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， \<LCp;- K
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， -H-U8/W C
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？ -/Q5?0z
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ D#g -mqar:
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function z = zernfun(n,m,r,theta,nflag) 8Y`Lq$u
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. F]$ Nu
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N m%HT)`>bg
% and angular frequency M, evaluated at positions (R,THETA) on the 2f,8Jnia
% unit circle. N is a vector of positive integers (including 0), and dN{A t-
% M is a vector with the same number of elements as N. Each element VE |:k:};
% k of M must be a positive integer, with possible values M(k) = -N(k) noZbsI4
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, O=0p}{3l
% and THETA is a vector of angles. R and THETA must have the same b fxE}>
% length. The output Z is a matrix with one column for every (N,M) Y 6a`{'
% pair, and one row for every (R,THETA) pair. Kr}RFJ"d
% r&u1-%%9[
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike
|Xso}Y{
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), m eF7[>!U
% with delta(m,0) the Kronecker delta, is chosen so that the integral W5|{A])N
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, t~+M>Fjm?d
% and theta=0 to theta=2*pi) is unity. For the non-normalized =M\yh,s!
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. fv;Q*; oC&
% V6g*"e/8
% The Zernike functions are an orthogonal basis on the unit circle. QQJGqM3a2
% They are used in disciplines such as astronomy, optics, and Ai q Kf =
% optometry to describe functions on a circular domain.
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% XcJ5KTn
% The following table lists the first 15 Zernike functions. N 63?4'_W
% #VQZ"7nI@
% n m Zernike function Normalization *p{p.%Qs:
% --------------------------------------------------
|~9rak,
% 0 0 1 1 vXJs.)D7
% 1 1 r * cos(theta) 2 Jf^3nBZ
% 1 -1 r * sin(theta) 2 zEQ]5>mG
% 2 -2 r^2 * cos(2*theta) sqrt(6) ^tw yy9VR
% 2 0 (2*r^2 - 1) sqrt(3) /X}1%p
% 2 2 r^2 * sin(2*theta) sqrt(6) ql?w6qFs]
% 3 -3 r^3 * cos(3*theta) sqrt(8) {v"f){
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) ZU\$x <,
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) z teu{0
% 3 3 r^3 * sin(3*theta) sqrt(8) v^9eTeFO
% 4 -4 r^4 * cos(4*theta) sqrt(10) Es= G' au
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ][
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% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) [v1$L p
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 4
]oe`yx
% 4 4 r^4 * sin(4*theta) sqrt(10) `,O7S9]R+
% -------------------------------------------------- 1jC85^1Taq
% )<x9 t@$
% Example 1: |~9jO/&r
% 2C C"Z
% % Display the Zernike function Z(n=5,m=1) M+t)# O4
% x = -1:0.01:1; z_c-1iXCW
% [X,Y] = meshgrid(x,x); PMQTc Q^
% [theta,r] = cart2pol(X,Y); '/GB 8L
% idx = r<=1; p{E(RsA
% z = nan(size(X)); 8@3=SO
% z(idx) = zernfun(5,1,r(idx),theta(idx)); `^#Rwn#
% figure ;MfqI/B{
% pcolor(x,x,z), shading interp }s2CN D
% axis square, colorbar ^B.Z 3Y
% title('Zernike function Z_5^1(r,\theta)') -Mo4`bN
% 4~
x> ]
% Example 2: eC/{c1C
% cqU6 Y*n
% % Display the first 10 Zernike functions 4K cEJ lK5
% x = -1:0.01:1; Zb o 4{.#
% [X,Y] = meshgrid(x,x); a`Bp^(f}
% [theta,r] = cart2pol(X,Y); 9Qyc!s`
% idx = r<=1; bK "I9T #
% z = nan(size(X)); B7Ket8<J
% n = [0 1 1 2 2 2 3 3 3 3]; +}jzge"
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; 0 \i\G|5
% Nplot = [4 10 12 16 18 20 22 24 26 28]; <MEm+8e/s6
% y = zernfun(n,m,r(idx),theta(idx)); 3[#^$_96b
% figure('Units','normalized') tM LiG4
|7
% for k = 1:10 MJX
ny4n
% z(idx) = y(:,k); .# y#u={{l
% subplot(4,7,Nplot(k)) x& _Y( bHA
% pcolor(x,x,z), shading interp WrP+n
% set(gca,'XTick',[],'YTick',[]) xWLZlUHEu
% axis square :V(C+bm *
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) /5 z+N(RFC
% end U<Oc&S{]*
% WX Fm'5V r
% See also ZERNPOL, ZERNFUN2. /CALX wL
p;4FZ $
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% Paul Fricker 11/13/2006 + 1#;s!e
< xBL/e
%
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% Check and prepare the inputs: )EMl GM'2q
% ----------------------------- {"jtR<{)
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) (6c/)M H
error('zernfun:NMvectors','N and M must be vectors.') q?frt3o
end gZHgL7@
p#c41_?'e
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if length(n)~=length(m) P^o@x,V!&
error('zernfun:NMlength','N and M must be the same length.') t7-r YY(
end 5[2kk5,
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n = n(:); bV:<%l]
m = m(:); e R[B0;c
if any(mod(n-m,2))
[<r.M<3
error('zernfun:NMmultiplesof2', ... 2KO`+
'All N and M must differ by multiples of 2 (including 0).') x7B;\D#`i/
end j hRr!
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if any(m>n) NJJ=ch
error('zernfun:MlessthanN', ... zw'%n+5m
'Each M must be less than or equal to its corresponding N.') [ 1G wcXr
end 4SUzR\
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if any( r>1 | r<0 ) 7r:&%?2:g
error('zernfun:Rlessthan1','All R must be between 0 and 1.') R KzO$T
end z}}P+P/
{KDN|o+%
I[r R-4.F]
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) /7#MJH5b6
error('zernfun:RTHvector','R and THETA must be vectors.') _KloX{a
end Qu<6X@+5
AP z "k?D0
#Fo#f<b p
r = r(:); %J'/ cmR&
theta = theta(:); qu# xc0?
length_r = length(r); >r X$E<B\
if length_r~=length(theta) h#Rza-?"\
error('zernfun:RTHlength', ... W3ms8=z
'The number of R- and THETA-values must be equal.') Q(A$ >A
end IkmEctAU
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h+Lpj^<2a
% Check normalization: 8wO4;
% -------------------- q%M~gp1
if nargin==5 && ischar(nflag) P )oNNY6}
isnorm = strcmpi(nflag,'norm'); ic}TiTK
if ~isnorm & tbAXU5$
error('zernfun:normalization','Unrecognized normalization flag.') tf54EIy5Y
end S;t`C~l\
else M^OYQf
isnorm = false; xC5Pv">
end )^P54_2
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5"G-r._
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% J:'_S `J
% Compute the Zernike Polynomials bL WY Tj
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% #: [F=2@,A
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9BlpqS:P&
% Determine the required powers of r: \7h>9}wGf
% ----------------------------------- E ,ilJl\
m_abs = abs(m); $;(@0UDE
rpowers = []; H;<>uE Lie
for j = 1:length(n) :B=Gb8?
rpowers = [rpowers m_abs(j):2:n(j)]; e*`ht+
end
PPy~dp
rpowers = unique(rpowers); -B/'ArOo]
[%yj'
)R/
[;yH.wn#5
% Pre-compute the values of r raised to the required powers, _ U LzA
% and compile them in a matrix: `<~=6H
% ----------------------------- 9fs-|E[ 5
if rpowers(1)==0 SAitufS
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 4 7mT
rpowern = cat(2,rpowern{:}); Ri AMW|M"C
rpowern = [ones(length_r,1) rpowern]; s 8's(*]
else c R!M {U.q
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); /zXOta G
rpowern = cat(2,rpowern{:}); 7f
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end P RUl-v
\}4*}Lr
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% Compute the values of the polynomials: $n W9VMa
% -------------------------------------- f|_\GVW
y = zeros(length_r,length(n)); fwA8=o SZd
for j = 1:length(n) 8oI|Z=
s = 0:(n(j)-m_abs(j))/2; x'\C'zeF
pows = n(j):-2:m_abs(j); du ~V=%9
for k = length(s):-1:1 S[7^#O.)
p = (1-2*mod(s(k),2))* ... ig YYkt
prod(2:(n(j)-s(k)))/ ... NZZ y^p&O
prod(2:s(k))/ ... |,=^P` #%
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... :
qK-Rku
prod(2:((n(j)+m_abs(j))/2-s(k))); hi$AZ+
idx = (pows(k)==rpowers); N2HD=[*cr
y(:,j) = y(:,j) + p*rpowern(:,idx); iFI+W<QR
end _x""-X~OL
l(87s^_
if isnorm XC; Icr)
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); }.'rh R+
end ^ Lc\{,m
end KiI+ V;o
% END: Compute the Zernike Polynomials ]&P\|b1*g
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + a nsN~3
H#V&5|K%
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% Compute the Zernike functions: w1-P6cf
% ------------------------------ N >*+Wg$Ne
idx_pos = m>0; XKws_
idx_neg = m<0; Pf,@U'f|
b+:J ?MR;}
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z = y; H
I_uR$m
if any(idx_pos) = &pLlG
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); -L]-u6kC[
end Mh~}RA"H
if any(idx_neg) &V~l(1
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); j-R*!i
end |BZrV3;H
2'- "&d+ O
*IWW,@0
% EOF zernfun %aw.o*@:
