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, 5U$0F$BBp
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, ?Z/V~,
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? xi}skA
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? /y}xX
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function z = zernfun(n,m,r,theta,nflag) 3f{3NzN
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. +cN8Y}V
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N )+DmOsH
% and angular frequency M, evaluated at positions (R,THETA) on the M .mfw#*
% unit circle. N is a vector of positive integers (including 0), and vl:KF7:#m
% M is a vector with the same number of elements as N. Each element UP,c |
% k of M must be a positive integer, with possible values M(k) = -N(k) DB}eA N/
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, u'BaKWPS
% and THETA is a vector of angles. R and THETA must have the same _q-*7hCQ`
% length. The output Z is a matrix with one column for every (N,M) jNk%OrP]
% pair, and one row for every (R,THETA) pair. i8]S:4 9
% SwMc
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% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 2J BR)P
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), S<Xf>-8w
% with delta(m,0) the Kronecker delta, is chosen so that the integral &%J08l6
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ( a#BV}=
% and theta=0 to theta=2*pi) is unity. For the non-normalized k{-Cwo
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. $=4QO
% ^ [@,
% The Zernike functions are an orthogonal basis on the unit circle. zTU0HR3A
% They are used in disciplines such as astronomy, optics, and }qD\0+`qi
% optometry to describe functions on a circular domain. >z@0.pN]7
% ]h5tgi?_l
% The following table lists the first 15 Zernike functions. oUlVI*~ND
% 5r^(P
% n m Zernike function Normalization G"A#Q"
% -------------------------------------------------- F:S}w
% 0 0 1 1 o`-msz
% 1 1 r * cos(theta) 2 UkFC~17P
% 1 -1 r * sin(theta) 2 LKDO2N
% 2 -2 r^2 * cos(2*theta) sqrt(6) A.w.rVDD
% 2 0 (2*r^2 - 1) sqrt(3)
Z *x'+X
% 2 2 r^2 * sin(2*theta) sqrt(6) 7@W>E;go
% 3 -3 r^3 * cos(3*theta) sqrt(8) ;aVZ"~a+\
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) l.M0`Cn-%
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) 4o5t#qP5$S
% 3 3 r^3 * sin(3*theta) sqrt(8) CU!Dhm/U
% 4 -4 r^4 * cos(4*theta) sqrt(10)
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% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 1?l1:}^L
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 3ckclO\|>
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) KMax$
% 4 4 r^4 * sin(4*theta) sqrt(10) \s\?l(ooq"
% -------------------------------------------------- ;!Fn1|)
% 5|)W.*Q
% Example 1: =Dj#gV
% %8v\FS
% % Display the Zernike function Z(n=5,m=1) 6_B]MN!(
% x = -1:0.01:1; B%68\
% [X,Y] = meshgrid(x,x); ]6j{@z?{
% [theta,r] = cart2pol(X,Y); o)/ 0a
% idx = r<=1; j1<Yg,_.p
% z = nan(size(X)); )boE/4
% z(idx) = zernfun(5,1,r(idx),theta(idx)); J<lW<:!3]
% figure cU
% pcolor(x,x,z), shading interp {P-):
% axis square, colorbar apn*,7ps65
% title('Zernike function Z_5^1(r,\theta)') UPGtj"2v-
% |DwZ{(R"W
% Example 2: +b6v!7_
% Q,Eo mt
% % Display the first 10 Zernike functions [nh>vqum
% x = -1:0.01:1; /x *3}oI
% [X,Y] = meshgrid(x,x); o4WDh@d5S
% [theta,r] = cart2pol(X,Y); 8{ I|$*nB
% idx = r<=1; @O~pV`_tD
% z = nan(size(X)); dc'Y`e
% n = [0 1 1 2 2 2 3 3 3 3]; qxc[M8s
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; # f\rt
% Nplot = [4 10 12 16 18 20 22 24 26 28]; lEBLZ}}\
% y = zernfun(n,m,r(idx),theta(idx)); NHE18_v5
% figure('Units','normalized') G#$-1"!`
% for k = 1:10 J .<F"r>
% z(idx) = y(:,k); ~.|_ RdN
% subplot(4,7,Nplot(k)) vih9KBT
% pcolor(x,x,z), shading interp W%w~ah|/]
% set(gca,'XTick',[],'YTick',[]) CvdN"k
% axis square J.%IfN
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) Ho]su?
% end Zwx%7l;C
% B-mowmJ3dg
% See also ZERNPOL, ZERNFUN2. (;,sc$H]
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% Paul Fricker 11/13/2006 G+m }MOQP7
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[N'h%1]\
% Check and prepare the inputs: O".=r}
% ----------------------------- qxj(p o
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) wgA_38To
error('zernfun:NMvectors','N and M must be vectors.') !`r$"}g
end GN>@ZdVG}#
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if length(n)~=length(m) Z30A{6}
error('zernfun:NMlength','N and M must be the same length.') *K;~!P
end {c0`Um3&>
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n = n(:); 6w7 7YTJ
m = m(:); cc3 4e
if any(mod(n-m,2)) LH6vLuf
error('zernfun:NMmultiplesof2', ... P93@;{c(
'All N and M must differ by multiples of 2 (including 0).') @o.I ;}*N
end L:x-%m%w
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if any(m>n) BJ0?kX@
error('zernfun:MlessthanN', ... IRbfNq^:
'Each M must be less than or equal to its corresponding N.') ,z?':TZ
end bPMhfK2 %
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if any( r>1 | r<0 ) !!y a
error('zernfun:Rlessthan1','All R must be between 0 and 1.') ~)'k 9?0
end Xm&L
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) -$@h1Y
error('zernfun:RTHvector','R and THETA must be vectors.') L0]_X#s>#
end 9!tW.pK5
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,
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r = r(:); ?%kV?eu'
theta = theta(:); A)~6Im
length_r = length(r); QCJM&
if length_r~=length(theta) H[|~/0?K
error('zernfun:RTHlength', ... -Qe Z#w|
'The number of R- and THETA-values must be equal.') y?!"6t7&
end -^wl>}#*T3
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% Check normalization: YoE3<[KD(
% -------------------- ~;] d"'
if nargin==5 && ischar(nflag) @|)Z"m7
isnorm = strcmpi(nflag,'norm'); H:\k}*w
if ~isnorm Ct|A:/z(
error('zernfun:normalization','Unrecognized normalization flag.') 4/)k)gLI
end J-4:H
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else y!%CffF2
isnorm = false; 3mni>*q7d
end h1(4Ic
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% )_NO4`ejs/
% Compute the Zernike Polynomials BPHW}F]X
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% E!AE4B1bd
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% Determine the required powers of r: xwty<?dRW1
% ----------------------------------- 4`R(?
m_abs = abs(m); W'.m'3#z
rpowers = []; l@:0e]8|o
for j = 1:length(n) KG5>]_GH
rpowers = [rpowers m_abs(j):2:n(j)]; ]:\dPw`A
end ?' je)F
rpowers = unique(rpowers); b u"!jHPB
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% Pre-compute the values of r raised to the required powers, W}1
;Z(.*
% and compile them in a matrix: fxIf|9Qi`
% ----------------------------- 8x{'@WCG%
if rpowers(1)==0 2Hv+W-6v
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 2:=
rpowern = cat(2,rpowern{:}); 9)=ctoZ'
rpowern = [ones(length_r,1) rpowern]; <Ok3FE.K
else y)gKxRaCS
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); cs'{5!i]
rpowern = cat(2,rpowern{:}); ?0,Ngrbe
end zv"Z DRW
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% Compute the values of the polynomials:
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% -------------------------------------- !Rt>xD
y = zeros(length_r,length(n)); :/Qq@]O>
for j = 1:length(n) I!?}jo3
s = 0:(n(j)-m_abs(j))/2; ]g&TKm
pows = n(j):-2:m_abs(j); !v0LBe4
for k = length(s):-1:1 Wxe0IXq3Nn
p = (1-2*mod(s(k),2))* ... O7IJ%_A&
prod(2:(n(j)-s(k)))/ ... w+{LAS
prod(2:s(k))/ ... vZoaT|3
G]
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... v}Fr@0%
prod(2:((n(j)+m_abs(j))/2-s(k))); m9Hit8f@Q
idx = (pows(k)==rpowers); L,@lp
y(:,j) = y(:,j) + p*rpowern(:,idx); bY0|N[g
end @y&bw9\
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if isnorm #lW`{i
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); "FKOaQ%IH
end #YOA`m,'
end Z)aUt
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% END: Compute the Zernike Polynomials e)O4^#i
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0_t`%l=
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% Compute the Zernike functions: xK\d4"
% ------------------------------ j,dR,N d
idx_pos = m>0; (*)hD(C5
idx_neg = m<0; ^]-6u:J!
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z = y; igR";OQk
if any(idx_pos) FG*r'tC~r
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); A$:U'ZG_
end >&5DsV.B
if any(idx_neg) 0=E]cQwh
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 1PV'?tXp(
end s}% M4
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% EOF zernfun 2V;PYI