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, E\iK_'#
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, d9{lj(2P
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&-`+V}U
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? #Xg;E3BM
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function z = zernfun(n,m,r,theta,nflag) [-(^>Y
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. LnR>!0:c
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N & SXw=;B
% and angular frequency M, evaluated at positions (R,THETA) on the M=,pn+}y>
% unit circle. N is a vector of positive integers (including 0), and k*1Lr\1
% M is a vector with the same number of elements as N. Each element z5@XFaQ
% k of M must be a positive integer, with possible values M(k) = -N(k) rWht},-|1
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, 9#DXA}
% and THETA is a vector of angles. R and THETA must have the same Sca"LaW1
% length. The output Z is a matrix with one column for every (N,M) Nd0tR3gi7
% pair, and one row for every (R,THETA) pair. Tm"H9
% ~,lt^@a
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike Q<sqlh!h
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), V%-hP~nyBx
% with delta(m,0) the Kronecker delta, is chosen so that the integral fe\lSGmf
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, Us`=^\
% and theta=0 to theta=2*pi) is unity. For the non-normalized F5?S8=i
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 93*csO?Db
% J3yK^@&&
% The Zernike functions are an orthogonal basis on the unit circle. Y"FV#<9@7E
% They are used in disciplines such as astronomy, optics, and eo+<@83
% optometry to describe functions on a circular domain. -XYvjW,|
% )+
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% The following table lists the first 15 Zernike functions. 7|}4UXr7y
% #*h\U]=VS
% n m Zernike function Normalization '!m6^*m|c
% -------------------------------------------------- GDLw_usV
% 0 0 1 1 SVU>q:ab
% 1 1 r * cos(theta) 2 <8WFaP3,
% 1 -1 r * sin(theta) 2 UytMnJ88
% 2 -2 r^2 * cos(2*theta) sqrt(6) 7I3_$uF
% 2 0 (2*r^2 - 1) sqrt(3) oc1BOW z
% 2 2 r^2 * sin(2*theta) sqrt(6) dN2JOyS
% 3 -3 r^3 * cos(3*theta) sqrt(8) :^7w
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) sVyV|!K
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) fRS;6Jc
% 3 3 r^3 * sin(3*theta) sqrt(8) 0? {ADQz
% 4 -4 r^4 * cos(4*theta) sqrt(10) bZ*=fdh
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) b
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% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) +O/b[O'0
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) V 20h\(\\
% 4 4 r^4 * sin(4*theta) sqrt(10) W
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% -------------------------------------------------- sK2N3B&6
% wR%Ta -
% Example 1: um,f!ho-U
% cC~RW71
% % Display the Zernike function Z(n=5,m=1) B4.:
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% x = -1:0.01:1; J&8KIOz14Z
% [X,Y] = meshgrid(x,x); wOAR NrPx2
% [theta,r] = cart2pol(X,Y); m. pm,
% idx = r<=1; a=2.Y?
% z = nan(size(X)); Mj@2=c
% z(idx) = zernfun(5,1,r(idx),theta(idx)); =|oi0
% figure C|ZPnm>f30
% pcolor(x,x,z), shading interp $a_y-lY
% axis square, colorbar !!C/($
% title('Zernike function Z_5^1(r,\theta)') Z- feMM
% [=K
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% Example 2: &M13F>!
% 6H |1IrG
% % Display the first 10 Zernike functions cx[^D,usf~
% x = -1:0.01:1; ^_]ZZin
% [X,Y] = meshgrid(x,x); ( d_z\U7l
% [theta,r] = cart2pol(X,Y); 8?Zhh.
% idx = r<=1;
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% z = nan(size(X)); qb? <u
% n = [0 1 1 2 2 2 3 3 3 3]; .!3e$mhV
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; 6?a`'&
% Nplot = [4 10 12 16 18 20 22 24 26 28]; hl1IG
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% y = zernfun(n,m,r(idx),theta(idx)); GRcPzneiz
% figure('Units','normalized') a{`hAI${
% for k = 1:10 ~nA k-toJ
% z(idx) = y(:,k); |.k'?!
% subplot(4,7,Nplot(k)) .\
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% pcolor(x,x,z), shading interp OjGI
!
% set(gca,'XTick',[],'YTick',[]) -Q20af-
% axis square G^.N$wcv
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) RqA>" [L
% end $cSUB
% ,iV%{*p]
% See also ZERNPOL, ZERNFUN2. ?~o`mg
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% Paul Fricker 11/13/2006 3-/|G-4k7
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% Check and prepare the inputs: 9L>?N:%5
% ----------------------------- WZ' Z"'
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 7DAP_C
error('zernfun:NMvectors','N and M must be vectors.') BA h'H&;V
end YYQvt
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if length(n)~=length(m) t'eaR-
error('zernfun:NMlength','N and M must be the same length.') cQEUHhRg!
end B<d=;V
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%Y&48''"
n = n(:); 0x<ASfka
m = m(:); {T8;-H0H
if any(mod(n-m,2)) I# tlaz#
error('zernfun:NMmultiplesof2', ... z|>TkCW6
'All N and M must differ by multiples of 2 (including 0).') "W(D0oy
end h`6 (Oo|
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if any(m>n) <H^jbK
error('zernfun:MlessthanN', ... v6 5C
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'Each M must be less than or equal to its corresponding N.') s, Gl{
end AMyg>n!
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if any( r>1 | r<0 ) <1*kXTN(
error('zernfun:Rlessthan1','All R must be between 0 and 1.') E^)FnXe5
end vbmt0df
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) "}3sL#|z
error('zernfun:RTHvector','R and THETA must be vectors.') k7U.]#5V
end IP`lx
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r = r(:); ^os_j39N9
theta = theta(:); as@8L|i*
length_r = length(r); 1WtE ]
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if length_r~=length(theta) @V
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error('zernfun:RTHlength', ... %2:UsI
'The number of R- and THETA-values must be equal.') +QN4hJK
end 0BXr[%{`
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% Check normalization: Ovx
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% -------------------- JL,Y9G*]s
if nargin==5 && ischar(nflag) S})f`X9_}
isnorm = strcmpi(nflag,'norm'); 6)1PDlB
if ~isnorm }F]Z1('
error('zernfun:normalization','Unrecognized normalization flag.') U$5x#{AFp
end fnX[R2KZ
else IT NFmD
isnorm = false; x{;{fMN1
end 7I
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ku,{NY
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% Compute the Zernike Polynomials V<
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% /%cDX:7X
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% Determine the required powers of r: f:M^q ;
% ----------------------------------- JLm3qIC
m_abs = abs(m); \HB
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rpowers = []; :Fhk$?/r
for j = 1:length(n) ^1 ){
@(
rpowers = [rpowers m_abs(j):2:n(j)]; +Kgl/Wg%
end Y%/RGYKh
rpowers = unique(rpowers); Un8' P8C
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% Pre-compute the values of r raised to the required powers, [CEV&B
% and compile them in a matrix: .QP`Qn6 (P
% ----------------------------- =+_nVO*
if rpowers(1)==0 /}1|'?P
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); B!mHO*g
rpowern = cat(2,rpowern{:}); j)/Vtf
rpowern = [ones(length_r,1) rpowern]; pmP~1=3
else V(Pw|u"
e
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); !%$[p'
rpowern = cat(2,rpowern{:}); Y*@7/2,
end sq=EL+=j
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% Compute the values of the polynomials: A o*IshVh
% -------------------------------------- [NE!
y = zeros(length_r,length(n)); S$SCW<LuN
for j = 1:length(n) rL\}>VC)
s = 0:(n(j)-m_abs(j))/2; @Nb/n
pows = n(j):-2:m_abs(j); hRXnig{;3
for k = length(s):-1:1 J t.<Z&
p = (1-2*mod(s(k),2))* ... 7[=G;2<
prod(2:(n(j)-s(k)))/ ... ZNH-0mk
prod(2:s(k))/ ... ^; /~$
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... !F s$W
prod(2:((n(j)+m_abs(j))/2-s(k))); 5@l5exuG*m
idx = (pows(k)==rpowers); *i*\dl
y(:,j) = y(:,j) + p*rpowern(:,idx); ~hq\XQX
end >&HW6 c
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if isnorm W'9{2h6u(
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); }o]}R#|
end &wU"6E
end #NyO'
% END: Compute the Zernike Polynomials "3jTU
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% kj2qX9Ms
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% Compute the Zernike functions: 4w|t|?
% ------------------------------ W2h*t"5W
idx_pos = m>0; fahQ^#&d`
idx_neg = m<0; zATOFV
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z = y; F
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if any(idx_pos) [3=Y 9P:
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); i<m)
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end q;R&valn
if any(idx_neg) b`%u}^B {
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); RCa1S^.
end 6{d?3Jk
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% EOF zernfun _LgP