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 fA?v\'Qq/
function z = zernfun(n,m,r,theta,nflag) $pAVTz
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. e8wPEDN*4
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N E
i>GhvRM
% and angular frequency M, evaluated at positions (R,THETA) on the d!}oS<6
% unit circle. N is a vector of positive integers (including 0), and Jc}6kFgO6
% M is a vector with the same number of elements as N. Each element n-],!pL^
% k of M must be a positive integer, with possible values M(k) = -N(k) ]];pWlo!
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, IbL'Z
% and THETA is a vector of angles. R and THETA must have the same Yb_HvP
% length. The output Z is a matrix with one column for every (N,M) h(~/JW[
% pair, and one row for every (R,THETA) pair. njZ vi}m~
% 'UxI-Lt
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike %#~wFW|]x
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), XqUQ{^;aI
% with delta(m,0) the Kronecker delta, is chosen so that the integral 0'.z|Jg=
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, .-mIU.Nwi
% and theta=0 to theta=2*pi) is unity. For the non-normalized mCk_c
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. |e+3d3T35
%
U#K4)(C
% The Zernike functions are an orthogonal basis on the unit circle. <H-kR\HF
% They are used in disciplines such as astronomy, optics, and DTM(SN8R+n
% optometry to describe functions on a circular domain. oYA"8ei =
%
89GW!
% The following table lists the first 15 Zernike functions. &!O?h/&X3
% 1#7|au%:)
% n m Zernike function Normalization WAR!#E#J7
% -------------------------------------------------- mAGD qz>f
% 0 0 1 1 X=Ar"Dx}}s
% 1 1 r * cos(theta) 2 pX*E(Q)@!
% 1 -1 r * sin(theta) 2 Q&w_kz.
% 2 -2 r^2 * cos(2*theta) sqrt(6) DEhR\Z!
% 2 0 (2*r^2 - 1) sqrt(3) %e0X-tXcmX
% 2 2 r^2 * sin(2*theta) sqrt(6) UR=s=G|
% 3 -3 r^3 * cos(3*theta) sqrt(8) ';8 ,RTe
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) :p@jslD
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) T,uF^%$@AQ
% 3 3 r^3 * sin(3*theta) sqrt(8) fp\mBei
% 4 -4 r^4 * cos(4*theta) sqrt(10) :AFU5mR4&
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) s-'~t#h
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) "DGap*=J
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 9+@z:j
% 4 4 r^4 * sin(4*theta) sqrt(10) &8Vh3QLEx
% -------------------------------------------------- }` H{;A
h
% C9MK3vtD.
% Example 1: !jU{ }RCR
% Bhx.q,X
% % Display the Zernike function Z(n=5,m=1) ohyq/u+y~A
% x = -1:0.01:1; ^>!&]@
% [X,Y] = meshgrid(x,x); vO~w~u5
% [theta,r] = cart2pol(X,Y); "nfi:A1
% idx = r<=1; \o2l;1~
% z = nan(size(X)); zA+0jhuG
% z(idx) = zernfun(5,1,r(idx),theta(idx)); lX2:8$?X
% figure &=M4Z/Ao
% pcolor(x,x,z), shading interp &Z!y>k%6
% axis square, colorbar mbX'*up
% title('Zernike function Z_5^1(r,\theta)') \),f?f-m
% dMsS OP0E
% Example 2: iHc(e(CB<
% K;rgLj0m
% % Display the first 10 Zernike functions >@cBDS<6R
% x = -1:0.01:1; p^q/u
% [X,Y] = meshgrid(x,x); }Rh%bf7,
% [theta,r] = cart2pol(X,Y); CMbID1M3
% idx = r<=1; st)v'ce,
% z = nan(size(X)); OgQ8yKfDB
% n = [0 1 1 2 2 2 3 3 3 3]; 6'e^np
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; -zJV(`
% Nplot = [4 10 12 16 18 20 22 24 26 28];
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% y = zernfun(n,m,r(idx),theta(idx)); m;rr7{7X
% figure('Units','normalized') C@]D*k
% for k = 1:10 B=%%3V)2
% z(idx) = y(:,k); [bX^_ Y
% subplot(4,7,Nplot(k)) <&+jl($"
% pcolor(x,x,z), shading interp B<-("P(q
% set(gca,'XTick',[],'YTick',[]) NT5##XOB
% axis square f_LXp$n
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) !t~tIJ>6
% end V9Mr&8{S4
% us1$
% See also ZERNPOL, ZERNFUN2. W-|CK&1
LD
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% Paul Fricker 11/13/2006 g{sp<w0
2^Im~p~ByE
4Y3@^8h&=
% Check and prepare the inputs: T95FoA
% ----------------------------- VB4V[jraCF
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) o$% KbfXO]
error('zernfun:NMvectors','N and M must be vectors.') hS &H*
end $0P16ZlPC
#
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if length(n)~=length(m) Tmu2G/yi
error('zernfun:NMlength','N and M must be the same length.') s 72yu}
end JBOU$A~
k'&1,78[l
n = n(:); =N\$$3m?
m = m(:); 3*j1v:x`
if any(mod(n-m,2)) ThW9=kzQW
error('zernfun:NMmultiplesof2', ... L>WxAeyu1K
'All N and M must differ by multiples of 2 (including 0).') Q"eqql<h#
end L8'4d'N+>
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if any(m>n) cp~6\F;c
error('zernfun:MlessthanN', ... l :u1P
'Each M must be less than or equal to its corresponding N.') $RF.LVc
end f>cUdEPBb
NM),2% <
if any( r>1 | r<0 ) :@E^oNKa0
error('zernfun:Rlessthan1','All R must be between 0 and 1.') :2NV;7Wke6
end %"
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+ `'wY?
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) | a
i#rU
error('zernfun:RTHvector','R and THETA must be vectors.') d!Y%7LmSE@
end 3d1xL+
Zm++5b`W/[
r = r(:); l& sEdEA
theta = theta(:); &"T7KXx
length_r = length(r); GyxLzrp
if length_r~=length(theta) OtQ]\:p7
error('zernfun:RTHlength', ... o>d0R
w4h
'The number of R- and THETA-values must be equal.') QKvaTy#
end %t1Z!xv_
Y:Lkh>S1Q
% Check normalization: ]w]BKpU=
% -------------------- H|j]uLZ
if nargin==5 && ischar(nflag) n4XkhY|
isnorm = strcmpi(nflag,'norm'); |pMP-
if ~isnorm P@5-3]m=
error('zernfun:normalization','Unrecognized normalization flag.') Y Kp@n8A
end G\k&sF
else 3^q9ll7Op
isnorm = false; .),9a,
end 'h~IbP
eW3?3l`fvt
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \7xc*v [
% Compute the Zernike Polynomials :U'n0\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% nDckT+eJ
XknNb{. r
% Determine the required powers of r: QL2LIs
% ----------------------------------- XPt>klf
m_abs = abs(m); }> C?Zx*
rpowers = []; D( TfW
for j = 1:length(n) efHCPj
rpowers = [rpowers m_abs(j):2:n(j)]; ,?%Y*?v
end MOB'rPIUI
rpowers = unique(rpowers); "?
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gr.G']9lNq
% Pre-compute the values of r raised to the required powers, rXTdhw?+
% and compile them in a matrix: tN.BI1nB
% ----------------------------- CJ)u#PmkJ
if rpowers(1)==0 l_+q a6C*
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); r,vSDHb`j
rpowern = cat(2,rpowern{:}); h.- o$+Sa
rpowern = [ones(length_r,1) rpowern]; }I`o%GL
else =R9`to|
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); e(DuJ-
rpowern = cat(2,rpowern{:}); /9P7;1?
end 7Ot&]M
?h#F& y
% Compute the values of the polynomials: Z~|%asjFE
% -------------------------------------- fG.6S"|M
y = zeros(length_r,length(n));
~Z#\f5yv@
for j = 1:length(n) SwrzW'%A
s = 0:(n(j)-m_abs(j))/2; _qt
pows = n(j):-2:m_abs(j); QT1oU P#*
for k = length(s):-1:1 q_>=| b
p = (1-2*mod(s(k),2))* ... 4m~p(r
prod(2:(n(j)-s(k)))/ ... 7(LB}
prod(2:s(k))/ ... cauKG@:2F
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... %/s+-j@s:
prod(2:((n(j)+m_abs(j))/2-s(k))); pg<cvok
idx = (pows(k)==rpowers); EF 8rh
y(:,j) = y(:,j) + p*rpowern(:,idx); 'Q*lp!2>
end ~_-+Q=3
4}YHg&@\d%
if isnorm 8N#.@\'kz.
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); jcxeXp|00
end poqNiOm4%
end sN1I+X
% END: Compute the Zernike Polynomials 2Aa
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% YQO9$g0%
~
*;T HD>
% Compute the Zernike functions: |hu9)0P
% ------------------------------ scd}{Y
idx_pos = m>0; =}SC .E\
idx_neg = m<0; LN'})CI8m
T^X um2Ec
z = y; JVPLE*T
if any(idx_pos) <2I<Z'B,e
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); g9=O<u#
end ~}K$z
if any(idx_neg) D r6u0rx8
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); P&Hhq>@Z
end 79'N/:.
a)/ }T
% EOF zernfun