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， yQ+#Tlji
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， m^T$H_*;
这个，跟我们用zygo干涉仪直接拟合出的36项zernike系数，有何关系呢？ K
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那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ YQ37P?u@
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function z = zernfun(n,m,r,theta,nflag) K~,,xsy,G&
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. |EU}& k2
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N H E+VanY![
% and angular frequency M, evaluated at positions (R,THETA) on the l*'jqR')h^
% unit circle. N is a vector of positive integers (including 0), and qI;k2sQR
% M is a vector with the same number of elements as N. Each element +-ieaF
% k of M must be a positive integer, with possible values M(k) = -N(k) _vU,avw
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, ,*S?L
qv^
% and THETA is a vector of angles. R and THETA must have the same do=VPqy
% length. The output Z is a matrix with one column for every (N,M) Y60ld7H
% pair, and one row for every (R,THETA) pair. #|$i H kVY
% {,s:vPoiA
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 3O#7OL68v
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 2U|Nkm
% with delta(m,0) the Kronecker delta, is chosen so that the integral mW8CqW\Q5
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, L?9Vz&8]
% and theta=0 to theta=2*pi) is unity. For the non-normalized 8%
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% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. KM0#M'dXy
% sSD(mO<(
% The Zernike functions are an orthogonal basis on the unit circle. VIi| :k
% They are used in disciplines such as astronomy, optics, and LDPo}o gs
% optometry to describe functions on a circular domain. @4$F%[g
h
% # M`ijN!Y
% The following table lists the first 15 Zernike functions. }clFaT>m?
% 7)_0jp~2
% n m Zernike function Normalization Nb]qY>K
% -------------------------------------------------- XkdNWR0
% 0 0 1 1 te:"1:e
% 1 1 r * cos(theta) 2 -'q=oT Z
% 1 -1 r * sin(theta) 2 7ky( g'
% 2 -2 r^2 * cos(2*theta) sqrt(6) *^.b}K%
% 2 0 (2*r^2 - 1) sqrt(3) E >'pMw
% 2 2 r^2 * sin(2*theta) sqrt(6) &LL81u6=S
% 3 -3 r^3 * cos(3*theta) sqrt(8) 6f1;4Jfp
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) ?;{A@icr
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) 3tLh{S?uJ
% 3 3 r^3 * sin(3*theta) sqrt(8) t1ZZru'r
% 4 -4 r^4 * cos(4*theta) sqrt(10) }#E 4t3
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 4-3B"
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) bQ
.y,+
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) SxH}/I|W
% 4 4 r^4 * sin(4*theta) sqrt(10) X^xu$d6
% -------------------------------------------------- OH)SdSBz
% "lmiGR* u
% Example 1: qHHWe<}OT
% (1jkZ^7
% % Display the Zernike function Z(n=5,m=1) v"$; aJ
% x = -1:0.01:1; {L4^IK I
% [X,Y] = meshgrid(x,x); slQKkx \Dn
% [theta,r] = cart2pol(X,Y); n .A[Z
% idx = r<=1; 9d2$F9]:o
% z = nan(size(X)); 9MXauTKI
% z(idx) = zernfun(5,1,r(idx),theta(idx)); s{iYf :
% figure eq4<
% pcolor(x,x,z), shading interp <M \Z}2 d
% axis square, colorbar ekAGzu
% title('Zernike function Z_5^1(r,\theta)') vNtbb]')m
% %pg*oX1VK6
% Example 2: 2G'G4 5Q
% ^ WD [>E~
% % Display the first 10 Zernike functions \h0e09& I
% x = -1:0.01:1; ^ul `b
% [X,Y] = meshgrid(x,x); ]}5j X^j
% [theta,r] = cart2pol(X,Y); KY"~Ta`
% idx = r<=1; =_,OucKkYG
% z = nan(size(X)); K1+,y1c
% n = [0 1 1 2 2 2 3 3 3 3]; #Ta@A~ .L
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; ix$+NM<n
% Nplot = [4 10 12 16 18 20 22 24 26 28]; (wA|lK3
% y = zernfun(n,m,r(idx),theta(idx)); {u5)zVYC,U
% figure('Units','normalized') sY#K=5R
% for k = 1:10 u>? VD%
% z(idx) = y(:,k); ~I^]O \?
% subplot(4,7,Nplot(k)) H+: $ 7;
% pcolor(x,x,z), shading interp a\MU5%}\
% set(gca,'XTick',[],'YTick',[]) m[8#h(s*t
% axis square =o HJ_
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) <A6<q&g|E
% end (PB|.`_<H
%
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% See also ZERNPOL, ZERNFUN2. j<gnh
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% Paul Fricker 11/13/2006 RE7[bM3a
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% Check and prepare the inputs: +5HO T{wj
% ----------------------------- |LjCtm)@+
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) )
:?^(&3;
error('zernfun:NMvectors','N and M must be vectors.') AzQ}}A;TSx
end M,{F/Yu
#".{i+3E
+,KuYa{lu
if length(n)~=length(m) `}=Fw0
error('zernfun:NMlength','N and M must be the same length.') sy#Gb #=#
end L NE]#8ue
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on mO>q*
n = n(:); vLC&C-f
m = m(:); h FjW.~B
if any(mod(n-m,2)) r94BEC 2
error('zernfun:NMmultiplesof2', ... Y}[r`}={
'All N and M must differ by multiples of 2 (including 0).') I2 /wu(~>
end }@#e D
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if any(m>n) Iq52rI}
error('zernfun:MlessthanN', ... lWdE^-
'Each M must be less than or equal to its corresponding N.') }i[jJb`bY
end VEn%_9(]
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if any( r>1 | r<0 ) Op0
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error('zernfun:Rlessthan1','All R must be between 0 and 1.') _1hqD EM
end dEL>Uly
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) V]8fn MH
error('zernfun:RTHvector','R and THETA must be vectors.') 4 I~,B[|
end (+U!# T]'D
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r = r(:); 8[Qw8z5-
theta = theta(:); ox*Ka]
length_r = length(r); W=b5{
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if length_r~=length(theta) zz9.OnZ~
error('zernfun:RTHlength', ... ?L
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'The number of R- and THETA-values must be equal.') ,yT4(cMBk?
end Twk zX|
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% Check normalization: b
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_
% -------------------- Ry(!< w,
if nargin==5 && ischar(nflag) ~<eiWDf
isnorm = strcmpi(nflag,'norm'); (0][hdI~B
if ~isnorm Ug"B/UUFd
error('zernfun:normalization','Unrecognized normalization flag.') B%t^QbU #\
end v
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else RWoiV10
isnorm = false; 1zM`g_(#
end B;@yOm=
" pH+YqJ$
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ^\CQWgY(
% Compute the Zernike Polynomials (I+-wki"e
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LY|h*a6Ym
x}roPhZ
*=
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% Determine the required powers of r: eGE,zkj
FY
% ----------------------------------- tsk}]@W
m_abs = abs(m); khIh<-s!
rpowers = []; }l/md/C0
for j = 1:length(n) f; b(W
rpowers = [rpowers m_abs(j):2:n(j)]; hZFbiGQr\
end 6 Ym[^U
rpowers = unique(rpowers); @M,_m X
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'[HFIJ0K!
% Pre-compute the values of r raised to the required powers, X=JSqO6V9
% and compile them in a matrix: m$ o|s1t
% ----------------------------- w&H
?; 1
if rpowers(1)==0 w7
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rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); \<P W_'6
rpowern = cat(2,rpowern{:}); 8'?e4;O
rpowern = [ones(length_r,1) rpowern]; }Orc;_)r
else 06ueE\@Sg
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); [$./'-I]
rpowern = cat(2,rpowern{:}); Ve=0_GR0
end ;6]+/e7O
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% Compute the values of the polynomials: mg;qG@?
% -------------------------------------- _W!g'HP-D
y = zeros(length_r,length(n)); ="u(o(j"
for j = 1:length(n) a@ lK+ t
s = 0:(n(j)-m_abs(j))/2; ,wq.C6;&
pows = n(j):-2:m_abs(j); 1{,WY(,c
for k = length(s):-1:1 , :#prT[P"
p = (1-2*mod(s(k),2))* ... 9(iJ=ao (
prod(2:(n(j)-s(k)))/ ... R1S Ev$
prod(2:s(k))/ ... D~1nh%x_
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... R=IeAuZR4k
prod(2:((n(j)+m_abs(j))/2-s(k))); D8h~?ph K
idx = (pows(k)==rpowers); |S4yol
y(:,j) = y(:,j) + p*rpowern(:,idx); n,0}K+}
end y /PEm)=Tt
K=~h1qV:
if isnorm lP@9%L
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); J>+Dv?Ni$
end ny MA%9,B
end eag$i.^aS
% END: Compute the Zernike Polynomials <oR Nd3d
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vI+ PL(T@
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l&C %oW
% Compute the Zernike functions: ;bZ)q
% ------------------------------ :H?p^ d
e
idx_pos = m>0; {o]OxqE@
idx_neg = m<0; a. gu
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z = y; 5%W3&F6 %
if any(idx_pos) N.VzA
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z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)');
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end 9(H8MUF0{
if any(idx_neg) %;zA_Wg
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); R {*p \;
end G q:7d]c~T
'[A> eC++
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% EOF zernfun nGP>M#F
