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, GDYFhH7H
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, Q!{,^Qb
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? YI?tmqzt
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? `\yQn7 Oq
RMlx[nsq
.* &F
|O{kv}YZ
3|BB#;
function z = zernfun(n,m,r,theta,nflag) (BGflb
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. *g"Xhk
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N MH/bJtNq
% and angular frequency M, evaluated at positions (R,THETA) on the $l_\9J913
% unit circle. N is a vector of positive integers (including 0), and BG/RNem
% M is a vector with the same number of elements as N. Each element %R P\,|
% k of M must be a positive integer, with possible values M(k) = -N(k) L[tq@[(IJ
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, up2wkc8
% and THETA is a vector of angles. R and THETA must have the same !+(H(,gI
% length. The output Z is a matrix with one column for every (N,M) g\pLQH
% pair, and one row for every (R,THETA) pair. gwDVWhq
% {R?VB!dR
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike *iJ>@vew
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 6*:U1{Gl)
% with delta(m,0) the Kronecker delta, is chosen so that the integral vF9fXY=
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, NUh+ &M
% and theta=0 to theta=2*pi) is unity. For the non-normalized WmuYHE U
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. y>0Gmr
% VjJ}q*/3e
% The Zernike functions are an orthogonal basis on the unit circle. >cH}sNHy
% They are used in disciplines such as astronomy, optics, and )!Zm*(
% optometry to describe functions on a circular domain. -'T^gEd)c
% Z6#(83G4
% The following table lists the first 15 Zernike functions. D~)bAPAD
% 8aTo
TA7JA
% n m Zernike function Normalization "Ug+#;}p$
% -------------------------------------------------- ,6aF~p;wI|
% 0 0 1 1 mXI'=Vo!S
% 1 1 r * cos(theta) 2 x{S2
% 1 -1 r * sin(theta) 2 9yp'-RKjw
% 2 -2 r^2 * cos(2*theta) sqrt(6) JZ/T:Hsh4
% 2 0 (2*r^2 - 1) sqrt(3) d(C5i8d
% 2 2 r^2 * sin(2*theta) sqrt(6) $V;0z~&!'
% 3 -3 r^3 * cos(3*theta) sqrt(8) q^6l`JJ
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) V`}u:t7r
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) w[#*f?at~
% 3 3 r^3 * sin(3*theta) sqrt(8) gw<udhk
% 4 -4 r^4 * cos(4*theta) sqrt(10) 1cD
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ] -%B4lT
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 6Oy6r
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 36}&{A
% 4 4 r^4 * sin(4*theta) sqrt(10) c9Q _Qr0'
% -------------------------------------------------- Y?:"nhN
% T>w;M?`9K
% Example 1: d'[q2y?6N
% lS?#(}a1)
% % Display the Zernike function Z(n=5,m=1) P?Kg7m W
% x = -1:0.01:1; E+J +fi
% [X,Y] = meshgrid(x,x); Wm/0Pi
% [theta,r] = cart2pol(X,Y); 3g5D[>J'
% idx = r<=1; h]&o)%{4
% z = nan(size(X)); =oTj3+7
% z(idx) = zernfun(5,1,r(idx),theta(idx)); o_&Qb^W
% figure WTu!/J<\
% pcolor(x,x,z), shading interp L&&AK`Ur3l
% axis square, colorbar 1V-si bE
% title('Zernike function Z_5^1(r,\theta)') s3=slWY=
% }j{Z
&(K
% Example 2: '`j MNKn\
% zZP&`#TAy
% % Display the first 10 Zernike functions XW:%YTv
% x = -1:0.01:1; BzTzIo5
% [X,Y] = meshgrid(x,x); ptlag&Z
% [theta,r] = cart2pol(X,Y); B^/(wHBp
% idx = r<=1; S2EV[K8#
% z = nan(size(X)); y 7z)lBy\
% n = [0 1 1 2 2 2 3 3 3 3]; [Xww`OUsh
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; zEDN^K '
% Nplot = [4 10 12 16 18 20 22 24 26 28]; t:n$9WB)
% y = zernfun(n,m,r(idx),theta(idx)); p,14'HS%@
% figure('Units','normalized') NG:
f>R
% for k = 1:10 ~|{_Go{
Q
% z(idx) = y(:,k); h$p}/A
% subplot(4,7,Nplot(k)) AI-ZZ6lzR
% pcolor(x,x,z), shading interp %-
Ga^[
% set(gca,'XTick',[],'YTick',[]) =.qPjp_Qd
% axis square O$X^Ea7~
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) *vT Abk$
% end Uurpho_~
% ,r;E[k@
% See also ZERNPOL, ZERNFUN2. K4b2)8
l@`n4U.Gwl
S~M/!Xb
% Paul Fricker 11/13/2006 kArF Gb2c
2Hk21y\
=69sWcC8
?(M]'ia{
$?On,U
% Check and prepare the inputs: lU.aDmy<
% ----------------------------- /sSM<r]5j
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) .Dy2O*`
error('zernfun:NMvectors','N and M must be vectors.') <g64N
end %!R\-Vej
V!s#xXD }
Y6a9S`o
if length(n)~=length(m) k=X)axt1
error('zernfun:NMlength','N and M must be the same length.') +No Ve#
end &D&U!3~(
pL: r\Y:R
TJ|do`fw>
n = n(:); $|VdGRZ1
m = m(:); gp+@+i>b+[
if any(mod(n-m,2)) 10_>EY`
error('zernfun:NMmultiplesof2', ... pv^: G;
'All N and M must differ by multiples of 2 (including 0).') p{&o{+c
end 2#Qw
L }R-|
DH uUEv<
if any(m>n) =[0|qGzg
error('zernfun:MlessthanN', ... @B!gxW\C
'Each M must be less than or equal to its corresponding N.') VRg
y
end cDz^jC
Ob]J!.
E9]*!^=/
if any( r>1 | r<0 ) \
*A!@T
error('zernfun:Rlessthan1','All R must be between 0 and 1.') oVd7ucnK
end M2nUY`%#v
-X-sykDm
'@1C$0tx
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) )GOio+{H
error('zernfun:RTHvector','R and THETA must be vectors.') I(j$^DA.
end "O_)~u
f,Am;:\ |
!1l~'/r
r = r(:); v3wq-
theta = theta(:); [@@EE>
y
length_r = length(r); ]$U A5/a
if length_r~=length(theta) W9S6
SO^\
error('zernfun:RTHlength', ... H% FP!03
'The number of R- and THETA-values must be equal.') (^58$IW71
end P!lfk:M^;
<&tdyAT?&
TV/ EC#48
% Check normalization: NMzq10M=6
% -------------------- 27MwZz
if nargin==5 && ischar(nflag) Xm<|m#
isnorm = strcmpi(nflag,'norm'); r-a0XNS*
if ~isnorm T&j:gg
error('zernfun:normalization','Unrecognized normalization flag.') UYzNaw4/x
end BCX2C
else BadnL<cj]
isnorm = false; p[}~Z|(
end >[Tt'.S!?
3Te&w9K
R`,|08E
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %zO>]f&
% Compute the Zernike Polynomials tD,I7%|@
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% uvi&! )x
5yjG\~
#X0Xc2}{f
% Determine the required powers of r: 0qW"b`9R
% ----------------------------------- arvKJmD
m_abs = abs(m); dV
/Es
rpowers = []; V;hO1xfR3&
for j = 1:length(n) N$u: !
rpowers = [rpowers m_abs(j):2:n(j)]; 2s}G6'xE]P
end _>_"cKS
rpowers = unique(rpowers); t_+owiF)M
9U3 .=J
f/"IC;<~t>
% Pre-compute the values of r raised to the required powers, -?-yeJP2
% and compile them in a matrix: iu2O/l#r
% ----------------------------- t
;fJ`.
if rpowers(1)==0 COj^pdE3
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); h-U]?De5\
rpowern = cat(2,rpowern{:}); E&)o.l<h|
rpowern = [ones(length_r,1) rpowern]; PJ,G_+b!
else ^2i$AM1t
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); x@D>JG
rpowern = cat(2,rpowern{:}); 3,J{!
end 2'Raj'2S4
n*caP9B
WkV0,_(P
% Compute the values of the polynomials: P G
zwS
% -------------------------------------- 4}Lui9
y = zeros(length_r,length(n)); X.ZG-TC
for j = 1:length(n) n6 wx/:
s = 0:(n(j)-m_abs(j))/2; -h=wLYl@0i
pows = n(j):-2:m_abs(j); /JIVp_-p
for k = length(s):-1:1 !E,|EdIr
p = (1-2*mod(s(k),2))* ... tH:?aP*2
prod(2:(n(j)-s(k)))/ ... \\C!{}+
prod(2:s(k))/ ... F2Gg_u@7M
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... B] Koi1B
prod(2:((n(j)+m_abs(j))/2-s(k))); SJE!14|e
idx = (pows(k)==rpowers); )JU`Z@?8
y(:,j) = y(:,j) + p*rpowern(:,idx); V7vojm4O
end }N:QB}7'_
|)mUO:*
if isnorm 5@n|uJA
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); U
!%IC7@
end w^:@g~
end .(s@{=
% END: Compute the Zernike Polynomials <3Rq!w/
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Z{2QDjAI;
/=QsZ,~xo
HUbXJsSP
% Compute the Zernike functions: 3wQ\L=
% ------------------------------ s !II}'Je
idx_pos = m>0; M&e=LV
idx_neg = m<0; 0*j\i@
q?8#D
lq?N>~PG
z = y; BF"eVKA
if any(idx_pos) ob/HO(h3
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ;KG}Yr72
end d
<zD@ z
if any(idx_neg) '1zC|:,
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); zLPCWP.u
end |BO5<`&I
}S%}%1pG7
$?9u;+jIR
% EOF zernfun MfA%Xep