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， JDT`C2-Q
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 mg}N
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？ *(DV\. l`
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ c9h6C
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function z = zernfun(n,m,r,theta,nflag) XZd,&YiaG
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. B^^#D0<
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N [.wYdv35
% and angular frequency M, evaluated at positions (R,THETA) on the ?gGHj-HYJ
% unit circle. N is a vector of positive integers (including 0), and 5$C-9
% M is a vector with the same number of elements as N. Each element \bw2u!
% k of M must be a positive integer, with possible values M(k) = -N(k) R8'RA%O9J
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, -n V9:opD
% and THETA is a vector of angles. R and THETA must have the same h~zT ydnH
% length. The output Z is a matrix with one column for every (N,M) YUk\Q%
% pair, and one row for every (R,THETA) pair. ZPYS$Ydy
% vx5Zl& 6r
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike [d ]9Oa4
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), /mzlH
% with delta(m,0) the Kronecker delta, is chosen so that the integral Z4ImV~m
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, {I't]Qj_e
% and theta=0 to theta=2*pi) is unity. For the non-normalized e$ rZ 5X
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. Mb*?5 R6;
% g[4WzDF*
% The Zernike functions are an orthogonal basis on the unit circle. }@d @3
% They are used in disciplines such as astronomy, optics, and lrIe"H@
% optometry to describe functions on a circular domain. --BW9]FW
% h <<v^+m
% The following table lists the first 15 Zernike functions. ^^ixa1H<
% 8YSAf+{FtK
% n m Zernike function Normalization pTLCWbF?
% -------------------------------------------------- uoh7Sz5!^
% 0 0 1 1 4BpZJ~(p
% 1 1 r * cos(theta) 2 - 1gVeT&
% 1 -1 r * sin(theta) 2 u Q KT
% 2 -2 r^2 * cos(2*theta) sqrt(6) AH~ E )S
% 2 0 (2*r^2 - 1) sqrt(3) S3Jo>jXS "
% 2 2 r^2 * sin(2*theta) sqrt(6) b@hqz!)l`
% 3 -3 r^3 * cos(3*theta) sqrt(8) SXP]%{@ R/
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) :gFx{*xN/9
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) X 0+vXz{~g
% 3 3 r^3 * sin(3*theta) sqrt(8) S{T >}'y
% 4 -4 r^4 * cos(4*theta) sqrt(10) ,r_Gf5c
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ,Ma^ &ypH
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) +9sQZB# (
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) dioGAai '
% 4 4 r^4 * sin(4*theta) sqrt(10) e~"U @8xk~
% -------------------------------------------------- 1 [Bk%G@D&
% xr^LFn)
% Example 1:
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% ysnx3(+|
% % Display the Zernike function Z(n=5,m=1) !0 <,@v"
% x = -1:0.01:1; +] {G@pn
% [X,Y] = meshgrid(x,x); >Y@H4LF;1x
% [theta,r] = cart2pol(X,Y); h^P#{W!e\
% idx = r<=1; {(Es(Sb}c
% z = nan(size(X)); ^E>3|du]O
% z(idx) = zernfun(5,1,r(idx),theta(idx)); aV0"~5
% figure B/Ws_Kv
% pcolor(x,x,z), shading interp uHRsFlw
% axis square, colorbar +k R4E 23:
% title('Zernike function Z_5^1(r,\theta)') N?`' /e
% > 9Vn.S
% Example 2: l,aay-E
% *wjrR1#81x
% % Display the first 10 Zernike functions -jm Y)(\
% x = -1:0.01:1; +R75v )
% [X,Y] = meshgrid(x,x); TIg3` Fon
% [theta,r] = cart2pol(X,Y); sU^1wB
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% idx = r<=1; M&M 6;Ph
% z = nan(size(X)); ]A_`0"m.U
% n = [0 1 1 2 2 2 3 3 3 3]; =:U`k0rn!
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; goWuw}?
% Nplot = [4 10 12 16 18 20 22 24 26 28]; -m#)B~)
% y = zernfun(n,m,r(idx),theta(idx)); DzRFMYBR
% figure('Units','normalized') VuZr:-K/
% for k = 1:10 E2+`4g@{8<
% z(idx) = y(:,k); 9%obq/Lb
% subplot(4,7,Nplot(k)) \o3gKoL%
% pcolor(x,x,z), shading interp 7F~ X,Dk_
% set(gca,'XTick',[],'YTick',[]) E' uZA
% axis square 8z q=N#x
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) wVtwx0|1
% end E _|<jy$`
% *lJxH8 \
% See also ZERNPOL, ZERNFUN2. [()koU#w.
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% Paul Fricker 11/13/2006 1Ai^cf :S
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% Check and prepare the inputs: |JsZJ9W+J
% ----------------------------- GTxk%
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) &BSn?
error('zernfun:NMvectors','N and M must be vectors.') RT8 ?7xFc
end bcz:q/f}@
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if length(n)~=length(m)
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error('zernfun:NMlength','N and M must be the same length.') Vi |#@tC'
end 3PF_H$`oJ
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n = n(:); %@Jsal'
m = m(:); 1{.9uw"2S
if any(mod(n-m,2)) DVeE 1Q
error('zernfun:NMmultiplesof2', ... |5 ]X| v
'All N and M must differ by multiples of 2 (including 0).') ,`sv1xwd
end ?\n >
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if any(m>n) ^=*;X;7
error('zernfun:MlessthanN', ... !p/goqT~dY
'Each M must be less than or equal to its corresponding N.') -t U'yKhn
end lk =<A"^S
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if any( r>1 | r<0 ) dgP3@`YS
error('zernfun:Rlessthan1','All R must be between 0 and 1.') J9 I :Q<;
end (w zQ2Dk
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) -ze J#B)C
error('zernfun:RTHvector','R and THETA must be vectors.') %]7d`/
end BL4 -7
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r = r(:); O.JN ENZf
theta = theta(:); 5E
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length_r = length(r); . c cp
if length_r~=length(theta) ;9'OOz|+1
error('zernfun:RTHlength', ... Zgb!E]V[
'The number of R- and THETA-values must be equal.') IUct
end *n"{J(Jt`
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B|X!>Q<g
% Check normalization: |+"(L#wk
% -------------------- .tr!( O],h
if nargin==5 && ischar(nflag) 9Gz=lc[!7
isnorm = strcmpi(nflag,'norm');
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if ~isnorm 7o}J%z
error('zernfun:normalization','Unrecognized normalization flag.') Yoll?_k+
end uvS)8-o&F
else ] }X
isnorm = false; 6d~'$<5on
end Q=dy<kg']
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {(?4!rh
% Compute the Zernike Polynomials -H-~;EzU
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 7cMv/g^ h@
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2 8 u_!f[
% Determine the required powers of r: UR5`ue ;
% ----------------------------------- {+ b7 sA3
m_abs = abs(m); r:TH]hs12+
rpowers = []; Qe(:|q _
for j = 1:length(n) l}M !8:UzU
rpowers = [rpowers m_abs(j):2:n(j)]; S$X Sei_q
end G .4X'
rpowers = unique(rpowers); 5Jnlz@P9
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% Pre-compute the values of r raised to the required powers, oe~b}:
% and compile them in a matrix: #A8sLkY
% ----------------------------- ( &x['IR
if rpowers(1)==0 6;5Ss?ep
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); "5$B>S(Q
rpowern = cat(2,rpowern{:}); Ny)X+2Ae
rpowern = [ones(length_r,1) rpowern]; Z ;)%%V %o
else 1[-tD 0{H
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); IV) j1
rpowern = cat(2,rpowern{:}); LBP`hK:>W~
end sdmT
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% Compute the values of the polynomials: ;i +#fQO7Q
% -------------------------------------- x'R`.
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y = zeros(length_r,length(n)); 'H <\x
for j = 1:length(n) 63 B?.
s = 0:(n(j)-m_abs(j))/2;
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pows = n(j):-2:m_abs(j); Aq7osU1B
for k = length(s):-1:1 >b4eL59
p = (1-2*mod(s(k),2))* ... %H"47ZFxAs
prod(2:(n(j)-s(k)))/ ... sCHJ&>m5-
prod(2:s(k))/ ... @U} 1EC{A
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... -z(+/ /K:#
prod(2:((n(j)+m_abs(j))/2-s(k))); DM>eVS3}
idx = (pows(k)==rpowers); g eCM<]
y(:,j) = y(:,j) + p*rpowern(:,idx); FaJ &GOM,
end .#pU=v#/[
v/=}B(TDF
if isnorm jRV/A!4
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); SasJic2M
end q> C'BIr
end >[*qf9$
% END: Compute the Zernike Polynomials _:27]K:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @f_+=}|dc
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% Compute the Zernike functions: ; 4~hB
% ------------------------------ Y:a]00&)#Y
idx_pos = m>0; 6!FQzFCZq
idx_neg = m<0; ~&bq0 (
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*;*r 8[U}q
z = y; y'*K|a TG
if any(idx_pos) !C: $?o U
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 0lR5<^B
end ~q Oa\#x_
if any(idx_neg) [cp+ i^f
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); L;I]OC^J
end CeC 6hGR5
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% EOF zernfun @<&m|qtMsz
