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}W
function z = zernfun(n,m,r,theta,nflag) qu dY9_
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 1s(]@gt
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N "PO8 Q
% and angular frequency M, evaluated at positions (R,THETA) on the D6+3f#k6
% unit circle. N is a vector of positive integers (including 0), and yNn=r;FZQ
% M is a vector with the same number of elements as N. Each element x?0K'
% k of M must be a positive integer, with possible values M(k) = -N(k) .XiO92d9
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, z,7;+6*=L
% and THETA is a vector of angles. R and THETA must have the same U{LS_VI~
% length. The output Z is a matrix with one column for every (N,M) *" C9F/R
% pair, and one row for every (R,THETA) pair. -)3+/4Q(
% ^FBu|eAkE
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike _)!*,\*`{
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), Dj'?12Onu=
% with delta(m,0) the Kronecker delta, is chosen so that the integral &}7R\co3
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 0GeL">v,:=
% and theta=0 to theta=2*pi) is unity. For the non-normalized VBF:MAA
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. JX[]u<h?
% _KxR~k^
% The Zernike functions are an orthogonal basis on the unit circle. )oz2V9X{
% They are used in disciplines such as astronomy, optics, and T]tu#h{
a
% optometry to describe functions on a circular domain. rKI<!
% un -h%-e|
% The following table lists the first 15 Zernike functions. ID!S}D
% Zv=pS
(9
% n m Zernike function Normalization D1 v0`od'
% -------------------------------------------------- J5HK1
% 0 0 1 1 [u2t1^#Ol
% 1 1 r * cos(theta) 2 8F`8=L NO
% 1 -1 r * sin(theta) 2 `BG>%#
% 2 -2 r^2 * cos(2*theta) sqrt(6) X;GU#8W
% 2 0 (2*r^2 - 1) sqrt(3) 2;s[ m3
% 2 2 r^2 * sin(2*theta) sqrt(6) OY:rcGc`t
% 3 -3 r^3 * cos(3*theta) sqrt(8) q/ 54=8*h0
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) -WF((s;<#
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) ]4c+{
% 3 3 r^3 * sin(3*theta) sqrt(8) r<!nU&FPD:
% 4 -4 r^4 * cos(4*theta) sqrt(10) *?HoN;^
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Fb8d=Zc
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) {z0iWY2Xw
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) X#JUorGp
% 4 4 r^4 * sin(4*theta) sqrt(10) 4
l-UrnZ
% -------------------------------------------------- j3/6hE>
% Og1vD5a
% Example 1: 5V =mj+X?
% hCr,6nc C
% % Display the Zernike function Z(n=5,m=1) =RRv&
"2r
% x = -1:0.01:1; 6vE#$(n#a&
% [X,Y] = meshgrid(x,x); OW\vbWX
% [theta,r] = cart2pol(X,Y); M|%bxG^l
% idx = r<=1; 0 D
'^:
% z = nan(size(X)); 7Vh
% z(idx) = zernfun(5,1,r(idx),theta(idx)); x< 2]UB`
% figure HB'9&
% pcolor(x,x,z), shading interp Z@&%"nO
% axis square, colorbar Pvi2j&W84
% title('Zernike function Z_5^1(r,\theta)') .IdbaH
_a
% !3k-' ),z&
% Example 2: ``:[Jr&
% K|-m6!C!7
% % Display the first 10 Zernike functions ]3f[v:JQ
% x = -1:0.01:1; v G\J8s
% [X,Y] = meshgrid(x,x); U), HrI>;
% [theta,r] = cart2pol(X,Y); M80Q6K
% idx = r<=1; WH1" HO
% z = nan(size(X)); Y3&,U
% n = [0 1 1 2 2 2 3 3 3 3]; \OFmd!Cz
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; W4 d32+V
% Nplot = [4 10 12 16 18 20 22 24 26 28]; 9cP{u$
% y = zernfun(n,m,r(idx),theta(idx)); `P !idg*
% figure('Units','normalized') *9kg\#
% for k = 1:10 O)VcW/
% z(idx) = y(:,k); O$m &!J
% subplot(4,7,Nplot(k)) xi "3NF%=
% pcolor(x,x,z), shading interp Kd+E]$F_OH
% set(gca,'XTick',[],'YTick',[]) sfn^R+x4,9
% axis square ~B"HI+:\L
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) np6G~0Y`
% end C{uT1`
% IBJNs$
% See also ZERNPOL, ZERNFUN2. !s1<)%Jt
!0Nf`iCQ(
% Paul Fricker 11/13/2006 }Cw,m0KV/
g%S/)R,,ct
PN]hG,q*4O
% Check and prepare the inputs: hZ
e{Ri
% ----------------------------- M NwY
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) _%D7D~2r|
error('zernfun:NMvectors','N and M must be vectors.') sZ&|omN
end $G"\@YC<
#vyf*jPr
if length(n)~=length(m) aaY AS"/:
error('zernfun:NMlength','N and M must be the same length.') lD[@D9
end Fovah4q%V
<zn)f@W
n = n(:); ,v8e7T
m = m(:); H<i!C|AF
if any(mod(n-m,2)) ZJ)Z
error('zernfun:NMmultiplesof2', ... 2 >O [Y1
'All N and M must differ by multiples of 2 (including 0).') @#,/6s7?
end -`\rDPGf
,Owk;MV@
if any(m>n) 67Pmnad
error('zernfun:MlessthanN', ... p+]S)K GZw
'Each M must be less than or equal to its corresponding N.') JnK<:]LcK
end Q?>r:vMi
q%kCTw
if any( r>1 | r<0 ) l%GArH`
error('zernfun:Rlessthan1','All R must be between 0 and 1.') 0/f|ZH ~!
end Bv@p9 ]
n
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) TU~y;:OJ
error('zernfun:RTHvector','R and THETA must be vectors.') N^oP,^+U
end zi6J|u
v0 :n:q
r = r(:); SEzjc ~@3
theta = theta(:); "*X\'LPs=
length_r = length(r); UG`~RO
if length_r~=length(theta) y<- ]'Yts
error('zernfun:RTHlength', ... v\?J=|S+
'The number of R- and THETA-values must be equal.') IW<rmP=R&
end A)n_ST0
A~vx,|I
% Check normalization: "M
iJM+,
% -------------------- U~ a\v8l~
if nargin==5 && ischar(nflag) \D z? h
isnorm = strcmpi(nflag,'norm'); 2H9hN4N
if ~isnorm ^|Fy!kp
error('zernfun:normalization','Unrecognized normalization flag.') fG>3gS6&
end 8TB|Y
else d9TTAaf
isnorm = false; (jU_lsG
end A? B+
Q<V1`e
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 6?M/71
% Compute the Zernike Polynomials 5"57F88Y1
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (n B[aM
SceHdx(]
% Determine the required powers of r: y-.{){uaD
% ----------------------------------- (y!bvp[" m
m_abs = abs(m); s;oe Qa}TB
rpowers = []; w" [T
for j = 1:length(n) Sq,>^|v4&e
rpowers = [rpowers m_abs(j):2:n(j)]; s1cu5eCt
end t6+W
rpowers = unique(rpowers); xP_%d,
y'^U4# (
% Pre-compute the values of r raised to the required powers, rMIX{K)'f
% and compile them in a matrix: l@GJcCufE
% ----------------------------- W3UxFs]$
if rpowers(1)==0 3)W_^6>bM
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false);
V^Z5i]zT
rpowern = cat(2,rpowern{:}); #OM'2@
rpowern = [ones(length_r,1) rpowern]; Q+Q"J U
else *\'t$se+
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); z~`X4Segw
rpowern = cat(2,rpowern{:}); jcj8w
end d*Mqs}8
8~Zw"
% Compute the values of the polynomials: oCkG
% -------------------------------------- {c3FJ5:
y = zeros(length_r,length(n)); Gu$J;bXVj
for j = 1:length(n) Hm'fK$y(
s = 0:(n(j)-m_abs(j))/2; s/hWhaS<
pows = n(j):-2:m_abs(j); 9b=0
4aWHm
for k = length(s):-1:1 MQw}R7
p = (1-2*mod(s(k),2))* ... D['J4B
prod(2:(n(j)-s(k)))/ ... HEFgEYlO
prod(2:s(k))/ ... [8Y7Q5Had
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... )_C>hWvo_
prod(2:((n(j)+m_abs(j))/2-s(k))); IYq#|^)5+
idx = (pows(k)==rpowers); Fl($0}ER
y(:,j) = y(:,j) + p*rpowern(:,idx); ldp9+7n~
end a"YVr'|
zOSUYn
if isnorm ?q4`&";{3
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); I^f|U
end [N~7PNd S
end Xux[
% END: Compute the Zernike Polynomials pm=O.)g4`
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% n[!QrEeR},
XZk%5t|t
% Compute the Zernike functions: x^)?V7[t
% ------------------------------ {:"<E?+
idx_pos = m>0; \PT!mbB?
idx_neg = m<0; &uE )Vr4 R
Dx /w&v
z = y; ?/MkH0[G =
if any(idx_pos) _I; hM
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); V2?{ebx`
end )?radg
if any(idx_neg) p2l@6\m\
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); (Q||5
end g,WTXRy
-eK0 +beQ
% EOF zernfun