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 ,nHz~Xi1t
function z = zernfun(n,m,r,theta,nflag) G)< k5U4
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. $S,Uoh
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N cK-!Evv
% and angular frequency M, evaluated at positions (R,THETA) on the ,{oP`4\Lm
% unit circle. N is a vector of positive integers (including 0), and (O`=$e
% M is a vector with the same number of elements as N. Each element u'32nf?
% k of M must be a positive integer, with possible values M(k) = -N(k) -3 W4
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, l}O`cC
% and THETA is a vector of angles. R and THETA must have the same i"e)LJz
% length. The output Z is a matrix with one column for every (N,M) `J-"S<c?_
% pair, and one row for every (R,THETA) pair. ]/$tt@h
% |LNXu
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike '6 /uc:zv
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), S0yPg9v
% with delta(m,0) the Kronecker delta, is chosen so that the integral t?0=;.D
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, YF:NRY[i
% and theta=0 to theta=2*pi) is unity. For the non-normalized fA3
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. zP9 HYS
% 6@I7UL >
% The Zernike functions are an orthogonal basis on the unit circle. uWfse19
% They are used in disciplines such as astronomy, optics, and -y/?w*Cx
% optometry to describe functions on a circular domain. |f>y"T+1
% Y7{|EI+@
% The following table lists the first 15 Zernike functions. sdO;vp^:b
% C*78ZwZ
% n m Zernike function Normalization yRgo1o w]
% -------------------------------------------------- Gf%o|kX]
% 0 0 1 1 v5 9>
% 1 1 r * cos(theta) 2 N%?o-IY
% 1 -1 r * sin(theta) 2 Ffhbs D
% 2 -2 r^2 * cos(2*theta) sqrt(6) S3J6P2P
% 2 0 (2*r^2 - 1) sqrt(3) jr9ZRHCU
% 2 2 r^2 * sin(2*theta) sqrt(6) +s S*EvF
% 3 -3 r^3 * cos(3*theta) sqrt(8) tNUcmiY
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 2i>xJMW
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) C$(t`G
% 3 3 r^3 * sin(3*theta) sqrt(8) F)%; gzs
% 4 -4 r^4 * cos(4*theta) sqrt(10) {T^'&W>8G8
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 9
/zz@
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) NeK:[Q@je
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) jkdNisq37
% 4 4 r^4 * sin(4*theta) sqrt(10) w{r->Phe
% -------------------------------------------------- Tbwq_3fK
% t|y4kM
% Example 1: .xk<7^ZD
% 7"[lWC!As5
% % Display the Zernike function Z(n=5,m=1) UwM}!K7)G
% x = -1:0.01:1; 9iOlR=-*
% [X,Y] = meshgrid(x,x); .(hb8 rCM
% [theta,r] = cart2pol(X,Y); 9M!_D?+P?
% idx = r<=1; Xt7'clr
% z = nan(size(X)); txgGL'
% z(idx) = zernfun(5,1,r(idx),theta(idx)); qB=pp!zQ
% figure b1&{%.3[
% pcolor(x,x,z), shading interp lZua"Ju
% axis square, colorbar cj8r-Vu/N
% title('Zernike function Z_5^1(r,\theta)') hZ#tB
% 5m bs0GL
% Example 2: YVaQ3o|!
% ^twv0>vEo
% % Display the first 10 Zernike functions $yc,D=*Isi
% x = -1:0.01:1; :^*V[77
% [X,Y] = meshgrid(x,x); @t2 Q5c
% [theta,r] = cart2pol(X,Y); o|}%pc3
% idx = r<=1; hKT:@l*
% z = nan(size(X)); 6X jUb
% n = [0 1 1 2 2 2 3 3 3 3]; +Va?wAnr
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; XnY}dsSO
% Nplot = [4 10 12 16 18 20 22 24 26 28]; 'w!gQ#De
% y = zernfun(n,m,r(idx),theta(idx)); i&30n#
% figure('Units','normalized') ^GAdl}
% for k = 1:10 SB'YV#--
% z(idx) = y(:,k); bOFLI#p&
% subplot(4,7,Nplot(k)) E*I]v
% pcolor(x,x,z), shading interp f|G7L5-
% set(gca,'XTick',[],'YTick',[]) 87Uv+((H
% axis square .;F+ QP0
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) \I'Zc]
% end X @Bpjg
% 7E5Dz7
% See also ZERNPOL, ZERNFUN2. b(yO
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% Paul Fricker 11/13/2006 ]T l\9we
"@?|Vv,vn
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% Check and prepare the inputs: bSR<d
% ----------------------------- b c4x"]!
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) (p?3#|^
error('zernfun:NMvectors','N and M must be vectors.') < t (Pw
end ~76.S
?xo,)``
if length(n)~=length(m) @r]s9~Lx9
error('zernfun:NMlength','N and M must be the same length.') yki
k4MeB
end 5muW*7
nMa^Eq#
n = n(:); vg.%. ~!9
m = m(:); M$W#Q\<*#r
if any(mod(n-m,2)) .fsk DW
error('zernfun:NMmultiplesof2', ... bZ9NnSuH
'All N and M must differ by multiples of 2 (including 0).') j>Z]J'P
end LA?\~rh!
\l:g{GnoT
if any(m>n) ThlJhTh<%4
error('zernfun:MlessthanN', ... ^'fKey`
'Each M must be less than or equal to its corresponding N.') u#M)i30j
end sBb.Y
k
+.lWck
if any( r>1 | r<0 ) 4ufLP DH
error('zernfun:Rlessthan1','All R must be between 0 and 1.') 9sCk\`n
end ?R]y}6P$
=.X?LWKY
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ^!<7#kX
error('zernfun:RTHvector','R and THETA must be vectors.') T"H)g
end $vLV<
y07
v|I5Gz$qpa
r = r(:); 3NN'E$"3
theta = theta(:); 2E2}|:
||&
length_r = length(r); y,&M\3A
if length_r~=length(theta) = b!J)]
error('zernfun:RTHlength', ... @,4%8E5
'The number of R- and THETA-values must be equal.') SO<m(o)G2
end kNj3!u$
<gdgcvd
% Check normalization: K~8tN,~&
% -------------------- dl6v
<
if nargin==5 && ischar(nflag) daIL> c"
isnorm = strcmpi(nflag,'norm'); 8KtgSash
if ~isnorm MgQU6O<
error('zernfun:normalization','Unrecognized normalization flag.') ewrWSffe
end MXF"F:-Kn
else b_jZL'en
isnorm = false; j 3MciQ`
end R5eB,FN
_`_IUuj$E
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 8q [c
% Compute the Zernike Polynomials 3rdfg
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% p$nK@t}
2-V)>98
% Determine the required powers of r: XLmMK{gs
% ----------------------------------- %Sn 6*\z
m_abs = abs(m); *fl{Y(_OO
rpowers = []; dA}
72D?
for j = 1:length(n) qX+gG",8
rpowers = [rpowers m_abs(j):2:n(j)]; ;:4P'FWm^
end v"r9|m~ '
rpowers = unique(rpowers); T]6c9_
[9O~$! <%
% Pre-compute the values of r raised to the required powers, ,![Du::1
% and compile them in a matrix: ,=Nw(GI
% -----------------------------
`cP'~OT
if rpowers(1)==0 C5+`<
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); nF<y7XkO
rpowern = cat(2,rpowern{:}); #t@x6Vt
rpowern = [ones(length_r,1) rpowern]; M7DLs;sD
else %A62xnX
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); S&3X~jD(1
rpowern = cat(2,rpowern{:}); &QTeGn
end V(2,\+ t
`(,*IK a
% Compute the values of the polynomials: ?7uKP}1|
% -------------------------------------- ~zxwg+:QO
y = zeros(length_r,length(n)); >&;>PZBPCO
for j = 1:length(n) H=&/ Q
s = 0:(n(j)-m_abs(j))/2; icPp8EwH
pows = n(j):-2:m_abs(j); `pi-zE)
for k = length(s):-1:1 aZj J]~bO
p = (1-2*mod(s(k),2))* ... ;tp]^iB#
prod(2:(n(j)-s(k)))/ ... QtY hg$K3
prod(2:s(k))/ ... 0\'Q&oTo
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 1BMB?I
prod(2:((n(j)+m_abs(j))/2-s(k))); -XVEV
idx = (pows(k)==rpowers); wb6 L?t
y(:,j) = y(:,j) + p*rpowern(:,idx); @VC .>
end ,\lYPx\P[
VW9>xVd4
if isnorm a|QE *s.
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); xHJ8?bD p
end .Iwur;/\
end :}@C9pqr2
% END: Compute the Zernike Polynomials dG\U)WA(p
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +Y>"/i.
N
QqiJun_m
% Compute the Zernike functions: 7m:|u*ij2~
% ------------------------------ u C,"5C
idx_pos = m>0; 0]T.Lh$3
idx_neg = m<0; uu}`warW
R:'Ou:Mh
z = y; +@emX$cFV
if any(idx_pos) 'tb(J3ZP
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); qoC]#M$oo#
end EBoGJ_l
if any(idx_neg) 8 ]q
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); H2qf'
end ~+O `9&
jR{-
% EOF zernfun