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 IMM+g]#e
function z = zernfun(n,m,r,theta,nflag) hi(e%da
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. ZI4dD.B
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N "1XTgCu\
% and angular frequency M, evaluated at positions (R,THETA) on the .x] pJ9
% unit circle. N is a vector of positive integers (including 0), and 0Ntvd7"`}
% M is a vector with the same number of elements as N. Each element _OJfd
% k of M must be a positive integer, with possible values M(k) = -N(k) PJ&L7
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, =_j<x$,b-
% and THETA is a vector of angles. R and THETA must have the same \b6{u6?+
% length. The output Z is a matrix with one column for every (N,M) +e.w]\}
% pair, and one row for every (R,THETA) pair. WrRY3X
% zN;P_@U
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike br TP}A
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), VR1[-OE
% with delta(m,0) the Kronecker delta, is chosen so that the integral 'Q7^bF^
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 8lDb<i
% and theta=0 to theta=2*pi) is unity. For the non-normalized ZNDi;6e
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. /:{4,aX2
% IsJx5GO
% The Zernike functions are an orthogonal basis on the unit circle. n'q:L(`M
% They are used in disciplines such as astronomy, optics, and sSwY!";
% optometry to describe functions on a circular domain. Ahba1\,N$
% sV5") /~
% The following table lists the first 15 Zernike functions. [MKG5=kaE
% <]DUJuF-M
% n m Zernike function Normalization d-m.aP)y:
% -------------------------------------------------- $%M]2_W(
% 0 0 1 1 hosY`"X
% 1 1 r * cos(theta) 2 34"PtWbV>
% 1 -1 r * sin(theta) 2 %{3q=9ii
% 2 -2 r^2 * cos(2*theta) sqrt(6) z$~F9Es9
% 2 0 (2*r^2 - 1) sqrt(3) n53c}^
% 2 2 r^2 * sin(2*theta) sqrt(6) '+vmC*-I(
% 3 -3 r^3 * cos(3*theta) sqrt(8) @OFxnF`
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) xs Pt
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) {,*vMQ<^
% 3 3 r^3 * sin(3*theta) sqrt(8) h~CLJoK<
% 4 -4 r^4 * cos(4*theta) sqrt(10) q
&{<HcP
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) IoK/ 2Gp
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) $)X8'1%6
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) YHu]\'Ff
% 4 4 r^4 * sin(4*theta) sqrt(10) >mR8@kob<
% -------------------------------------------------- L@zhbWY
% VlL%dN;
0
% Example 1: 3Z me?o*bY
% *TI?tD
% % Display the Zernike function Z(n=5,m=1) |</) 6r
% x = -1:0.01:1; dT?3Q;>B?
% [X,Y] = meshgrid(x,x); PXJ7Ek*/
% [theta,r] = cart2pol(X,Y); pWv1XTs@t:
% idx = r<=1; %.$7-+:7A
% z = nan(size(X)); 5U+4vV/*
% z(idx) = zernfun(5,1,r(idx),theta(idx)); ]{\M,txo8
% figure ]b sabS?
% pcolor(x,x,z), shading interp [2Nux0g
% axis square, colorbar 7:b.c
% title('Zernike function Z_5^1(r,\theta)') <LXx_{=:
% :lvBcFw
% Example 2: ^eO/?D8~h
% p nI=
% % Display the first 10 Zernike functions <Up?w/9
% x = -1:0.01:1; GQCdB>
% [X,Y] = meshgrid(x,x); iI7ocyUv
% [theta,r] = cart2pol(X,Y); NsM`kZM4H
% idx = r<=1; Vr( Z;YO
% z = nan(size(X)); {]dtA&8(
% n = [0 1 1 2 2 2 3 3 3 3]; PR$;*|@
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; udLI AV*
% Nplot = [4 10 12 16 18 20 22 24 26 28]; gM=:80
% y = zernfun(n,m,r(idx),theta(idx)); &{+ 0a[rN
% figure('Units','normalized') qdv O>k3
% for k = 1:10 yrfV&C%=n
% z(idx) = y(:,k); n;N79`mZC
% subplot(4,7,Nplot(k)) 4sn\UuKyL
% pcolor(x,x,z), shading interp Bi :!"Nw[X
% set(gca,'XTick',[],'YTick',[]) i-5,*0e6m
% axis square e3 :L]4t
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) {$yju _[
% end uh2_Rzln
% <.gDg?'3
% See also ZERNPOL, ZERNFUN2. "2sk1
Q1?*+]
% Paul Fricker 11/13/2006 9jEH"`qqk
2@GizT*mA
N1Ag.
% Check and prepare the inputs: bP#!U'b" =
% ----------------------------- Q!U}
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 7>F{.\Z
error('zernfun:NMvectors','N and M must be vectors.') 1hGj?L0m.
end NM![WvtjW
&s(&B>M
if length(n)~=length(m) je2_.^
error('zernfun:NMlength','N and M must be the same length.') flFdoEV.U)
end 15<? [`:6
sTlel&
n = n(:); PMB4]p%o
m = m(:); tS]
if any(mod(n-m,2)) }F _c0zM
error('zernfun:NMmultiplesof2', ... $Emu*'
'All N and M must differ by multiples of 2 (including 0).') 5Q"w{ n
end |.UY'B
!+^'Ej)z
if any(m>n) /+SLq`'u)
error('zernfun:MlessthanN', ... ~S\L(B(
'Each M must be less than or equal to its corresponding N.') =huV(THU
end +W*~=*h|
`;;l {8
if any( r>1 | r<0 ) Hn(1_I%zF
error('zernfun:Rlessthan1','All R must be between 0 and 1.') 'Uf?-t*LT@
end k<^M >` $
X4!7/&
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) #H>{>0q
error('zernfun:RTHvector','R and THETA must be vectors.') )XK\[tL
end @q/g%-WNz
SXOAa<u5
r = r(:); l_+@Xpl
theta = theta(:); d"#Zp
length_r = length(r); Q]xkDr?
if length_r~=length(theta) .=#jdc/
error('zernfun:RTHlength', ... K -rR)-rI
'The number of R- and THETA-values must be equal.') Ytlzn%
end YoKyiO!
H,X|-B
% Check normalization: K?!qNK
% -------------------- &HM-UC|
if nargin==5 && ischar(nflag) ;J 5z
isnorm = strcmpi(nflag,'norm'); 5h#h>0F
if ~isnorm cu0IFNF}[
error('zernfun:normalization','Unrecognized normalization flag.') XTJD>
end e}e8WR=B
else <s'de$[
isnorm = false; `)n4I:)2
end ?W'p&(;
ilL0=[2
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% d7X&3L%Oq
% Compute the Zernike Polynomials <'I["Um
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% .9qK88fU R
Fr8GGN~/
% Determine the required powers of r: RU:Rt'
% ----------------------------------- (G>[A}-
m_abs = abs(m); |:=o\eu&
rpowers = []; ijF_
KP'
for j = 1:length(n) LSJ?;Zg(=z
rpowers = [rpowers m_abs(j):2:n(j)]; 6@J=n@J$p
end c0@8KW[,
rpowers = unique(rpowers); ~.m<`~u
m.e]tTe
% Pre-compute the values of r raised to the required powers, 6gg8h>b
% and compile them in a matrix: AC)
M2;
% ----------------------------- q!5:M\
if rpowers(1)==0 I#M3cI!X?
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); >d 2Fa4u3
rpowern = cat(2,rpowern{:}); az(<<2=
rpowern = [ones(length_r,1) rpowern]; Wp=3heCa6
else 2@D`^]]
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 9B: 3Ha=
rpowern = cat(2,rpowern{:}); +$,Re.WnP
end %t9C
LwH#|8F
% Compute the values of the polynomials: 'x!\pE-
% -------------------------------------- m|@H`=`d
y = zeros(length_r,length(n)); $7S"4rou
for j = 1:length(n) pN%&`]Wev
s = 0:(n(j)-m_abs(j))/2; nVb@sI{{k
pows = n(j):-2:m_abs(j); |W">&Rb<t#
for k = length(s):-1:1 K9lgDk"i
p = (1-2*mod(s(k),2))* ... 4>hHUz[_
prod(2:(n(j)-s(k)))/ ... i--t
?@#
prod(2:s(k))/ ... j9+$hu#a
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 11[lc2
prod(2:((n(j)+m_abs(j))/2-s(k))); :S+K\
idx = (pows(k)==rpowers); #<im?
y(:,j) = y(:,j) + p*rpowern(:,idx); o\IMYT
end x *Lt]]A
)h!cOEt
if isnorm N@q}eGe
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); dj0; tQ=C
end kmI0V[Y
end 7 F^d-
% END: Compute the Zernike Polynomials RK>Pe3<
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% `2s!%/
z ^gJy,T
% Compute the Zernike functions: E9HMhUe
% ------------------------------ kSQ8kU_w+
idx_pos = m>0; <B"sp r&1
idx_neg = m<0; [VCC+_
rH+OXGoB
z = y; c7Z4u|G
if any(idx_pos) _FLEz|%~
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); hRcb}>pr
end o`?rj!\
if any(idx_neg) )*,/L <
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); &1xCPKIr
end T(4d5 fY
K"}fD;3
% EOF zernfun