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 ZAJ~Tbm[f
function z = zernfun(n,m,r,theta,nflag) V=gu'~
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. $~e55X'!+
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 63`5A3rii
% and angular frequency M, evaluated at positions (R,THETA) on the g-pEt#
% unit circle. N is a vector of positive integers (including 0), and }wB!Bx2
% M is a vector with the same number of elements as N. Each element '2qbIYanh
% k of M must be a positive integer, with possible values M(k) = -N(k) Qo/pz2N
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, HCKoc L/]h
% and THETA is a vector of angles. R and THETA must have the same /H?) qk
% length. The output Z is a matrix with one column for every (N,M) FwE<_hq//
% pair, and one row for every (R,THETA) pair. U:AB%gr[
% 5d;(D i5z
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike %H[~V
f?d
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), j/8q
% with delta(m,0) the Kronecker delta, is chosen so that the integral ?7#{#sj
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, %x./>-[t
% and theta=0 to theta=2*pi) is unity. For the non-normalized C).+h7{nd
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ^V~^[Yp
% >u\'k+=
% The Zernike functions are an orthogonal basis on the unit circle. },=ORIB B:
% They are used in disciplines such as astronomy, optics, and )gx*;z@
% optometry to describe functions on a circular domain. SB|Cr:wM
% RDU 'l^
% The following table lists the first 15 Zernike functions. HXN. ,[
% QFB2,k6jN
% n m Zernike function Normalization |['SiO$)
% -------------------------------------------------- as73/J6
% 0 0 1 1 3!h 3flE
% 1 1 r * cos(theta) 2 th{ie2$
% 1 -1 r * sin(theta) 2 l*Q OM
% 2 -2 r^2 * cos(2*theta) sqrt(6) [s+FX5' K
% 2 0 (2*r^2 - 1) sqrt(3) uF ;8B]"
% 2 2 r^2 * sin(2*theta) sqrt(6) Nx zAlu
% 3 -3 r^3 * cos(3*theta) sqrt(8) |kF"p~s
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) -
i{1h"
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) g7w#;E
% 3 3 r^3 * sin(3*theta) sqrt(8) =eR#]d
% 4 -4 r^4 * cos(4*theta) sqrt(10) tI
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10)
T4J
WZ
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) /eBcPu"[Vb
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 5Z(q|nn7P
% 4 4 r^4 * sin(4*theta) sqrt(10) -M+o;
% -------------------------------------------------- |RBL5,t^
% gk}.LE
% Example 1:
]D^zTl3=q
% =I9hGj6
% % Display the Zernike function Z(n=5,m=1) a
*bc#!e
% x = -1:0.01:1; /GO((v+J
% [X,Y] = meshgrid(x,x); -^*8D(j*
% [theta,r] = cart2pol(X,Y); p`S~UBcL.
% idx = r<=1; Gx|/
Jq
% z = nan(size(X)); 29W`L2L
% z(idx) = zernfun(5,1,r(idx),theta(idx)); -j^G4J
% figure @7sHFwtar?
% pcolor(x,x,z), shading interp ,!^g8zO
% axis square, colorbar ;[7#h8
% title('Zernike function Z_5^1(r,\theta)') 8SBa w'a
% PKev)M;C+
% Example 2: @sRb1+nn
% CX 7eCo
% % Display the first 10 Zernike functions "Z"`X3,-z
% x = -1:0.01:1; rm<`H(cT
% [X,Y] = meshgrid(x,x); sDwE,f0h
% [theta,r] = cart2pol(X,Y); ;`Sn66&
% idx = r<=1; V.!z9AQ
% z = nan(size(X)); orEb+
% n = [0 1 1 2 2 2 3 3 3 3]; wh3Wuh?x
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; t&C0V|s79$
% Nplot = [4 10 12 16 18 20 22 24 26 28]; F3nPQw{;
% y = zernfun(n,m,r(idx),theta(idx)); -}5dZ;
% figure('Units','normalized') (OG>=h8?
% for k = 1:10 ai)?RF
% z(idx) = y(:,k); @ 3b-
% subplot(4,7,Nplot(k)) gT|&tTS1@
% pcolor(x,x,z), shading interp P!/:yWd
% set(gca,'XTick',[],'YTick',[]) PkK#HD
% axis square 602=qb
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) AVp"<Uv
% end E;r~8^9)
% &RlYw#*1.
% See also ZERNPOL, ZERNFUN2. \qbEC.-K
6}_J;g\|
% Paul Fricker 11/13/2006 (k %0|%eR
0[s<!k9=
!_:|mu'
% Check and prepare the inputs: ^p~ 3H
% ----------------------------- sv*xO7D.
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) rzKn5Z
error('zernfun:NMvectors','N and M must be vectors.') e)4L}a
end f q*V76F
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if length(n)~=length(m) HTP~5J
error('zernfun:NMlength','N and M must be the same length.') j2:A@a6
end \fC}l
Ll
q%FXox~b
n = n(:); BeM|1pe.
m = m(:); R(A"6a8*
if any(mod(n-m,2)) YfH+kDT
error('zernfun:NMmultiplesof2', ... I=V]_Ik4N
'All N and M must differ by multiples of 2 (including 0).') }/z\%Y
end SG3qNM: g
M+\LH
if any(m>n) o(5
(]bJ
error('zernfun:MlessthanN', ... #]Q.B\\
'Each M must be less than or equal to its corresponding N.') "cX*GTNi8
end UyOoyyd.
6H!"oC&
if any( r>1 | r<0 ) dRLvej,
error('zernfun:Rlessthan1','All R must be between 0 and 1.') ZSW`/}Dp;
end yl~h
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u}KEH@yv
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) LwIX&\Ub
error('zernfun:RTHvector','R and THETA must be vectors.') 4 Yl:1rz
end Edav }z
w77"?kJ9X
r = r(:); C AF{7 `{
theta = theta(:); 3.I:`>;EO
length_r = length(r); iLG~_Ob:
if length_r~=length(theta) o*|j}hnbv
error('zernfun:RTHlength', ... Qtn%h:i
S~
'The number of R- and THETA-values must be equal.') WUqfY?5
end 38O_PK
ZIM 5$JdCv
% Check normalization: Kg;1%J>ee
% -------------------- 0~j0x#
if nargin==5 && ischar(nflag) ZfN%JJOz(
isnorm = strcmpi(nflag,'norm'); Tg.}rNA4
if ~isnorm 9!oNyqQ
error('zernfun:normalization','Unrecognized normalization flag.') NX:i]t
end q/yL={H?
else '#0'_9}
isnorm = false; )}jXC4
end _8"%nV
v}\Nx[}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% xA2"i2k9
% Compute the Zernike Polynomials >~k"C,6
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% QWV12t$v
S-k:+ 4
% Determine the required powers of r: .`K<Iug1
% ----------------------------------- Ox1#}7`0>
m_abs = abs(m); X,8]g.<
rpowers = []; =%V(n{7=
for j = 1:length(n) qA6;Q$
rpowers = [rpowers m_abs(j):2:n(j)]; pT` oC&
end aM|^t:
rpowers = unique(rpowers); YCd[s[
11(:#4Y,
% Pre-compute the values of r raised to the required powers, qE&R.I!o
% and compile them in a matrix: 3@/\j^U
% ----------------------------- 0xYPK7a=L\
if rpowers(1)==0 <wZ2S3RNA
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); F"<TV&xf
rpowern = cat(2,rpowern{:}); %nfaU~IqK
rpowern = [ones(length_r,1) rpowern]; ]V K%6PQ0
else i#Y[I"'
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 9c@."O`
rpowern = cat(2,rpowern{:}); ?W(>Yefk
end D-tm'APq
%`[Oz[V
% Compute the values of the polynomials: lU[" ZFP
% -------------------------------------- 58@YWvAk
y = zeros(length_r,length(n)); plRBfw>]N
for j = 1:length(n) +(-L
s = 0:(n(j)-m_abs(j))/2; 9=J+5V^qD<
pows = n(j):-2:m_abs(j); #DI%l`B
for k = length(s):-1:1 eZMDt B
p = (1-2*mod(s(k),2))* ... ;5bzXW#U
prod(2:(n(j)-s(k)))/ ... .A. VOf_
prod(2:s(k))/ ... +I {ZW}rA
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... %9!,PeRe
prod(2:((n(j)+m_abs(j))/2-s(k))); vO#=]J8`
idx = (pows(k)==rpowers); NM;0@ o
y(:,j) = y(:,j) + p*rpowern(:,idx); .MzVc42<
end '*~_!lE5
5DEK`#*
if isnorm 69{BJ]q
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 1@)kNg)*$
end mu[:b
end ,u1Yn}
% END: Compute the Zernike Polynomials /Jjub3>Q
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +EZ Lic
G'5p /:
% Compute the Zernike functions: {WE1^&Vk-}
% ------------------------------ Pde|$!Jo
idx_pos = m>0; q*|H*sS
idx_neg = m<0; aeQvIob@
w@&4dau
z = y; `5V=U9zdE
if any(idx_pos) K\7\
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); avmuI^LLs
end f.%mp$~T
if any(idx_neg) 6fozc2h@x%
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); -_bnGY%,
end 7S_rN!E1i*
7<<-\7`
% EOF zernfun