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, IM~2=+
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, g,5Tr_
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? -|&&lxrwh
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? QetyuhS~
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function z = zernfun(n,m,r,theta,nflag) &X
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%ZERNFUN Zernike functions of order N and frequency M on the unit circle. /~;om\7r
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 59M\uVWR
% and angular frequency M, evaluated at positions (R,THETA) on the Y`!Zk$8
% unit circle. N is a vector of positive integers (including 0), and (<xl _L:*.
% M is a vector with the same number of elements as N. Each element /}$D&KwYg
% k of M must be a positive integer, with possible values M(k) = -N(k) 4:Id8rzz
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, _T.k/a
% and THETA is a vector of angles. R and THETA must have the same ._US8
% length. The output Z is a matrix with one column for every (N,M) Hn!13+fS
% pair, and one row for every (R,THETA) pair. 4,qhWe`/
% ppK`7J>Z
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 9._owKj
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 0}I aWd^4
% with delta(m,0) the Kronecker delta, is chosen so that the integral 4b:q84
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, [,/~*L;7
% and theta=0 to theta=2*pi) is unity. For the non-normalized bGe@yXId5
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. *VgiJ
% n]fMl:77
% The Zernike functions are an orthogonal basis on the unit circle. {#4F}@Q
% They are used in disciplines such as astronomy, optics, and 3DS&-rN
% optometry to describe functions on a circular domain. g.T:72"
% ^K'@W
% The following table lists the first 15 Zernike functions. yJ?S7+b
% \*5${[
% n m Zernike function Normalization E8]kd
% -------------------------------------------------- :2(U3~3:
% 0 0 1 1 -|_MC^)
% 1 1 r * cos(theta) 2 gis;)al
% 1 -1 r * sin(theta) 2 zX}t1:nc
% 2 -2 r^2 * cos(2*theta) sqrt(6) 20A`]-D
% 2 0 (2*r^2 - 1) sqrt(3) V(3=j)#
% 2 2 r^2 * sin(2*theta) sqrt(6) w0`8el;
% 3 -3 r^3 * cos(3*theta) sqrt(8) fm1yZX?`
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 6g&Ev'
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) S>V+IKW;(
% 3 3 r^3 * sin(3*theta) sqrt(8) b.|k j
% 4 -4 r^4 * cos(4*theta) sqrt(10) XsbYWJdds
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) =C 7 WQ
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) YML]pNB
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) qPF`=#
% 4 4 r^4 * sin(4*theta) sqrt(10) 5)iOG#8qJ
% -------------------------------------------------- omzG/)M:O
% 2R];Pv
% Example 1: ce:p*
% HvzXAd
% % Display the Zernike function Z(n=5,m=1) x>$e*
% x = -1:0.01:1; wGg_ vAn
% [X,Y] = meshgrid(x,x); V;29ieE!
% [theta,r] = cart2pol(X,Y); +o-jMvK9
% idx = r<=1; 7m:ZG
% z = nan(size(X)); 'M!M$<j
% z(idx) = zernfun(5,1,r(idx),theta(idx)); IRyZ0$r:e\
% figure cPy/}A
% pcolor(x,x,z), shading interp Mqv[7.|
% axis square, colorbar I>JBGR`j
% title('Zernike function Z_5^1(r,\theta)') }\0ei(%H
% *WaqNMD[%
% Example 2: qs Wy
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% LY;FjbyU
% % Display the first 10 Zernike functions zd|n!3;
% x = -1:0.01:1; 0TWd.+
% [X,Y] = meshgrid(x,x); HT
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% [theta,r] = cart2pol(X,Y); sOVU>tb\'
% idx = r<=1; TyhO+;
% z = nan(size(X)); Kv9Z.DY
% n = [0 1 1 2 2 2 3 3 3 3]; 0p]v#z}
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; z3I
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% Nplot = [4 10 12 16 18 20 22 24 26 28]; r4m z
% y = zernfun(n,m,r(idx),theta(idx)); _Wqy,L;J
% figure('Units','normalized') v=d16
% for k = 1:10 )M><09
% z(idx) = y(:,k); gCq'#G\Z
% subplot(4,7,Nplot(k)) D$N;Qb
% pcolor(x,x,z), shading interp =;"=o5g_
% set(gca,'XTick',[],'YTick',[]) V]NCFG
% axis square QQJf;p7
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) d}Q%I
% end YD;G+"n?T
% <*(^QOM
% See also ZERNPOL, ZERNFUN2. jn(%v]
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% Paul Fricker 11/13/2006 _"*}8{|
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% Check and prepare the inputs: #a9O3C/MP
% ----------------------------- Al=ByX @
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) $,P:B%]
error('zernfun:NMvectors','N and M must be vectors.') XBoq/kbw!
end w2db=9
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if length(n)~=length(m) $<yhEvv
error('zernfun:NMlength','N and M must be the same length.') P0pBR_:o
end "([/G?QAG
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n = n(:); xHMFYt+0$G
m = m(:); M*f]d`B
if any(mod(n-m,2)) s VHk;:e>x
error('zernfun:NMmultiplesof2', ... 7Ja*T@ ! h
'All N and M must differ by multiples of 2 (including 0).') z0OxJ e
end yM~bUmSg
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if any(m>n) YZ`SF"Bd(
error('zernfun:MlessthanN', ... GC:q6}
'Each M must be less than or equal to its corresponding N.') ES?*w@x
end `XpQR=IOMb
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if any( r>1 | r<0 ) X'-Yz7J?o
error('zernfun:Rlessthan1','All R must be between 0 and 1.') aydNSgu
end G:p85k`
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) )
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error('zernfun:RTHvector','R and THETA must be vectors.') JJM<ywPGp
end Px&_6}YWy
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r = r(:); 3 ye
theta = theta(:); Rq%Kw> {&
length_r = length(r); |?ssHW
if length_r~=length(theta) ?*%_:fB
error('zernfun:RTHlength', ... bi^?SH\
'The number of R- and THETA-values must be equal.') ,T`,OZm
end #K6cBfqI
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% Check normalization: F,{mF2U*$
% -------------------- o$buoGSPc
if nargin==5 && ischar(nflag) C!a1.&HHZ7
isnorm = strcmpi(nflag,'norm'); bD{k=jum
if ~isnorm mr^3Y8$s
error('zernfun:normalization','Unrecognized normalization flag.') @(~:JP?KNC
end 80wzn,o
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else ##*]2Dy
isnorm = false; 4G?^#+|^
end (rd
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {VL@U$'oI
% Compute the Zernike Polynomials yjg&/6
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Pps-,*m
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% Determine the required powers of r: v$W[(
% ----------------------------------- dy&UF,l6
m_abs = abs(m); $KO2+^%y
rpowers = []; w_xca(
for j = 1:length(n) odsFgh
rpowers = [rpowers m_abs(j):2:n(j)]; :Ko6.|
end q.VYPkEib
rpowers = unique(rpowers); u]};QR
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% Pre-compute the values of r raised to the required powers, OegeZV
% and compile them in a matrix: kkF)Tro\
% ----------------------------- >sfg`4
if rpowers(1)==0 {P]C>
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 6:]N%
rpowern = cat(2,rpowern{:}); X,7y| tb
rpowern = [ones(length_r,1) rpowern]; &)%+DUV|
else S{rltT-
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 6'r8.~O
rpowern = cat(2,rpowern{:}); ViPC Yt`of
end IW3k{z
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% Compute the values of the polynomials: ?m
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% -------------------------------------- '4[=*!hs!
y = zeros(length_r,length(n)); l@4_D;b3o"
for j = 1:length(n) vxOqo)yO
s = 0:(n(j)-m_abs(j))/2; xc:E>-
pows = n(j):-2:m_abs(j); <Kd(fFe
for k = length(s):-1:1 qN)y-N.LI(
p = (1-2*mod(s(k),2))* ... YAr6cl
prod(2:(n(j)-s(k)))/ ... _rT\?//B
prod(2:s(k))/ ... %9J@##+
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ;*<tU
n^t
prod(2:((n(j)+m_abs(j))/2-s(k))); T{k
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4
idx = (pows(k)==rpowers); MzJCiX^
y(:,j) = y(:,j) + p*rpowern(:,idx); G*fo9eu5$
end oJz2-PmX
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if isnorm 5Q?Jm~H9
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); B `~EA] d
end W$rWg>4>
end 0
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% END: Compute the Zernike Polynomials GXtMX ha,
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Dcl$?
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% Compute the Zernike functions: ;c(a)_1
% ------------------------------ n~N>;mP
idx_pos = m>0; 9DxHdpOk
idx_neg = m<0; 2/LSB8n|
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z = y; 2ym(fk.6{
if any(idx_pos) rFRcK>X\L
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 5)k8(kH
end Xwm3# o.&)
if any(idx_neg) Da=EAG-{7
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 8Bf>
end BG>Y[u\N
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% EOF zernfun )dXa:h0RZ