下面这个函数大家都不会陌生，计算zernike函数值的，并根据此可以还原出图像来， O_=2{k ~s0
我输入10阶的n、m，r,theta为38025*1向量，最后得到的z是29525*10阶的矩阵， (j<FS>##
这个，跟我们用zygo干涉仪直接拟合出的36项zernike系数，有何关系呢？ Ub[SUeBGH
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ <46> v<
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function z = zernfun(n,m,r,theta,nflag) a
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%ZERNFUN Zernike functions of order N and frequency M on the unit circle. >- S? rXO
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N jGm`Qg{<
% and angular frequency M, evaluated at positions (R,THETA) on the QjT$.pU d
% unit circle. N is a vector of positive integers (including 0), and P} "=67$
% M is a vector with the same number of elements as N. Each element zEM c)
% k of M must be a positive integer, with possible values M(k) = -N(k) d `MTc
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, |h^[/
% and THETA is a vector of angles. R and THETA must have the same D;?cf+6$
% length. The output Z is a matrix with one column for every (N,M) '%Fg+cZN\
% pair, and one row for every (R,THETA) pair. \NZ(Xk
% # <?igtUO
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike OdKfU ^
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), :zA/~/Wo
% with delta(m,0) the Kronecker delta, is chosen so that the integral L
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% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 5r2A ^<)
% and theta=0 to theta=2*pi) is unity. For the non-normalized y
J|/^qs
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. L<D<3g|4
%
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% The Zernike functions are an orthogonal basis on the unit circle. `GDWy^-Q+!
% They are used in disciplines such as astronomy, optics, and srbES6
% optometry to describe functions on a circular domain. z:Y
Z]
% w]@H]>sHd
% The following table lists the first 15 Zernike functions. ^U q%-a
% I3I1<}>]Z
% n m Zernike function Normalization gDN7ly]6M
% -------------------------------------------------- #}xw
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% 0 0 1 1 o:wI{?%-3
% 1 1 r * cos(theta) 2 QG1+*J76b@
% 1 -1 r * sin(theta) 2 gPE` mE
% 2 -2 r^2 * cos(2*theta) sqrt(6) 6y+_ x'
% 2 0 (2*r^2 - 1) sqrt(3) {<}9r6k ;f
% 2 2 r^2 * sin(2*theta) sqrt(6) ! V^wq]D2
% 3 -3 r^3 * cos(3*theta) sqrt(8) 42oW]b%P{;
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) XJZ\ ss
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) M&[bb $00j
% 3 3 r^3 * sin(3*theta) sqrt(8) !{1 ;wC(b
% 4 -4 r^4 * cos(4*theta) sqrt(10) #}p@+rkg2
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) | V: 9 ][\
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) v:F_ ! Q
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) W5*Kq^6Pd
% 4 4 r^4 * sin(4*theta) sqrt(10) 1a gNwFd~
% -------------------------------------------------- 11^.oa+`
% 8P? p
% Example 1: oBS m>V
% ]qd$rX
% % Display the Zernike function Z(n=5,m=1) A+=K<e
% x = -1:0.01:1; ?S<`*O
+
% [X,Y] = meshgrid(x,x); h}y] Pt?
% [theta,r] = cart2pol(X,Y); Q]{ `m
% idx = r<=1; wi/qI(O!
% z = nan(size(X)); 3<x1s2U
% z(idx) = zernfun(5,1,r(idx),theta(idx)); ;7>k[?'e
% figure x%'5 r nm|
% pcolor(x,x,z), shading interp |1pD n7
% axis square, colorbar `nCVO;B
% title('Zernike function Z_5^1(r,\theta)') f6,?Yex8B
% =OeLF
% Example 2: gs"w
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% p:NIRs
% % Display the first 10 Zernike functions OQ&'3hv{
% x = -1:0.01:1; "h5.^5E6
% [X,Y] = meshgrid(x,x); e? 7O om
% [theta,r] = cart2pol(X,Y); ^)E#
c
% idx = r<=1; 60R]Q
% z = nan(size(X)); %a:>3!
+
% n = [0 1 1 2 2 2 3 3 3 3]; X \BxRgl},
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; *e25!#o1
% Nplot = [4 10 12 16 18 20 22 24 26 28]; 8|{d1dy
% y = zernfun(n,m,r(idx),theta(idx)); vlq L
% figure('Units','normalized') l3xI\{jn
% for k = 1:10 :+_
% z(idx) = y(:,k); ~f:"Q(f+
% subplot(4,7,Nplot(k)) y 2C Jk~
% pcolor(x,x,z), shading interp hLr\;Swyp
% set(gca,'XTick',[],'YTick',[]) udOdXz 6K?
% axis square FEO /RMh
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) /E-s g,
k
% end ?J@P0(M#
% f+lPQIB
% See also ZERNPOL, ZERNFUN2. cN:d y #
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% Paul Fricker 11/13/2006 mpNS}n6
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% Check and prepare the inputs: h0?w V5H
% ----------------------------- 4" p U\g
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) {TZV^gT4
error('zernfun:NMvectors','N and M must be vectors.') jp7cPpk:LG
end s6QD^[
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if length(n)~=length(m) ffMk.SqI
error('zernfun:NMlength','N and M must be the same length.') vSy[lB|)24
end mqtYny'
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n = n(:); SZUo RWx
m = m(:); ZfXgV TJ`
if any(mod(n-m,2)) V KxuK0{
error('zernfun:NMmultiplesof2', ... q8!]x-5$6j
'All N and M must differ by multiples of 2 (including 0).') Ae%AG@L
end [1mEdtqf*
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if any(m>n) $*:g~#bh
error('zernfun:MlessthanN', ... q+} \ (|
'Each M must be less than or equal to its corresponding N.') !X9^ L^v}
end n]6-`fpD
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if any( r>1 | r<0 ) 6OB3%R'p
error('zernfun:Rlessthan1','All R must be between 0 and 1.') dQz#&&s-
end IL1iTR H
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) K
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error('zernfun:RTHvector','R and THETA must be vectors.') G^';9 UK
end OIIA^QyV
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r = r(:); !L@<?0x LW
theta = theta(:); W 4bN']?
length_r = length(r); "lrQC`?
if length_r~=length(theta) 0cDP:EzR;
error('zernfun:RTHlength', ... da i+"
'The number of R- and THETA-values must be equal.') NT EN
end 7xFZJ#
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% Check normalization: +&dkJ 4g[
% -------------------- ddN G :
if nargin==5 && ischar(nflag) do*a E
isnorm = strcmpi(nflag,'norm'); :[CEHRc7x
if ~isnorm h#c7v !g
error('zernfun:normalization','Unrecognized normalization flag.') Uu52uR
end 'tDUPm38
else f7 |Tp m
isnorm = false; .
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end &po !X )
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% IT)3Et@Y
% Compute the Zernike Polynomials 1J72*`4O K
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% I~6 o<HO
!{ {gL=_@
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% Determine the required powers of r: pp@B]We
% ----------------------------------- yn"4qC#Z
m_abs = abs(m); AW E ab
rpowers = []; ;4(}e{
for j = 1:length(n) b dLi _k
rpowers = [rpowers m_abs(j):2:n(j)]; L`e19I$
end d S'J @e=#
rpowers = unique(rpowers); Nu OxEyC
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% Pre-compute the values of r raised to the required powers, #vhN$H :&q
% and compile them in a matrix: N'-[>w7vK2
% ----------------------------- znPh7{|<
if rpowers(1)==0 u>G9r#~`k
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); =Xg/[J%
rpowern = cat(2,rpowern{:}); h5pfmN\-5
rpowern = [ones(length_r,1) rpowern]; @g4o8nH}
else hF$qH^-c*A
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); N>,`TsUwW
rpowern = cat(2,rpowern{:}); zsd 1n`r
end A)V*faD
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% Compute the values of the polynomials: @K+gh#
% -------------------------------------- tXH;4K@
y = zeros(length_r,length(n)); D/~1?p
for j = 1:length(n) xb<|m2<)H
s = 0:(n(j)-m_abs(j))/2; ,[t? $Cy ;
pows = n(j):-2:m_abs(j); $B_%MfI
for k = length(s):-1:1 ajtH 1Z #
p = (1-2*mod(s(k),2))* ... 9cUa@;*1
prod(2:(n(j)-s(k)))/ ... =*j Faj
prod(2:s(k))/ ... # {{p4/:
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... zL9~gJ
prod(2:((n(j)+m_abs(j))/2-s(k))); eBs.RR
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idx = (pows(k)==rpowers); y(MB _B7j
y(:,j) = y(:,j) + p*rpowern(:,idx); Eu: /U*j
end 80_w_i +
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if isnorm gwQMy$
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); <n 06(9BF
end fZ5 UFq_~s
end Su"Z3gm5Kw
% END: Compute the Zernike Polynomials c9fz x
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% u~j
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\20} /&
% Compute the Zernike functions: Zfcf?&><
% ------------------------------ A8{ xZsH
idx_pos = m>0; !CcDA/0
idx_neg = m<0; MO0NNVVi%U
WV.hQX9P
%" 7UYLX
z = y; bTm hz
if any(idx_pos) #`
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z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); L;u 5
end W
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if any(idx_neg) ~{tO8
]
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Vjv6d&Q
end q%e'WM G~n
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% EOF zernfun JJ
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