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 \{ G1d"n
function z = zernfun(n,m,r,theta,nflag) czf|c
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. u@$C i/J*
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 8L<GAe
% and angular frequency M, evaluated at positions (R,THETA) on the JYB<};,
% unit circle. N is a vector of positive integers (including 0), and \P_1@sH=
% M is a vector with the same number of elements as N. Each element ;$\d^i{N
% k of M must be a positive integer, with possible values M(k) = -N(k) )MZQ\8,)]
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, FU|c[u|z
% and THETA is a vector of angles. R and THETA must have the same KN;b+`x;M
% length. The output Z is a matrix with one column for every (N,M) PXk+Vi,%k
% pair, and one row for every (R,THETA) pair. {%5tqF
% (!U5B
Hnd
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 37@_"
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), X#mp pMU
% with delta(m,0) the Kronecker delta, is chosen so that the integral
zF2GW
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ttPa[h{!
% and theta=0 to theta=2*pi) is unity. For the non-normalized NGlX%j4j
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. >g@;`l.Z#
% E{*~>#+
% The Zernike functions are an orthogonal basis on the unit circle. V11Zl{uOl
% They are used in disciplines such as astronomy, optics, and Kd _tjWS
% optometry to describe functions on a circular domain. Brh<6Btl
% !tT$}?Ano
% The following table lists the first 15 Zernike functions. (ROurq"
% >uuP@j
% n m Zernike function Normalization "|S \J5-%
% -------------------------------------------------- 0.-2FHc9L
% 0 0 1 1 2 fX-J
% 1 1 r * cos(theta) 2 H/p<lp
% 1 -1 r * sin(theta) 2 "]ow1{
% 2 -2 r^2 * cos(2*theta) sqrt(6) }MDu QP]
% 2 0 (2*r^2 - 1) sqrt(3) /YWoDHL
% 2 2 r^2 * sin(2*theta) sqrt(6) zF'{{7o
% 3 -3 r^3 * cos(3*theta) sqrt(8) dwKre#4F
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) *K6 V$_{S
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) vIrLG1EK
% 3 3 r^3 * sin(3*theta) sqrt(8) 7CzZHkTg
% 4 -4 r^4 * cos(4*theta) sqrt(10)
] }XK
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ;SF0}51
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) Cyxt EzPp
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) O&=?,zLO[
% 4 4 r^4 * sin(4*theta) sqrt(10) 'g8~539{&
% -------------------------------------------------- W;coi4
% UB]}j^
% Example 1: ^.F@yo2}
% 2jf-vWV_
% % Display the Zernike function Z(n=5,m=1) ti)foam
% x = -1:0.01:1; AG2iLictv
% [X,Y] = meshgrid(x,x); ,qak_bP
% [theta,r] = cart2pol(X,Y); gOZ$rv^g
% idx = r<=1; IBY3QG
% z = nan(size(X)); b+\jFGC%6=
% z(idx) = zernfun(5,1,r(idx),theta(idx)); z]> 0A
% figure XB-pOtVm
% pcolor(x,x,z), shading interp kIV/o
% axis square, colorbar 12aAO|]/~
% title('Zernike function Z_5^1(r,\theta)') :cop0;X:Wm
% MN|y5w}$u
% Example 2: g6$X {
% qtTys gv
% % Display the first 10 Zernike functions |QJ!5nb
% x = -1:0.01:1; 8w~I(2S:#
% [X,Y] = meshgrid(x,x); !}^c.<38Q
% [theta,r] = cart2pol(X,Y); }`4o+
% idx = r<=1; %-|Po:6
% z = nan(size(X)); 0 ]U
;5
% n = [0 1 1 2 2 2 3 3 3 3]; Xvm.Un<N
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; Gd`qZqx#
% Nplot = [4 10 12 16 18 20 22 24 26 28]; A5tY4?|
% y = zernfun(n,m,r(idx),theta(idx)); Deq~"
% figure('Units','normalized') {j[[E/8N!y
% for k = 1:10 5.#r\' Z#
% z(idx) = y(:,k); to^ &:
% subplot(4,7,Nplot(k)) B=#rp*vwL
% pcolor(x,x,z), shading interp UXoaUW L
% set(gca,'XTick',[],'YTick',[]) dfGdY"&
% axis square 9u lJZ\cQ
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) =L9sb!
% end ;Aiuy{<
% &h-d\gMJ
% See also ZERNPOL, ZERNFUN2. r80w{[S$
(F]f{8
% Paul Fricker 11/13/2006 Ooz+V;#Q
uh%%MhTjv
_L(6F
TJ
% Check and prepare the inputs: 4hg]/X"H#
% ----------------------------- gQgG_&xkC
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) dl@
error('zernfun:NMvectors','N and M must be vectors.') m;lwMrY\7>
end I)V2cOrXM
+q"d=
if length(n)~=length(m) CTbdY,=B
error('zernfun:NMlength','N and M must be the same length.') j/{F#auI
end Miqu
gAC}
n = n(:); >IC.Zt@
m = m(:); MftW^7W-
if any(mod(n-m,2)) ~!&WK,k6
error('zernfun:NMmultiplesof2', ... Z,qo
jtw
'All N and M must differ by multiples of 2 (including 0).') /OK.n3Tt
end K0yTHX?(.
]nhLv!Co
if any(m>n) 7 w_`<b6
error('zernfun:MlessthanN', ... }XWic88!~
'Each M must be less than or equal to its corresponding N.') GptJQ=pV
end 3_B .W
Lg[*P8wE
if any( r>1 | r<0 ) ]y@9z b
error('zernfun:Rlessthan1','All R must be between 0 and 1.') p@/!+$^{
end a Umcs!@
NQ !t `
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) )
FAJ\9
error('zernfun:RTHvector','R and THETA must be vectors.') C;}~C:aJ
end THWT\3~,
v2G_p|+O
r = r(:); !m^;Apuy
theta = theta(:); C,hs!v6
length_r = length(r); QK<sibDI
if length_r~=length(theta) :h=];^/E
error('zernfun:RTHlength', ... &OK(6o2m;
'The number of R- and THETA-values must be equal.') sbZ)z#Tr
end F(^vD_G
\$T
% Check normalization: mMjY I1F
% -------------------- XU5/7
.
if nargin==5 && ischar(nflag) HvN!_}[
isnorm = strcmpi(nflag,'norm'); Bjq1za
if ~isnorm 63QMv[`,
error('zernfun:normalization','Unrecognized normalization flag.')
YH&`+ +
end )7Gm<r
else D3$PvX[f
isnorm = false; )9 5&-Hs
end kjfZ*V=-
&Vg+n0
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 4 FW~Y
% Compute the Zernike Polynomials hU3c;6]3
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% >K1)XP
W~aVwO'(
% Determine the required powers of r: AIR,XlD
% ----------------------------------- O;9u1,%w
m_abs = abs(m); I!dA{INN
rpowers = []; G)]'>m<y
for j = 1:length(n) b4ZZyw
rpowers = [rpowers m_abs(j):2:n(j)]; A&jkc '
end cKdn3 2Y4
rpowers = unique(rpowers); 0z #'=XWk
>A|(mc
% Pre-compute the values of r raised to the required powers, IP7j)SM!
% and compile them in a matrix: 2Hw&}8
% ----------------------------- !qS~YA
if rpowers(1)==0 KPSFy<
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); UBzX%:A
rpowern = cat(2,rpowern{:}); J:Ea|tXK^
rpowern = [ones(length_r,1) rpowern]; 0f&B;?)!
else D+P(
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Ci4`,
rpowern = cat(2,rpowern{:}); #3>o^cN~8k
end H<#M)8
JGOry \
% Compute the values of the polynomials: <{GpAf8-
% -------------------------------------- dIg/g~ t"
y = zeros(length_r,length(n)); nICc}U?k
for j = 1:length(n) Oq@+/UWX
s = 0:(n(j)-m_abs(j))/2; 7DDd1"jE
pows = n(j):-2:m_abs(j); }(A`aB_
for k = length(s):-1:1 ukpbx;O:hc
p = (1-2*mod(s(k),2))* ... "3.v(GVr
prod(2:(n(j)-s(k)))/ ... 3}(6z"r
prod(2:s(k))/ ... 3)88B"E
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 5.5<.")
prod(2:((n(j)+m_abs(j))/2-s(k))); FM;NA{
idx = (pows(k)==rpowers); WHeyE3}p
y(:,j) = y(:,j) + p*rpowern(:,idx); z/!LC;(
end nNz1gV:0X
^MIF+/bQ
if isnorm cWjb149@)
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 7rQwn2XD{
end =!)Ye:\Q
end k>E^FB=
% END: Compute the Zernike Polynomials a?jUm.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% YbtsJ
<w
:dq.@:+<R
% Compute the Zernike functions: L#O1>
% ------------------------------ waI?X2
idx_pos = m>0; dp#JvZb
idx_neg = m<0; ?C)a0>L
SW5V:|/
z = y; 3}aKok"k
if any(idx_pos) VzfaUAIZl
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); [ )3rc}:1
end b.I_
if any(idx_neg) N8x[8Rp
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); -]el_:H
end 2[~|#0x
~MWI-oK
% EOF zernfun