)*p.^4.*cos(4*fai); end end
结果如下:
泽尼克图像:
拟合结果对比,浅色曲面为原高斯曲面,深色为拟合曲面: