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 cPF<D$B
function z = zernfun(n,m,r,theta,nflag) C2F0tr|
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 5z/Er".P
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N m:CTPzAt
% and angular frequency M, evaluated at positions (R,THETA) on the .p6+l!"
% unit circle. N is a vector of positive integers (including 0), and 0Bolv_e
% M is a vector with the same number of elements as N. Each element 3smM,fi
% k of M must be a positive integer, with possible values M(k) = -N(k) 9@VO+E$7L
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, +C{p%`<
% and THETA is a vector of angles. R and THETA must have the same 6LUC!Sh
% length. The output Z is a matrix with one column for every (N,M) `sHuM*
% pair, and one row for every (R,THETA) pair. I4_d[O9
% LLAa1Wq
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike t-e5ld~a
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), =[tSd)D,y
% with delta(m,0) the Kronecker delta, is chosen so that the integral c|~6Ie
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, @e2}BhB2
% and theta=0 to theta=2*pi) is unity. For the non-normalized viaJblYj(f
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. udqS'g&
% @9G- m(?*
% The Zernike functions are an orthogonal basis on the unit circle. e;95a
% They are used in disciplines such as astronomy, optics, and Xa9TS"
% optometry to describe functions on a circular domain. $0Yh!L ?\
% omX?Bl
% The following table lists the first 15 Zernike functions. |QZ58)>
% >v5k{Cbp0
% n m Zernike function Normalization u:gtOjk2
% -------------------------------------------------- fZWGn6$
% 0 0 1 1 5i So8*9}
% 1 1 r * cos(theta) 2 A2H4k|8
% 1 -1 r * sin(theta) 2 F@<0s&)1
% 2 -2 r^2 * cos(2*theta) sqrt(6) gPC@Yy
% 2 0 (2*r^2 - 1) sqrt(3) ~%y @Xsot>
% 2 2 r^2 * sin(2*theta) sqrt(6) ]dPZ .r
% 3 -3 r^3 * cos(3*theta) sqrt(8) *JCQu0
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) .V'V:;BE%
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) sKaE-sbJY
% 3 3 r^3 * sin(3*theta) sqrt(8) s4= "kT]
% 4 -4 r^4 * cos(4*theta) sqrt(10) ,w)p"[^b
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ~|+zJ5
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) rnm03 '{
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) MQ/
A]EeL
% 4 4 r^4 * sin(4*theta) sqrt(10) Q[ieaL6&
% -------------------------------------------------- v Y|!
% &~DTZgY
% Example 1: HRa@
% ]rBM5~
% % Display the Zernike function Z(n=5,m=1) ><?BqRm+
% x = -1:0.01:1; Jc*XXu)
% [X,Y] = meshgrid(x,x); CZ{k@z`r
% [theta,r] = cart2pol(X,Y); Q}AE.Ef@<
% idx = r<=1; h rN%
% z = nan(size(X)); w=b(X
q+:
% z(idx) = zernfun(5,1,r(idx),theta(idx)); 2h^WYpCm
% figure ,t$,idcT+
% pcolor(x,x,z), shading interp JN3cg
% axis square, colorbar 5ua?I9fY
% title('Zernike function Z_5^1(r,\theta)') b
B
% *e"a0
% Example 2: {==pZpyyh
% "E!mva*NU
% % Display the first 10 Zernike functions Tp%(I"H'_;
% x = -1:0.01:1; =H]F`[B=
% [X,Y] = meshgrid(x,x);
:S
%lv
% [theta,r] = cart2pol(X,Y); 1qdZc_x
% idx = r<=1; D #2yIec
% z = nan(size(X)); \&xl{64
% n = [0 1 1 2 2 2 3 3 3 3]; PFSLyV*
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; %Q|eiXD
% Nplot = [4 10 12 16 18 20 22 24 26 28]; /L=(^k=a.;
% y = zernfun(n,m,r(idx),theta(idx)); (il0M=M
% figure('Units','normalized') *tQk;'/A]
% for k = 1:10 p
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% z(idx) = y(:,k); E;\M1(\u
% subplot(4,7,Nplot(k)) 7()?C}Ni-
% pcolor(x,x,z), shading interp j#A%q"]8
% set(gca,'XTick',[],'YTick',[]) 5CYo7mJ6+
% axis square N"5fmY<
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) / l>.mK()
% end j}HFs0<L
% 8pZ<9t'
% See also ZERNPOL, ZERNFUN2. Y0uvT7+[hi
d 4{FDqto
% Paul Fricker 11/13/2006 | FM
}
#} ,x @]p
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% Check and prepare the inputs: mS=r(3#
% ----------------------------- - Xupq/[,
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) %FkLQ+v/<
error('zernfun:NMvectors','N and M must be vectors.') w:=V@-S8
end F}?<v8#z0
NC23Z0y
if length(n)~=length(m) +JdZPb
error('zernfun:NMlength','N and M must be the same length.') T3J'fjY
end #XIc
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O_,O,1
n = n(:); GY!C|7kN
m = m(:); P~$<X
if any(mod(n-m,2)) V-W'RunnW
error('zernfun:NMmultiplesof2', ... t=wXTK5"
'All N and M must differ by multiples of 2 (including 0).') nL`9l1
end -$8.3\6h
bi[7!VQf
if any(m>n) uGtV}-t:
error('zernfun:MlessthanN', ... I+?hG6NM
'Each M must be less than or equal to its corresponding N.') _]>JB0IY
end C*~aSl7
%IZ)3x3l
if any( r>1 | r<0 ) '?Bg;Z'L %
error('zernfun:Rlessthan1','All R must be between 0 and 1.') 1JS2SxF
end TR vZ
`^F: -
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) @s*,xHE
error('zernfun:RTHvector','R and THETA must be vectors.') E)p9eU[#
end $^% N U
ETw]!
br
r = r(:); $xW**&
theta = theta(:); o
\L!(hm
length_r = length(r); 0irr7Y
if length_r~=length(theta) S q@H
error('zernfun:RTHlength', ... b Y8GA
'The number of R- and THETA-values must be equal.') -$k>F#
end XX; 6 P
jZ9[=?
% Check normalization: gT52G?-
% -------------------- dSK0h(8
if nargin==5 && ischar(nflag) f?UzD#50D
isnorm = strcmpi(nflag,'norm'); Di(9]:+
if ~isnorm 440FhDMj
error('zernfun:normalization','Unrecognized normalization flag.') 7!4V>O8@
end #a~"K|'G
else pa/9F[
isnorm = false; b)}+>Wx
end Lk,+Tfk"
b5`KB75sbo
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% v548ysE)
% Compute the Zernike Polynomials Zr/r2
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% C8b''9t.
C7"HQQ
% Determine the required powers of r: .Ao0;:;(2-
% ----------------------------------- !vqC+o>@
m_abs = abs(m); LsTffIP
rpowers = []; s@@1
*VQ
for j = 1:length(n) Eu<r$6Q0}o
rpowers = [rpowers m_abs(j):2:n(j)]; Bq}x9C&<
end F+aQ $pQ
rpowers = unique(rpowers); wyQb5n2`;~
K&`Awv
% Pre-compute the values of r raised to the required powers,
ZXXiL#^
% and compile them in a matrix: &d^=siL
% ----------------------------- ?S`>>^
if rpowers(1)==0 \HSicV#i
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Ol+Kp!ocY
rpowern = cat(2,rpowern{:}); DdjCn`jqlf
rpowern = [ones(length_r,1) rpowern]; uH{'gd,q8
else 3)E(RyQA3
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); F@SG((`
rpowern = cat(2,rpowern{:}); ,x#ztdvr
end zB)%lb
Lo`F
% Compute the values of the polynomials: \Ow,CUd
% -------------------------------------- (cV
y = zeros(length_r,length(n)); v*TeTA
%
for j = 1:length(n) zy)i1d
s = 0:(n(j)-m_abs(j))/2; ejcwg*i
pows = n(j):-2:m_abs(j); \r-N(;m
for k = length(s):-1:1 7'j9rmTXs
p = (1-2*mod(s(k),2))* ... SGf9U^ds
prod(2:(n(j)-s(k)))/ ... 4XG]z_+I
prod(2:s(k))/ ... #x)}29%e#
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Jt=>-Spj
prod(2:((n(j)+m_abs(j))/2-s(k))); UxqWnHH.`
idx = (pows(k)==rpowers); $WaZ_kt
y(:,j) = y(:,j) + p*rpowern(:,idx); n<R \w''x
end Yn<)k_kp
a@W7<9fY;
if isnorm .E<Dz
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Uf2:gLrF
end G11cNr>*
end Q_}n%P:u
% END: Compute the Zernike Polynomials K2|7%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \y~)jq:d"
'lQYJ0
% Compute the Zernike functions: [x_s/"Md;
% ------------------------------ *zQOJsg"e
idx_pos = m>0; ,)$Wm-
idx_neg = m<0; Mq+<mX7
BjZ>hhs!*
z = y; %$9:e
J?
if any(idx_pos) otnV-7)@
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)');
`ue?Z%p|
end ~CFMIQ et
if any(idx_neg) 1n3$V:00
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Xp^$
E6YFy
end [=~!w_
!R{em4 8D
% EOF zernfun