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 C'/M/|=Q#
function z = zernfun(n,m,r,theta,nflag) $H-D9+8 7
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. |4.o$*0Y
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N q'F_j"
% and angular frequency M, evaluated at positions (R,THETA) on the ynZ[c8.
% unit circle. N is a vector of positive integers (including 0), and 3
9{"T0
% M is a vector with the same number of elements as N. Each element $;uWj|
% k of M must be a positive integer, with possible values M(k) = -N(k) }<ONx g6Kb
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, S"TMsi
% and THETA is a vector of angles. R and THETA must have the same yF5
% length. The output Z is a matrix with one column for every (N,M) *C@[5#CA2z
% pair, and one row for every (R,THETA) pair. DJYXC,r
% N~;
khS]
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike &U$8zn~[k
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 9id~NNr7
% with delta(m,0) the Kronecker delta, is chosen so that the integral cbCE
$
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, MAe<.DHY
% and theta=0 to theta=2*pi) is unity. For the non-normalized @=NVOJy}c
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 5m.KtnT)
% G:c8`*5Q
% The Zernike functions are an orthogonal basis on the unit circle. \W`} L
% They are used in disciplines such as astronomy, optics, and .aismc`=
% optometry to describe functions on a circular domain. >}DjHLTW\
% zz 'dg-F
% The following table lists the first 15 Zernike functions. AIl$qPKj&
% hG~]~ )
% n m Zernike function Normalization O<dZA=Oez
% -------------------------------------------------- \gp,Txueb
% 0 0 1 1 VUy)4*
% 1 1 r * cos(theta) 2 w
<#*O:
% 1 -1 r * sin(theta) 2 $]%<r?MUb-
% 2 -2 r^2 * cos(2*theta) sqrt(6) n`m_S
% 2 0 (2*r^2 - 1) sqrt(3) O:,2OMB}B`
% 2 2 r^2 * sin(2*theta) sqrt(6) a(ux?V)E.
% 3 -3 r^3 * cos(3*theta) sqrt(8) !/4V^H
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) YR|(;B
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) u1;e*ty
% 3 3 r^3 * sin(3*theta) sqrt(8) o7Cnyy#:
% 4 -4 r^4 * cos(4*theta) sqrt(10) iVKbGgA
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) n4vXm
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) N{^>MRK=5
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ,"N3k(g
% 4 4 r^4 * sin(4*theta) sqrt(10) ^3WIl]
% -------------------------------------------------- sm2p$3v
% UN *dU
% Example 1: lbKv
% 6)#- 5m
% % Display the Zernike function Z(n=5,m=1) g<2lPH
% x = -1:0.01:1; S< EB&P
% [X,Y] = meshgrid(x,x); fXu~69_
% [theta,r] = cart2pol(X,Y); 9B+ zJ Vte
% idx = r<=1; 7O8V1Tt
% z = nan(size(X));
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% z(idx) = zernfun(5,1,r(idx),theta(idx)); JA?,0S
% figure y\)G7
(
% pcolor(x,x,z), shading interp |D;"D
% axis square, colorbar S2'`|uI
% title('Zernike function Z_5^1(r,\theta)') +EST58
% ' 1P=^
% Example 2: ,5eH2W
% nE]~E xr
% % Display the first 10 Zernike functions `z-H]fU
% x = -1:0.01:1; vh|Tb5W<
% [X,Y] = meshgrid(x,x); u=@h`5-fp
% [theta,r] = cart2pol(X,Y); [GR]!\!%~
% idx = r<=1; jh 7p62R
% z = nan(size(X)); {?EEIfg
% n = [0 1 1 2 2 2 3 3 3 3]; y:g7'+c
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; ]RH=s7L
% Nplot = [4 10 12 16 18 20 22 24 26 28]; 8zQ_xE
% y = zernfun(n,m,r(idx),theta(idx)); i{tTUA
% figure('Units','normalized') gx!*O<|e4
% for k = 1:10 1u"R=D9p,=
% z(idx) = y(:,k); ^8?j~&u$F
% subplot(4,7,Nplot(k)) wJ80};!
% pcolor(x,x,z), shading interp 1<LC8?wt
% set(gca,'XTick',[],'YTick',[]) \LO_Nu9
% axis square r{K\(UT]!
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) s{(ehP.Dd
% end H$~M`Y9I~
% WF ?/GN
% See also ZERNPOL, ZERNFUN2. Lnh':7FQJx
2
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% Paul Fricker 11/13/2006 CrnB{Z4L
*.kj]BoO
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% Check and prepare the inputs: sg3h i"Im
% ----------------------------- KI Ek/]<H
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) o"'iXUJ
error('zernfun:NMvectors','N and M must be vectors.') PHQ{-b?4t
end R&6n?g6@/V
Ms.PO{wb
if length(n)~=length(m) wrH7 pd
error('zernfun:NMlength','N and M must be the same length.') vP3K7En
end =E;=+eqt
a`7%A H)
n = n(:); #V<`U:.
m = m(:); /a@ k S
if any(mod(n-m,2)) CnabD{uTf
error('zernfun:NMmultiplesof2', ... y._'K+nl
'All N and M must differ by multiples of 2 (including 0).') Z:I*y7V-
end %z(9lAe
Px'R`1^
if any(m>n) $Llta,ULE
error('zernfun:MlessthanN', ... OI~}e,[2z
'Each M must be less than or equal to its corresponding N.') V3##
B}2[Y
end J1.qhy>
W;^N8ap%
if any( r>1 | r<0 ) 4Z*|Dsw
error('zernfun:Rlessthan1','All R must be between 0 and 1.') %/P=m-K
end O[; +i
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Oc7 >S.1
error('zernfun:RTHvector','R and THETA must be vectors.') Af`z/:0<
end ;xL67e%?
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r = r(:); >OT\~C
theta = theta(:); V?=TVI*k
length_r = length(r); ^rL,&rk
if length_r~=length(theta) Was'A+GZ
error('zernfun:RTHlength', ... zCBplb
'The number of R- and THETA-values must be equal.') f:xUPH?+
end Z,3 CC \
f7Yz>To
% Check normalization: -<6v:Z
% -------------------- d;{y`4p)s
if nargin==5 && ischar(nflag) EY]a6@;
isnorm = strcmpi(nflag,'norm'); p:B
]Ft
if ~isnorm qB+n6y%
error('zernfun:normalization','Unrecognized normalization flag.') pqJ)G;%9
end Z
#EvRC
else P2Onkl
isnorm = false; CQ<8P86gt
end VO9XkA7
8zAg;b[
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% JfkTw~'R
% Compute the Zernike Polynomials =:4?>2)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% r]9 e^
3)y{n%3L
% Determine the required powers of r: _D-5}a"
% ----------------------------------- D%A@lMru
m_abs = abs(m); d4J<,
rpowers = []; zHV|-R
for j = 1:length(n) > =Jsv
rpowers = [rpowers m_abs(j):2:n(j)];
P&mtA2
end sW?B7o?
rpowers = unique(rpowers); [g +y_@9s
~Yl<S(/4
% Pre-compute the values of r raised to the required powers, z`OkHX*+2|
% and compile them in a matrix: QTYYghz
% ----------------------------- 9Fk4|+OJ
if rpowers(1)==0 Yc
d3QRB
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false);
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rpowern = cat(2,rpowern{:}); v{.\iIg N
rpowern = [ones(length_r,1) rpowern]; o_O+u%y
else )
oxIzF
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); E3f9<hm
rpowern = cat(2,rpowern{:}); z>|)ieL
end -?5$ PH
*2=W5LaK.
% Compute the values of the polynomials: {S*!B
% -------------------------------------- Mb/L~gd"
y = zeros(length_r,length(n)); gH'_ymT=
3
for j = 1:length(n) /1[gn8V691
s = 0:(n(j)-m_abs(j))/2; UQ~4c,
pows = n(j):-2:m_abs(j); /$Z
m~Mp
for k = length(s):-1:1 k-Fdj5/
p = (1-2*mod(s(k),2))* ... <raG07{!*
prod(2:(n(j)-s(k)))/ ... "XhOsMJ
prod(2:s(k))/ ... k}zd'
/b
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... xg} ug[
prod(2:((n(j)+m_abs(j))/2-s(k))); 5>P7]?U.]
idx = (pows(k)==rpowers); @zrNN>
y(:,j) = y(:,j) + p*rpowern(:,idx); waCboK'
end d&u7]<yDA
(zC
if isnorm }/p/pVz
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); .H2qs{N!
end $/paEn"
end } L <,eV
% END: Compute the Zernike Polynomials .LObOR5J7
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +]c}rWm
5B{k\H;
% Compute the Zernike functions: qm'b'!gq~
% ------------------------------ .T$D^?G!D
idx_pos = m>0; g4wZvra6%)
idx_neg = m<0; {a@>6)
0[)VO[
z = y; |l7%l&!
if any(idx_pos) 2tf6GX:
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); KDD@%E
end Sl>>SP
if any(idx_neg) jV^C19
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Hbk&6kS
end ?'sXgo.}
!5UfWk\G
% EOF zernfun