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 _If?&KJ r
function z = zernfun(n,m,r,theta,nflag) R!qrb26k
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. N+75wtLy&
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N +eXfT*=u5
% and angular frequency M, evaluated at positions (R,THETA) on the Acv{XnB
% unit circle. N is a vector of positive integers (including 0), and rv%[?Ml
% M is a vector with the same number of elements as N. Each element d~8~RT2m
% k of M must be a positive integer, with possible values M(k) = -N(k) ptQ(7N
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, (*BW/.Fq
% and THETA is a vector of angles. R and THETA must have the same -=IM8Dny
% length. The output Z is a matrix with one column for every (N,M) uJ\Nga<?
% pair, and one row for every (R,THETA) pair. XCriZ|s
% ~Xw?>&
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike Uroj%xN
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), #wiP{+%b
% with delta(m,0) the Kronecker delta, is chosen so that the integral r ngw6?`n-
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, -P&e4sV{
% and theta=0 to theta=2*pi) is unity. For the non-normalized IBh~(6
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. -rlX<(pl)
% ?Fpl.t~
% The Zernike functions are an orthogonal basis on the unit circle. 1?\ Y,+
% They are used in disciplines such as astronomy, optics, and 0&@pX~h:
% optometry to describe functions on a circular domain. Am
$L
%
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% The following table lists the first 15 Zernike functions. "M &4c:cz
% a6P.Zf7
% n m Zernike function Normalization fk1f'M)/8
% -------------------------------------------------- V p{5Kxq
% 0 0 1 1 Y cpO;md
% 1 1 r * cos(theta) 2 T%/w^27E
% 1 -1 r * sin(theta) 2 Q$j48,e
% 2 -2 r^2 * cos(2*theta) sqrt(6) tvRy8u;
% 2 0 (2*r^2 - 1) sqrt(3) 1bkUT_
% 2 2 r^2 * sin(2*theta) sqrt(6) hh&y2#Io
% 3 -3 r^3 * cos(3*theta) sqrt(8) pa-4|)qY
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 1+($"$ZC&B
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) edx'p`%d5
% 3 3 r^3 * sin(3*theta) sqrt(8) [^~9wFNtd
% 4 -4 r^4 * cos(4*theta) sqrt(10) y@_?3m7B=
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) RiG!TTa
b
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) w-Fk&dC69
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) A!yLwkc:5
% 4 4 r^4 * sin(4*theta) sqrt(10) 'bPo 5V|
% -------------------------------------------------- k)Wz b
% @x
+#ZD(
% Example 1: e~?]F0/
% G. TX1
% % Display the Zernike function Z(n=5,m=1) cU|jT8Q4H
% x = -1:0.01:1; #jiqRhm
% [X,Y] = meshgrid(x,x); #"-^;Z
% [theta,r] = cart2pol(X,Y); S
'+"+%^tj
% idx = r<=1; *'-^R9dN.S
% z = nan(size(X)); i{qU RP}.
% z(idx) = zernfun(5,1,r(idx),theta(idx)); G[j79o
% figure BxYA[#fd}
% pcolor(x,x,z), shading interp V}+;bbUc-
% axis square, colorbar krc!BK`V
% title('Zernike function Z_5^1(r,\theta)') Ypj)6d
% mC(t;{
% Example 2: b0 `9wn
% 7!wnx.
% % Display the first 10 Zernike functions k]pD3.QJ
% x = -1:0.01:1; x`i`]6q
% [X,Y] = meshgrid(x,x); XtdLKYET
% [theta,r] = cart2pol(X,Y); e8<nPt`C
% idx = r<=1; uf]$@6)
% z = nan(size(X)); ;tiUOixJ
% n = [0 1 1 2 2 2 3 3 3 3]; r0
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% m = [0 -1 1 -2 0 2 -3 -1 1 3]; f om"8iL1
% Nplot = [4 10 12 16 18 20 22 24 26 28]; >]8.xkQq
% y = zernfun(n,m,r(idx),theta(idx)); >irT|VTf
% figure('Units','normalized') 1G.gPx[
% for k = 1:10 tta0sJ8i
% z(idx) = y(:,k); Nn1^#kc
% subplot(4,7,Nplot(k)) DNBpIC5&6
% pcolor(x,x,z), shading interp I]1Hi?A2
% set(gca,'XTick',[],'YTick',[]) Gi4dgMVei
% axis square ,8nZzVo
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) @rE)xco
% end :=v{inN
% ?Zp!AV
% See also ZERNPOL, ZERNFUN2. @6'E8NFl
/,$\H
% Paul Fricker 11/13/2006 wQB{K3
?u!AHSr(
X>8?p'*
% Check and prepare the inputs: G>>u#>0
% ----------------------------- V_622~Tc/[
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) w1(06A}/
error('zernfun:NMvectors','N and M must be vectors.') }h h^U^ia
end _rd j,F8
}(EOQ2TI
if length(n)~=length(m) dU^<7 K:S
error('zernfun:NMlength','N and M must be the same length.') g_c)Ts(
end \&)W#8V
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n = n(:); |ZlT>u
m = m(:); YKOO(?lv
if any(mod(n-m,2)) ?$4R <
error('zernfun:NMmultiplesof2', ... .|`=mx
'All N and M must differ by multiples of 2 (including 0).') (ul-J4E\O
end qpqz. {\
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if any(m>n) 8'WMspX
error('zernfun:MlessthanN', ... q)xl$*g
'Each M must be less than or equal to its corresponding N.') ;Jn0e:x`E
end ^|i\d\
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if any( r>1 | r<0 ) TM}'XZ&
error('zernfun:Rlessthan1','All R must be between 0 and 1.') gLMea:
end fB,1s}3Hn
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) =C>`}%XT}
error('zernfun:RTHvector','R and THETA must be vectors.') EZumJ."
end 1(7.V-(G
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r = r(:); &W<>^C2v
theta = theta(:); 39aCwhh7v
length_r = length(r); Q>a7Ps@~
if length_r~=length(theta) nf.:5I.
error('zernfun:RTHlength', ... Y\Qxdq
'The number of R- and THETA-values must be equal.') 8w8I:*
end .>64h H
v&b.Q:h*'
% Check normalization: }-q`&1!t
% -------------------- VIYksv
if nargin==5 && ischar(nflag) }A)36
isnorm = strcmpi(nflag,'norm'); KD"&_PX
if ~isnorm ={E!8"
error('zernfun:normalization','Unrecognized normalization flag.') p@7i=hyt`p
end >yA,@%X
else oD#<?h)(
isnorm = false; u ?G\b{$m
end y.*=Ww+
%6IlE.*,
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ,*nZf|
% Compute the Zernike Polynomials ]%<Q:+38
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 1u"*09yZd
P
5qa:<
% Determine the required powers of r: x\J;ZiWwW
% -----------------------------------
M o"JV
m_abs = abs(m); x
!:9c<
rpowers = []; q
5v?`c
for j = 1:length(n) bxhg*A
rpowers = [rpowers m_abs(j):2:n(j)]; f*T)*R_
end B=gsd0^]
rpowers = unique(rpowers); NrJ_6sjF0g
)}Rfa}MD
% Pre-compute the values of r raised to the required powers, P7wqZ?
% and compile them in a matrix: wsJ%*
eYf
% ----------------------------- N;x<| %peL
if rpowers(1)==0 oWx_O-_._
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); WE.$a t{*h
rpowern = cat(2,rpowern{:}); .mT#%ex
rpowern = [ones(length_r,1) rpowern]; G_^iR-
else dm,}Nbc91(
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); JIP+ !2
rpowern = cat(2,rpowern{:}); j
FPU
zB"
end oGJ*Rn)Z
T}t E/
% Compute the values of the polynomials: =CKuiO.j
% -------------------------------------- #6N+5Yx_[
y = zeros(length_r,length(n)); {C/L5cZ]J
for j = 1:length(n) xMNNXPz(
s = 0:(n(j)-m_abs(j))/2; .L^pMU+!^
pows = n(j):-2:m_abs(j); YXX36
for k = length(s):-1:1 YA"Ti9-EV
p = (1-2*mod(s(k),2))* ... >d{dZD}
prod(2:(n(j)-s(k)))/ ... ws>WA{]gq
prod(2:s(k))/ ... b.R!2]T]i^
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... g=FDm*
prod(2:((n(j)+m_abs(j))/2-s(k))); +HOCVqx
idx = (pows(k)==rpowers); )+n,5W
y(:,j) = y(:,j) + p*rpowern(:,idx); qY$*#*Q
end hgweNRTh!
15xd~V?ai:
if isnorm Q%& _On
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 80?6I%UB<
end x)ddRq
l
end t;.^K\S4
% END: Compute the Zernike Polynomials RIy5ww}3|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {Ax)[<i
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% Compute the Zernike functions: :!JpP
R5
% ------------------------------ n#+%!HTh
idx_pos = m>0; qIbg
4uE
idx_neg = m<0; .3lGX`d{
[j)\v^m
z = y; {W5ydHXy
if any(idx_pos) W.,% 0cZ
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 1R=)17'O
end =tr1*s{
if any(idx_neg) `z|=~
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); bZNIxkc[Dh
end { OB-J\7Y
Em e'Gk
% EOF zernfun