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, 9Z|jxy
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, ]GMe\n
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? u7Y
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那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? wVX[)E\J
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function z = zernfun(n,m,r,theta,nflag) #pT"BSz]
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. c'^?/$H|
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N l>2E (Y|
% and angular frequency M, evaluated at positions (R,THETA) on the ({
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% unit circle. N is a vector of positive integers (including 0), and %<)2/|lCd
% M is a vector with the same number of elements as N. Each element Lco~,OE
% k of M must be a positive integer, with possible values M(k) = -N(k) @GPCwE1
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, spGb!Y`mR
% and THETA is a vector of angles. R and THETA must have the same 9`T)@Uj2n
% length. The output Z is a matrix with one column for every (N,M) XR8,Vt)=
% pair, and one row for every (R,THETA) pair. ]jtK I4
% Y4OPEo 5o
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike qt"G[9;
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 'OE&/
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% with delta(m,0) the Kronecker delta, is chosen so that the integral
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% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, T%x}Y#U'`
% and theta=0 to theta=2*pi) is unity. For the non-normalized zE336
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. %r<rcY
% Z EXc%-M
% The Zernike functions are an orthogonal basis on the unit circle. Um}
% They are used in disciplines such as astronomy, optics, and ob+b<HFv
% optometry to describe functions on a circular domain. qPWP&k
% +s"hqm
% The following table lists the first 15 Zernike functions. [8.c8-lZ^
% 6}Vf\j~
% n m Zernike function Normalization kj|6iG
% -------------------------------------------------- rR$h*
% 0 0 1 1 *]. 7dec/
% 1 1 r * cos(theta) 2 4ae`pAu
% 1 -1 r * sin(theta) 2 ,oORW/0iS
% 2 -2 r^2 * cos(2*theta) sqrt(6) Z_PNI#h*
% 2 0 (2*r^2 - 1) sqrt(3) :lX!\(E2
% 2 2 r^2 * sin(2*theta) sqrt(6) ~9?cn
% 3 -3 r^3 * cos(3*theta) sqrt(8) Eou~P h*t
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) gMv.V{vD
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) efSM`!%j
% 3 3 r^3 * sin(3*theta) sqrt(8) ZWii)0'PV
% 4 -4 r^4 * cos(4*theta) sqrt(10) w:??h4lt
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 'WMh8)
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) JHW"-b
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 4]rnY~
% 4 4 r^4 * sin(4*theta) sqrt(10) 'UkxS b
% -------------------------------------------------- zUDg&-J3
% Hh%I0#
% Example 1: &d9{k5/+\
% Y}@&h!
% % Display the Zernike function Z(n=5,m=1) R7]l{2V#^
% x = -1:0.01:1; zqd@EF6/bz
% [X,Y] = meshgrid(x,x); +QB"8-
% [theta,r] = cart2pol(X,Y); +~St !QV%
% idx = r<=1; 6T>mW#E&
% z = nan(size(X)); B*qi_{Gp
% z(idx) = zernfun(5,1,r(idx),theta(idx)); pb^i^tA+A
% figure ke{8 ^X~#
% pcolor(x,x,z), shading interp ZjT,pOSyb
% axis square, colorbar iz5CAxm
% title('Zernike function Z_5^1(r,\theta)') 9*$t!r{B@
% 3NZK*!@'
% Example 2: M])ZK
% w;D+y*2
% % Display the first 10 Zernike functions J%8(kWQ|
% x = -1:0.01:1; ::o lN
% [X,Y] = meshgrid(x,x); wWgWWXGT}
% [theta,r] = cart2pol(X,Y); k2E0/ @f{k
% idx = r<=1; "vA}FV%tRq
% z = nan(size(X)); s.EI`*xylY
% n = [0 1 1 2 2 2 3 3 3 3]; O[# 27_dH
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; M-\Y"]sW
% Nplot = [4 10 12 16 18 20 22 24 26 28]; nv ca."5 y
% y = zernfun(n,m,r(idx),theta(idx)); yh^!'!I6u[
% figure('Units','normalized') R[Ll59-
% for k = 1:10 "X2 Vrn'
% z(idx) = y(:,k); YpQ7)_s?
% subplot(4,7,Nplot(k)) ,/[6e\0~
% pcolor(x,x,z), shading interp h"lX4
% set(gca,'XTick',[],'YTick',[]) QpZ:gM_
% axis square =5aDM\L$&
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) >O1[:%Z1
% end +
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% YZP(tn
% See also ZERNPOL, ZERNFUN2. @HT% n
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% Paul Fricker 11/13/2006 z
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% Check and prepare the inputs: Z.d7U~_
% ----------------------------- )iq-yjO6
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Z1zVwHa_
error('zernfun:NMvectors','N and M must be vectors.') H|,Oswk~-
end 5>VY LI
%R1 tJ( /
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if length(n)~=length(m) u?>B)PW
error('zernfun:NMlength','N and M must be the same length.') Ny_lrfh) [
end l6(-I
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n = n(:); VQY&g;[d
m = m(:); Q=BZ N]g2
if any(mod(n-m,2)) (E/lIou
error('zernfun:NMmultiplesof2', ... ANvR i+ _
'All N and M must differ by multiples of 2 (including 0).') YRv&1!VLE
end ;g6M%;1-
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if any(m>n) u4m,'XR
error('zernfun:MlessthanN', ... H1I{/g
'Each M must be less than or equal to its corresponding N.') fKp#\tCc y
end (* 1v\Q
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if any( r>1 | r<0 ) :q
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error('zernfun:Rlessthan1','All R must be between 0 and 1.') u583_k%
end 6``'%S'#
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) " .SJ~`S
error('zernfun:RTHvector','R and THETA must be vectors.') <F'X<Bau
end .P.z B}0=
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r = r(:); \(VTt|}By$
theta = theta(:); uMut=ja(U
length_r = length(r); 4VHqBQ4
if length_r~=length(theta) 76wc ,+
error('zernfun:RTHlength', ... hj
'The number of R- and THETA-values must be equal.') /R~1Zj2&
end (
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Dw=gs{8D
% Check normalization: 6&DX] [G
% -------------------- $BkubWM
if nargin==5 && ischar(nflag) uA,>a>xYI
isnorm = strcmpi(nflag,'norm'); ;l&4V
if ~isnorm `Q+(LBP
error('zernfun:normalization','Unrecognized normalization flag.') I#m-g-J
end MS>t_C(
else * 5
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isnorm = false; fBgEnz/
end 8~9030>Q
%YSpCI
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @Ys!DScY,
% Compute the Zernike Polynomials \%/#x V
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ]Pry>N3G5
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% Determine the required powers of r: UbEb&9}
% ----------------------------------- bV edFm
m_abs = abs(m); =8r 0 (c
rpowers = []; &FH2fMLQ
for j = 1:length(n)
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rpowers = [rpowers m_abs(j):2:n(j)]; Vw#_68EybM
end N2oRJ,:B
rpowers = unique(rpowers); $ e\h}A6
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% Pre-compute the values of r raised to the required powers, G2]4n T
% and compile them in a matrix: +Vo}F
% ----------------------------- : p{+G
if rpowers(1)==0 j.*VJazb;
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); c9kzOQ2n
rpowern = cat(2,rpowern{:}); QCH}-q)
rpowern = [ones(length_r,1) rpowern]; <&&SX;
else FP0G]=ME
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); R+nMy=I%8
rpowern = cat(2,rpowern{:}); MZTx:EN!
end R)M_|ca
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% Compute the values of the polynomials: }j2Y5
% -------------------------------------- a-"k/P#
y = zeros(length_r,length(n)); N[<H7_/3
for j = 1:length(n) 6`0mta Q
s = 0:(n(j)-m_abs(j))/2; Nru7(ag1~
pows = n(j):-2:m_abs(j); B|C/
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for k = length(s):-1:1 sp7*_&'J
p = (1-2*mod(s(k),2))* ... MZpK~c1`
prod(2:(n(j)-s(k)))/ ... v1|Bf8
prod(2:s(k))/ ... ,h{A^[yl
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... N0K){
prod(2:((n(j)+m_abs(j))/2-s(k))); _bzqd"
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idx = (pows(k)==rpowers); Vs)--t
y(:,j) = y(:,j) + p*rpowern(:,idx); S@}1t4Ls:
end Iq# ZhAk
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if isnorm kaxvPv1
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); oT{@_U{*J
end 2+cNo9f
end 1VF
% END: Compute the Zernike Polynomials 5aBAr
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Tx1vL
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% Compute the Zernike functions: op"$E1+
% ------------------------------ hY*0aZ|(
idx_pos = m>0; zVi15P$
idx_neg = m<0; Z1ALq5
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z = y; j}BHj.YuP
if any(idx_pos) +&X%<S
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z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ]Ni;w]KE
end Nrah;i+H\o
if any(idx_neg) !Oj)B1gc6&
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Z2Zq'3*
end _TUk(Qe
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% EOF zernfun 9'DtaTmGW