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, )uazB!X
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, u Tvck6
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? zrE Dld9
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? &pN/+,0E
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function z = zernfun(n,m,r,theta,nflag) TKrh3
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. {Ax{N
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N cwBf((~
% and angular frequency M, evaluated at positions (R,THETA) on the pa2cM%48
% unit circle. N is a vector of positive integers (including 0), and p^X
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% M is a vector with the same number of elements as N. Each element |P`: NAf2
% k of M must be a positive integer, with possible values M(k) = -N(k) B`/p[ U5
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, MB!$s_~o#L
% and THETA is a vector of angles. R and THETA must have the same woyeKOr
% length. The output Z is a matrix with one column for every (N,M) ZuVes?&j
% pair, and one row for every (R,THETA) pair. Xw]L'+V=
% gQlL0jAV
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike .!yw@kg
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), yGX"1Fb?;x
% with delta(m,0) the Kronecker delta, is chosen so that the integral FWl'='5L
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, RJ~I?{yR0[
% and theta=0 to theta=2*pi) is unity. For the non-normalized kdp- |9
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. +@jX|
% 'J(B{B7|
% The Zernike functions are an orthogonal basis on the unit circle. 65AG#O5R
% They are used in disciplines such as astronomy, optics, and D>m!R[!o
% optometry to describe functions on a circular domain. {/K_NSg+h
% y)D7!s
% The following table lists the first 15 Zernike functions. oa:30@HSb
% Qv/Kb w
N{
% n m Zernike function Normalization \zv?r:1t
% -------------------------------------------------- [RFF&uy
% 0 0 1 1 qb?9i-(
% 1 1 r * cos(theta) 2 d,*#yzO
% 1 -1 r * sin(theta) 2 " twq#Alx
% 2 -2 r^2 * cos(2*theta) sqrt(6) 1jkMje
% 2 0 (2*r^2 - 1) sqrt(3) WJF#+)P:Y
% 2 2 r^2 * sin(2*theta) sqrt(6) D/Hob
% 3 -3 r^3 * cos(3*theta) sqrt(8) L>{p>
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) WbH#@]+DN
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) mrId`<L5l{
% 3 3 r^3 * sin(3*theta) sqrt(8) sEm064
% 4 -4 r^4 * cos(4*theta) sqrt(10) I+g[
p
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) E'wJ+X9 +
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) _NkbB"+L
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10)
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% 4 4 r^4 * sin(4*theta) sqrt(10) \&.]!!Q
% -------------------------------------------------- #t?tt,nc}
% eZk4$y
% Example 1: GEQ3r'B|
% L0dj 76'M
% % Display the Zernike function Z(n=5,m=1)
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% x = -1:0.01:1; \SQ wIM
% [X,Y] = meshgrid(x,x); b@m\ca
% [theta,r] = cart2pol(X,Y); t-3y`31i.
% idx = r<=1; p7eRAQ\'
% z = nan(size(X));
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% z(idx) = zernfun(5,1,r(idx),theta(idx)); &(t/4)IZox
% figure 3gNVnmZG
% pcolor(x,x,z), shading interp `D9AtN] R
% axis square, colorbar RT$.r5l_@
% title('Zernike function Z_5^1(r,\theta)') 'v:%} qMv
% Fg<rz&MR
% Example 2: SxWK@)tP
% Ed +"F{!eQ
% % Display the first 10 Zernike functions +*vg)F:
% x = -1:0.01:1; E[E7Gsmq V
% [X,Y] = meshgrid(x,x); Cp[
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% [theta,r] = cart2pol(X,Y); 5\\a49k.p
% idx = r<=1; 568qdD`PS
% z = nan(size(X)); RJO40&Z<Z
% n = [0 1 1 2 2 2 3 3 3 3]; ]v,>!~8r
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; Vi o ~2
% Nplot = [4 10 12 16 18 20 22 24 26 28]; E"[h20`\/
% y = zernfun(n,m,r(idx),theta(idx)); Mpu8/i
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% figure('Units','normalized') #CY Dh8X<i
% for k = 1:10 l1MVC@'pvP
% z(idx) = y(:,k); Ln
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% subplot(4,7,Nplot(k)) di5>aAJ)D
% pcolor(x,x,z), shading interp $bd2TVNV:
% set(gca,'XTick',[],'YTick',[]) %}0B7_6B+@
% axis square 0}d^UGD
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) H(WRm1i"G
% end Ccx1#^`
% 2qkZ B0[
% See also ZERNPOL, ZERNFUN2. g7 r_jj%ow
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% Paul Fricker 11/13/2006 GoTJm}[NP
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% Check and prepare the inputs: %pk'YA{M)q
% ----------------------------- {ICW"Rlcs
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ~qP_1()
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error('zernfun:NMvectors','N and M must be vectors.') h.ln%6:d
end j68_3zpl
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if length(n)~=length(m) 7Fd`MTo
error('zernfun:NMlength','N and M must be the same length.') dW`!/OaQD
end n^P~]1i
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n = n(:); 4`M7
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m = m(:); wTw)GV4
if any(mod(n-m,2)) ~ WO
error('zernfun:NMmultiplesof2', ... AZgeu$:7p<
'All N and M must differ by multiples of 2 (including 0).') ccPTJ/%$
end CfMCc:8mL
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if any(m>n) :psP|7%|
error('zernfun:MlessthanN', ... i3[%]_eP.
'Each M must be less than or equal to its corresponding N.') RL|d-A+;
end ^KRe(
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if any( r>1 | r<0 ) yJRqX]MLA
error('zernfun:Rlessthan1','All R must be between 0 and 1.') <jwQ&fm)/R
end g,61'5\
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 2 f8\Osn>m
error('zernfun:RTHvector','R and THETA must be vectors.') DY(pU/q
end suF<VJ)&s
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r = r(:); R|u2ga~
theta = theta(:); )d$FFTH
length_r = length(r); \a7caT{
if length_r~=length(theta) r\."=l
error('zernfun:RTHlength', ... _yN&+]c
'The number of R- and THETA-values must be equal.') M8{J
end z?I"[M
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% Check normalization: y`Nprwb
% -------------------- CAT{)*xc
if nargin==5 && ischar(nflag) W_bp~Wu
isnorm = strcmpi(nflag,'norm'); p-o8Ctc?V
if ~isnorm KKcajN
error('zernfun:normalization','Unrecognized normalization flag.') \0,8?S
end Hq;*T3E
else &)ED||r,
isnorm = false; 2K
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end ~4V-{-=0a7
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% /:. p{y
% Compute the Zernike Polynomials "969F(S$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% N eC]MW
>*]dB| 2
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% Determine the required powers of r: s/\<;g:u^
% ----------------------------------- k((kx:
m_abs = abs(m); f!K{f[aDa
rpowers = []; m8,jV R
for j = 1:length(n) "%rzL.</
rpowers = [rpowers m_abs(j):2:n(j)]; [R(d Cq>
end nJ2910"<
rpowers = unique(rpowers); me
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% Pre-compute the values of r raised to the required powers, ,:6.Gi)|
% and compile them in a matrix: @*&`1
% ----------------------------- #9rCF 3P
if rpowers(1)==0 AK//]
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); "BA&
rpowern = cat(2,rpowern{:}); fi
rpowern = [ones(length_r,1) rpowern]; :/\KVz'fw}
else gHox>r6.A
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); )u=46EU_
rpowern = cat(2,rpowern{:}); '>:%n
end `1i\8s&O6@
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% Compute the values of the polynomials: ?.g="{5X
% -------------------------------------- jP31K{G?
y = zeros(length_r,length(n)); 4&<zkAMR
for j = 1:length(n) M Ui#3o\f
s = 0:(n(j)-m_abs(j))/2; Sd *7jW?
pows = n(j):-2:m_abs(j); 'NN3XyD
for k = length(s):-1:1 p>1Klh:8.'
p = (1-2*mod(s(k),2))* ... TUX:[1~Nf[
prod(2:(n(j)-s(k)))/ ... i;<K)5Z
prod(2:s(k))/ ... 7e:7RAX
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... us )NgG
prod(2:((n(j)+m_abs(j))/2-s(k))); #&Fd16ov
idx = (pows(k)==rpowers); {k)H.zwe
y(:,j) = y(:,j) + p*rpowern(:,idx); I#-T/1N
end o@qI!?p&
asJ t6C
if isnorm (G 9Ku 8Y
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); tN_~zP
end fiQ/ &]|5
end \79aG3MyK
% END: Compute the Zernike Polynomials 2#Y5*r's\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -ze@~Z@
X=[`+=
tg;AF<VI
% Compute the Zernike functions: rW[7
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% ------------------------------ _/5xtupxE
idx_pos = m>0; DG/<#SCF
idx_neg = m<0; Q32GI,M%B
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7!q.MOYm
z = y; mU;\,96#
if any(idx_pos) `r+`vJ$
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ,%]xT>kH
end Z`yW2ON$'
if any(idx_neg) k-8$43
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); | (: PX
end #4{9l
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% EOF zernfun 2KQpmNN