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, (>/Dw|,m
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, Uu<sntyv
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? @eD2<e
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? l'X?S(fiV
sRI=TE]s
yyYbB ]D
k1
txY
Wn)A/Z ^r
function z = zernfun(n,m,r,theta,nflag) 1 bx^Pt)
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 75cr!+
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 3_['[}
% and angular frequency M, evaluated at positions (R,THETA) on the 1F%*k &R
% unit circle. N is a vector of positive integers (including 0), and _O'rZ5}&
% M is a vector with the same number of elements as N. Each element nHL>}Yg
% k of M must be a positive integer, with possible values M(k) = -N(k) E?W!.hbA
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, y#SD-#I-
% and THETA is a vector of angles. R and THETA must have the same >w'?DV>u|
% length. The output Z is a matrix with one column for every (N,M) XwqfWd_
% pair, and one row for every (R,THETA) pair. fxCPGj
% a}8>(jtSt
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike oz7udY=]0
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), gl4
f9Ff
% with delta(m,0) the Kronecker delta, is chosen so that the integral j-\^
}K.&
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, xn#I7]]G
% and theta=0 to theta=2*pi) is unity. For the non-normalized t7&
GCZ
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 5|H(N}S_
% Ib<+m%Ac
% The Zernike functions are an orthogonal basis on the unit circle. 6j.(l4}
% They are used in disciplines such as astronomy, optics, and K0bmU(Xxp
% optometry to describe functions on a circular domain. vVRCM
% 9n2%7dLQ*
% The following table lists the first 15 Zernike functions. jfhDi6N
% i7E7%~S
% n m Zernike function Normalization [ Sa
C
% -------------------------------------------------- ;C@^wI
% 0 0 1 1 X|0`$f
% 1 1 r * cos(theta) 2 'g,
x}6
% 1 -1 r * sin(theta) 2 yru}f;1
% 2 -2 r^2 * cos(2*theta) sqrt(6) D+nj[8y
% 2 0 (2*r^2 - 1) sqrt(3) }Z%{QJ$z
% 2 2 r^2 * sin(2*theta) sqrt(6) &_TjRj"
% 3 -3 r^3 * cos(3*theta) sqrt(8) 15yV4wHr
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) T_
#oMXZ/
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) iK23`@&%_
% 3 3 r^3 * sin(3*theta) sqrt(8) JN|#
% 4 -4 r^4 * cos(4*theta) sqrt(10) Tp.iRFFkP
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) U0=zuRr n
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) =Qq^=3@h
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) tWy<9TF
% 4 4 r^4 * sin(4*theta) sqrt(10) hndRgCo
% -------------------------------------------------- JHOBg{Wg
% Nv#, s_hG
% Example 1: {dH<Un(4Z
% ]qTr4`.
% % Display the Zernike function Z(n=5,m=1) L{ ^@O0S
% x = -1:0.01:1; xVo)!83+Q
% [X,Y] = meshgrid(x,x); QE6-(/
% [theta,r] = cart2pol(X,Y); +m},c-,=$w
% idx = r<=1; E^ti!4{<
% z = nan(size(X)); !!pi\J?sk
% z(idx) = zernfun(5,1,r(idx),theta(idx)); uw&,pq
% figure d|HM
% pcolor(x,x,z), shading interp e;Z`&
% axis square, colorbar _Pm}]Y:_
% title('Zernike function Z_5^1(r,\theta)') lBC-G*#
% _
q1|\E%`h
% Example 2: c.> (/
% lt"*y.%@b
% % Display the first 10 Zernike functions Q";eyYdOL
% x = -1:0.01:1; +u)$o
% [X,Y] = meshgrid(x,x); )}lV41u
% [theta,r] = cart2pol(X,Y); M- A}(r +J
% idx = r<=1; I=-;*3g6
% z = nan(size(X)); K?I&,t_*R
% n = [0 1 1 2 2 2 3 3 3 3]; =f|a?j,f~
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; Um|Tf]q
% Nplot = [4 10 12 16 18 20 22 24 26 28]; XOFaS '.
% y = zernfun(n,m,r(idx),theta(idx)); SZ){1Hu
% figure('Units','normalized') +Enff0 =+
% for k = 1:10 &1Iy9&y
% z(idx) = y(:,k); cW%O-
% subplot(4,7,Nplot(k)) Ez-o*&
% pcolor(x,x,z), shading interp 0_.hU^fP
% set(gca,'XTick',[],'YTick',[]) U /Fomu
% axis square {bEEQCweNJ
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ApBThW*E
% end J8'zvH&I
% +.uk#K0o
% See also ZERNPOL, ZERNFUN2. =hlu,
B y
&d;$k
a^,RbV/
% Paul Fricker 11/13/2006 M] *pBc(o0
TR20{8"
?Ea"%z*c5
7mBL#T2
%q^]./3p
% Check and prepare the inputs: /ep~/#Ia
% ----------------------------- xnOlV
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Z;s-t\C
error('zernfun:NMvectors','N and M must be vectors.') b>WT-.b0
end vL0Ol-Vt
7F~+z7(h
Y6a|\K|
if length(n)~=length(m) tTt~W5lo
error('zernfun:NMlength','N and M must be the same length.') b<7f:drVC
end Awh)@iTL
E(#2/E6
@xEQ<g
n = n(:); ]={Hq9d@
m = m(:); H'>
if any(mod(n-m,2)) w (1a{m?ht
error('zernfun:NMmultiplesof2', ... q4oZJ -`
'All N and M must differ by multiples of 2 (including 0).') ^eii
4
end P $S P4F
Q!v[b{]8
NBX/V^
if any(m>n) nc)`ISI
error('zernfun:MlessthanN', ... TH &B9
'Each M must be less than or equal to its corresponding N.') d\M
!o*U
end a6_`V;
%b9M\
,?+yu6eLb
if any( r>1 | r<0 ) 3}+
\&[
error('zernfun:Rlessthan1','All R must be between 0 and 1.') ,d#4Ib
end I5]zOKlVR
)3
KMZ`Wn=
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) {'%=tJ[YX
error('zernfun:RTHvector','R and THETA must be vectors.') %<t/xAge
end @$]h[
|Oe6OCPf
;nE}%lT
r = r(:); HFOp4
theta = theta(:); V5+a[`]
length_r = length(r);
8z"Yo7no
if length_r~=length(theta) c/RG1w
error('zernfun:RTHlength', ... |a+8-@-Tj
'The number of R- and THETA-values must be equal.') MZ'HMYed
end 2X`M&)"X
|wx1
[xZ
{;U:0BPI3
% Check normalization: szHUHW~;J
% -------------------- &n|gPp77$
if nargin==5 && ischar(nflag) *6L^A`_1]
isnorm = strcmpi(nflag,'norm'); @Klj!2cv$
if ~isnorm 0dW1I|jR
error('zernfun:normalization','Unrecognized normalization flag.') ~gN'";1i
end c1[;a>
else E2MpMR
isnorm = false; -84Z8?_
end ?jbam!A
Iu8=[F>
|dK-r
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Hg4Ut/0
% Compute the Zernike Polynomials 2k_Bo~.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% c1i7Rc{q
f[w$3
`8I&(k<wLe
% Determine the required powers of r: 0^=S:~G
% ----------------------------------- ?k#%AM
m_abs = abs(m); #p]On87>
rpowers = []; hY!G>d{J
for j = 1:length(n) dn Xc- <
rpowers = [rpowers m_abs(j):2:n(j)]; aozk,{9-
end (&S v$L@
rpowers = unique(rpowers); kQ +
4[?Q*f!
Vr'Z5F*@
% Pre-compute the values of r raised to the required powers, N|DY)W
% and compile them in a matrix: ;$Y?j8g
% ----------------------------- m#Cp.|>kP4
if rpowers(1)==0 )~6974
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); NoMC*",b>
rpowern = cat(2,rpowern{:}); 3]'3{@{}H
rpowern = [ones(length_r,1) rpowern]; SNQ+ XtoO
else %UmE=V
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); zMa`olTZ
rpowern = cat(2,rpowern{:}); o}T]f(>}
end m2;%|QE(
fqcyCu7Ep
YXvKDw'95
% Compute the values of the polynomials: KksbhN{AB
% -------------------------------------- \sk,3b-&'
y = zeros(length_r,length(n)); ;j$84o{
for j = 1:length(n) f:TW<
s = 0:(n(j)-m_abs(j))/2; m>iuy:ti
pows = n(j):-2:m_abs(j); R{T4AZ@,'
for k = length(s):-1:1 _7Z$"
p = (1-2*mod(s(k),2))* ... *
08LW|:,
prod(2:(n(j)-s(k)))/ ... !%Ak15o
prod(2:s(k))/ ... KT3[{lr
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 0xC!d-VIJ
prod(2:((n(j)+m_abs(j))/2-s(k))); b`^$2RM&
idx = (pows(k)==rpowers); ,yB-jk?
y(:,j) = y(:,j) + p*rpowern(:,idx); 2uB.0
end @-hy:th#
LcF0: h'
if isnorm })J]D~!p
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); X3nwA#If1
end -/h$Yb
end DU;]Q:r{
% END: Compute the Zernike Polynomials 2W}RXqV<
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Y$(G)Fs
&P\T{d2"
9<R:)Df
% Compute the Zernike functions: * 6uiOtH
% ------------------------------ zP5H TEz
idx_pos = m>0; &=f%(,+
idx_neg = m<0; UOa{J|k>h
77)C`]0(
QII>XJ9
z = y; P|G:h&
if any(idx_pos) il:+O08_
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); *{XbC\j
end ?f a/}|T
if any(idx_neg) U7{,
*
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); RlpW)\{j?
end %cBJ haR{(
wt-)5f'{
I+}h+[W
% EOF zernfun &ed.%: