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, 9YD\~v;x
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, 802H$P^ps
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? zEj#arSE4
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? {{\ce;hN
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function z = zernfun(n,m,r,theta,nflag) O#)jr-vXdV
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. cLG6(<L
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 8#w)X/
% and angular frequency M, evaluated at positions (R,THETA) on the ?F_)-
% unit circle. N is a vector of positive integers (including 0), and lNz]HiD
% M is a vector with the same number of elements as N. Each element FH8k'Hxg
% k of M must be a positive integer, with possible values M(k) = -N(k) 22&;jpL'?
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, YHB9mZi
% and THETA is a vector of angles. R and THETA must have the same 1Ipfw
% length. The output Z is a matrix with one column for every (N,M) E"6X|I n
% pair, and one row for every (R,THETA) pair. nn+_TMu
% I-kWS4
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike .XS9,/S
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), rQb7?O@-
% with delta(m,0) the Kronecker delta, is chosen so that the integral V%*b@zv
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, wP<07t[-g
% and theta=0 to theta=2*pi) is unity. For the non-normalized @ }&_Dvf
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ?s2^zT
% VL\t>n
% The Zernike functions are an orthogonal basis on the unit circle. lyv4fP
% They are used in disciplines such as astronomy, optics, and '#.#$8l
% optometry to describe functions on a circular domain. d|lpec
% cE\>f8 I
% The following table lists the first 15 Zernike functions. i{Ds&{
% \~~ }N4
% n m Zernike function Normalization wNYg$d0M
% -------------------------------------------------- ;j9\b9m
% 0 0 1 1 @1:0h9%
% 1 1 r * cos(theta) 2 2YlH}fnH
% 1 -1 r * sin(theta) 2 9t$]X>}
% 2 -2 r^2 * cos(2*theta) sqrt(6) D+RiM~LH8
% 2 0 (2*r^2 - 1) sqrt(3) oyvKag
% 2 2 r^2 * sin(2*theta) sqrt(6) /?*]lH.
% 3 -3 r^3 * cos(3*theta) sqrt(8) kXrlSaIc
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) +?dl`!rE
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) %JyXbv3m,
% 3 3 r^3 * sin(3*theta) sqrt(8) 2VoKr)
% 4 -4 r^4 * cos(4*theta) sqrt(10) M{mSd2
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) (Un_!)
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) m@Rtlb
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) =0
% 4 4 r^4 * sin(4*theta) sqrt(10) ;j%BK(5
% -------------------------------------------------- k[kju%i4
% Vsnuy8~k
% Example 1: :O= \<t
% }`\/f
% % Display the Zernike function Z(n=5,m=1) /.z;\=;[n!
% x = -1:0.01:1; g(|{')8?d
% [X,Y] = meshgrid(x,x); 6"f}O<M5H
% [theta,r] = cart2pol(X,Y); ~Z'w)!h
% idx = r<=1; 8|%^3O 0X
% z = nan(size(X)); >e,mg8u6$
% z(idx) = zernfun(5,1,r(idx),theta(idx)); Wwujh2g"0|
% figure cC'x6\a
% pcolor(x,x,z), shading interp UVQ7L9%?f
% axis square, colorbar 7 msAhz
% title('Zernike function Z_5^1(r,\theta)') T0z n,ej
% ._O
% Example 2: hrGH}CU"
% T r0B[QF
% % Display the first 10 Zernike functions $*R/tJ.
% x = -1:0.01:1; U}k9 Py
% [X,Y] = meshgrid(x,x); \ZU1Jb1c
% [theta,r] = cart2pol(X,Y); A:l@_*C..
% idx = r<=1; jPZaD>!
% z = nan(size(X)); cWyW~Ek
% n = [0 1 1 2 2 2 3 3 3 3]; ^vilgg~
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; j _L@U2i
% Nplot = [4 10 12 16 18 20 22 24 26 28]; 3&&9_`r&_
% y = zernfun(n,m,r(idx),theta(idx)); ={>Lrig:l
% figure('Units','normalized') &0zT I?c
% for k = 1:10 jz58E}
% z(idx) = y(:,k);
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% subplot(4,7,Nplot(k)) 03E4cYxt5
% pcolor(x,x,z), shading interp 9d[5{"2j
% set(gca,'XTick',[],'YTick',[]) { FZ=olZ
% axis square rE9I>|tX
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) Z[__"^}
% end V-'K6mn;
% w }^ I
% See also ZERNPOL, ZERNFUN2. o6
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% Paul Fricker 11/13/2006 YW u cvw&
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% Check and prepare the inputs: ,C%eBna4Iq
% ----------------------------- 26T "XW'_
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) )
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error('zernfun:NMvectors','N and M must be vectors.') Q'_z<V
end A+hT3;lp
b)(?qfXWP
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if length(n)~=length(m) m3Il3ZY.
error('zernfun:NMlength','N and M must be the same length.') hW!)w
end mU}F!J#6
!,V{zTR
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n = n(:); rV08ad
m = m(:); (=~&+z
if any(mod(n-m,2)) !uQPc
error('zernfun:NMmultiplesof2', ... .9Y)AtJTS
'All N and M must differ by multiples of 2 (including 0).') y~()|L[
end yR(x+Gs{]
o,|[GhtHqs
lz1wO5%h
if any(m>n) ~ vqa7~}m
error('zernfun:MlessthanN', ... OS8q( 2z?s
'Each M must be less than or equal to its corresponding N.') r@ZJ{4\Q
end W`c'=c
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if any( r>1 | r<0 ) BcI|:qv|
error('zernfun:Rlessthan1','All R must be between 0 and 1.') +TXX$)3%
end !.d@L6
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) y*X_T,K8
error('zernfun:RTHvector','R and THETA must be vectors.') F_CYYGZ
end Yk=PS[f
M![J2=
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r = r(:); hP15qKy
theta = theta(:); `]%|f
length_r = length(r); AM!G1^c
if length_r~=length(theta) H)n9O/u
error('zernfun:RTHlength', ... 8YbE`32
'The number of R- and THETA-values must be equal.') EY tQw(!Q
end M3q|l7|9
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% Check normalization: 6fw(T.Pe
% -------------------- 0\e IQp
if nargin==5 && ischar(nflag) lv04g} W
isnorm = strcmpi(nflag,'norm'); |j7,Mu+
if ~isnorm 13>0OKg`#
error('zernfun:normalization','Unrecognized normalization flag.') 5k.oW=
end ^0 -:G6H
else J@u;H$@/y
isnorm = false; >6?__v]9G
end 2 O%`G+\)
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&wV]"&-
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ajEjZ6
% Compute the Zernike Polynomials n^g|Ja
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ]iUxp+
9?SZNL['V
x U4 +|d
% Determine the required powers of r: k=jk`c{<[
% ----------------------------------- V{!J-nO
m_abs = abs(m); xsD($_
rpowers = []; =o$sxb
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for j = 1:length(n) LA}Syt\F
rpowers = [rpowers m_abs(j):2:n(j)]; B\o Mn
end T:=lz:}I
rpowers = unique(rpowers); \hx1o\
A|<jX}
s*-n^o-
% Pre-compute the values of r raised to the required powers, H<PtAYFS
% and compile them in a matrix: r2,.abo
% ----------------------------- U`2e{>'4t
if rpowers(1)==0 bAx-"Lu
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); oY933i@l)P
rpowern = cat(2,rpowern{:}); _I:/ZF5
rpowern = [ones(length_r,1) rpowern]; FG.em
else Q$zO83
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); aWR}R>E
rpowern = cat(2,rpowern{:}); Hl{S]]z
end *GL/aEI<$
KbA?7^zo`
Z$/xy"
% Compute the values of the polynomials: ,F,X
,
% -------------------------------------- 8Djc
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z
y = zeros(length_r,length(n)); AP'*Nh@Ik(
for j = 1:length(n) R#%(5-Zu#R
s = 0:(n(j)-m_abs(j))/2; 7/I, HxXp!
pows = n(j):-2:m_abs(j); iOW#>66d
for k = length(s):-1:1 Brf5dT49
p = (1-2*mod(s(k),2))* ... (>nGQS]H
prod(2:(n(j)-s(k)))/ ... H|3:6x
prod(2:s(k))/ ... `erV$( M
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... jIC_[
prod(2:((n(j)+m_abs(j))/2-s(k))); [XEkz#{
idx = (pows(k)==rpowers); ~?d Nd
y(:,j) = y(:,j) + p*rpowern(:,idx); >7jbgHB
end &,{fw@#)_
;$.J3!
if isnorm _Xk.p_uh
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Rwz0poG`WG
end CDQW !XHc
end f4 P8Oz
% END: Compute the Zernike Polynomials ywGd> @
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% }`%*W`9b
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% Compute the Zernike functions: 7z JRJ*NB
% ------------------------------ pwL;A3$|
idx_pos = m>0; WW4vn|0v
idx_neg = m<0; gQ Fjr_IS#
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z = y; 6e9,PS
if any(idx_pos) B-ngn{Yc
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); X' H[7 ^W
end l;R%= P?'F
if any(idx_neg) <D<4BnZ(
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Pg,b-W?n*
end oHd FMD@
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n}F&1Z
% EOF zernfun \<JSkr[h!"