非常感谢啊,我手上也有zernike多项式的拟合的源程序,也不知道对不对,不怎么会有 $R^"~|m3M
function z = zernfun(n,m,r,theta,nflag) N\p3*#M
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. Bg3^BOT
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N n4:WM+f4
% and angular frequency M, evaluated at positions (R,THETA) on the :{sX8U%
% unit circle. N is a vector of positive integers (including 0), and WN0^hDc-
% M is a vector with the same number of elements as N. Each element ZK;HW
% k of M must be a positive integer, with possible values M(k) = -N(k) k~?@~xm,R
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, >Nov9<p
% and THETA is a vector of angles. R and THETA must have the same (YR1ML3N
% length. The output Z is a matrix with one column for every (N,M) xGA%/dy,;
% pair, and one row for every (R,THETA) pair. 2@ad! h
% i^n&K:6
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ]t,ppFC#
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), | o?@Eh
% with delta(m,0) the Kronecker delta, is chosen so that the integral ;%U`P8b!
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, qvT9d7x
% and theta=0 to theta=2*pi) is unity. For the non-normalized ,
w_ Ew
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. al5?w{us
% ";jhj:Xj
% The Zernike functions are an orthogonal basis on the unit circle. 8T%z{ A1T
% They are used in disciplines such as astronomy, optics, and `527vK
6
% optometry to describe functions on a circular domain. 2sXWeiJy;
% EZ$m4:{e
% The following table lists the first 15 Zernike functions. SDot0`s>
% %9M_*]
% n m Zernike function Normalization ^@N@gB
% -------------------------------------------------- K(_nfE{
% 0 0 1 1 O=yUAAD$
% 1 1 r * cos(theta) 2 KQEn C`Nz
% 1 -1 r * sin(theta) 2 k:c)|2
% 2 -2 r^2 * cos(2*theta) sqrt(6) N~a?0x
% 2 0 (2*r^2 - 1) sqrt(3) N[AX29
% 2 2 r^2 * sin(2*theta) sqrt(6) 8&3G|m1-2
% 3 -3 r^3 * cos(3*theta) sqrt(8) gHTo|2 Q{
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) lc*<UZR
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) f#[Fqkmj
% 3 3 r^3 * sin(3*theta) sqrt(8) /N~.,vf
% 4 -4 r^4 * cos(4*theta) sqrt(10) E")82I
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Fd3V5h
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) VPf=LSxJe
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) .aNy)Yu8
% 4 4 r^4 * sin(4*theta) sqrt(10) b,I$.&BD
% -------------------------------------------------- :sJVklK
%
B[8
% Example 1: Jz3u r)|
% `,xKK+~YG-
% % Display the Zernike function Z(n=5,m=1) xFgY#F
% x = -1:0.01:1; 8E|S`I
% [X,Y] = meshgrid(x,x); >d_O0a*W-
% [theta,r] = cart2pol(X,Y); hH%@8'1v
% idx = r<=1; :dB6/@fW
% z = nan(size(X)); kvKbl;<
% z(idx) = zernfun(5,1,r(idx),theta(idx)); <D=U= 5
% figure $+-2/=>Xk
% pcolor(x,x,z), shading interp f~t*8rG~m
% axis square, colorbar u>d,6
!
% title('Zernike function Z_5^1(r,\theta)') lLl^2[4k5
% ]M#_o]
% Example 2: FL-sXg
% U#-89.x
% % Display the first 10 Zernike functions >=$( ,8"
% x = -1:0.01:1; U }xRvNz
% [X,Y] = meshgrid(x,x); LLCMp3qBz
% [theta,r] = cart2pol(X,Y); [$f
% idx = r<=1; Eqnc("m)
% z = nan(size(X)); jo/-'Lf{?
% n = [0 1 1 2 2 2 3 3 3 3]; kbiMqiPG
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; jgbE@IA@!'
% Nplot = [4 10 12 16 18 20 22 24 26 28]; ~:v" TuuK
% y = zernfun(n,m,r(idx),theta(idx)); !Yd7&#s
% figure('Units','normalized') XJ.bK
% for k = 1:10 &E0P`F,GQA
% z(idx) = y(:,k); 83e{rcs
% subplot(4,7,Nplot(k)) ,~>A>J
% pcolor(x,x,z), shading interp 7ZqC1
% set(gca,'XTick',[],'YTick',[]) CB:G4VqOT
% axis square 8 Zhx&
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) [ lW~v:W
% end gWL'Fl}H
% C/U^8,6\n
% See also ZERNPOL, ZERNFUN2. |aIY
*\L\Bzm
% Paul Fricker 11/13/2006 3%p^>D\
h`;w/+/Zr
OLg=kF[[
% Check and prepare the inputs: #+>8gq^5
% ----------------------------- +a0q?$\
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) TldqF BX
error('zernfun:NMvectors','N and M must be vectors.') unY+/p $
end oF7o"NHaWa
Db3#;
if length(n)~=length(m) fq-e2MCX5
error('zernfun:NMlength','N and M must be the same length.') Yi:@>A<#
end H$^IT#
* `1W})
n = n(:); OXAr..
m = m(:); s"gNHp.oF
if any(mod(n-m,2)) 1CXO=Q
error('zernfun:NMmultiplesof2', ... `o4alK\
'All N and M must differ by multiples of 2 (including 0).') cdY|z]B
end P+K< /i
DPqk~ KCM
if any(m>n) RE6dN
error('zernfun:MlessthanN', ... ROqz$yY
'Each M must be less than or equal to its corresponding N.') %zsY=qT
end 3V2dN)\
!g=4\C`mY
if any( r>1 | r<0 ) 1<766
error('zernfun:Rlessthan1','All R must be between 0 and 1.') xL&M8:
end s_:7dD
OpWTw&B"+
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) qD!qSM
error('zernfun:RTHvector','R and THETA must be vectors.') Pk)>@F<
end jjLx60|{
]l/ PyX
r = r(:); `-yo-59E[
theta = theta(:); hc#Sy:T>
length_r = length(r); 9+S$,|9
if length_r~=length(theta) ;D'6sd"
error('zernfun:RTHlength', ... cCa+UTxaJ
'The number of R- and THETA-values must be equal.') EIdEXAC(
end 'ip2| UG
rlMahY"C
% Check normalization: VO
u/9]a
% -------------------- '/O >#1
if nargin==5 && ischar(nflag) L/*D5k%J
isnorm = strcmpi(nflag,'norm'); /hF@Xh%hY
if ~isnorm w&F.LiX^
error('zernfun:normalization','Unrecognized normalization flag.') p#;I4d G
end {$AwG#kt
else mZ_643|
isnorm = false; \k
9EimT}
end dBRK6hFC
z}.Q~4 f0D
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [[FDt[ l4
% Compute the Zernike Polynomials Ar{7H)V:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% <ddXvUCX
4J5 RtK
% Determine the required powers of r: 0)Nu
% ----------------------------------- M1HGXdN* B
m_abs = abs(m); ^K3Bn
rpowers = []; i0q<,VSl$_
for j = 1:length(n) 9@3cz_[J
rpowers = [rpowers m_abs(j):2:n(j)]; 3%~c\naD?O
end K&'Vd@
rpowers = unique(rpowers); `En>o~ L;
m:-=K
% Pre-compute the values of r raised to the required powers, +Hd'*'c
% and compile them in a matrix: nI_UL
% ----------------------------- 4"^v]&I
if rpowers(1)==0 Yx[B*] 2
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 5do49H_
rpowern = cat(2,rpowern{:}); ZVIlVuZ}
rpowern = [ones(length_r,1) rpowern]; pOq9J7BS
else 4ux^K:z
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); _ci8!PP
rpowern = cat(2,rpowern{:}); 2H,n"-9+
end SX1w5+p$C
;s\ck:Xg
% Compute the values of the polynomials: c9O0YQ3&8
% -------------------------------------- vw2yOLRX
y = zeros(length_r,length(n)); iy-~CPNB_
for j = 1:length(n) @V =HY
s = 0:(n(j)-m_abs(j))/2; LS%;ZKJ
pows = n(j):-2:m_abs(j); ]5a,%*f+
for k = length(s):-1:1 e|
Sw+fhy<
p = (1-2*mod(s(k),2))* ... #Y<QEGb(
prod(2:(n(j)-s(k)))/ ... p >h&SD?b
prod(2:s(k))/ ... ]j: aO
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... /LC!|-1E
prod(2:((n(j)+m_abs(j))/2-s(k))); W&=F<n`
idx = (pows(k)==rpowers); <wTD}.n
y(:,j) = y(:,j) + p*rpowern(:,idx); 3)*Twqt
end
s;W1YN
I?OnEw
if isnorm HDQH7Bs
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 'U*Kb
end tlyDXB~+
end @)x8<
% END: Compute the Zernike Polynomials uRnSwJ"hE
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% IA~wmOF
}@TtX\7(D
% Compute the Zernike functions: gJYX
% ------------------------------ Jty/gjK+
idx_pos = m>0; zlhI \jRdc
idx_neg = m<0; d>hLnz1O
cyXnZs ?|
z = y; /SKgN{tWe
if any(idx_pos) wS;hC&~2
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ><w=
end k.6(Q_TS
if any(idx_neg) dkAY%z two
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); cr>"LAi
end eb=#{
u&Cu"-%=M
% EOF zernfun