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, wUcp_)aE|
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, _q3|Ddm2LN
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? 9\KMU@Ne
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? b]Z>P{ j
t B Kra
'uU{.bq
;'4HR+E"
=SLCG.
function z = zernfun(n,m,r,theta,nflag) "D?:8!\!
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. K#4Toc#=V
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N d2(3 ,
% and angular frequency M, evaluated at positions (R,THETA) on the Rg~F[j$N
% unit circle. N is a vector of positive integers (including 0), and rxQ&N[r2
% M is a vector with the same number of elements as N. Each element R>dd#`r"
% k of M must be a positive integer, with possible values M(k) = -N(k) `u#N
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, o6A1;e
% and THETA is a vector of angles. R and THETA must have the same Bf{c4YiF
% length. The output Z is a matrix with one column for every (N,M) ZCz#B2Sf8
% pair, and one row for every (R,THETA) pair. &M*f4PeXb
% eD?f|bif
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike :XeRc"m<
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), (I\qTfN4
% with delta(m,0) the Kronecker delta, is chosen so that the integral hLF ;MH@
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, jC_m0Iwc
% and theta=0 to theta=2*pi) is unity. For the non-normalized klSA Y
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ?"L ^0%
% *g!7PzJ'
% The Zernike functions are an orthogonal basis on the unit circle. )l[bu6bM
% They are used in disciplines such as astronomy, optics, and 5Za%EaW%G
% optometry to describe functions on a circular domain. .l +yK-BZ
% .+$ox-EK8
% The following table lists the first 15 Zernike functions. p@iU9K\,
% c!dc`R
% n m Zernike function Normalization JpC_au7CX
% -------------------------------------------------- 2tI ,`pSU
% 0 0 1 1 jCp`woV
% 1 1 r * cos(theta) 2 S0mzDLgE
% 1 -1 r * sin(theta) 2 0=Mu|G|Z
% 2 -2 r^2 * cos(2*theta) sqrt(6) IHcR/\mz
% 2 0 (2*r^2 - 1) sqrt(3) ,#Mt10e{
% 2 2 r^2 * sin(2*theta) sqrt(6) OS sYmF
% 3 -3 r^3 * cos(3*theta) sqrt(8) sglH=0MP
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 9N V.<&~
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) #9CLIYJAd
% 3 3 r^3 * sin(3*theta) sqrt(8) 2i)vT)~
% 4 -4 r^4 * cos(4*theta) sqrt(10) #8@o%%Fd
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ^j]_MiA4
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 'ocPG.PaU
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) d34BJ<
% 4 4 r^4 * sin(4*theta) sqrt(10) tzrvIVD
% -------------------------------------------------- ]oxi~TwY^
% xA SH-9
% Example 1: &AP`k
% MZ"|Jn
% % Display the Zernike function Z(n=5,m=1) ,v_NrX=f?
% x = -1:0.01:1; Aqo90(jffx
% [X,Y] = meshgrid(x,x);
e"&QQ-q
% [theta,r] = cart2pol(X,Y); 3oBR
% idx = r<=1; 1"UHe*2
% z = nan(size(X)); ;bRyk#
% z(idx) = zernfun(5,1,r(idx),theta(idx)); :s>x~t8g#n
% figure oMHTB!A=2
% pcolor(x,x,z), shading interp =Hx]K8N )
% axis square, colorbar P$5K[Y4f
% title('Zernike function Z_5^1(r,\theta)') '^%k TNn
% aM YtWj
% Example 2: ;"|QW?>$D
% ~}RfepM
% % Display the first 10 Zernike functions RAj>{/E#W
% x = -1:0.01:1; 9nSfFGu
% [X,Y] = meshgrid(x,x); fs0EbVDF
% [theta,r] = cart2pol(X,Y); %uDH_J|^
% idx = r<=1; +F+M[ef<ws
% z = nan(size(X)); <h%I-e6
% n = [0 1 1 2 2 2 3 3 3 3]; {Bz E
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; ;Q,,i
% Nplot = [4 10 12 16 18 20 22 24 26 28]; <.hutU*1
% y = zernfun(n,m,r(idx),theta(idx)); _
o.j({S
% figure('Units','normalized') |dhKeg_
% for k = 1:10 9J$-E4G.M
% z(idx) = y(:,k); 2]=`^rC*
% subplot(4,7,Nplot(k)) bX>R9i$
% pcolor(x,x,z), shading interp ym_p49
% set(gca,'XTick',[],'YTick',[]) H{hzw&dZ<P
% axis square *USG
p<iH
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) {r'+icvLX
% end ^09-SUl^
% `IT]ZAem`/
% See also ZERNPOL, ZERNFUN2. 5GbC}y>
!cW!zP-B*p
($-m}UF\/
% Paul Fricker 11/13/2006 dozC[4mF
)6(|A$~C+
.FG%QF F~
1Eb2X}XC
y/+IPR
% Check and prepare the inputs: bvS6xU-
J
% ----------------------------- \,pObWm
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) }$i/4?dYsQ
error('zernfun:NMvectors','N and M must be vectors.') O L 9(~p
end _!,Ees=b
*/2nh%>$
p>B-Ubu
if length(n)~=length(m) 9{
#5~WP
error('zernfun:NMlength','N and M must be the same length.') 54=*vokX_
end -e"A)Bpl(
<~P!yL r
pQ>|dH+.
n = n(:); b0Dco0U(
m = m(:); [iZH[7&j
if any(mod(n-m,2)) RL3*fRlb
error('zernfun:NMmultiplesof2', ... 4w)>}
'All N and M must differ by multiples of 2 (including 0).')
1D_&n@
end Cz
&3=),G
E^A S65%bL
+lb&_eD
if any(m>n) B<i(Y1n[
error('zernfun:MlessthanN', ... LI].*n/v
'Each M must be less than or equal to its corresponding N.') v3]5`&3~
end W^)mz,%x
`QtkC>[
\*[DR R0
if any( r>1 | r<0 ) qsQ{`E0
error('zernfun:Rlessthan1','All R must be between 0 and 1.') 7hTpjox2
end +abb[
7Mk>`4D'c
V~p01f"J
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 4XAs^>N+
error('zernfun:RTHvector','R and THETA must be vectors.') ]6M,s0
end c g)>A
==Xy'n9'
JOJuGB-d
r = r(:); 3dlY_z=0
theta = theta(:); 3!$+N\ #w
length_r = length(r); .]s? 01Z
if length_r~=length(theta) ZZ
Hjv
error('zernfun:RTHlength', ... -+Ot'^
'The number of R- and THETA-values must be equal.') e ^oGiL~
end I=:"Fqj'N
6VVxpDAi:
r}es_9*~Z
% Check normalization: FSm.o?>
% -------------------- 3n)$\aBE
if nargin==5 && ischar(nflag) P;o{t
isnorm = strcmpi(nflag,'norm'); ^RO<r}Bu
if ~isnorm 6<T:B[a-
error('zernfun:normalization','Unrecognized normalization flag.') @HPr;m!
end Cf9{lhE8
else Arm'0)B>
isnorm = false; 0|.jIix;
end oyr b.lu/
3E^qh03(
l5.k2{'
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% _ xTpW
% Compute the Zernike Polynomials }X?#"JFX?
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
y*ZA{
ox%j_P9@:
3}!u8,P
% Determine the required powers of r: R?{xs
% ----------------------------------- !+A%`m
m_abs = abs(m); |9=A"092{
rpowers = []; \pfa\,rW
for j = 1:length(n) q&J5(9]O|L
rpowers = [rpowers m_abs(j):2:n(j)]; #>("(euXMF
end yvj /u
c
rpowers = unique(rpowers); ]J'TebP=L5
IdN3Ea]
rJkJ/9s
% Pre-compute the values of r raised to the required powers, z=) m6\
% and compile them in a matrix: Ak,JPzT
% ----------------------------- (Hj[9[=
if rpowers(1)==0 A&)2m
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); +Wg/O
-
rpowern = cat(2,rpowern{:}); M:GpyE%
rpowern = [ones(length_r,1) rpowern]; ]95VMyN
else pB\:.?.pd
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); '/NpmNY:L
rpowern = cat(2,rpowern{:}); bj}Lxc ],
end X!K> .r_Dg
""jW'%wR
h?p_jI
% Compute the values of the polynomials: v}N\z2A
% -------------------------------------- `
PQQU~^
y = zeros(length_r,length(n)); oe] *Q
for j = 1:length(n) mjWU0.
s = 0:(n(j)-m_abs(j))/2; NI#]#yM+
pows = n(j):-2:m_abs(j); _%=CW'
B
for k = length(s):-1:1 OPDT:e86Y=
p = (1-2*mod(s(k),2))* ... 'I&0$<
prod(2:(n(j)-s(k)))/ ... ,c|MB
prod(2:s(k))/ ... 8 5X}CCQ
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... w(&EZDe
prod(2:((n(j)+m_abs(j))/2-s(k)));
R%RxF=@
idx = (pows(k)==rpowers);
Ao8ua|:
y(:,j) = y(:,j) + p*rpowern(:,idx); >fzyD(>
end c>K]$;}
l;0([_>*j
if isnorm $uDgBZA\
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); X':FFD4h
end Z::I3 Q
end eZAMV/]jH
% END: Compute the Zernike Polynomials ,\iHgsZ
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% TLVsTM8P
QF/_?Tm4
G|KA!q
% Compute the Zernike functions: i,r:R
g~
% ------------------------------ `
= O
idx_pos = m>0; =yZq]g6Q
idx_neg = m<0; fV|uKs(W
x)Bbo9J
0>Snps3*Z
z = y; > v%.q]E6n
if any(idx_pos) kEnGr6e
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); dEtjcId
end H?];8wq$G
if any(idx_neg) jeWv~JA%L|
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); (T#$0RFq
end Cjr]l!
;,[0 bmL
{WrEe7dLy
% EOF zernfun qx5`lm~L