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 kJlRdt2
function z = zernfun(n,m,r,theta,nflag) zRD{"uqi
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 1
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% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N enO5XsIc
% and angular frequency M, evaluated at positions (R,THETA) on the :p=IZY
% unit circle. N is a vector of positive integers (including 0), and <S6|$7{1
% M is a vector with the same number of elements as N. Each element `V$i*{c:#
% k of M must be a positive integer, with possible values M(k) = -N(k) DKF`uRvGN:
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, mmu{K$9}I
% and THETA is a vector of angles. R and THETA must have the same wX<)Fj'
% length. The output Z is a matrix with one column for every (N,M) cmZ39pjBJ
% pair, and one row for every (R,THETA) pair. W.HM!HQp
% R3jhq3F\Y
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike =Mc*~[D/
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), <I&X[Sqp
% with delta(m,0) the Kronecker delta, is chosen so that the integral [_^K}\/+
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, (m|p|rL
% and theta=0 to theta=2*pi) is unity. For the non-normalized eXc`"T,C.
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 8)\ ?6C
% 38tRb"3zP
% The Zernike functions are an orthogonal basis on the unit circle. G9 ;X=c
% They are used in disciplines such as astronomy, optics, and NJI-8qTGI
% optometry to describe functions on a circular domain. `&LPqb
% Z0`Bn5
% The following table lists the first 15 Zernike functions. dli?/U@hO
% 4@u*#Bp`|
% n m Zernike function Normalization 7ykpDl^ @
% -------------------------------------------------- kOfbO'O9
% 0 0 1 1 LS}u6\(
% 1 1 r * cos(theta) 2 MXh0 a@*]
% 1 -1 r * sin(theta) 2 >OgA3)X
% 2 -2 r^2 * cos(2*theta) sqrt(6) [1F.
% 2 0 (2*r^2 - 1) sqrt(3) pV9$Vg?-H
% 2 2 r^2 * sin(2*theta) sqrt(6) (oBvpFP33
% 3 -3 r^3 * cos(3*theta) sqrt(8) [i==
Tp
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) *?zmo@-
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) TTZb.
% 3 3 r^3 * sin(3*theta) sqrt(8) <'>c`80@\*
% 4 -4 r^4 * cos(4*theta) sqrt(10) 1Mn=m w
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) i+
]3J/J
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) SP?~i@H
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 4@AY~"dq
% 4 4 r^4 * sin(4*theta) sqrt(10) n0bm 'qw
% -------------------------------------------------- +DmfqKKbd
% !nQ_<
% Example 1: v*iD)k:|t
% pX8TzmIB0
% % Display the Zernike function Z(n=5,m=1) RZoSP(6
% x = -1:0.01:1; (HbA?Aja
% [X,Y] = meshgrid(x,x); -N
$4\yp
% [theta,r] = cart2pol(X,Y); >o9tlO)
% idx = r<=1; MKPxF@N(
% z = nan(size(X)); NOM6},rp
% z(idx) = zernfun(5,1,r(idx),theta(idx)); p{X?_ F
% figure UCV1 {
% pcolor(x,x,z), shading interp GVR/p
% axis square, colorbar ]s_,;PG U
% title('Zernike function Z_5^1(r,\theta)') eocq Hwbv
% /|Z_Dy
% Example 2: Y\75cfD
% _}+Aw{7!r
% % Display the first 10 Zernike functions f$1&)1W[
% x = -1:0.01:1; CGw, RNV
% [X,Y] = meshgrid(x,x); *Tc lcu
% [theta,r] = cart2pol(X,Y); eFKF9m
% idx = r<=1; H j [!F%
% z = nan(size(X)); F3nYMf
% n = [0 1 1 2 2 2 3 3 3 3]; $ /`X7a{
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; pLj[b4p9
% Nplot = [4 10 12 16 18 20 22 24 26 28]; >|zMN$:
% y = zernfun(n,m,r(idx),theta(idx)); R*0]*\C z
% figure('Units','normalized') "`Q&s
% for k = 1:10 ~(*2:9*0
% z(idx) = y(:,k); Op()`x
m
% subplot(4,7,Nplot(k)) (yrN-M4~t
% pcolor(x,x,z), shading interp boS=
% set(gca,'XTick',[],'YTick',[]) (vP<}
% axis square }TQa<;Q
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 0\zY?UUww
% end AjVX
% \uPyvA=
% See also ZERNPOL, ZERNFUN2. CKI.\o
=j~BAS*"
% Paul Fricker 11/13/2006 -\<\OV:c*
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xO[V>Ud
% Check and prepare the inputs: <*Nd%Ca
% ----------------------------- C19}Y4r:
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) PctXh, =
error('zernfun:NMvectors','N and M must be vectors.') GJ5R <f9I
end J6J">
.af+h<RG4$
if length(n)~=length(m) r=-b@U.fk>
error('zernfun:NMlength','N and M must be the same length.') A!cY!aQ
end N TcojA{V$
U ,NGV0
n = n(:); fUMjLA|*I<
m = m(:); f$76p!pDa
if any(mod(n-m,2)) Yt[LIn-v:
error('zernfun:NMmultiplesof2', ... 1etT."
'All N and M must differ by multiples of 2 (including 0).') ZIN1y;dJ
end +T\<oj%}2
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if any(m>n) IaZmN.k*
error('zernfun:MlessthanN', ... b(oe^jeGz
'Each M must be less than or equal to its corresponding N.') 4a0Ud !Qcs
end X J`*dgJ
Mz.C`Z>o
if any( r>1 | r<0 ) et2;{Tb,5
error('zernfun:Rlessthan1','All R must be between 0 and 1.') %~I&T".iC
end #+QJ5VI:
~!S/{Un
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) @F]w]d
error('zernfun:RTHvector','R and THETA must be vectors.') hraR:l
D
end ht*N[Pi4;
0BNH~,0u
r = r(:); Tw djBMte
theta = theta(:);
)ut$644R
length_r = length(r); 4[i 3ckFT,
if length_r~=length(theta) 9N `WT=
error('zernfun:RTHlength', ... #]dq^B~~
'The number of R- and THETA-values must be equal.') oP`:NCj\9
end Mq#m;v$E
o{>4PZ}=g
% Check normalization: 5kGQf
% -------------------- &c 2Qa
if nargin==5 && ischar(nflag) r95,X!
isnorm = strcmpi(nflag,'norm'); e/cHH34
if ~isnorm <o9AjASv\,
error('zernfun:normalization','Unrecognized normalization flag.') k,$/l1D
end 1 $1>cuu
else `-%dHvB^R
isnorm = false; IqV" 4
end -8l(eDm"m
[ )
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% i3mw.`7
% Compute the Zernike Polynomials uB^"A ;0v
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% g%tUk M
1TX3/]:
% Determine the required powers of r: f,i2U|1pbj
% ----------------------------------- FAL#p$y}
m_abs = abs(m); B8eZ}9X
rpowers = []; ~"0{<mMcX
for j = 1:length(n) 'zav%}b]L
rpowers = [rpowers m_abs(j):2:n(j)]; p2Gd6v.t
end (&NLLrsio
rpowers = unique(rpowers); H>D sAHS
cLp_\\
% Pre-compute the values of r raised to the required powers, pY-!NoES
% and compile them in a matrix: JBA{i45x
% ----------------------------- 8\9W:D@"x
if rpowers(1)==0 "!(@MfjT
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 0LXu!iix
rpowern = cat(2,rpowern{:}); ~CHcbEWk)W
rpowern = [ones(length_r,1) rpowern]; n:B){'S
else <m^a
?q^
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); pGD-K41O]
rpowern = cat(2,rpowern{:}); f+ZOE?"
end JL!^R_b&c
*g
%bdO
% Compute the values of the polynomials: ^7ID |uMr
% -------------------------------------- x^c,cV+*
y = zeros(length_r,length(n)); yPT o,,ca=
for j = 1:length(n) ]@cI _n
s = 0:(n(j)-m_abs(j))/2; (=WbLNBS
pows = n(j):-2:m_abs(j); N.+A-[7,W
for k = length(s):-1:1 9>0OpgvC(
p = (1-2*mod(s(k),2))* ... Jw}&[
prod(2:(n(j)-s(k)))/ ... nC
!NZ
prod(2:s(k))/ ... Cq7 uy
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ]l h=ZC
prod(2:((n(j)+m_abs(j))/2-s(k))); rN7JJHV
idx = (pows(k)==rpowers); "M+I$*]
y(:,j) = y(:,j) + p*rpowern(:,idx); )(yaX
end OGLA1}k4
qhG2j;
if isnorm 4;)t\9cy_
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ^8bc<c:P
end 3!cenyE
end G9xO>Xp^Al
% END: Compute the Zernike Polynomials js;YSg{m
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ,Xn%0]
>ySO.S
% Compute the Zernike functions: 9bRUN<
% ------------------------------ =aQlT*n%3
idx_pos = m>0; p:$v,3:
idx_neg = m<0; {/N8[?zML
pRxVsOb
z = y; %jf|efxo
if any(idx_pos) T*Ge67
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); -G? IXgG
end GV) "[O
if any(idx_neg) xT* 3QwK
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ME!P{ _/
end P_mP ^L
90Sras>F
% EOF zernfun