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， WR=e$ ;
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， ? &ew$%
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？ w+bQpIP M
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ uYAPGs#k
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function z = zernfun(n,m,r,theta,nflag) f_X]2i n
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 6|3$43J,F
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N "; tl>Ot
% and angular frequency M, evaluated at positions (R,THETA) on the NvWwj%6 ]
% unit circle. N is a vector of positive integers (including 0), and k2l(!0o|;
% M is a vector with the same number of elements as N. Each element =NwmhV
% k of M must be a positive integer, with possible values M(k) = -N(k) vRY Q4B4o
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, 4lH$BIAW
% and THETA is a vector of angles. R and THETA must have the same K:fK! /
% length. The output Z is a matrix with one column for every (N,M) >I Aw Nr
% pair, and one row for every (R,THETA) pair. $ QmP'
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% :^FOh*H
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ipnvw4+
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), -Y%#z'^-
% with delta(m,0) the Kronecker delta, is chosen so that the integral O.CRF-` t
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, Ia$&SS)K
% and theta=0 to theta=2*pi) is unity. For the non-normalized )Ac+5bs
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. MjN Cn&c
% Ce}wgKzr
% The Zernike functions are an orthogonal basis on the unit circle. h=um t<&D
% They are used in disciplines such as astronomy, optics, and ~hPp)- A
% optometry to describe functions on a circular domain. h|"98PI
% .P.TqT@)r
% The following table lists the first 15 Zernike functions. 4;W eB
% 'WkDp a
% n m Zernike function Normalization :\x53-&hO4
% -------------------------------------------------- d9h"Q
% 0 0 1 1 Gd1%6}<~
% 1 1 r * cos(theta) 2 qlmz@kTb
% 1 -1 r * sin(theta) 2 8;/`uB:zV
% 2 -2 r^2 * cos(2*theta) sqrt(6) 7!.%HhU0
% 2 0 (2*r^2 - 1) sqrt(3) @ $z/=g sy
% 2 2 r^2 * sin(2*theta) sqrt(6) "knSc0 ,u
% 3 -3 r^3 * cos(3*theta) sqrt(8) lG,/tMy
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 'CsD [<
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) r 11:T 3
% 3 3 r^3 * sin(3*theta) sqrt(8) B5pM cw
% 4 -4 r^4 * cos(4*theta) sqrt(10) 6?Ul)'
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) n}P K0
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) )vO;=% GQ
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ~` v 7
% 4 4 r^4 * sin(4*theta) sqrt(10) .g_B KeU
% -------------------------------------------------- z|[#6X6 tT
% fRC(Yyx
% Example 1: EU.vw0}u8
% Z W`
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% % Display the Zernike function Z(n=5,m=1) `W< 7.
% x = -1:0.01:1; #=UEx
% [X,Y] = meshgrid(x,x); p "f=[awp
% [theta,r] = cart2pol(X,Y); 3/mVdU?U
% idx = r<=1; mz;S*ONlV
% z = nan(size(X)); uhvmh
% z(idx) = zernfun(5,1,r(idx),theta(idx)); \dSMF,E
% figure rMAH Y H9
% pcolor(x,x,z), shading interp [ ,)yc/{*
% axis square, colorbar |xy r6gY
% title('Zernike function Z_5^1(r,\theta)') | iEhe
% mz@`*^7?
% Example 2: XH&F n+
% ysD @yM,
% % Display the first 10 Zernike functions 6z@OGExmd#
% x = -1:0.01:1; rRy BG Ej
% [X,Y] = meshgrid(x,x); h"/FqO
% [theta,r] = cart2pol(X,Y); pvM;2
% idx = r<=1; `'9Kj9}
% z = nan(size(X)); w_|R.T\7
% n = [0 1 1 2 2 2 3 3 3 3]; yaV=e1W
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; g9(zJ
% Nplot = [4 10 12 16 18 20 22 24 26 28]; c0jdZ#H
% y = zernfun(n,m,r(idx),theta(idx)); xevG)m
% figure('Units','normalized') -Qx:-,.a
% for k = 1:10 j|gv0SI_
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% z(idx) = y(:,k); } r^ @Xh
% subplot(4,7,Nplot(k)) 'bp*hqG[
% pcolor(x,x,z), shading interp Vzf{gr?
% set(gca,'XTick',[],'YTick',[]) CZyOAoc<
% axis square ;V ]EF
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) & P{
% end %\#s@8=2u
% ;m $F~!Y
% See also ZERNPOL, ZERNFUN2. ]z`Y'wSxd
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% Paul Fricker 11/13/2006 6dQa|ACX_
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% Check and prepare the inputs: f2Frb
% ----------------------------- INSI$tA~
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) |VM c,_D
error('zernfun:NMvectors','N and M must be vectors.') NfcY30}:
end A3ad9?LR[R
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if length(n)~=length(m) < y5V],-U
error('zernfun:NMlength','N and M must be the same length.') iK{q_f\"
end 0-cqux2U
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n = n(:); ZxkX\gl91
m = m(:); *9)7.} uY
if any(mod(n-m,2)) AH`D&V
error('zernfun:NMmultiplesof2', ... ]4SnOSV?S
'All N and M must differ by multiples of 2 (including 0).') p'1n'|$e
end -'+|r]
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if any(m>n) }CZ,W Jz=
error('zernfun:MlessthanN', ... EB jiSQw
'Each M must be less than or equal to its corresponding N.') @<Au|l`
end 1)
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if any( r>1 | r<0 ) FV^CSaN[R
error('zernfun:Rlessthan1','All R must be between 0 and 1.') 6"Q/Y[y
end fEc}c.!5
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) SPW @TF1
error('zernfun:RTHvector','R and THETA must be vectors.') `Yp\.K z
end %lNWaA
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r = r(:); GyJp!
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theta = theta(:); :/ns/~5xa:
length_r = length(r); ~jAOGo/&6
if length_r~=length(theta) b6 _*ljM
error('zernfun:RTHlength', ... C3-l(N1O{
'The number of R- and THETA-values must be equal.') 65 AXUTg
end [ YP8 z~
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% Check normalization: 8
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% -------------------- gVCkj!{
if nargin==5 && ischar(nflag) q:#,b0|bv
isnorm = strcmpi(nflag,'norm'); ;5# P?
if ~isnorm *{tn/ ro6a
error('zernfun:normalization','Unrecognized normalization flag.') FOpOS?Cr'
end S]ZO *+
else &Th/Qv}[
isnorm = false; @;_r `AT7
end lJoMJS;S]}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |tz1'YOB
% Compute the Zernike Polynomials ',8]vWsl
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 1JgnuBX"
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% Determine the required powers of r: m }6GVQ'Q
% ----------------------------------- C]*9:lK
m_abs = abs(m); %^ ^2
rpowers = []; xuO5|{ h
for j = 1:length(n) {.SN
rpowers = [rpowers m_abs(j):2:n(j)]; hU5[k/ q
end hF+YZU]rT
rpowers = unique(rpowers); tc@v9`^_
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\A:m<::
% Pre-compute the values of r raised to the required powers, VJD$nh
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% and compile them in a matrix: L':;Vv~-
% ----------------------------- / fA:Fnv
if rpowers(1)==0 BMU~1[ r
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); e`4OlM]
rpowern = cat(2,rpowern{:}); jnt0,y A
rpowern = [ones(length_r,1) rpowern]; 9C[3w[G~ C
else Cst\ _j
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ^Ot+,l)
rpowern = cat(2,rpowern{:}); *Au4q<
end 82Nh;5T r
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Z4}Yw{=f
% Compute the values of the polynomials: B9iH+
]W
% -------------------------------------- wED~^[] f
y = zeros(length_r,length(n)); W>dS@;E
for j = 1:length(n) Slq=;TDp
s = 0:(n(j)-m_abs(j))/2;
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pows = n(j):-2:m_abs(j); y&lj+j
for k = length(s):-1:1 B^U5= L[:p
p = (1-2*mod(s(k),2))* ... EU ThH.
prod(2:(n(j)-s(k)))/ ... !fwLC"QC
prod(2:s(k))/ ... + %eMm.(
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Cv{rd##Y8
prod(2:((n(j)+m_abs(j))/2-s(k))); IyOujdKa
idx = (pows(k)==rpowers); gsc/IUk
y(:,j) = y(:,j) + p*rpowern(:,idx); U ?>P6p
end ZNFn^i uQ
l+kI4B7--
if isnorm qOZe\<.V<
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); "6
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end +;`Cm.Iu
end ub}t3#
% END: Compute the Zernike Polynomials p(Y'fd}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% mY(~94{d
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% Compute the Zernike functions: ],>Z' W
% ------------------------------ eXnMS!g%Z
idx_pos = m>0; cli P+#
idx_neg = m<0; \M=" R-&b
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z = y; }_3<Q\j
if any(idx_pos) zjM+F{P8
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 5Tb93Q@c
end `P)atQ
if any(idx_neg) m]=|%a6
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); &D qg<U
end 3tS~/o+]
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% EOF zernfun fseHuL=~
