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 U ]7;K>.T
function z = zernfun(n,m,r,theta,nflag) +F~B"a
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. l=L(pS3 ~
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N :jJ0 +Q
% and angular frequency M, evaluated at positions (R,THETA) on the jW{bP_,"
% unit circle. N is a vector of positive integers (including 0), and xwj{4fzpk{
% M is a vector with the same number of elements as N. Each element +UiJWO
% k of M must be a positive integer, with possible values M(k) = -N(k) .LGA0
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, w,j;XPp
% and THETA is a vector of angles. R and THETA must have the same }@~+%_;
% length. The output Z is a matrix with one column for every (N,M) g>g*1oS
% pair, and one row for every (R,THETA) pair. UgD)O:xaU
% zYM0?O8pJ~
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike j8%Y[:~D
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 5lyHg{iqD
% with delta(m,0) the Kronecker delta, is chosen so that the integral wRZS+^hx
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, /]of@
% and theta=0 to theta=2*pi) is unity. For the non-normalized u
$B24Cy.
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. xEv?2n@A
% a`zHx3Yg
% The Zernike functions are an orthogonal basis on the unit circle. eIOMW9Ivt
% They are used in disciplines such as astronomy, optics, and $W9dUR0
% optometry to describe functions on a circular domain. C}ASVywc,1
% z/nW;ow
% The following table lists the first 15 Zernike functions. |E;+j\
% 30<_`
% n m Zernike function Normalization 6!8uZ>u%Vg
% -------------------------------------------------- ""m/?TZq'
% 0 0 1 1 ,t!I%r
% 1 1 r * cos(theta) 2 Oc-ia)v1G
% 1 -1 r * sin(theta) 2 oi8M6l
% 2 -2 r^2 * cos(2*theta) sqrt(6) Ua4P@#cU
% 2 0 (2*r^2 - 1) sqrt(3) E= .clA
% 2 2 r^2 * sin(2*theta) sqrt(6) L* ScSxw
% 3 -3 r^3 * cos(3*theta) sqrt(8) |XMWi/p
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 7I*rtc&Kb
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) >qUD_U3A
% 3 3 r^3 * sin(3*theta) sqrt(8) pD }b $
% 4 -4 r^4 * cos(4*theta) sqrt(10) g?K? Fn.}
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) m}]QP\
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 2`> (LH
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) c7R&/JV
% 4 4 r^4 * sin(4*theta) sqrt(10) jUDE)~h
% -------------------------------------------------- uJ8FzS>[V
% \FF|b"E_=
% Example 1: cQsSJBZ[v5
% y'n<oSB}
% % Display the Zernike function Z(n=5,m=1) GIfs]zVr`
% x = -1:0.01:1; [^XD@
% [X,Y] = meshgrid(x,x); FC
% [theta,r] = cart2pol(X,Y); L0w2qF
% idx = r<=1; Pn L?zae
% z = nan(size(X)); G&`5o*).bb
% z(idx) = zernfun(5,1,r(idx),theta(idx)); R^]a<g,
% figure [{#n?BT
% pcolor(x,x,z), shading interp rDu?XJA
% axis square, colorbar g|h;*
% title('Zernike function Z_5^1(r,\theta)') n57mh5mixM
% WI.+9$1:P
% Example 2: s@Loax6@B
% a&dP@)
% % Display the first 10 Zernike functions nFe
% x = -1:0.01:1; ;iJ}[HUo
% [X,Y] = meshgrid(x,x); kBY#=e).
% [theta,r] = cart2pol(X,Y); 3>=G-AH/$K
% idx = r<=1; !3o/c w9
% z = nan(size(X)); P7REE_<1
% n = [0 1 1 2 2 2 3 3 3 3]; b,'rz04^
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; um\A
% Nplot = [4 10 12 16 18 20 22 24 26 28]; ]7RK/Zu i
% y = zernfun(n,m,r(idx),theta(idx)); 9*Fc+/
% figure('Units','normalized')
bjN"H`Q
% for k = 1:10 )Y"t$Iw"
% z(idx) = y(:,k); )i\foSbB`V
% subplot(4,7,Nplot(k)) +ZV?yR2yn
% pcolor(x,x,z), shading interp )bpdj,
% set(gca,'XTick',[],'YTick',[]) J7~Kjl
% axis square KXUJ*l-5
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) #qJ6iA6{
% end |uX&T`7?-
% ''k}3o.K[
% See also ZERNPOL, ZERNFUN2. Uo[`AzD3
VTi;y{
% Paul Fricker 11/13/2006 t+j dV
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Tz7 R:S.
% Check and prepare the inputs: ,S~A]uH'
% ----------------------------- 'b+
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if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) I;9DG8C&v*
error('zernfun:NMvectors','N and M must be vectors.') Fl"LK:)
end 6\%#=GG
zE7)4!
if length(n)~=length(m) A-eCc#I
error('zernfun:NMlength','N and M must be the same length.') O<XNI(@
end L:jv%;DM
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n = n(:); h*sL' fJ]
m = m(:); 5j _[z|W2
if any(mod(n-m,2)) w"A>mEex<
error('zernfun:NMmultiplesof2', ... .e}`n)z
'All N and M must differ by multiples of 2 (including 0).') \tdYTb.
end ;)sC{ "Jb
2#'"<n,G
if any(m>n) ENf(E9O
error('zernfun:MlessthanN', ... :%U
lNk
'Each M must be less than or equal to its corresponding N.') Xj:\B] v]
end (D\`:1g
mk6>}z*
if any( r>1 | r<0 ) u0$}VO5/a
error('zernfun:Rlessthan1','All R must be between 0 and 1.') *O-m:M!eA
end (&/~q:a>
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) j'UWgwB
error('zernfun:RTHvector','R and THETA must be vectors.') c{jTCkzq
end 4=|oOIhgb
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r = r(:); a
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theta = theta(:); xDmwiVy
length_r = length(r); X"T)X#:)
if length_r~=length(theta) )xTu|V
error('zernfun:RTHlength', ... 0X%#9s~
'The number of R- and THETA-values must be equal.') p,\(j
end gNh4c{Al9
F_V/&OV
% Check normalization: f6#1sO4"
% -------------------- ]YB,K)WQ
if nargin==5 && ischar(nflag) X C'|
isnorm = strcmpi(nflag,'norm'); qi8~bQ{rH
if ~isnorm ;]2d%Qt
error('zernfun:normalization','Unrecognized normalization flag.') Gk|T1%
end MnptC 1N
else a%wa3N=v
isnorm = false; lK#uyag
end MhN8'y(
+@\=v}:
F
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% EsLtC5]
% Compute the Zernike Polynomials `V?NS,@$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 85+w\KuEY
#?bOAWAwLh
% Determine the required powers of r: !Eb!y`jK
% ----------------------------------- DWU(ld:_
m_abs = abs(m); :n oZ
p:a
rpowers = []; H8!lSRq
for j = 1:length(n) $XFFNE`%
rpowers = [rpowers m_abs(j):2:n(j)]; Vv>hr+e
end uecjR8\e
rpowers = unique(rpowers); <@qJsRbhK
?lIh&C8]X
% Pre-compute the values of r raised to the required powers, 8ZDWaq8^2N
% and compile them in a matrix: gy/bA
% ----------------------------- qn `
\g
if rpowers(1)==0 qvRs1yr?q
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 4n2*2
yTg
rpowern = cat(2,rpowern{:}); 8b+%:eJ
rpowern = [ones(length_r,1) rpowern]; l D]?9K29
else ;oRgg'k<
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 4aG}ex-s|
rpowern = cat(2,rpowern{:}); ='HLA-uT
end Ewo6Q){X
DXfQy6k'
% Compute the values of the polynomials: 7:OF>**
% -------------------------------------- [<#`@Kr
y = zeros(length_r,length(n)); l/bZE.GJ
for j = 1:length(n) ,uS}wJAX
s = 0:(n(j)-m_abs(j))/2; kT&GsR/
pows = n(j):-2:m_abs(j); 2Vg+Aly4D
for k = length(s):-1:1 r6}-EYq=
p = (1-2*mod(s(k),2))* ... u:\DqdlU`
prod(2:(n(j)-s(k)))/ ... ]DI%7kw'
prod(2:s(k))/ ... !A"-9OS2
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... M V~3~h8
prod(2:((n(j)+m_abs(j))/2-s(k))); n*N`].r#{=
idx = (pows(k)==rpowers); CSMx]jbb
y(:,j) = y(:,j) + p*rpowern(:,idx); \2)~dV:6+
end _Ns_$_
AJt4I
W@
if isnorm E^V4O l<
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); dxF)) Z
end 2;YL+v2
end ] U[4r9V
% END: Compute the Zernike Polynomials / U"3LX
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 2sT\+C&H
BE," lX
% Compute the Zernike functions: 9 H~OC8R:
% ------------------------------ fb|lWEw5h.
idx_pos = m>0; P64<O5l/
idx_neg = m<0; 6"jV>CNc@
f15n ~d
z = y; I>spJ5ls
if any(idx_pos) -&r A<j
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); . AX6xc6
end 76EMS?e
if any(idx_neg) -2*Pm1\Z
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); UN`O*(k[
end >/DlxYG?
R"[U<^
% EOF zernfun