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, b+#~N>|
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, 3b/vyZF
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? 8J(zWV7 r
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? kk7:A0._
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function z = zernfun(n,m,r,theta,nflag) o%JIJ7M
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. V$F.`O!hfi
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Ak-7}i
% and angular frequency M, evaluated at positions (R,THETA) on the FoXQ]X7"
% unit circle. N is a vector of positive integers (including 0), and EF^=3
% M is a vector with the same number of elements as N. Each element 0*M}QXt
% k of M must be a positive integer, with possible values M(k) = -N(k) umn~hb5O
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, qO3BQ]UF
% and THETA is a vector of angles. R and THETA must have the same 1kw4'#J8
% length. The output Z is a matrix with one column for every (N,M) A-`J!xj#/
% pair, and one row for every (R,THETA) pair. T-8nUo}i
% "^e?E:( 3
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike "}aM*(l+\
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), B]}V$*$\?
% with delta(m,0) the Kronecker delta, is chosen so that the integral imq(3?
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, Q>c6ouuJ
% and theta=0 to theta=2*pi) is unity. For the non-normalized !l~aRj-WZ
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 7?WBzo!!L
% kxf=%<l
% The Zernike functions are an orthogonal basis on the unit circle. 6zZR:ej
% They are used in disciplines such as astronomy, optics, and g-gBg\y{v
% optometry to describe functions on a circular domain. %~(i[Ur;
% {hP&P
% The following table lists the first 15 Zernike functions. =v=!x
% ]<z(Rmn`Q
% n m Zernike function Normalization +((31l
% -------------------------------------------------- =9@yJ9c-
% 0 0 1 1 "fJ|DE&@<i
% 1 1 r * cos(theta) 2 ~"0X,APR5
% 1 -1 r * sin(theta) 2 O9&:(2'f
% 2 -2 r^2 * cos(2*theta) sqrt(6) G")EE#W$}
% 2 0 (2*r^2 - 1) sqrt(3) U+M?<4J)"
% 2 2 r^2 * sin(2*theta) sqrt(6) QNwAuH T
% 3 -3 r^3 * cos(3*theta) sqrt(8) F@K;A%us)
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) sBI%lrO
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) 5kNs@FP
% 3 3 r^3 * sin(3*theta) sqrt(8) RYaofW
% 4 -4 r^4 * cos(4*theta) sqrt(10) eE_XwLE
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) w o9f99
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) -)+DVG.t
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) <&Xq`i/(
% 4 4 r^4 * sin(4*theta) sqrt(10) uL AXN
% -------------------------------------------------- 3m7V6##+
% l;kZS
% Example 1: -s "$I:v
% o_m.MMEU
% % Display the Zernike function Z(n=5,m=1) -RDs{c`y%N
% x = -1:0.01:1; 6+#cyKj
% [X,Y] = meshgrid(x,x); k(+u"T
% [theta,r] = cart2pol(X,Y); ?tQv|x
% idx = r<=1; A6.'1OD
% z = nan(size(X)); !\4FIs&Qv
% z(idx) = zernfun(5,1,r(idx),theta(idx)); ha~s<
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% figure n9-[z2n
% pcolor(x,x,z), shading interp N\&;R$[9:
% axis square, colorbar 6\@, Lb
% title('Zernike function Z_5^1(r,\theta)') r0bPaAKw
% @ xr
% Example 2: PaJwM%s)L
% - Sgp,"a
% % Display the first 10 Zernike functions X+@,vCC
% x = -1:0.01:1; 1R9/AP
% [X,Y] = meshgrid(x,x); E=trJge
% [theta,r] = cart2pol(X,Y); !2I wuru
% idx = r<=1; @'4D9A
% z = nan(size(X)); 3s`3}DKK
% n = [0 1 1 2 2 2 3 3 3 3]; *4y r7~S5
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; Jj:4@p:
% Nplot = [4 10 12 16 18 20 22 24 26 28]; j-|0&X1C
% y = zernfun(n,m,r(idx),theta(idx)); '|T=
% figure('Units','normalized') zxdO3I
% for k = 1:10 ZW%`G@d"H-
% z(idx) = y(:,k); 3zHiu*2/!
% subplot(4,7,Nplot(k)) DL_\luh
% pcolor(x,x,z), shading interp eO G%6C%a
% set(gca,'XTick',[],'YTick',[]) CU_06A|}
% axis square .x%SbG<k{
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ]Jqe)o
% end a| cD{d
% TD7ONa-,
% See also ZERNPOL, ZERNFUN2. &r%3)Z8Et
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% Paul Fricker 11/13/2006 g2q=&eI"
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% Check and prepare the inputs: 6B{Awm@v}X
% ----------------------------- p.|;
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if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) m
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error('zernfun:NMvectors','N and M must be vectors.') UtrbkuT
end A>puk2 s
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if length(n)~=length(m) i$bBN$<b<
error('zernfun:NMlength','N and M must be the same length.') y[rLk
end _T$\$v$ {
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%54![-@
n = n(:); rge/jE,^~Z
m = m(:); ,}0pK\Y>$
if any(mod(n-m,2)) M<Mr (z
error('zernfun:NMmultiplesof2', ... +|;IIwo
'All N and M must differ by multiples of 2 (including 0).') b&1@rE-
end Zpmy)W]1
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if any(m>n) k|5nu-B0v
error('zernfun:MlessthanN', ... ,R+u%bmn#
'Each M must be less than or equal to its corresponding N.') U9w*x/Swb
end 0"N %Vm
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if any( r>1 | r<0 ) 4SYN$?.Mp
error('zernfun:Rlessthan1','All R must be between 0 and 1.') MR}\fw$(.
end RAC-;~$WB
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) d9(F wmE
error('zernfun:RTHvector','R and THETA must be vectors.') +,lD_{}_
end -)@.D>HsOt
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r = r(:); _,^sI%
theta = theta(:); H &JKja}`
length_r = length(r); DYS(ZY)4
if length_r~=length(theta) sAN#j
{
error('zernfun:RTHlength', ... !NCT) #G`
'The number of R- and THETA-values must be equal.') HD ~9EK~
end qU}DOL|
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% Check normalization: abND#t
% -------------------- AZa3!e/1
if nargin==5 && ischar(nflag) C N"c
isnorm = strcmpi(nflag,'norm'); >BX_Bou
if ~isnorm m"*:XfOL
error('zernfun:normalization','Unrecognized normalization flag.') Ij+zR>P8=\
end pqe**`z@y
else pGIeW}2'9
isnorm = false; fh~&&f