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， ?5%|YsJP_
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， ~ "] 6
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？ .Jt&6N
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ SOyE$GoOsx
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function z = zernfun(n,m,r,theta,nflag) FuZ7xM,
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. M~/%V NX
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N HqW|
% and angular frequency M, evaluated at positions (R,THETA) on the {-sy,EYc w
% unit circle. N is a vector of positive integers (including 0), and w%no6 ;
% M is a vector with the same number of elements as N. Each element N {]|!#
% k of M must be a positive integer, with possible values M(k) = -N(k) w,\#)<boyb
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, yTDlDOmV!
% and THETA is a vector of angles. R and THETA must have the same <uugT9By
% length. The output Z is a matrix with one column for every (N,M) |]5g+sd
% pair, and one row for every (R,THETA) pair. ,3k"J4|d
% * q8L$D
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike x,\PV>
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), hCX}*
% with delta(m,0) the Kronecker delta, is chosen so that the integral y[*Bw)F\N
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, -ISI!EU$
% and theta=0 to theta=2*pi) is unity. For the non-normalized %bnDxCj"
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. nj*B-M\p
% eCY gi7?
% The Zernike functions are an orthogonal basis on the unit circle.
#'Q_eBX
% They are used in disciplines such as astronomy, optics, and +"!,rZ7,A
% optometry to describe functions on a circular domain. t@Qs&DZ7k
% _MZq H8
% The following table lists the first 15 Zernike functions. PrIS L[@
% N#')Qz: P
% n m Zernike function Normalization Hnwir!=7
% -------------------------------------------------- yfS`g-j{~
% 0 0 1 1 C:n55BE9
% 1 1 r * cos(theta) 2 y ?FKou'
% 1 -1 r * sin(theta) 2 3A_ 7R-sQ
% 2 -2 r^2 * cos(2*theta) sqrt(6) R qS2Qo]
% 2 0 (2*r^2 - 1) sqrt(3) 0k I.d X)
% 2 2 r^2 * sin(2*theta) sqrt(6) TxY xB 1C)
% 3 -3 r^3 * cos(3*theta) sqrt(8) $cri"G
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) ~y+QL{P4~
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) o$4n D#P3
% 3 3 r^3 * sin(3*theta) sqrt(8) Vcg$H 8m
% 4 -4 r^4 * cos(4*theta) sqrt(10) ,TTt<& c
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) NTk"W!<Cl2
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) n&=3Knbd@d
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) L$7
NT}L
% 4 4 r^4 * sin(4*theta) sqrt(10) [-cYF dt"V
% -------------------------------------------------- L &F0 ^
% dA[Z\
% Example 1: v\#69J5.>)
% d18%z Y>
% % Display the Zernike function Z(n=5,m=1) Nhv~f0
% x = -1:0.01:1; U}7 a;4?
% [X,Y] = meshgrid(x,x); NZ/>nNs
% [theta,r] = cart2pol(X,Y); ~A+D H
% idx = r<=1; x68$?CD
% z = nan(size(X)); tY<D\T
% z(idx) = zernfun(5,1,r(idx),theta(idx)); !tGXh9g
% figure C6=7zYhR
% pcolor(x,x,z), shading interp A-vK0l+
% axis square, colorbar 95;q ] =U
% title('Zernike function Z_5^1(r,\theta)') ~xqRCf{8
% 5V\\w ~&/
% Example 2: Z |u II#lq
% '{ j.5~4y
% % Display the first 10 Zernike functions w{3
B
% x = -1:0.01:1; %ci/(wL
% [X,Y] = meshgrid(x,x); PuAc sYQhN
% [theta,r] = cart2pol(X,Y); Dh0`t@
% idx = r<=1; ;"=a-$vm
% z = nan(size(X)); DG&14c>g
% n = [0 1 1 2 2 2 3 3 3 3]; P ?dE\Po7
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; $VYMAk&\
% Nplot = [4 10 12 16 18 20 22 24 26 28]; t%<nS=u
% y = zernfun(n,m,r(idx),theta(idx)); a_/\.
% figure('Units','normalized') X62h7?'Pd
% for k = 1:10 {w.rcObIw+
% z(idx) = y(:,k); bNR}Mk]?
% subplot(4,7,Nplot(k)) |a #4
% pcolor(x,x,z), shading interp CRvUD.D
% set(gca,'XTick',[],'YTick',[]) _>B0q|]j4'
% axis square EoqUFa,
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 8
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% end O:3pp8
% q
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% See also ZERNPOL, ZERNFUN2. j QOY \1SR
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% Paul Fricker 11/13/2006 '/kSUvd
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% Check and prepare the inputs: Me[T=Tt`@w
% ----------------------------- -J4?Km
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) #Yi,EwD
error('zernfun:NMvectors','N and M must be vectors.') 7f_4qb8
end #q40 >)]
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if length(n)~=length(m) &*RJh'o|N(
error('zernfun:NMlength','N and M must be the same length.') ma>{((N
end Ok[y3S
r Ip84}
@*'|8%
n = n(:); *xXa4HB
m = m(:); 7 %L%dyN
if any(mod(n-m,2)) ,T?8?? bZ
error('zernfun:NMmultiplesof2', ... . Y[sQO~%
'All N and M must differ by multiples of 2 (including 0).') ZurQr}
end ]kx)/n-K
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if any(m>n) ;LNFPo
error('zernfun:MlessthanN', ... -8; ,#
'Each M must be less than or equal to its corresponding N.') s2L|J[Y"s
end iD#HB o
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if any( r>1 | r<0 ) 48Y5ppcS
error('zernfun:Rlessthan1','All R must be between 0 and 1.') X *VHi
end Q[`J=
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) y-O#
+{7
error('zernfun:RTHvector','R and THETA must be vectors.') *IUw$|Z6z)
end Px5ArSS
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r = r(:); l j*ELy
theta = theta(:); dHc 38zp
length_r = length(r); I^ sWf3'db
if length_r~=length(theta) |\"vHt?@G
error('zernfun:RTHlength', ... Ffk$8"
'The number of R- and THETA-values must be equal.') h[72iVn
end ork/:y9*y
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% Check normalization: z.n`0 `^
% -------------------- x nWCio>M
if nargin==5 && ischar(nflag) SHS: >V
isnorm = strcmpi(nflag,'norm'); =( b;Cow
if ~isnorm |&+g ,A _w
error('zernfun:normalization','Unrecognized normalization flag.') XbdoTri E
end e|u|b
else ).@8+}`
isnorm = false; J"'2zg1&
end .f
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Va,M9)F
% Compute the Zernike Polynomials uZ][#[u
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ~Fv&z'R
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% Determine the required powers of r: c'?4*O
% ----------------------------------- 4Z>hP]7
m_abs = abs(m); &WAO.*:y
rpowers = [];
E ;\XZ<E
for j = 1:length(n) B
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rpowers = [rpowers m_abs(j):2:n(j)]; 0mo^I==J1
end k .? aq
rpowers = unique(rpowers); B~oSKM%8R
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hW pn~q
% Pre-compute the values of r raised to the required powers, ^/\OS@CT\
% and compile them in a matrix: V_jVVy30Ji
% ----------------------------- _l,?Y;OF
if rpowers(1)==0 -G&>b
D
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); T677d.zaT
rpowern = cat(2,rpowern{:}); ^p(t*%LM
rpowern = [ones(length_r,1) rpowern]; rks+\e}^Z
else 7qSl qA<Hs
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); bHE'R!*
rpowern = cat(2,rpowern{:}); 3?I^D /K^
end GgkljF@{}
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% Compute the values of the polynomials: s#om
% -------------------------------------- %
INRds
y = zeros(length_r,length(n)); H6?Z E
for j = 1:length(n) :Z( ?Ct&8
s = 0:(n(j)-m_abs(j))/2; d!/@+i
pows = n(j):-2:m_abs(j); ?Z%Ja_}8ma
for k = length(s):-1:1 s mub > V
p = (1-2*mod(s(k),2))* ... [o8a(oC
prod(2:(n(j)-s(k)))/ ... jq(3y|6,
prod(2:s(k))/ ... OD<0,r0f,
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ^c{}G<U^
prod(2:((n(j)+m_abs(j))/2-s(k))); 2%\Nq:; T
idx = (pows(k)==rpowers); ZxkX\gl91
y(:,j) = y(:,j) + p*rpowern(:,idx); @!6eRp>Z
end {H s" "/sb
k7P~*ll$
if isnorm 6W$ #`N>
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); <$Q\vCR
end I b.`2@ o&
end kb1{ ; c:
% END: Compute the Zernike Polynomials |8}f
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Frn#?n)S9
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p.i$[ 6M
% Compute the Zernike functions: )l *H $8
% ------------------------------ SzkF-yRd
idx_pos = m>0; Yf Udpa0
idx_neg = m<0; _`Ey),c _
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z = y; L^rtypkJ
if any(idx_pos) ~ J! a?]
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); x-+[gNc
6
end pWH8ex+
if any(idx_neg) hABC
rd Em
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); E (tdL,m'
end !OM9aITv[
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% EOF zernfun wqBGJ
