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 MI8f(ZJK5
function z = zernfun(n,m,r,theta,nflag) J)(KG dk
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. @o4+MQFn
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N m#
y`
% and angular frequency M, evaluated at positions (R,THETA) on the NQBa+N
% unit circle. N is a vector of positive integers (including 0), and Y2lBQp8'|
% M is a vector with the same number of elements as N. Each element ~)J]`el,Q
% k of M must be a positive integer, with possible values M(k) = -N(k) 7oUecyoj
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, $ glt%a
% and THETA is a vector of angles. R and THETA must have the same 3JhT
% length. The output Z is a matrix with one column for every (N,M) ,Sz`$'^c
% pair, and one row for every (R,THETA) pair. OYnxEdo7
% kH;DAphk
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike }~B @Z\`O
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), U(xN}Y?
% with delta(m,0) the Kronecker delta, is chosen so that the integral VTS7K2lBvX
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 7Iz%Jty
% and theta=0 to theta=2*pi) is unity. For the non-normalized U`)\|\NY
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. _:N+mEF
% FYwMmb
~3
% The Zernike functions are an orthogonal basis on the unit circle. g+(Cs
% They are used in disciplines such as astronomy, optics, and sR~D3-
% optometry to describe functions on a circular domain. 1gK<dg
% 4lM)ZDg
% The following table lists the first 15 Zernike functions. bQ%6z}r
% \@\r`=WgB
% n m Zernike function Normalization 2SjH7
'
% -------------------------------------------------- egXHp<bqw
% 0 0 1 1 R #f*QXv
% 1 1 r * cos(theta) 2 Xu.Wdl/{Ra
% 1 -1 r * sin(theta) 2 yr;~M{{4
% 2 -2 r^2 * cos(2*theta) sqrt(6) kv!QO^;^Y
% 2 0 (2*r^2 - 1) sqrt(3) v|K,
% 2 2 r^2 * sin(2*theta) sqrt(6) Ru&>8Ln0
% 3 -3 r^3 * cos(3*theta) sqrt(8) z`Jcpt
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) @V\u<n
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) k^H&IS!
% 3 3 r^3 * sin(3*theta) sqrt(8) 6D\$K
% 4 -4 r^4 * cos(4*theta) sqrt(10) *dAQ{E(rO
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) /VmtQ{KTt+
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 0I v(ioB=
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) CD!Aa
% 4 4 r^4 * sin(4*theta) sqrt(10) 2iWSk6%R
% -------------------------------------------------- ?lPn{oB9"
% M1mx {<]A
% Example 1: ;_K3/:
% "y9]>9:$-
% % Display the Zernike function Z(n=5,m=1) XsEotW
% x = -1:0.01:1; 1PN!1= F}
% [X,Y] = meshgrid(x,x); {i^F4A@=Z
% [theta,r] = cart2pol(X,Y); o#Viz:
% idx = r<=1; |Wg!>g!
% z = nan(size(X)); Ql1J?9W
% z(idx) = zernfun(5,1,r(idx),theta(idx)); A"}Ib'
% figure 0'ge}2^
% pcolor(x,x,z), shading interp h}U>K4BJ
% axis square, colorbar BO,xA -+
% title('Zernike function Z_5^1(r,\theta)') 'lMDlTU O
% r
[E4/?_
% Example 2: *%ta5a
% 8Q(A1U
% % Display the first 10 Zernike functions u_=^Bd
% x = -1:0.01:1; <'N~|B/yZ
% [X,Y] = meshgrid(x,x); ;U&~tpd
% [theta,r] = cart2pol(X,Y); | Kq<}R
% idx = r<=1; DP.Y<V)B
% z = nan(size(X));
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% n = [0 1 1 2 2 2 3 3 3 3]; C({r1l4[D
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; %w;wQ_
% Nplot = [4 10 12 16 18 20 22 24 26 28]; yTR5*{?j
% y = zernfun(n,m,r(idx),theta(idx)); 717OzrF}A?
% figure('Units','normalized') v[\GhVb
% for k = 1:10 +TC##}Zmb
% z(idx) = y(:,k); Ov;q]Vn>
% subplot(4,7,Nplot(k)) b=kY9!GN,v
% pcolor(x,x,z), shading interp %RIlu[J
% set(gca,'XTick',[],'YTick',[]) w$0*5n>)
% axis square (7C$'T-ZK
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) |)OC1=As
% end cp&1yB
% u/apnAW@M
% See also ZERNPOL, ZERNFUN2. 6Z5$cR_vC7
rrSFmhQUk
% Paul Fricker 11/13/2006 bQ-n<Lx
? CU;
W/.n
R[!
% Check and prepare the inputs: ma6Wr !J
% ----------------------------- }_D{|!!!T
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) N}Or+:"O:q
error('zernfun:NMvectors','N and M must be vectors.') P6)d#M
end \Rw^&;\1
G_}oI|B
if length(n)~=length(m) ~i0>[S3'
error('zernfun:NMlength','N and M must be the same length.') D7Y?$=0ycb
end L7"<a2J
l-2lb&n
n = n(:); & j*Ylj}
m = m(:); %reW/;)l{
if any(mod(n-m,2)) AMN`bgxW
error('zernfun:NMmultiplesof2', ... 3}B-n!|*
'All N and M must differ by multiples of 2 (including 0).') p2gu@!
end ,=2)1I]
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if any(m>n) ]d*O>Pm
error('zernfun:MlessthanN', ... GL^
j
|1
'Each M must be less than or equal to its corresponding N.') }Qh%Z)
end yMOYTN@]
_)~|Z~
if any( r>1 | r<0 ) X^C $|:
error('zernfun:Rlessthan1','All R must be between 0 and 1.') z'zC
end )O\l3h"
~]BR(n
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) @^`5;JiUk
error('zernfun:RTHvector','R and THETA must be vectors.') y*8;T v|
end ^.M_1$-
{X W>3 "
r = r(:); 0.#%KfQ
theta = theta(:); !9^GkFR6n
length_r = length(r); YGi_7fTyc=
if length_r~=length(theta) 7A
error('zernfun:RTHlength', ... VKi3z%kwK
'The number of R- and THETA-values must be equal.') kEg~yN
end Q8DKU
`U;V-
% Check normalization: d%Ku'Jy
% -------------------- l4OPzNc'
if nargin==5 && ischar(nflag) vf`]
isnorm = strcmpi(nflag,'norm'); ~'):1}KN]
if ~isnorm }U b "Vb
error('zernfun:normalization','Unrecognized normalization flag.') ^Cg@'R9
end & aF'IJC
else 1'5!")r
isnorm = false; Z8pZm`g)T
end ,=P0rbtK
cr{dl\Na
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% B^hK
% Compute the Zernike Polynomials U4Pk^[,p1G
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [oH,FSuO!2
6.4,Qae9E
% Determine the required powers of r: +Jc-9Ko\c;
% ----------------------------------- 16I(S
m_abs = abs(m); b$FXRR\G
rpowers = []; gwYTOs^
for j = 1:length(n) ,]?l(H $x'
rpowers = [rpowers m_abs(j):2:n(j)]; q{.~=~
end tQ4{:WPG
rpowers = unique(rpowers); zyI4E\
l1RFn,Tzr
% Pre-compute the values of r raised to the required powers, Jaf=qwZ/`
% and compile them in a matrix: &S#bLE
% ----------------------------- \y/+H
if rpowers(1)==0 t{/
EN)J
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); J15$P8J
rpowern = cat(2,rpowern{:}); $E@ke:
rpowern = [ones(length_r,1) rpowern]; B?_ujH80m
else E9[8th,t
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); jdVdz,Y
rpowern = cat(2,rpowern{:}); Eb9M;u
end ?Qs>L~
?r~](l
% Compute the values of the polynomials: 9$'Edi=6
% -------------------------------------- g:c
@
y = zeros(length_r,length(n)); 3!B3C(g
for j = 1:length(n) BcoE&I?[m|
s = 0:(n(j)-m_abs(j))/2; FdJC@Y-#uA
pows = n(j):-2:m_abs(j); ?)5M3lV3k
for k = length(s):-1:1 |m7`:~ow
p = (1-2*mod(s(k),2))* ... RwwX;I"o%
prod(2:(n(j)-s(k)))/ ... Qod2m$>wp}
prod(2:s(k))/ ... QfM*K.7Sl
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... E0S[TEDa]
prod(2:((n(j)+m_abs(j))/2-s(k))); N:/$N@"Ge
idx = (pows(k)==rpowers); ) uyh
y(:,j) = y(:,j) + p*rpowern(:,idx); Wkv**X}
end I!Za2?
IN]bAd8"
if isnorm )O%lh
8fI
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Lp*T=]C]
end JGD{cr[S
end Jq`fD~(7
% END: Compute the Zernike Polynomials am05>c9
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (;h]'I@
j|(bDa4\
% Compute the Zernike functions: XT_BiZ%l5O
% ------------------------------ ?-'Q-\j
idx_pos = m>0; |qNrj~n@
idx_neg = m<0; V2]S{!p}k
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z = y; /ADxHw`k
if any(idx_pos) 0KT{K(
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); S8vmXlD
end emS +%6U
if any(idx_neg) 90aPIs-
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); r5iO%JFg
end cmN0ya
"x$S%:p
% EOF zernfun