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, E^.
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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, yf6&'Y{
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? l=JK+uZ
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? 'H,l\i@"
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function z = zernfun(n,m,r,theta,nflag) /P|jHK|{
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. !P0Oq)q
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N SLc'1{
% and angular frequency M, evaluated at positions (R,THETA) on the {GiR-q{t
% unit circle. N is a vector of positive integers (including 0), and -.E<~(fad
% M is a vector with the same number of elements as N. Each element r
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% k of M must be a positive integer, with possible values M(k) = -N(k) ^T|~L<A3
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, qcfLA~y
% and THETA is a vector of angles. R and THETA must have the same Io&F0~Z;;(
% length. The output Z is a matrix with one column for every (N,M) r 6STc,%5
% pair, and one row for every (R,THETA) pair. <&rvv4*H
% /P0%4aWu=
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike DFt1{qS8@u
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), uIvE~<
% with delta(m,0) the Kronecker delta, is chosen so that the integral R@r"a&{/
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, `=Hh5;ep
% and theta=0 to theta=2*pi) is unity. For the non-normalized O=St}B\!m
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. #\$R^u]!
% xGeRoW(X
% The Zernike functions are an orthogonal basis on the unit circle. pemb2HQ'4j
% They are used in disciplines such as astronomy, optics, and P-QZ=dm
% optometry to describe functions on a circular domain. {e?D6`#x
% b#^UP
% The following table lists the first 15 Zernike functions. pRj1b^F5y
% fNx3\<~V=
% n m Zernike function Normalization v>71?te
% -------------------------------------------------- o84!$2P+w
% 0 0 1 1 <gKT 7ONtg
% 1 1 r * cos(theta) 2 fG5 U' Vw
% 1 -1 r * sin(theta) 2 q8.K-"f(Q
% 2 -2 r^2 * cos(2*theta) sqrt(6) A@EeX4N
% 2 0 (2*r^2 - 1) sqrt(3) lM5Xw
% 2 2 r^2 * sin(2*theta) sqrt(6) .4~n|d>z
% 3 -3 r^3 * cos(3*theta) sqrt(8) VZ;ASA?;
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) ^l6q
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) +)FB[/pXk
% 3 3 r^3 * sin(3*theta) sqrt(8) Cv|ya$}a
% 4 -4 r^4 * cos(4*theta) sqrt(10) kQ~*iY
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) `Q*L!/K+
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) ".eD&oX{
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 2mbZ6'p {
% 4 4 r^4 * sin(4*theta) sqrt(10) ?*a:f"vQ
% -------------------------------------------------- %uyRpG3,
% 40oRO0p
% Example 1: ajW[}/)
% vO"Sy{)Z>
% % Display the Zernike function Z(n=5,m=1) 2*5Z|
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% x = -1:0.01:1; _rK}~y=0
% [X,Y] = meshgrid(x,x); \&J7>vu^y
% [theta,r] = cart2pol(X,Y); [C)-=.Xx)j
% idx = r<=1; C9U~lcIS
% z = nan(size(X)); <5A(rDij
% z(idx) = zernfun(5,1,r(idx),theta(idx)); keEyE;O}u
% figure !h{qO&ZH=
% pcolor(x,x,z), shading interp |Gb"%5YD
% axis square, colorbar B]q
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% title('Zernike function Z_5^1(r,\theta)') J
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% S\
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% Example 2: _YY:}'+
% UfSWdR)
% % Display the first 10 Zernike functions ^PfFW
% x = -1:0.01:1; ` a5$VV%J
% [X,Y] = meshgrid(x,x); ]n0kO&
% [theta,r] = cart2pol(X,Y); rE-Xv.
|
% idx = r<=1; 1y l2i|m+
% z = nan(size(X)); Tz1St{s\
% n = [0 1 1 2 2 2 3 3 3 3]; h&||Ql1
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; %<JjftNQ
% Nplot = [4 10 12 16 18 20 22 24 26 28]; 67Z|=B!7
% y = zernfun(n,m,r(idx),theta(idx)); 16[>af0<g
% figure('Units','normalized') _* ] ~MQ=
% for k = 1:10 %8tlJQvu
% z(idx) = y(:,k); 0x'>}5`5
% subplot(4,7,Nplot(k))
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% pcolor(x,x,z), shading interp Y!v `0z
% set(gca,'XTick',[],'YTick',[]) X~GnK>R
% axis square cpp0Y^
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) BDyOX6
% end )R+@vh#Q<$
% J_OIU#-B
% See also ZERNPOL, ZERNFUN2. r>sk@[4h
l=[<gPE
#[C|%uq
% Paul Fricker 11/13/2006 |_8-3
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% Check and prepare the inputs: .HF+JHIUu
% ----------------------------- mF[w-<:.d
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) _` |Hk2O
error('zernfun:NMvectors','N and M must be vectors.') 3Ln~"HwP
end &F*s.gL
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)V%xbDd S
if length(n)~=length(m) IL?3>$,
error('zernfun:NMlength','N and M must be the same length.') 0F6^[osqtl
end 7^#f<m;Ar!
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n = n(:); wIz<Y{HA=
m = m(:); Z!60n{T79c
if any(mod(n-m,2)) (EGsw o
error('zernfun:NMmultiplesof2', ... y!;rY1
'All N and M must differ by multiples of 2 (including 0).')
;?1H&
end \@vR*E
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if any(m>n) *g]q~\b/;
error('zernfun:MlessthanN', ... +^YXqOXU
'Each M must be less than or equal to its corresponding N.') t&^9o$
end s\,F6c
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if any( r>1 | r<0 ) 6uW?xB9
error('zernfun:Rlessthan1','All R must be between 0 and 1.') LCx{7bN1ro
end @*e|{;X]hy
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) j|TcmZGO
error('zernfun:RTHvector','R and THETA must be vectors.') b26#0;i
end w d2GKq!
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%>Z^BM<e
r = r(:); AHc:6v^
theta = theta(:); bO>q`%&
length_r = length(r); :2iNw>z1
if length_r~=length(theta) NypM+y
error('zernfun:RTHlength', ... >Lx,<sE
'The number of R- and THETA-values must be equal.') G=/a>{
end 3
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% Check normalization: b}< T<
% -------------------- 5A
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if nargin==5 && ischar(nflag) O*xC}$OOn
isnorm = strcmpi(nflag,'norm'); >=BH$4Ce
if ~isnorm =/Pmi_
error('zernfun:normalization','Unrecognized normalization flag.') \fIGMoy!
end U>?q|(u
else q&OF?z7H
isnorm = false; u:AKp<'
end NC'+-P'y
/|V!2dQs"
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% al/Mgo
% Compute the Zernike Polynomials XG FjqZr`
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% P1KXvc}JGe
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t&}Z~Zp
% Determine the required powers of r: S`g:zb_
% ----------------------------------- 5Z"IM8?
m_abs = abs(m); I,;@\
rpowers = []; )@+lfIE(l
for j = 1:length(n) vFKX@wV S
rpowers = [rpowers m_abs(j):2:n(j)]; /{@^h#4M1
end QP/%+[E.
rpowers = unique(rpowers); 7R9.g6j
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8w1TX [b
% Pre-compute the values of r raised to the required powers, p|fSPSz
% and compile them in a matrix: 8>^(-ca_
% ----------------------------- i4;`dCT|A
if rpowers(1)==0 I3sH8/*
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); {p3VHd#
rpowern = cat(2,rpowern{:}); xuBXOr4"P
rpowern = [ones(length_r,1) rpowern]; 4Ufx,]
else GVS-_KP\
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); +/
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rpowern = cat(2,rpowern{:}); N6K%Wkz
end 74f3a|vx/
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% Compute the values of the polynomials: 8/q6vk><
% -------------------------------------- oVi_X98R
y = zeros(length_r,length(n)); 0zH^yx:ma
for j = 1:length(n) j{}-zQ]n
s = 0:(n(j)-m_abs(j))/2; x~1.;dBF
pows = n(j):-2:m_abs(j); *;^!FBT
for k = length(s):-1:1 fDe4 [QQ8
p = (1-2*mod(s(k),2))* ... 5WhR|
prod(2:(n(j)-s(k)))/ ... ~9#x/EG/
prod(2:s(k))/ ... _D{zB1d\0
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 9J>b6
prod(2:((n(j)+m_abs(j))/2-s(k))); [t)omPy<c
idx = (pows(k)==rpowers); 2hB';Dv
y(:,j) = y(:,j) + p*rpowern(:,idx); 85;hs
end $BIQ#T>qK
\1`L-lz
if isnorm Y)D~@|D,
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 4a'O#;ho
end ?Q$LIoR
end do3 BI4Q
% END: Compute the Zernike Polynomials ;=rM Ii
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% mX@Un9k
NpmPm1Ix .
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% Compute the Zernike functions: fwR3=:5~
% ------------------------------ _-NS-E
idx_pos = m>0; r 5$(
idx_neg = m<0; `b(y 5 Z
:V)W?~Z7B
#3uBq(-Z
z = y; w1zI"G~4/Q
if any(idx_pos) {?a9>g-BW
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); p(2j7W-/
end fVR:m`'Iq_
if any(idx_neg) GPqF>
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); F~Kd5-I@
end &&1q@m,cP
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% EOF zernfun 1[k~*QS