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.bi05l
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, siDh="{s
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? Y/ot3[
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? =WZqQq{
yL4 T
kzmQm
"Ml&[Oge
*u6Y8IL1
function z = zernfun(n,m,r,theta,nflag) T GB_~Bqe
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. D('2p8;2"7
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N mog[pu:!,
% and angular frequency M, evaluated at positions (R,THETA) on the SlLw{Yb7\.
% unit circle. N is a vector of positive integers (including 0), and s)
O[t
% M is a vector with the same number of elements as N. Each element lK'Rn~
% k of M must be a positive integer, with possible values M(k) = -N(k) owpWz6k7
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, Ty(@+M~-
% and THETA is a vector of angles. R and THETA must have the same D#A~Nbc
% length. The output Z is a matrix with one column for every (N,M) #:x4DvDkR
% pair, and one row for every (R,THETA) pair. -5l6&Y
% f$HH:^#
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike qo6y %[
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), &hIRd,1#
% with delta(m,0) the Kronecker delta, is chosen so that the integral S"m cUU}}
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, -D^A:}$
% and theta=0 to theta=2*pi) is unity. For the non-normalized 8e~|.wOL
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. qGmNz}4D5
% aA`/E
% The Zernike functions are an orthogonal basis on the unit circle. qB]i6*
% They are used in disciplines such as astronomy, optics, and =,!\~`^
% optometry to describe functions on a circular domain. cXMhq<GkAA
% nR>r2wMk@
% The following table lists the first 15 Zernike functions. b
IW'c_
,
% w9RS)l2FQ
% n m Zernike function Normalization E`H$YS3o
% -------------------------------------------------- q@5K6yE
% 0 0 1 1 2f`nMW
% 1 1 r * cos(theta) 2 DmVP
% 1 -1 r * sin(theta) 2 }ov&.,vQ
% 2 -2 r^2 * cos(2*theta) sqrt(6) ]'~vI/p
% 2 0 (2*r^2 - 1) sqrt(3) KfCoe[Vv
% 2 2 r^2 * sin(2*theta) sqrt(6) &5{xXWJK
% 3 -3 r^3 * cos(3*theta) sqrt(8) . v@>JZC
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) lOwS&4UT
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) S\6[EQ65
% 3 3 r^3 * sin(3*theta) sqrt(8) Nr<`Z
% 4 -4 r^4 * cos(4*theta) sqrt(10) Si9Z>MR
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) L(>=BK*
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) +|Hioq*,t
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 'D1A}X
% 4 4 r^4 * sin(4*theta) sqrt(10) ;< )~Y-
% -------------------------------------------------- gk BdR +
% w6dFb6~R
% Example 1: [
=x s4=
% v4miU;|\
% % Display the Zernike function Z(n=5,m=1) C${S^v
% x = -1:0.01:1; 9mc!bj^811
% [X,Y] = meshgrid(x,x); >>Ts??
% [theta,r] = cart2pol(X,Y); p,pR!qC>
% idx = r<=1; )? M9|u
% z = nan(size(X)); K
]OK:hY4
% z(idx) = zernfun(5,1,r(idx),theta(idx)); (KZHX5T=
% figure /N>e&e[35\
% pcolor(x,x,z), shading interp @;xMs8@
% axis square, colorbar <WXzh5D2
% title('Zernike function Z_5^1(r,\theta)') 1
Q-bYJG
% C'=k<-
% Example 2: &0TVi
% +bK.NcS
% % Display the first 10 Zernike functions GSoZx0
% x = -1:0.01:1; ffXyc2o
% [X,Y] = meshgrid(x,x); 8E&XbqP+
% [theta,r] = cart2pol(X,Y); C.^Ven
% idx = r<=1; .O*bILU
% z = nan(size(X)); &Lt[WT$
% n = [0 1 1 2 2 2 3 3 3 3]; gw`B "c|
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; m+{K^kr[
% Nplot = [4 10 12 16 18 20 22 24 26 28]; cWGDee(
% y = zernfun(n,m,r(idx),theta(idx)); }),w1/#5u8
% figure('Units','normalized') b96%")
% for k = 1:10 <D&)OxEn\
% z(idx) = y(:,k); iVFkYx%}
% subplot(4,7,Nplot(k)) 3QSZ ZJ
% pcolor(x,x,z), shading interp DcMJ^=r8O:
% set(gca,'XTick',[],'YTick',[]) ]`g<w#
% axis square 3Y)PU=
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) @cRZk`|1n
% end xR"M*%{@0
% ]%uZ\Q;9p
% See also ZERNPOL, ZERNFUN2. Uw-p758dD
+9O5KI?P
4ww]9J
% Paul Fricker 11/13/2006 6OiSK@<Hk
M L7\BT
`G1&Z]z
j7FN\
cz
;o/>JHGj
% Check and prepare the inputs: S~qZr
% ----------------------------- b,P ]9$Ut
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) }7{t^>;D
error('zernfun:NMvectors','N and M must be vectors.') Obw?_@X
end ky>wOaTmN6
bW#@OrsS
KtS)'jf
if length(n)~=length(m) ?Y:x[pOe
error('zernfun:NMlength','N and M must be the same length.') 5#3W5z
end C=uZ1xg*,
1tCQpf
Z'^U ad6
n = n(:); y5= `ap
m = m(:); 5_0(D;Q
if any(mod(n-m,2)) /$n ~lf
error('zernfun:NMmultiplesof2', ... ~zm7?_"@]
'All N and M must differ by multiples of 2 (including 0).') dk
QaM@
end _qvK*nE
A392=:N+Q
q0%
if any(m>n) S1n3(U:m
error('zernfun:MlessthanN', ... c4e_6=Iv
'Each M must be less than or equal to its corresponding N.') ^^i6|l1
end *O:r7_ Y0
1\RGM<q$f
9 7%0;a8
if any( r>1 | r<0 ) K.C>
a:J
error('zernfun:Rlessthan1','All R must be between 0 and 1.') sUl6hX4
end ?#0snlah|
s#h8%['
oMcK`%ydm
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) YL
jHt\
error('zernfun:RTHvector','R and THETA must be vectors.') QQk{\PV
end w.Ezg j
UX`]k{Mz
* v8Ts
r = r(:); -71dN0hWh
theta = theta(:);
qLncn}oNM
length_r = length(r); d{et8N
if length_r~=length(theta) ?%Rw(E
error('zernfun:RTHlength', ...
|{g+Y
'The number of R- and THETA-values must be equal.') 0,*%vG?Q
end ;TQf5|R\K
D+ V7hpH-
>idBS
% Check normalization: ;vhyhP.oM
% -------------------- wI M{pK
if nargin==5 && ischar(nflag) [#" =yzR<3
isnorm = strcmpi(nflag,'norm'); O^LTD#}$a)
if ~isnorm DPe]daF
error('zernfun:normalization','Unrecognized normalization flag.') d
"BW/%m|g
end iK;dU2h
else ?:^mBb)T
isnorm = false; -@^Zq}
end HQ!Xj.y
J MX6yV
t<uYM
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% SEQ%'E5-'
% Compute the Zernike Polynomials jD){I
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% DG(7|`(aY
#Z=tJ
kI*(V[i
% Determine the required powers of r: >,C4rC+:XN
% ----------------------------------- G
DSfT{kK\
m_abs = abs(m); .F&9.#>
rpowers = []; lM\LN^f5*
for j = 1:length(n) z;]CmR@Ki
rpowers = [rpowers m_abs(j):2:n(j)]; >Sk[vI0Y
end n9LGP2#!
rpowers = unique(rpowers); $
E1Tb{'
Ocg"M Gb
rgIrr5
% Pre-compute the values of r raised to the required powers, 2J;`m_oP
% and compile them in a matrix: \a"Ct'
% ----------------------------- { PlK@#UN
if rpowers(1)==0 (A k\Lm
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Qz<d~N
rpowern = cat(2,rpowern{:}); UIJx*
rpowern = [ones(length_r,1) rpowern]; %/"Oxi^G
else FHy76^h>e
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Itm8b4e9;
rpowern = cat(2,rpowern{:}); )G^TW'9
end `znB7VQ0
*E>.)B i
ofc.zwH
% Compute the values of the polynomials: U3;aLQ*
% -------------------------------------- -P=g3Q i
y = zeros(length_r,length(n)); $X`y%*<<v
for j = 1:length(n) TmRxKrRs
s = 0:(n(j)-m_abs(j))/2; @}F Awv^f
pows = n(j):-2:m_abs(j); wn+FTqj
for k = length(s):-1:1 yT OyDm-
p = (1-2*mod(s(k),2))* ... 4FeEGySow
prod(2:(n(j)-s(k)))/ ... >hMUr*j
prod(2:s(k))/ ... !&kL9A).
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 2H#N{>7
prod(2:((n(j)+m_abs(j))/2-s(k))); l1_X(Z._V
idx = (pows(k)==rpowers); \L!uHAE2a
y(:,j) = y(:,j) + p*rpowern(:,idx); qG8s;_G
end 4Wel[]
dLh6:Gh8_I
if isnorm `qpc*enf0
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ";3*?/uM
end UgHf*m
end 4|J[Jdj
% END: Compute the Zernike Polynomials hP?fMW$V
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% rp!
LP#*
s}x>J8hK
bPD)D'Hs
% Compute the Zernike functions: Ry;$^.7%
% ------------------------------ hAR?
t5c
idx_pos = m>0; ZwI
1* f
idx_neg = m<0; GrEs1M1]*
kka"C]!
:1fagaPg
z = y; =6nD0i9+
if any(idx_pos) #mc!Wt10
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); J07O:cjyu
end 'E]A.3-Mt
if any(idx_neg) ND]S(C"?
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); _uH9XGm
end 9V!-ZG
a_T,t'6
:Z`4j
% EOF zernfun iv%w!3#