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 .^eajb`:
function z = zernfun(n,m,r,theta,nflag) w~Aw?75t
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. tmKHT
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N eot%Th?[
% and angular frequency M, evaluated at positions (R,THETA) on the +JsMYv
% unit circle. N is a vector of positive integers (including 0), and +xp)la.
% M is a vector with the same number of elements as N. Each element 4S5U|n
% k of M must be a positive integer, with possible values M(k) = -N(k) 3VaL%+T$,
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, wt]onve}%
% and THETA is a vector of angles. R and THETA must have the same Zcjh
% length. The output Z is a matrix with one column for every (N,M) $i1$nc8
% pair, and one row for every (R,THETA) pair. T,r?% G{XE
% f}=>c|Do
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike .~u[rc|<
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), xa8;"Y~"bg
% with delta(m,0) the Kronecker delta, is chosen so that the integral |0OY>5
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, nvbzC tC
% and theta=0 to theta=2*pi) is unity. For the non-normalized ||D PIn]
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. \(_(pcl
% 5:|9pe)
% The Zernike functions are an orthogonal basis on the unit circle. ^p)#;$6b
% They are used in disciplines such as astronomy, optics, and 0RgE~x!hI
% optometry to describe functions on a circular domain. (1 (~r"4I
% z^vfha
% The following table lists the first 15 Zernike functions. .exBU1Yk@
% 8yk7d76Y
% n m Zernike function Normalization SSxp!E'
% -------------------------------------------------- P?p]sLrP
% 0 0 1 1 oR7[[H.4
% 1 1 r * cos(theta) 2 4M#i_.`z
% 1 -1 r * sin(theta) 2 ^hXm=r4ozR
% 2 -2 r^2 * cos(2*theta) sqrt(6) {]2^b )
% 2 0 (2*r^2 - 1) sqrt(3) auga`*
% 2 2 r^2 * sin(2*theta) sqrt(6) fV@[S
% 3 -3 r^3 * cos(3*theta) sqrt(8) @R%*; )*F
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) "AU.Eh"-1
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) .fbY2b([
% 3 3 r^3 * sin(3*theta) sqrt(8) LaO8)lqR
% 4 -4 r^4 * cos(4*theta) sqrt(10) .W^B(y(tA
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) yX4Vv{g
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) ^3[_4av
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) GF6 o
% 4 4 r^4 * sin(4*theta) sqrt(10) e8rZP(g&g
% -------------------------------------------------- 2TUV9Z
% C7ug\_,s
% Example 1: hs[x\:})/
% =WjHf8v;
% % Display the Zernike function Z(n=5,m=1) tPl 4'tW_
% x = -1:0.01:1; 1^LdYO?g'
% [X,Y] = meshgrid(x,x); jB8Q% {%
% [theta,r] = cart2pol(X,Y); 4XNheP;b
% idx = r<=1; `Jk0jj6Z
% z = nan(size(X)); /i3JP}
% z(idx) = zernfun(5,1,r(idx),theta(idx)); kL%ot<rt)w
% figure N]w_9p~=1
% pcolor(x,x,z), shading interp u Jqv@GFv
% axis square, colorbar +9w[/n ^,G
% title('Zernike function Z_5^1(r,\theta)') [EDX@Kdq)
% r5DRF4,7
% Example 2: l3sF/zkH
% 4d`YZNvZW/
% % Display the first 10 Zernike functions ID43s9
% x = -1:0.01:1; ~.aR=m\#
% [X,Y] = meshgrid(x,x); r|EN 5
% [theta,r] = cart2pol(X,Y); C<
9x\JY%
% idx = r<=1; G9f6'5 O
% z = nan(size(X)); HohCb4do
% n = [0 1 1 2 2 2 3 3 3 3]; @khFk.LBD
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; $AZYY\1
% Nplot = [4 10 12 16 18 20 22 24 26 28]; /?eVWCR
% y = zernfun(n,m,r(idx),theta(idx)); au{)5W4~
% figure('Units','normalized') 053bM)qW
% for k = 1:10 LH5Z@*0#
% z(idx) = y(:,k); (-gomn
% subplot(4,7,Nplot(k)) @|\9<S
% pcolor(x,x,z), shading interp ,X.[37
% set(gca,'XTick',[],'YTick',[]) 17 Ugz?
% axis square GGp.u@\r
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ; ~pgF_
% end C|V7ZL>W
% G&ck98
% See also ZERNPOL, ZERNFUN2. !;eE7xn &
ib=)N)l
% Paul Fricker 11/13/2006 Sc7 Ftb%
'z ?Hv
B*Tn@t W
% Check and prepare the inputs: KqK]R6>
% ----------------------------- PVI Oe}N
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) <tD,Uu{P
error('zernfun:NMvectors','N and M must be vectors.') <Ht"t]u*Bn
end ANhqS
i#'K7XM2
if length(n)~=length(m) KN}#8.'>3
error('zernfun:NMlength','N and M must be the same length.') .KrLvic
end ?()*"+N(ck
~/L:$
n = n(:); T#ls2UL*xh
m = m(:); CD&a_-'z$K
if any(mod(n-m,2)) )ros-dp`
error('zernfun:NMmultiplesof2', ... wW%b~JX
'All N and M must differ by multiples of 2 (including 0).') \D@j`o
end G"/;Cq=t
Z>g72I%X
if any(m>n) dla_uXtM6
error('zernfun:MlessthanN', ... //&3{B
'Each M must be less than or equal to its corresponding N.') s ~Eo]e
end pr<u
5
B
~v6_x
if any( r>1 | r<0 ) b7sfr!t_d
error('zernfun:Rlessthan1','All R must be between 0 and 1.') %r^tZ ;;l
end hi(b\ABx
9C7Npf?~M
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) =F\Xt "
error('zernfun:RTHvector','R and THETA must be vectors.') >g$iO`2
end RvR.t"8
I bD
u+~)
r = r(:); <-1:o*8:}
theta = theta(:); -53c0g@X
length_r = length(r); 0Z2XVq~T$
if length_r~=length(theta) ]WMzWt:L
error('zernfun:RTHlength', ... }XUL\6 U
'The number of R- and THETA-values must be equal.') #x.v)S
end X!|eRA~o
&<><4MQ
% Check normalization: >a975R*g
% -------------------- )xVf3l
pQ
if nargin==5 && ischar(nflag) ! VT$U6
isnorm = strcmpi(nflag,'norm'); 4rDVCXE
if ~isnorm u.A}&'H
error('zernfun:normalization','Unrecognized normalization flag.') 1L`V{\_0s
end (c0L@8L
else 29=ob("
isnorm = false; B*:I-5
end /SJ><
jwjLxt
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% )&E]
% Compute the Zernike Polynomials m=n79]b:N
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% </'n={+q
O)W+rmToI
% Determine the required powers of r: vw>(JCR
% ----------------------------------- ~0+<-T
m_abs = abs(m); .(/HU Qn
rpowers = []; rV\G/)xL
for j = 1:length(n) i%!<9D~n
rpowers = [rpowers m_abs(j):2:n(j)]; T}{zh
end 'C}ku>B_r
rpowers = unique(rpowers); [*u\ S
l1kHFeq
% Pre-compute the values of r raised to the required powers, 8VG}-
% and compile them in a matrix: 1?w=v|b:P)
% ----------------------------- 6Br^Ugy
if rpowers(1)==0 <V)z{uK
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 3f$n8>mq
rpowern = cat(2,rpowern{:}); @H$8;CRM
rpowern = [ones(length_r,1) rpowern]; /pkN=OBR
else CT_tJ
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); F3vywN1$,
rpowern = cat(2,rpowern{:}); J|hVD
end OYxYlUq
oQpGa>6U&
% Compute the values of the polynomials: @r[SqGa:
% -------------------------------------- @"h4S*U
y = zeros(length_r,length(n)); `%~}p7Zu
for j = 1:length(n) t$,G%micj
s = 0:(n(j)-m_abs(j))/2; }|/A &c
pows = n(j):-2:m_abs(j); 2%fzRXhu%
for k = length(s):-1:1 F5+FO^3E
p = (1-2*mod(s(k),2))* ... \IC^z
prod(2:(n(j)-s(k)))/ ... g]JJ!$*1
prod(2:s(k))/ ... &?Erkc~#
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 9i 9
,X^=
prod(2:((n(j)+m_abs(j))/2-s(k))); ud(0}[
idx = (pows(k)==rpowers); jP/Vqe%%8
y(:,j) = y(:,j) + p*rpowern(:,idx); [?:MIl#!
end 8a@k6OZ
Jlb{1B$7
if isnorm 'bLP#TAzf
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); >`&2]Wc)
end
e(0cz6
end M `q|GY
% END: Compute the Zernike Polynomials fsK=]~<g
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 'MX|=K!C
^S;{;c+'
% Compute the Zernike functions: OAiW8BAe
% ------------------------------ Q5dqn"?
idx_pos = m>0; A{-S )Z3}
idx_neg = m<0; Iv3yDL;
7neJV
z = y; &R.5t/x_
if any(idx_pos) ( sl{Rgxe*
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); urkuG4cY
end (||qFu9a
if any(idx_neg) - |DWPU!"
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); FsO-xG"@"
end vOCaru?~h
SX'NFdY
% EOF zernfun