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 lNtxM"G&
function z = zernfun(n,m,r,theta,nflag) \okv}x^L=Z
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. \NEk B&^n
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N c h((u(G
% and angular frequency M, evaluated at positions (R,THETA) on the X%kJ3{
% unit circle. N is a vector of positive integers (including 0), and UUb0[oy
% M is a vector with the same number of elements as N. Each element m^3j|'mG
% k of M must be a positive integer, with possible values M(k) = -N(k) X.[bgvm~C
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, AE~@F4MK
% and THETA is a vector of angles. R and THETA must have the same 56.JBBZZ
% length. The output Z is a matrix with one column for every (N,M) B3u/
y
% pair, and one row for every (R,THETA) pair. dNF_T?E\
% q-uzu !
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike nZ (wfNk
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), lSO$Q]!9
% with delta(m,0) the Kronecker delta, is chosen so that the integral w-xigm>{Z
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, f?ibyoXL
% and theta=0 to theta=2*pi) is unity. For the non-normalized kE8s])Z,+
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. \i@R5v=zL
% 3}&3{kt
% The Zernike functions are an orthogonal basis on the unit circle. VmN 7a6a
% They are used in disciplines such as astronomy, optics, and "PO8 Q
% optometry to describe functions on a circular domain. D6+3f#k6
% yNn=r;FZQ
% The following table lists the first 15 Zernike functions. _nEVmz!zg
% }Nwp{["}]L
% n m Zernike function Normalization O>a1S*mxP
% -------------------------------------------------- 3S2Alx!6
% 0 0 1 1 jYFmL_{
% 1 1 r * cos(theta) 2 !MOsP<2
% 1 -1 r * sin(theta) 2 96QY0
% 2 -2 r^2 * cos(2*theta) sqrt(6) _)!*,\*`{
% 2 0 (2*r^2 - 1) sqrt(3) Dj'?12Onu=
% 2 2 r^2 * sin(2*theta) sqrt(6) &}7R\co3
% 3 -3 r^3 * cos(3*theta) sqrt(8) 0GeL">v,:=
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) VBF:MAA
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) fjl9*
% 3 3 r^3 * sin(3*theta) sqrt(8) ->.9[|lIg
% 4 -4 r^4 * cos(4*theta) sqrt(10) #N>66!/V
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) o$Nhx_F
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) W6i9mER-
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) g1"ZpD
% 4 4 r^4 * sin(4*theta) sqrt(10) d|7LCW+HW
% -------------------------------------------------- Q^nfD
% i8-Y,&>V
% Example 1: e@TwZ6l
% 9+s&|XS*
% % Display the Zernike function Z(n=5,m=1) &z:bZH]DH
% x = -1:0.01:1; 8F`8=L NO
% [X,Y] = meshgrid(x,x); `BG>%#
% [theta,r] = cart2pol(X,Y); X;GU#8W
% idx = r<=1; 2;s[ m3
% z = nan(size(X)); JS%LJ_J
% z(idx) = zernfun(5,1,r(idx),theta(idx)); HiU)q
% figure uL1lB@G@
% pcolor(x,x,z), shading interp Zl3e=sg=
% axis square, colorbar CM++:Y vJ
% title('Zernike function Z_5^1(r,\theta)') t&q~ya/C
% oVn&L*H
% Example 2: PsXCpyY!s
% LD5`9-
% % Display the first 10 Zernike functions lN,a+S/'
% x = -1:0.01:1; ~wv$uL8y
% [X,Y] = meshgrid(x,x); q{f\_2[
% [theta,r] = cart2pol(X,Y); F`x_W;\
% idx = r<=1; /_{ZWLi(
% z = nan(size(X)); !bYVLFp=\_
% n = [0 1 1 2 2 2 3 3 3 3]; tp7$t#
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; tcv(<0
% Nplot = [4 10 12 16 18 20 22 24 26 28]; ckY#oRQ1
% y = zernfun(n,m,r(idx),theta(idx)); B>!mD{N
% figure('Units','normalized') aEIz,^3
% for k = 1:10 $`/UG0rdC
% z(idx) = y(:,k); ZCc23UwI
% subplot(4,7,Nplot(k)) tUc<ExvP,
% pcolor(x,x,z), shading interp *PL&CDu=)
% set(gca,'XTick',[],'YTick',[]) 4* >j:1
% axis square {4Kvr4)4
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) uyB 2
% end &,jUaC5I
% OQKg/1
% See also ZERNPOL, ZERNFUN2. 37a1O>A
qmFbq<&
% Paul Fricker 11/13/2006 2-8Dc4H]r
GF%/q :9
~//E'V-
% Check and prepare the inputs: 4}/gV)
% ----------------------------- ppvlU H5;
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ly[dV.<P
error('zernfun:NMvectors','N and M must be vectors.') :dULsl$Nz
end NFEr ,n
jmaw-Rx
if length(n)~=length(m) UhS:tT]7
error('zernfun:NMlength','N and M must be the same length.') K&NH?
end 0LL0\ly]
: q%1Vi
n = n(:); 0q-lyVZ^X
m = m(:); }k%6X@
if any(mod(n-m,2)) ^IuhHP
error('zernfun:NMmultiplesof2', ... FP=-
jf/
'All N and M must differ by multiples of 2 (including 0).') xlwf @XW
end ZZo<0kDk
"D_:`@V(
if any(m>n) PLs`Ci|`
error('zernfun:MlessthanN', ... `Tyd1!~
'Each M must be less than or equal to its corresponding N.') 1Xm>nF~
end ROQ]sQpk
j;_
if any( r>1 | r<0 ) +z?gf*G_W'
error('zernfun:Rlessthan1','All R must be between 0 and 1.') U~7udUR
end ?VE'!DW
A~a 3bCX+"
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) P*
0kz@
error('zernfun:RTHvector','R and THETA must be vectors.') [y'blCb
end zE$HHY2ovi
,v8e7T
r = r(:); H<i!C|AF
theta = theta(:); ZJ)Z
length_r = length(r); 2 >O [Y1
if length_r~=length(theta) 8Z\q)T
error('zernfun:RTHlength', ... [iq^'E
'The number of R- and THETA-values must be equal.') eQ/w
Mr
end CA`V)XIsP
zc)nDyn
% Check normalization: zytN leyc
% -------------------- IP#vfM
if nargin==5 && ischar(nflag) ]YhQQH1>]
isnorm = strcmpi(nflag,'norm'); vJ'22)n
if ~isnorm kGAgXtE
error('zernfun:normalization','Unrecognized normalization flag.') :K2
X~Ty
end 0O`Rh"O
else T2w4D!
isnorm = false; ff.k1%wr^
end Q34u>VkdQI
!vu-`u~86
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% j`.&4.7+
% Compute the Zernike Polynomials g*oX`K.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% qF bj~ec
dNt^lx
% Determine the required powers of r: uVU)LOx
% ----------------------------------- hfY/)-60o
m_abs = abs(m); \os"w "
rpowers = []; r7R'beiH
for j = 1:length(n) 4_QfM}Fyp
rpowers = [rpowers m_abs(j):2:n(j)]; /fT"WaTEK
end SQK82/
rpowers = unique(rpowers); #*CMf.OCh
O8\f]!O(
% Pre-compute the values of r raised to the required powers, &&C70+_po
% and compile them in a matrix: Q}B]b-c+E
% ----------------------------- 8h=m()Eu
if rpowers(1)==0 hizM}d-"C
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); )GG9[%H!
rpowern = cat(2,rpowern{:}); N80ogio_Tk
rpowern = [ones(length_r,1) rpowern]; )YEAk@h@
else +:jonN9d
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ya~;Of5
rpowern = cat(2,rpowern{:}); iKPgiL~
end KQ]sUNH
MhHh`WUGh
% Compute the values of the polynomials: sv%E5@
% -------------------------------------- @,sjM]
y = zeros(length_r,length(n)); lJFy(^KQG,
for j = 1:length(n) ^rq\kf*]
s = 0:(n(j)-m_abs(j))/2; `O2P&!9&
pows = n(j):-2:m_abs(j); Psx"[2iZm
for k = length(s):-1:1 \)uA:v
p = (1-2*mod(s(k),2))* ... a~LA&>@
prod(2:(n(j)-s(k)))/ ... wMCg`rk
prod(2:s(k))/ ... nm<VcCc
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... | VaJ70\o
prod(2:((n(j)+m_abs(j))/2-s(k))); ]}b
idx = (pows(k)==rpowers); F5x*#/af
y(:,j) = y(:,j) + p*rpowern(:,idx); ^TZmc{i
end dcmf~+T
zL+t&P[\
if isnorm UJqh~s
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); em,1Yn?
end %(&ja_oO
end Yu" Q
% END: Compute the Zernike Polynomials /Lr`Aka5
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% <v -YMk@
*Lz'<=DLoW
% Compute the Zernike functions: "TaLvworb4
% ------------------------------ l+2NA4s
idx_pos = m>0; Z|*#)<|~
idx_neg = m<0; ]3,9."^
L$O\fhO?
z = y; D
ON.)F
if any(idx_pos) :X}SuM?c
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); G.L}VpopM
end Z_bVCe{
if any(idx_neg) 0^V<,CAV
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); :t`W&z41
end +j F|8
S[$9_J f
% EOF zernfun