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， KGY bPty}
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， | gA@WV-%
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？ F#gA2VCm
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ 3uocAmY
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function z = zernfun(n,m,r,theta,nflag) pM?~AYWb
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. &{V |%u}v
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N hBjU(}\3
% and angular frequency M, evaluated at positions (R,THETA) on the t,?, T~#9
% unit circle. N is a vector of positive integers (including 0), and LUbj^iQ9
% M is a vector with the same number of elements as N. Each element `q c"JB
% k of M must be a positive integer, with possible values M(k) = -N(k) u]Ku96!
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, He(65ciT<O
% and THETA is a vector of angles. R and THETA must have the same )&@YRT\c?8
% length. The output Z is a matrix with one column for every (N,M) Y"H`+U V
% pair, and one row for every (R,THETA) pair. +@Qr GY
% C2}y#A I
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike +})QT FV
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 1'qXT{f/~
% with delta(m,0) the Kronecker delta, is chosen so that the integral :)~l3:O
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 1.du #w
% and theta=0 to theta=2*pi) is unity. For the non-normalized >qo!#vJc
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% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. h m RmU{(Y
% &DWSf`:Hx
% The Zernike functions are an orthogonal basis on the unit circle. QPVi& *8_
% They are used in disciplines such as astronomy, optics, and Uj7Y TB
% optometry to describe functions on a circular domain. 0]4X/u#N
% CP J21^
% The following table lists the first 15 Zernike functions. H~Uf2A)C
% 2Mt$Dah
% n m Zernike function Normalization ~#E&E%sJ
% -------------------------------------------------- ',r` )9 o
% 0 0 1 1 tnJ7m8JmC
% 1 1 r * cos(theta) 2 98Vv K?
% 1 -1 r * sin(theta) 2 p<
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% 2 -2 r^2 * cos(2*theta) sqrt(6) <[k3x8H'
% 2 0 (2*r^2 - 1) sqrt(3) I _KHQ&Z*
% 2 2 r^2 * sin(2*theta) sqrt(6) `
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% 3 -3 r^3 * cos(3*theta) sqrt(8) q`1tUd 4G
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) K=N&kda
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) @D;K&:~|N
% 3 3 r^3 * sin(3*theta) sqrt(8) ] `$6=) _X
% 4 -4 r^4 * cos(4*theta) sqrt(10) 9i\RdJv.
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) $`|h F[tv
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) ~^2w)-N
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) f6Y?),`
% 4 4 r^4 * sin(4*theta) sqrt(10) @rYZ 0`E9
% -------------------------------------------------- M2Nh3ijr
% PEI$1,z
% Example 1: 8xLQ"
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% |KhpF1/(
% % Display the Zernike function Z(n=5,m=1) bo=H-d|
% x = -1:0.01:1; p6- //0qb
% [X,Y] = meshgrid(x,x); MLV]+H[mt
% [theta,r] = cart2pol(X,Y); +y wz@ 0nx
% idx = r<=1; b$'%)\( 'g
% z = nan(size(X)); aH"d~Y^
% z(idx) = zernfun(5,1,r(idx),theta(idx)); @ym:@<D
% figure
v c: kY
% pcolor(x,x,z), shading interp 8XH;<z<oJ
% axis square, colorbar 2E-Kz?,:[
% title('Zernike function Z_5^1(r,\theta)') f! +d*9
% & `m.]RV
% Example 2: (]q
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% dEDhdF #f
% % Display the first 10 Zernike functions $*{,Z<|2
% x = -1:0.01:1; %Ik5|\ob?
% [X,Y] = meshgrid(x,x); 791v>h
% [theta,r] = cart2pol(X,Y); )j8'6tk )Z
% idx = r<=1; %1{S{FB
% z = nan(size(X)); lz`\Q6rZ
% n = [0 1 1 2 2 2 3 3 3 3]; ?*~
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% m = [0 -1 1 -2 0 2 -3 -1 1 3]; E/H9#
% Nplot = [4 10 12 16 18 20 22 24 26 28]; ()|e
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% y = zernfun(n,m,r(idx),theta(idx)); pss ')YP.
% figure('Units','normalized') i|h{ <X 7[
% for k = 1:10 y ;ey(
% z(idx) = y(:,k); S_sHwObFu|
% subplot(4,7,Nplot(k)) '{,JuX"n
% pcolor(x,x,z), shading interp |}77 'w :
% set(gca,'XTick',[],'YTick',[])
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% axis square W
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% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 3SY1>}(Y
% end ~[ !Tpq5
% -d?<t}a
% See also ZERNPOL, ZERNFUN2. @u+LF]MY
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% Paul Fricker 11/13/2006 x<t ?Yc9
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% Check and prepare the inputs: iw.F8[})
% ----------------------------- :2
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if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) #: EhGlq8
error('zernfun:NMvectors','N and M must be vectors.') \ $TM=Ykj
end xz`0V}dPl
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if length(n)~=length(m) ~&,S xQT
error('zernfun:NMlength','N and M must be the same length.') uaD+G:{ [
end c @lF*"4
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n = n(:); hXH+C-%{
m = m(:); FS7D
if any(mod(n-m,2)) rxx VLW
error('zernfun:NMmultiplesof2', ... hB 'rkjt
'All N and M must differ by multiples of 2 (including 0).') /?>W\bP<
end ht\_YiDg3
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if any(m>n) fda2dY;
error('zernfun:MlessthanN', ... pw))9~ XU
'Each M must be less than or equal to its corresponding N.') k-4z2qB
end ./ tZ*sP:
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if any( r>1 | r<0 ) {#t7l V'4
error('zernfun:Rlessthan1','All R must be between 0 and 1.') uKY1AC__
end 3W[||V[r]<
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ;<MHl[jJD
error('zernfun:RTHvector','R and THETA must be vectors.') O ZKZv,
end 8VpmcGvc3
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r = r(:); &G-dx ET]
theta = theta(:); 75h]# k9\
length_r = length(r); D=f$-rn
if length_r~=length(theta) k/U rz*O
error('zernfun:RTHlength', ... fHuWBC_YO
'The number of R- and THETA-values must be equal.') TC gW^iu
end XB[EJGa X
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% Check normalization: Eg4&D4TG p
% -------------------- tI0 D{Xrc
if nargin==5 && ischar(nflag) dF&@q,
isnorm = strcmpi(nflag,'norm'); ZlMS=<hgFx
if ~isnorm P-Gp^JX8
error('zernfun:normalization','Unrecognized normalization flag.') oB<!U%B N
end l >oJ^J
else '^Q$:P{G?
isnorm = false; e=!sMWx6
end -23sm~ `
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% N D3|wQ`M0
% Compute the Zernike Polynomials =Q# (2
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ;e 2D}
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R<r,&X?m
% Determine the required powers of r: uesIkJ^Q[
% ----------------------------------- a0k/R<4
m_abs = abs(m); d|sf2
rpowers = []; Nc^:v /(P
for j = 1:length(n) #A~7rH%hi
rpowers = [rpowers m_abs(j):2:n(j)]; Wq25, M'
end e\ZV^h}TQ
rpowers = unique(rpowers); GG4FS
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% Pre-compute the values of r raised to the required powers, Eh *u6K)Z
% and compile them in a matrix: F:Yp1Wrb <
% ----------------------------- 5^{2 g^jH6
if rpowers(1)==0 j^/^PUR
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); B?d +^sz]
rpowern = cat(2,rpowern{:}); y=}o|/5"
rpowern = [ones(length_r,1) rpowern]; , 9bu I='
else EO/TuKt
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); +~xzgaL
rpowern = cat(2,rpowern{:}); 5',&8
end ] $F%
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% Compute the values of the polynomials: h \6 t\_^\
% -------------------------------------- bW GMgC
y = zeros(length_r,length(n)); =>e>
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for j = 1:length(n) Jn\>S z(96
s = 0:(n(j)-m_abs(j))/2; "!#KQ' 'R
pows = n(j):-2:m_abs(j); 0J .]`kR
for k = length(s):-1:1 E iPOY'
p = (1-2*mod(s(k),2))* ... .p78
\ T
prod(2:(n(j)-s(k)))/ ... dp }zG+
prod(2:s(k))/ ... }(#;{_
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... O}z-g&e.U
prod(2:((n(j)+m_abs(j))/2-s(k))); 7!
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idx = (pows(k)==rpowers); w*7wSP
y(:,j) = y(:,j) + p*rpowern(:,idx); dlDO?T
end v|rBOv
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if isnorm ! !)NER-dv
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); X(;W Y^i!
end =GC,1WVEqV
end 4=l$wg~;
% END: Compute the Zernike Polynomials vSo ,,~ F
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% gAK"ShOhG=
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% Compute the Zernike functions: WS\Ir-B
% ------------------------------ I$ ?.9&.&
idx_pos = m>0; D0X!j,Kc
idx_neg = m<0; l-8rCaq& J
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z = y; 9bUFxSH
if any(idx_pos) }k @S mO8
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); w u0q.]
end +-Z `v
if any(idx_neg) =A_fL{ SM
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); zCmx 1Djz
end ^K: -r !v^
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% EOF zernfun U<6+2y P
