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 mKT>,M
function z = zernfun(n,m,r,theta,nflag) A<\JQ
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. Hg9CZMko
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N vsL[*OeI
% and angular frequency M, evaluated at positions (R,THETA) on the tX!nsm1
% unit circle. N is a vector of positive integers (including 0), and EwS!]h?
% M is a vector with the same number of elements as N. Each element ~+<olss_
% k of M must be a positive integer, with possible values M(k) = -N(k) @:tj<\G]
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, y7S4d~&
% and THETA is a vector of angles. R and THETA must have the same .XkMk|t8
% length. The output Z is a matrix with one column for every (N,M) % aUsOB-RV
% pair, and one row for every (R,THETA) pair. k<RZKw Qc
% j
F-v%?
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike -k(CJ5H9
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), Cda!Mk:
% with delta(m,0) the Kronecker delta, is chosen so that the integral .[u>V
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, |v[ Rp=?]
% and theta=0 to theta=2*pi) is unity. For the non-normalized bu&t'?zx!
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. pq:7F
% -dv%H{
% The Zernike functions are an orthogonal basis on the unit circle. ;f,c't@w
% They are used in disciplines such as astronomy, optics, and _5MNMVLwW
% optometry to describe functions on a circular domain. #{9G sD
% "lNzGi-H
% The following table lists the first 15 Zernike functions. 5'w^@Rs5
% QQe;1O
% n m Zernike function Normalization
`VQb-V
% -------------------------------------------------- 9'x)M?{8
% 0 0 1 1 )2DQ>cm
% 1 1 r * cos(theta) 2 \@}#Gez
% 1 -1 r * sin(theta) 2 CSV;+,Vv
% 2 -2 r^2 * cos(2*theta) sqrt(6) p"*y58
% 2 0 (2*r^2 - 1) sqrt(3) @<M*qK1h
% 2 2 r^2 * sin(2*theta) sqrt(6) Qp2I[Ioz3
% 3 -3 r^3 * cos(3*theta) sqrt(8) NNG}M(/V
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) ?EU\}N J
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) * MM[u75
% 3 3 r^3 * sin(3*theta) sqrt(8) y<XlRTy[}
% 4 -4 r^4 * cos(4*theta) sqrt(10) 24Z]%+b*E
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) {FN;'Uc
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) V@d)?T
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) IMnP[WA!
% 4 4 r^4 * sin(4*theta) sqrt(10) /D_+{dtE
% -------------------------------------------------- 1! p/6
% Wk^RA_
% Example 1: ^MD;"A<
% n:U>Fj>q
% % Display the Zernike function Z(n=5,m=1) w(1Gi$Z(Q)
% x = -1:0.01:1; bXYA5wG
% [X,Y] = meshgrid(x,x); E3a_8@ZB7
% [theta,r] = cart2pol(X,Y); .bf<<+'o
% idx = r<=1; Gjz[1d
% z = nan(size(X)); P6Bl
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% z(idx) = zernfun(5,1,r(idx),theta(idx)); qQ7w&9r.M
% figure j%<}jw[2
% pcolor(x,x,z), shading interp )a=/8ofe
% axis square, colorbar bg?"ILpk
% title('Zernike function Z_5^1(r,\theta)') xx*2?i
% BO.dz06(Rw
% Example 2: _SZ5P>GIU
% ]WJfgN4
% % Display the first 10 Zernike functions /?"8-0d
% x = -1:0.01:1; lH|LdlX
% [X,Y] = meshgrid(x,x); OMihXt[
% [theta,r] = cart2pol(X,Y); RV-h IdAU
% idx = r<=1; Fk^3a'/4KJ
% z = nan(size(X)); 8_uzpeRhJc
% n = [0 1 1 2 2 2 3 3 3 3]; 1 7hTr
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; \'19BAm'
% Nplot = [4 10 12 16 18 20 22 24 26 28]; *.f2VQ~H
% y = zernfun(n,m,r(idx),theta(idx)); 5;)*T6Y
% figure('Units','normalized') LT+3q%W.UC
% for k = 1:10 G>T')A
% z(idx) = y(:,k); %K 4
% subplot(4,7,Nplot(k)) oJ*1>7[ J
% pcolor(x,x,z), shading interp (#(Or
% set(gca,'XTick',[],'YTick',[]) TrE3S'EU#R
% axis square _-cK{
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ,D80/2U^
% end mnF}S5[9
% v4*rPGv
% See also ZERNPOL, ZERNFUN2. 3Rl,GWK
q]4pEip
% Paul Fricker 11/13/2006 myQ&%M
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Z&!!]"I
% Check and prepare the inputs: =G-N`
39
% ----------------------------- FE5Q?*Ea
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) FQeYx-7
error('zernfun:NMvectors','N and M must be vectors.') F=@i6ERi
end j!#OG
>tRHNB_
if length(n)~=length(m) `T!#@&+
error('zernfun:NMlength','N and M must be the same length.') x.DzViP/
end !ZtSbOC '
96|[}:+$&:
n = n(:); +6W(z3($
m = m(:); Ruh)^g
if any(mod(n-m,2)) p{;i& HNdp
error('zernfun:NMmultiplesof2', ... |qjZ38;6
'All N and M must differ by multiples of 2 (including 0).') K <`>O,
F
end 0.(<'!"y
eS!C3xC;J]
if any(m>n) 'u[%}S38
error('zernfun:MlessthanN', ... KI&:9j+M)
'Each M must be less than or equal to its corresponding N.') PjqeE,5
end }HZ{(?
HD # r0)
if any( r>1 | r<0 ) 2P~)I)3V
error('zernfun:Rlessthan1','All R must be between 0 and 1.') hCc0sRp
end |w)5;uQ&\
k&s; {|!
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) -6E K#!+
error('zernfun:RTHvector','R and THETA must be vectors.') [ x>
end $Tl<V/
}Zl"9A#K
r = r(:); oh}^?p
theta = theta(:); ]EL\)xCr
length_r = length(r); v|+5:jFOqb
if length_r~=length(theta) ZCiY,;c
error('zernfun:RTHlength', ... $iMC/Kym
'The number of R- and THETA-values must be equal.') o)]FtL:mm
end WfVMdwz=
Y)p4]>lT+8
% Check normalization: r+gjc?Ol
% -------------------- Lar r}o=
if nargin==5 && ischar(nflag) hLuJWjCV
isnorm = strcmpi(nflag,'norm'); (r F?If
if ~isnorm emWGIo
error('zernfun:normalization','Unrecognized normalization flag.') !EFBI+?&
end M9"Sgb`g
else ;L6Xs_L~
isnorm = false;
-0|K,k
end v}`1)BUeF
oX|?:MS:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0\ f-z6
% Compute the Zernike Polynomials 8M93cyX
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vl5){@
t .=Oj
% Determine the required powers of r: 1X@b?6
% ----------------------------------- YN#XmX%
m_abs = abs(m); ZgF/;8!~V-
rpowers = []; BlaJl[P iv
for j = 1:length(n) k^*$^;z
rpowers = [rpowers m_abs(j):2:n(j)]; YBylyVZ
end ,ep9V,+|
rpowers = unique(rpowers); _t.FL@3e
A'g,:8Ou
% Pre-compute the values of r raised to the required powers, w6U
@tW
% and compile them in a matrix: R+Lk~X^*l'
% ----------------------------- 0zV 4`y
if rpowers(1)==0 plku-O;]
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); tp+=0k2i
rpowern = cat(2,rpowern{:}); HDj$"pS
rpowern = [ones(length_r,1) rpowern]; $c9=mjwH
else l\aUresm
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); FfXZ|o$;
rpowern = cat(2,rpowern{:}); oc;VIK)g]c
end 1ZNNsB
,9vJtP+T+!
% Compute the values of the polynomials: vf_OQ4'G,
% -------------------------------------- k`@w(HhS
y = zeros(length_r,length(n)); 4WG=m}X
for j = 1:length(n) B(Y.`L? %E
s = 0:(n(j)-m_abs(j))/2; h#O"Q+J9n
pows = n(j):-2:m_abs(j); QK7e|M
for k = length(s):-1:1 msG3~@q
p = (1-2*mod(s(k),2))* ... |8'B/
p=
prod(2:(n(j)-s(k)))/ ... ~,Mr0
prod(2:s(k))/ ... 8r^j P.V
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... -mC:r&Y>[
prod(2:((n(j)+m_abs(j))/2-s(k))); K P6PQgc
idx = (pows(k)==rpowers); "OJr*B
y(:,j) = y(:,j) + p*rpowern(:,idx); `vX4!@Tw
end cuMc*i$w!
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if isnorm :p$EiR
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); TK %<a/
end &%:*\_2s
end -fQX4'3R
% END: Compute the Zernike Polynomials 3.~h6r5-
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% x
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N1s.3`
% Compute the Zernike functions: #'iPDRYy
% ------------------------------ c.-cpFk^L&
idx_pos = m>0; oB}K[3uB:t
idx_neg = m<0; '2xcce#
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z = y; /pU|ZA.z'2
if any(idx_pos) kU(kU2u%9
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 26}u4W$
end :@;6
if any(idx_neg) AtT"RG-6
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 59~FpjJ
end 6~3jn+K$1
$>(9~Yh0
% EOF zernfun