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, BK6oW3wD/
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, @#;~_?$?C
这个,跟我们用zygo干涉仪直接拟合出的36项zernike系数,有何关系呢? 0(HUy`]>
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? &@nI(PXv
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function z = zernfun(n,m,r,theta,nflag) 1;S@XC>
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 7oK!!Qd^w
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N "){"{~
% and angular frequency M, evaluated at positions (R,THETA) on the arRbq!mO
% unit circle. N is a vector of positive integers (including 0), and ?>DN7je
% M is a vector with the same number of elements as N. Each element E%2]c?N5
% k of M must be a positive integer, with possible values M(k) = -N(k) qy/xJ>:
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, :[,-wZiT~6
% and THETA is a vector of angles. R and THETA must have the same 8FU8E2zo
% length. The output Z is a matrix with one column for every (N,M) `Z0FQ( r_
% pair, and one row for every (R,THETA) pair. <U$x')W
% 1Sx2c
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike bRfac/:}
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), UM3}7|
% with delta(m,0) the Kronecker delta, is chosen so that the integral ?7*.S Lt
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, /*i[MB
% and theta=0 to theta=2*pi) is unity. For the non-normalized _?CyKk\I
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. (gQP_Oa(
% RG0kOw0
% The Zernike functions are an orthogonal basis on the unit circle. 2.qEy6
% They are used in disciplines such as astronomy, optics, and *3d+ !#;rG
% optometry to describe functions on a circular domain. O,x[6P54P
% ?^n),mR
% The following table lists the first 15 Zernike functions. Vo"Wr>F
% r roI
% n m Zernike function Normalization gE\&[;)DB
% -------------------------------------------------- 9$$dSN\&
% 0 0 1 1 h'jc4mu0
% 1 1 r * cos(theta) 2 )%dxfwd6
% 1 -1 r * sin(theta) 2 @>cz$##`
% 2 -2 r^2 * cos(2*theta) sqrt(6) _(l?gj
% 2 0 (2*r^2 - 1) sqrt(3) qHaH=g%
% 2 2 r^2 * sin(2*theta) sqrt(6) nl5A{ s
% 3 -3 r^3 * cos(3*theta) sqrt(8) XXPn)kmWR
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) <hvs{}TS
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) k<Qhw)M8
% 3 3 r^3 * sin(3*theta) sqrt(8) . |%n"{
% 4 -4 r^4 * cos(4*theta) sqrt(10) '
Dcj\=8
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) x{4{.s%+:
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) SO4?3wg7
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 6I2`oag
% 4 4 r^4 * sin(4*theta) sqrt(10) ^F,sV*
% -------------------------------------------------- G%iTL"6
% & 6'Rc#\P
% Example 1: x<5ARK6\=
% }@x!r=O)I
% % Display the Zernike function Z(n=5,m=1) u}3D'h
% x = -1:0.01:1; *IX<&u#
% [X,Y] = meshgrid(x,x); h?[|1.lJx(
% [theta,r] = cart2pol(X,Y); 6S`0<Z;;/
% idx = r<=1; ~jC+6v
% z = nan(size(X)); =' uePM")
% z(idx) = zernfun(5,1,r(idx),theta(idx)); *:bexD H
% figure bd]9kRq1K
% pcolor(x,x,z), shading interp 0vX4v)-^u
% axis square, colorbar >3ax `8
% title('Zernike function Z_5^1(r,\theta)') Xii>?sA5Z"
% "i#aII+T
% Example 2: 0civXZgj
% [ ?%q,>F
% % Display the first 10 Zernike functions Lq|>n[KY
% x = -1:0.01:1; Q2/65$nW
% [X,Y] = meshgrid(x,x); XeX\u3<D
% [theta,r] = cart2pol(X,Y); m/z,MT74*J
% idx = r<=1; mG"xo^1_H
% z = nan(size(X)); H2H`7 +I,
% n = [0 1 1 2 2 2 3 3 3 3]; X NgcBSD
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; +F-EgF+J
% Nplot = [4 10 12 16 18 20 22 24 26 28]; !O,Sq/=.
% y = zernfun(n,m,r(idx),theta(idx)); K!]a+M]>
% figure('Units','normalized') _f'v>"K
% for k = 1:10 >
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% z(idx) = y(:,k); gg>QXui
% subplot(4,7,Nplot(k)) DQT'OZ:w
% pcolor(x,x,z), shading interp 8Qo'[+4;
% set(gca,'XTick',[],'YTick',[]) d]poUN~x
% axis square h2 KI
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) nl
qn:[BU
% end NMe{1RM
% w
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% See also ZERNPOL, ZERNFUN2. (`S^6-^
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% Paul Fricker 11/13/2006 XEQTT D<
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2 /FQ;<L
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% Check and prepare the inputs: Hs*["zFc
% ----------------------------- ,Cb3R|L8
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) #8|LPfA
error('zernfun:NMvectors','N and M must be vectors.') ?u|@,tQ[
end ]I [~0PCSX
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if length(n)~=length(m)
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error('zernfun:NMlength','N and M must be the same length.') " !~o
end ^Jp,&
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n = n(:); =eDIvNps
m = m(:); .E<nQWz8
if any(mod(n-m,2)) J0?kEr
error('zernfun:NMmultiplesof2', ... Ut;`6t
'All N and M must differ by multiples of 2 (including 0).') Zz0e4C
end BH">#&j[
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if any(m>n) ZEAUoC1E1
error('zernfun:MlessthanN', ... M2O_kOeZ
'Each M must be less than or equal to its corresponding N.') u.ggN=Z
end xWxc1tT`
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if any( r>1 | r<0 ) c2y,zq|H
error('zernfun:Rlessthan1','All R must be between 0 and 1.') Ax;=Zh<DAv
end :OG I|[
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Y()"2CCV
error('zernfun:RTHvector','R and THETA must be vectors.') 1^!SuAA@
end T$I_nxh[)L
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/|IPBU 5
r = r(:); VPe0\?!d
theta = theta(:); FJ:^pROpm
length_r = length(r); *yu}e)(0
if length_r~=length(theta) l3>S{
error('zernfun:RTHlength', ... JZ:@iI5>+
'The number of R- and THETA-values must be equal.') >]\I:T
end ieFl4hh[G
]:P7}Kpb
_)M,p@!?=h
% Check normalization: =dmr,WE
% -------------------- c$O8Rhx
if nargin==5 && ischar(nflag) :?>7Z6
isnorm = strcmpi(nflag,'norm'); '<R>cN"
if ~isnorm ^"WVE["
error('zernfun:normalization','Unrecognized normalization flag.') e-nA>v
end 3v/B*M VI
else \^x{NV@v42
isnorm = false; =p+y$
end &mwd0%4
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ?sf<cFF
% Compute the Zernike Polynomials KdkA@>L!;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 9)Fx;GxL
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2 !s&|lI
% Determine the required powers of r: CXa[%{[n
% ----------------------------------- M/zO|-j&
m_abs = abs(m); Zf'*pp T&q
rpowers = []; IH]9%d)
for j = 1:length(n) *'%V}R[>
rpowers = [rpowers m_abs(j):2:n(j)]; %FO{:@CH
end (l{vlFWd
rpowers = unique(rpowers); i5 '&u:
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% Pre-compute the values of r raised to the required powers, #z\ub5um
% and compile them in a matrix: dzf2`@8#
% ----------------------------- Ql*zl
if rpowers(1)==0 T(b9b,ov)
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); EBj^4=b[
rpowern = cat(2,rpowern{:}); sV\_DP/l
rpowern = [ones(length_r,1) rpowern]; oBzl=N3<
else !wAT`0<94F
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); jvzioFCt
rpowern = cat(2,rpowern{:}); $v^hzC
end !?2)apM
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% Compute the values of the polynomials: }A#IBqf5
% -------------------------------------- _P>YG<*"kQ
y = zeros(length_r,length(n)); ;_<R +w3-
for j = 1:length(n) K7
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s = 0:(n(j)-m_abs(j))/2; ).T&fa"
pows = n(j):-2:m_abs(j); 6TtB3;5
for k = length(s):-1:1 xoaO=7\io
p = (1-2*mod(s(k),2))* ... @<.@X*#I
prod(2:(n(j)-s(k)))/ ... ,g*!NK_:5t
prod(2:s(k))/ ... \br!77
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... &V"oJ}M/a
prod(2:((n(j)+m_abs(j))/2-s(k))); _Nx
/<isdL
idx = (pows(k)==rpowers); V%Uj\cv
y(:,j) = y(:,j) + p*rpowern(:,idx); jr6_|(0
i6
end VYvfx
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if isnorm c}(WniR-"
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); t@q'm.:uw<
end &!!*xv-z
end Ndmt$(b
% END: Compute the Zernike Polynomials |)-kUu
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% nm'l}/Ug
+mQ5\14#
|P|B"I<?
% Compute the Zernike functions: ^^y eC|~N:
% ------------------------------ c_lHj#A(l
idx_pos = m>0; v^|U?
idx_neg = m<0; i\R0+O{
5]xuU.w'
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z = y; 3h@]cWp
if any(idx_pos) RNg?o[S
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Lvk}% ,S8t
end nJDGNm,
if any(idx_neg) fi+}hGj(r
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); j\>LJai"
end Xn7G2Yp
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<izQ]\kL
% EOF zernfun #&3,T1i`