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 "_UnN}Uk
function z = zernfun(n,m,r,theta,nflag) iM8Cw/DS
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. {qw'gJmX
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N G
`|7NL
% and angular frequency M, evaluated at positions (R,THETA) on the --]blP7
% unit circle. N is a vector of positive integers (including 0), and gxO~44"
% M is a vector with the same number of elements as N. Each element {gzQ/|}#z-
% k of M must be a positive integer, with possible values M(k) = -N(k) XuP%/\
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, %i\rw*f
% and THETA is a vector of angles. R and THETA must have the same M
%,\2!$
% length. The output Z is a matrix with one column for every (N,M) jsAx;Z:QT
% pair, and one row for every (R,THETA) pair. e;vI XJE
% uYeb RCdR
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike "7sv@I_j
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), @|(cr: (=H
% with delta(m,0) the Kronecker delta, is chosen so that the integral qq!ZYWy2
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, c%5P|R~g]p
% and theta=0 to theta=2*pi) is unity. For the non-normalized R_j.k3r4d
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 7NJl+*u
% J;>;K6pW
% The Zernike functions are an orthogonal basis on the unit circle. rT R$\ [C
% They are used in disciplines such as astronomy, optics, and !y@\w
% optometry to describe functions on a circular domain. ;\th.!'rn
% 2}<tzDI'
% The following table lists the first 15 Zernike functions. F(1E@xs
% p@78Xmu?q
% n m Zernike function Normalization (g;O,`|c,
% -------------------------------------------------- $x }R2
% 0 0 1 1 3sV$#l P
% 1 1 r * cos(theta) 2 ox SSEs
% 1 -1 r * sin(theta) 2 iJOoO"Ai
% 2 -2 r^2 * cos(2*theta) sqrt(6) ;8#6da,
% 2 0 (2*r^2 - 1) sqrt(3) N]yT/8
% 2 2 r^2 * sin(2*theta) sqrt(6) Ju>QQOxi|
% 3 -3 r^3 * cos(3*theta) sqrt(8) 9(fh+
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) OR&pGoW
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) 8@vq.z}
% 3 3 r^3 * sin(3*theta) sqrt(8) (w-"1(
% 4 -4 r^4 * cos(4*theta) sqrt(10) kt
Z~r. +
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) to13&#o
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) !43nL[]
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ->qRGUW
% 4 4 r^4 * sin(4*theta) sqrt(10) SkV pZh
% -------------------------------------------------- ~V(>L=\V;
% hg12NzbK
% Example 1: Jb{g{a/
% VP< zOk7
% % Display the Zernike function Z(n=5,m=1) t[k ['<G
% x = -1:0.01:1; Sy?^+JdM/
% [X,Y] = meshgrid(x,x); pKXSJ"Xo
% [theta,r] = cart2pol(X,Y); 3T(ft^~
% idx = r<=1; >? o5AdZ
% z = nan(size(X)); X,@nD@
% z(idx) = zernfun(5,1,r(idx),theta(idx)); At>e4t2@
% figure &5jc
&CS
% pcolor(x,x,z), shading interp #}.{|'L
% axis square, colorbar .\H-?6R^
% title('Zernike function Z_5^1(r,\theta)') 8r}tf3xMCM
% &pl)E$Y
% Example 2: ]l }v
% L]=mQo
% % Display the first 10 Zernike functions ?p6@uM\Q7
% x = -1:0.01:1; MuO(%.H
% [X,Y] = meshgrid(x,x); B_#M)d
O
% [theta,r] = cart2pol(X,Y); y< gRl/e
% idx = r<=1; ^Zpz@T>m
% z = nan(size(X)); Up-^km
% n = [0 1 1 2 2 2 3 3 3 3]; D7R;IA-w
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; 1$))@K-I
% Nplot = [4 10 12 16 18 20 22 24 26 28]; rSDI.m
% y = zernfun(n,m,r(idx),theta(idx)); Dg^s$2
% figure('Units','normalized') zKk=R6w
% for k = 1:10 x15&U\U
% z(idx) = y(:,k); 1_&W1o
% subplot(4,7,Nplot(k)) q8_E_s-U,
% pcolor(x,x,z), shading interp /hg^hF
% set(gca,'XTick',[],'YTick',[]) _7v4S/V
% axis square `-s]dq
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 0(5qVJ12
% end G{pF! q
% xxGQXW
% See also ZERNPOL, ZERNFUN2. ='I2&I,)
g:^Hex?Yfd
% Paul Fricker 11/13/2006 E08!a
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r"L:Mu
% Check and prepare the inputs: rk+s[Qi~
% ----------------------------- q%s<y+
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) DbI)tDi5D
error('zernfun:NMvectors','N and M must be vectors.') G"J
8i|~
end =J-&usX
abVEi[nP
if length(n)~=length(m) 5[6{o$I
error('zernfun:NMlength','N and M must be the same length.') L$Xkx03lz>
end +IGSOWL
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n = n(:); <@>icDFEHn
m = m(:); 4\U"e*
if any(mod(n-m,2)) 22d>\u+c
error('zernfun:NMmultiplesof2', ... !y1qd
'All N and M must differ by multiples of 2 (including 0).') TD ;u"
end aE]RVyG@L
RXO}mu]Iu
if any(m>n) m2%
error('zernfun:MlessthanN', ... ZV/g_i#
'Each M must be less than or equal to its corresponding N.') Rs]Y/9F;{
end !9S!zRy@
{- &wV
if any( r>1 | r<0 ) LK|rLoia:
error('zernfun:Rlessthan1','All R must be between 0 and 1.') x;&iLQZh
end QF.M%she+
[{F;4>g
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) \!)1n[N
error('zernfun:RTHvector','R and THETA must be vectors.') i:R_g]
end hs)_h^P
gE&83i"
r = r(:); ,PWMl[X
theta = theta(:); P1qnU
length_r = length(r); #9(iu S+BU
if length_r~=length(theta) EzU=q
E
error('zernfun:RTHlength', ... R"`<ZY6(Ou
'The number of R- and THETA-values must be equal.') H"JzTo8u
end @oRo6Y<-
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% Check normalization: T ^`R
% -------------------- 4n\O6$&.x
if nargin==5 && ischar(nflag) )k 6z
isnorm = strcmpi(nflag,'norm'); bmRp)CYd
if ~isnorm eeUEqM$7EX
error('zernfun:normalization','Unrecognized normalization flag.') l5Q-M{w0x
end a[BIY&/Q
else ;3O=lo:$~
isnorm = false; .gwT?O,
end ibuoq X`
UDgUbi^v|D
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% .Nd_p{
% Compute the Zernike Polynomials QL@}hw.F
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% D89(u.h
UTxqqcqEny
% Determine the required powers of r: YLNJ4nE
% ----------------------------------- JW=P}h
m_abs = abs(m); Z&Z=24q_
rpowers = []; D7,{p2<2T
for j = 1:length(n) V%w]HIhq
rpowers = [rpowers m_abs(j):2:n(j)]; X|pOw,"
end \ci[<CP
rpowers = unique(rpowers); :&=`xAX-
{r[g.@
% Pre-compute the values of r raised to the required powers, -]Q6Ril
% and compile them in a matrix: >KCnmi
% ----------------------------- D]5cijO6
if rpowers(1)==0 `< cn
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 5cSqo{|En
rpowern = cat(2,rpowern{:}); j
!rQa^
rpowern = [ones(length_r,1) rpowern]; a.G;s2>
else "D/\&1.&
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); y[~w2a&+
rpowern = cat(2,rpowern{:}); {edjvPlk
end l 1Ns~
#s]` jdc
% Compute the values of the polynomials: ,wH]|`w
% -------------------------------------- Xp_G9I,+
y = zeros(length_r,length(n)); MN. $a9m
for j = 1:length(n) Jbqm?Fy4X
s = 0:(n(j)-m_abs(j))/2; [f@[gE
pows = n(j):-2:m_abs(j); #BwkbOgr
for k = length(s):-1:1 gK>aR ^*
p = (1-2*mod(s(k),2))* ... k|F TT
prod(2:(n(j)-s(k)))/ ... \~@a/J
prod(2:s(k))/ ... i| OG#PsY-
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... <(dg^;
prod(2:((n(j)+m_abs(j))/2-s(k))); QMWDII&t
idx = (pows(k)==rpowers); 0%GQXiy
y(:,j) = y(:,j) + p*rpowern(:,idx); u+,
end &0%x6vea
Y.v. EZ
if isnorm 9/I|oh_
G
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); zQyt 1&!
end +OX:T) 4h6
end ;UDd4@3`S"
% END: Compute the Zernike Polynomials u(g0Ob
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Ga#5xAI{a
_|vY)4B4U
% Compute the Zernike functions: Q\[2BJo/
% ------------------------------ 72{Ce7J4
idx_pos = m>0; gOKF%Ej31T
idx_neg = m<0; )l"py9STF
w>Y!5RnO
z = y; Jde@Th
if any(idx_pos) A1V^Gi@i
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); }2~$"L,_
end %cFqD
& 6
if any(idx_neg) q[MZSg
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); qA5PIEvdq
end sAfNu~d
Y>2kOE
% EOF zernfun