利用前36阶zernike函数拟合曲面：
脚本程序
clc;clear;
load unwrap_ph.mat
unwrap_ph=max(max(unwrap_ph))-unwrap_ph;
unwrap_ph=unwrap_ph(:,81:560);
x=linspace(-1,1,size(unwrap_ph,1));
y=linspace(-1,1,size(unwrap_ph,2));
xy=[x;y];
a0=ones(1,36);
unwrap_ph(isnan(unwrap_ph)) = 0;
a=lsqcurvefit('SH',a0,xy,unwrap_ph)
FIT=SH(a,xy);
FIT(FIT==0)=nan;
figure(1)
imagesc(FIT)
colormap('jet')
grid off
shading interp
figure(2)
imagesc(unwrap_ph)
colormap('jet')
grid off
shading interp
zernike函数
function z = zernfun(n,m,r,theta,nflag)
m=m*-1;
if (~any(size(n)==1))||( ~any(size(m)==1))
error('zernfun:NMvectors','N and M must be vectors.')
end
if length(n)~=length(m)
error('zernfun:NMlength','N and M must be the same length.')
end
n=n(:);
m=m(:);
if any(mod(n-m,2))
error('zernfun:NMmultiplesof2', ...
'All N and M must differ by multiples of 2 (including 0).')
end
if any(m>n)
error('zernfun:MlessthanN', ...
'Each M must be less than or equal to its corresponding N.')
end
if any(r>1|r<0)
error('zernfun:Rlessthan1','All R must be between 0 and 1.')
end
if (~any(size(r)==1))||(~any(size(theta)==1))
error('zernfun:RTHvector','R and THETA must be vectors.')
end
r=r(:);
theta=theta(:);
length_r=length(r);
if length_r~=length(theta)
error('zernfun:RTHlength', ...
'The number of R- and THETA-values must be equal.')
end
% Check normalization:
% --------------------
if nargin==5&&ischar(nflag)
isnorm=strcmpi(nflag,'norm');
if ~isnorm
error('zernfun:normalization','Unrecognized normalization flag.')
end
else
isnorm=false;
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Compute the Zernike Polynomials
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Determine the required powers of r:
% -----------------------------------
m_abs=abs(m);
rpowers=[];
for j=1:length(n)
rpowers=[rpowers m_abs(j):2:n(j)];
end
rpowers=unique(rpowers);
% Pre-compute the values of r raised to the required powers,
% and compile them in a matrix:
% -----------------------------
if rpowers(1)==0
rpowern=arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false);
rpowern=cat(2,rpowern{:});
rpowern=[ones(length_r,1) rpowern];
else
rpowern=arrayfun(@(p)r.^p,rpowers,'UniformOutput',false);
rpowern=cat(2,rpowern{:});
end
% Compute the values of the polynomials:
% --------------------------------------
y=zeros(length_r,length(n));
for j=1:length(n)
s=0:(n(j)-m_abs(j))/2;
pows=n(j):-2:m_abs(j);
for k=length(s):-1:1
p=(1-2*mod(s(k),2))* ...
prod(2:(n(j)-s(k)))/ ...
prod(2:s(k))/ ...
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ...
prod(2:((n(j)+m_abs(j))/2-s(k)));
idx=(pows(k)==rpowers);
y(:,j)=y(:,j) + p*rpowern(:,idx);
end
if isnorm
y(:,j)=y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi);
end
end
% END: Compute the Zernike Polynomials
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Compute the Zernike functions:
% ------------------------------
idx_pos=m>0;
idx_neg=m<0;
z = y;
if any(idx_pos)
if (m>=0&&rem(n,2)==1)
z(:,idx_pos)=1-y(:,idx_pos).*sin(theta*m(idx_pos)');
else
z(:,idx_pos)=y(:,idx_pos).*sin(theta*m(idx_pos)');
end
end
if any(idx_neg)
if (m>=0&&rem(n,2)==1)
z(:,idx_neg)=1-y(:,idx_neg).*cos(theta*m(idx_neg)');
else
z(:,idx_neg)=y(:,idx_neg).*cos(theta*m(idx_neg)');
end
end
% EOF zernfun
拟合函数
function z=SH(c,data) x=data(1,:); y=data(2,:); [X,Y]=meshgrid(x,y); [theta,r]=cart2pol(X,Y); idx=r<=1; zz=zeros(size(X)); b=c(1:4); a=c(5:end); zz(idx)=b(1)*zernfun(0,0,r(idx),theta(idx))+b(2)*zernfun(1,1,r(idx),theta(idx))+b(3)*zernfun(1,-1,r(idx),theta(idx))+b(4)*zernfun(2,0,r(idx),theta(idx))+... a(1)*zernfun(2,2,r(idx),theta(idx))+a(2)*zernfun(2,-2,r(idx),theta(idx))+a(3)*zernfun(3,1,r(idx),theta(idx))+... a(4)*zernfun(3,-1,r(idx),theta(idx))+a(5)*zernfun(3,3,r(idx),theta(idx))+a(6)*zernfun(3,-3,r(idx),theta(idx))+... a(7)*zernfun(4,0,r(idx),theta(idx))+a(8)*zernfun(4,2,r(idx),theta(idx))+a(9)*zernfun(4,-2,r(idx),theta(idx))+... a(10)*zernfun(4,4,r(idx),theta(idx))+a(11)*zernfun(4,-4,r(idx),theta(idx))+a(12)*zernfun(5,1,r(idx),theta(idx))+... a(13)*zernfun(5,-1,r(idx),theta(idx))+a(14)*zernfun(5,3,r(idx),theta(idx))+a(15)*zernfun(5,-3,r(idx),theta(idx))+... a(16)*zernfun(5,5,r(idx),theta(idx))+a(17)*zernfun(5,-5,r(idx),theta(idx))+a(18)*zernfun(6,0,r(idx),theta(idx))+... a(19)*zernfun(6,2,r(idx),theta(idx))+a(20)*zernfun(6,-2,r(idx),theta(idx))+a(21)*zernfun(6,4,r(idx),theta(idx))+... a(22)*zernfun(6,-4,r(idx),theta(idx))+a(23)*zernfun(6,6,r(idx),theta(idx))+a(24)*zernfun(6,-6,r(idx),theta(idx))... +a(25)*zernfun(7,1,r(idx),theta(idx))+a(26)*zernfun(7,-1,r(idx),theta(idx))+a(27)*zernfun(7,3,r(idx),theta(idx))... +a(28)*zernfun(7,-3,r(idx),theta(idx))+a(29)*zernfun(7,5,r(idx),theta(idx))+a(30)*zernfun(7,-5,r(idx),theta(idx))... +a(31)*zernfun(7,7,r(idx),theta(idx))+a(32)*zernfun(7,-7,r(idx),theta(idx)); z=zz;
原图
拟合结果
应当注意的是zernike是圆对称函数，也就意味着其拟合的是一个圆，所以原图圆外的数值并未拟合。