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 S?\hbM]V-o
function z = zernfun(n,m,r,theta,nflag) nDB 2>J
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. kN|5
J
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N dyiEK)$h
% and angular frequency M, evaluated at positions (R,THETA) on the X2dc\v.x
% unit circle. N is a vector of positive integers (including 0), and r>~d[,^$m4
% M is a vector with the same number of elements as N. Each element 4:&qTY)H
% k of M must be a positive integer, with possible values M(k) = -N(k) RB7AI!'a?
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, )|y2Q
% and THETA is a vector of angles. R and THETA must have the same C]yQ "b
% length. The output Z is a matrix with one column for every (N,M) 7k=F6k0)
% pair, and one row for every (R,THETA) pair. MiH}VfI
% 7X{bB
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike Fiu!!M6
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), TT2cOw
% with delta(m,0) the Kronecker delta, is chosen so that the integral \!JS7!+
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, KU|BT.o8
% and theta=0 to theta=2*pi) is unity. For the non-normalized g(1B W#$
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. MATgJ`lsy
% P B(
% The Zernike functions are an orthogonal basis on the unit circle. ]TBtLU3
% They are used in disciplines such as astronomy, optics, and R'I_xjC
% optometry to describe functions on a circular domain. jc&/}o$K
% 7AO3-;
l]
% The following table lists the first 15 Zernike functions. J,0pe\5
% !/6\m!e|1R
% n m Zernike function Normalization ;EJPrDHTk
% -------------------------------------------------- 8pk#sJ51
% 0 0 1 1 P}hY{y'
% 1 1 r * cos(theta) 2 Ni!;-,H+E
% 1 -1 r * sin(theta) 2 _}zo
/kDA
% 2 -2 r^2 * cos(2*theta) sqrt(6) n</k/Mk}
% 2 0 (2*r^2 - 1) sqrt(3) c.(Ud`jc
% 2 2 r^2 * sin(2*theta) sqrt(6) J3~hzgY
% 3 -3 r^3 * cos(3*theta) sqrt(8) 0L:V#y-*
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 4`8IFK
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) \5Vp6^
% 3 3 r^3 * sin(3*theta) sqrt(8) >+:r '
% 4 -4 r^4 * cos(4*theta) sqrt(10) `10X5V@hP
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) qRPc%"
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) j|[(*i%7|
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) tw'hh@7-Y
% 4 4 r^4 * sin(4*theta) sqrt(10) ,u}<Ws8N
% -------------------------------------------------- e&$p-0DmT|
% z;dcAdz9
% Example 1: gX@nPZjg
% cBifZv*l
% % Display the Zernike function Z(n=5,m=1) L$1K7<i.
% x = -1:0.01:1; m~u|VgD
% [X,Y] = meshgrid(x,x); {*QvC
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% [theta,r] = cart2pol(X,Y); $%g\YdC
% idx = r<=1; ytjK++(T5
% z = nan(size(X)); rI0)F
% z(idx) = zernfun(5,1,r(idx),theta(idx)); >ik1]!j]Lv
% figure J3;Tm~KJ_
% pcolor(x,x,z), shading interp I*D<J$ 9N
% axis square, colorbar f}[H
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% title('Zernike function Z_5^1(r,\theta)') ?}vzLgp
% w`L~#yu
% Example 2: %p0b{P j_p
% Bk@)b`WR
% % Display the first 10 Zernike functions 1"}B]5!
% x = -1:0.01:1; p?Ed-
S
% [X,Y] = meshgrid(x,x); Hqvc7 -c6
% [theta,r] = cart2pol(X,Y); pT4qPta,2
% idx = r<=1; [%)@|^hw91
% z = nan(size(X)); !w q4EV
% n = [0 1 1 2 2 2 3 3 3 3]; Q[M (Wqg
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; '!!e+\h#
% Nplot = [4 10 12 16 18 20 22 24 26 28];
bRNK.[|
% y = zernfun(n,m,r(idx),theta(idx)); ~<n(y-P^
% figure('Units','normalized') h$70H ^r
% for k = 1:10 <B!'3C(P
% z(idx) = y(:,k); Y}ng_c
% subplot(4,7,Nplot(k)) -yoAxPDW
% pcolor(x,x,z), shading interp AHwG<k
% set(gca,'XTick',[],'YTick',[]) 0<g<GQ(E
% axis square U^[<
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) Va7c#P?
% end czI{qi5N
% n@
4@,
% See also ZERNPOL, ZERNFUN2. tQrS3Hz'nA
/|GT\X4o
% Paul Fricker 11/13/2006 &y70
8h|M!/&2
2{-!E ^g
% Check and prepare the inputs: Edw2W8
% ----------------------------- i'HPRY
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) F.<L>
G7{1
error('zernfun:NMvectors','N and M must be vectors.') zOB !(R
end IU|kNBo
r|
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if length(n)~=length(m) `?|Rc
error('zernfun:NMlength','N and M must be the same length.') xYI;V7
end 6\4Z\82
RNTa XR+Zn
n = n(:); GRT]aw
m = m(:); 8Atq,GcG
if any(mod(n-m,2)) WuM C^
error('zernfun:NMmultiplesof2', ... i@5)`<?
'All N and M must differ by multiples of 2 (including 0).') r<c #nD~K
end t<63 8`{kk
nIn2 *r
if any(m>n) @vRwzc\
error('zernfun:MlessthanN', ... pYo=oI
'Each M must be less than or equal to its corresponding N.') zrRFn `B
end NvJV</l6A
A1),el-^5
if any( r>1 | r<0 ) FI"HJwAs
error('zernfun:Rlessthan1','All R must be between 0 and 1.') =gjDCx$|
end sI,W%I':d
,%[4j9#!_
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) m\ S\3n
error('zernfun:RTHvector','R and THETA must be vectors.') *fnvZw?
end Bz /@c)
j6S"UwJjp
r = r(:); n2f6p<8A
theta = theta(:); /_t|Dry015
length_r = length(r); pKT2^Q}-h
if length_r~=length(theta) RWKH%C[Yd
error('zernfun:RTHlength', ... +G*JrwJ&=
'The number of R- and THETA-values must be equal.') ""% A'TZ
end ^/@jwZ
g/~XCC^F?
% Check normalization: 5~H#(d<oZ
% -------------------- S6xgiem
if nargin==5 && ischar(nflag) KxzYfH
isnorm = strcmpi(nflag,'norm'); =*Z5!W'd
if ~isnorm H8{ol6wc)6
error('zernfun:normalization','Unrecognized normalization flag.') ["3\eFg
end IiJZ5'{
else
y<:<$22O
isnorm = false; P7.' kX9
end 9'[ N1Un.=
\%0n}.A
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% j%IF2p2
% Compute the Zernike Polynomials ! RW
`3
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% fCWGAO2
V~@^`Gd
% Determine the required powers of r: z (?=Iv3
% ----------------------------------- YW/QC'_iC
m_abs = abs(m); `=lc<T^
rpowers = []; $za8"T*I
for j = 1:length(n) m908jI_So
rpowers = [rpowers m_abs(j):2:n(j)]; N$>^g"6o
end S!v(+|
rpowers = unique(rpowers); #S]ER907
q
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% Pre-compute the values of r raised to the required powers, %Dg0fL
% and compile them in a matrix: EJ@p-}I!
% ----------------------------- kE9esC3
if rpowers(1)==0 xG&)1sT#-\
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); .Z:zZ_Ev
rpowern = cat(2,rpowern{:}); o%9*B%HO/
rpowern = [ones(length_r,1) rpowern]; /1mW|O>0
else mpPdG
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); CCBfKp
rpowern = cat(2,rpowern{:}); /DQaGq/Ld
end CHrFM@CM
p$Ox'A4
% Compute the values of the polynomials: ojyIQk+
% -------------------------------------- .Asv%p[W
y = zeros(length_r,length(n)); [W%$qZlP
for j = 1:length(n) 8V^oP]Y
s = 0:(n(j)-m_abs(j))/2; -gSUjP
pows = n(j):-2:m_abs(j); h$4Hw+Yxs]
for k = length(s):-1:1 qlL`jWJ
p = (1-2*mod(s(k),2))* ... 3s/H2fz
prod(2:(n(j)-s(k)))/ ... T!Hb{Cg*
prod(2:s(k))/ ... Llr>9(|
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Tyvtmx M
prod(2:((n(j)+m_abs(j))/2-s(k))); o.0ci+z@
idx = (pows(k)==rpowers); ]]Cb$$Td
y(:,j) = y(:,j) + p*rpowern(:,idx); GGnpjwXeH
end tjupJ*Rt
J]nohICe
if isnorm su*'d:L
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); I'V4D[H5
end gbH<]?
end -$\+'
\
% END: Compute the Zernike Polynomials {q"OM*L(
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% W[Ls|<Q
qWQ/'M
% Compute the Zernike functions: j'A_'g'^
% ------------------------------ 7=;R& mqC
idx_pos = m>0; ILGMMA_2
idx_neg = m<0; _d5QbTe
9I}-[|`u
z = y; etTn_v
if any(idx_pos) D)L+7N0D~
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); \WxukYH
end o,_?^'@
if any(idx_neg) LDPUD'
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Lm%:K]X
end G3Z)Z)N
3kybLOG
% EOF zernfun