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, GkFNLM5'
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, {r)M@@[
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}1wZ*
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? JnDR(s4(E
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function z = zernfun(n,m,r,theta,nflag) hlZjk0ez
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. IYPLitT
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N psVRdluS
% and angular frequency M, evaluated at positions (R,THETA) on the ;21JM2JI8
% unit circle. N is a vector of positive integers (including 0), and }f}&|Vap
% M is a vector with the same number of elements as N. Each element T9A5L"-6T
% k of M must be a positive integer, with possible values M(k) = -N(k) (x@"Dp=MZW
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, </QSMs
% and THETA is a vector of angles. R and THETA must have the same x&d<IU)5
% length. The output Z is a matrix with one column for every (N,M) _G|6xlO
% pair, and one row for every (R,THETA) pair. p
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% C#R9Hlb
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike bOdD:=f
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), .B*)A.
% with delta(m,0) the Kronecker delta, is chosen so that the integral @[Th{HTc.G
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, mfvQ]tz_+
% and theta=0 to theta=2*pi) is unity. For the non-normalized AXCJFqk;
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. Z"jo
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% )j]RFt
% The Zernike functions are an orthogonal basis on the unit circle. uu>g(q?4II
% They are used in disciplines such as astronomy, optics, and `*a,8M%
% optometry to describe functions on a circular domain. 7vFqO;
% 8
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% The following table lists the first 15 Zernike functions. 21qhlkdc
% oS4ag
% n m Zernike function Normalization u(R`}C?P'
% -------------------------------------------------- 1tDN$rM5
% 0 0 1 1 I(.XK ucU
% 1 1 r * cos(theta) 2 JpDkf$kM
% 1 -1 r * sin(theta) 2 '};Xb|msU
% 2 -2 r^2 * cos(2*theta) sqrt(6) -vyC,A
% 2 0 (2*r^2 - 1) sqrt(3) n!p&.Mt
% 2 2 r^2 * sin(2*theta) sqrt(6) R~i<*
% 3 -3 r^3 * cos(3*theta) sqrt(8) Z&%61jGK
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) +vP1DXtj(
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) epnDvz\
% 3 3 r^3 * sin(3*theta) sqrt(8) b+3pu\w`
% 4 -4 r^4 * cos(4*theta) sqrt(10) G4i&:0
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 6T-(GHzfHJ
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) tua+R_"
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 7;XdTx
% 4 4 r^4 * sin(4*theta) sqrt(10) jHd~yCq
% -------------------------------------------------- AXyuXB
% QMIXz[9w
% Example 1: 2eNm2;
% *M="k 1P1
% % Display the Zernike function Z(n=5,m=1) ,MLPVDN*D
% x = -1:0.01:1; 6V)# Yf
% [X,Y] = meshgrid(x,x); v1}
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% [theta,r] = cart2pol(X,Y); ,=mn*
% idx = r<=1; {E9Y)Z9
% z = nan(size(X)); Zy'bX* s|
% z(idx) = zernfun(5,1,r(idx),theta(idx)); 7(jt:V6V
% figure 1G\ugLm
% pcolor(x,x,z), shading interp =Rui
% axis square, colorbar 6ul34\;
% title('Zernike function Z_5^1(r,\theta)') AOTI&v
% y]Y)?])
% Example 2: f.,-KIiF
% W>"i0p
% % Display the first 10 Zernike functions B *:6U+I
% x = -1:0.01:1; mJT7e
% [X,Y] = meshgrid(x,x); MW p^.
% [theta,r] = cart2pol(X,Y); .G^.kg ,
% idx = r<=1; $tb$gO
% z = nan(size(X)); _+UD>u{
% n = [0 1 1 2 2 2 3 3 3 3]; s?=J#WV1y
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; _-EHG
% Nplot = [4 10 12 16 18 20 22 24 26 28]; sl)_HA7G
% y = zernfun(n,m,r(idx),theta(idx)); @]A4{
% figure('Units','normalized') 2qN6{+]
% for k = 1:10 ^UJO(
% z(idx) = y(:,k); JK_sl>v.7
% subplot(4,7,Nplot(k)) GwpJxiFgk
% pcolor(x,x,z), shading interp vXyaOZ
% set(gca,'XTick',[],'YTick',[]) ><$hFrR!
% axis square JL]6o8x
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) Xh){W~-
% end "5vFa7y
% z7J#1q~:yY
% See also ZERNPOL, ZERNFUN2. )'nGuL-w!i
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% Paul Fricker 11/13/2006 +f|u5c
-[ F<u
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% Check and prepare the inputs: X1$0'usS
% ----------------------------- M7En%sBp
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) g,9o'fs`x
error('zernfun:NMvectors','N and M must be vectors.') c^I_~OwaE
end !x|Ok'izDL
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if length(n)~=length(m) )j!22tlL
error('zernfun:NMlength','N and M must be the same length.') |odl~juU
end ->:G+<
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n = n(:); P)ZGNtO9fG
m = m(:); 8D)2/$NsY}
if any(mod(n-m,2)) \,lgv
error('zernfun:NMmultiplesof2', ... Kp8!^os
'All N and M must differ by multiples of 2 (including 0).') BY72 fy#e
end X5'foFE'
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if any(m>n) h)fi9
error('zernfun:MlessthanN', ... m^% [
'Each M must be less than or equal to its corresponding N.') -#|J
end O\=3{
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if any( r>1 | r<0 ) O#uTwnW
error('zernfun:Rlessthan1','All R must be between 0 and 1.') 6m|j "m
end 0sLR5A
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) $RfM}!7?
error('zernfun:RTHvector','R and THETA must be vectors.') 49E<`f0
end '!I^Lfz-Z
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r = r(:); +'Ec)7m
theta = theta(:); `B}(Ln
length_r = length(r); s+8
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if length_r~=length(theta) za`
error('zernfun:RTHlength', ... JBo/<W#|
'The number of R- and THETA-values must be equal.') \cP\I5IW:s
end -^`]tF`M
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% Check normalization: XbsEO>_Z'A
% -------------------- T0J"Wr>WY
if nargin==5 && ischar(nflag) ;I1}g]
isnorm = strcmpi(nflag,'norm'); EbZRU65J}O
if ~isnorm Dm?>U1{
error('zernfun:normalization','Unrecognized normalization flag.') K+5S7wFDZ
end =\GuIH2
else NHG+l)y:
isnorm = false; uDJi2,|n
end tt2`N3Eu\
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% &^KmfT5C
% Compute the Zernike Polynomials O:cta/M
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% St}j^i
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z;yb;),
% Determine the required powers of r: ~0aWjMc(>
% ----------------------------------- Hg\+:}k&9
m_abs = abs(m); xs_l+/cZ
rpowers = []; ;O5p>o
for j = 1:length(n) ">PpC]Y1
rpowers = [rpowers m_abs(j):2:n(j)]; Nn5z
end JDrh-6Zgj
rpowers = unique(rpowers); qfE>N?/
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% Pre-compute the values of r raised to the required powers, JfS:K'
% and compile them in a matrix: VDq4n;p1
% ----------------------------- 6UOV,`:m+
if rpowers(1)==0 H-$ )@
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 3)ac
rpowern = cat(2,rpowern{:}); G66A]FIg
rpowern = [ones(length_r,1) rpowern]; jsL\{I^>
else ij&_>
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); !m)P*Lw
rpowern = cat(2,rpowern{:}); eV$pza
end eq+t%
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% Compute the values of the polynomials: ZS[(r-)$F
% -------------------------------------- Blv!%es
y = zeros(length_r,length(n)); \-3\lZ3qj
for j = 1:length(n) ma@3BiM
s = 0:(n(j)-m_abs(j))/2; 2]W"sT[
pows = n(j):-2:m_abs(j); c^0YuBps[
for k = length(s):-1:1 ip6$Z3[)
p = (1-2*mod(s(k),2))* ... `|@# ~
prod(2:(n(j)-s(k)))/ ... o;bK 7D
prod(2:s(k))/ ... n46A
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... )QS4Z{)U
prod(2:((n(j)+m_abs(j))/2-s(k))); k{_ Op/k}V
idx = (pows(k)==rpowers); %%J)@k^vH
y(:,j) = y(:,j) + p*rpowern(:,idx); ? ->:,I=<~
end J!r,ktO^U?
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if isnorm OL+dx`Y
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 3Jt_=!qlo
end |^&n\vXv
end pm$ZKM
% END: Compute the Zernike Polynomials ILdRN
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% i
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% Compute the Zernike functions: Cw`8[)=}o
% ------------------------------ g$C-G5/bjD
idx_pos = m>0; WmU5YZ(mAq
idx_neg = m<0; yU*upQ
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z = y; ke.{wh\0
if any(idx_pos) "-aak )7w
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 6{h+(|.(
end ]L0GIVIE
if any(idx_neg) Z9cg,#(D
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Hg)5c!F7
end GdZ_
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% EOF zernfun LxqK@Q<B