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 ,?~,"IQyi[
function z = zernfun(n,m,r,theta,nflag) irj}:f;!eF
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. :S6 <v0`Z
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N y s6"Q[B
% and angular frequency M, evaluated at positions (R,THETA) on the G)|HFcE
% unit circle. N is a vector of positive integers (including 0), and 8^i,M^f^{
% M is a vector with the same number of elements as N. Each element oioN0EuDk
% k of M must be a positive integer, with possible values M(k) = -N(k) _tJURk%
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, oYx
f((x
% and THETA is a vector of angles. R and THETA must have the same yN%Pe:R
% length. The output Z is a matrix with one column for every (N,M) A~SSu.L@
% pair, and one row for every (R,THETA) pair. W\Y
4%y}
% >&Lu0oHH
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike IQY#EyTb
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), !`E2O*g
% with delta(m,0) the Kronecker delta, is chosen so that the integral A1T;9`E
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, vG:,oB}
% and theta=0 to theta=2*pi) is unity. For the non-normalized u)>*U'bM
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ?IO/zkeXg
% (:]iHg3
% The Zernike functions are an orthogonal basis on the unit circle. 824%]i3
% They are used in disciplines such as astronomy, optics, and vtjG&0GSK
% optometry to describe functions on a circular domain. cu|q&
% e$I:[>
% The following table lists the first 15 Zernike functions. P^+>QJ1
% * OFT)S
% n m Zernike function Normalization Py<vN!
% -------------------------------------------------- e{S`iO
% 0 0 1 1 "+Rm4_
% 1 1 r * cos(theta) 2 xF0*q
% 1 -1 r * sin(theta) 2 PmTd+Gj$
% 2 -2 r^2 * cos(2*theta) sqrt(6) $"1&!
% 2 0 (2*r^2 - 1) sqrt(3) mz'8
% 2 2 r^2 * sin(2*theta) sqrt(6) 5OE?;PJ(
% 3 -3 r^3 * cos(3*theta) sqrt(8) 6Z:|"AwC2
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) .1M>KRSr,
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) wt,N<L
% 3 3 r^3 * sin(3*theta) sqrt(8) i/B"d,=<
% 4 -4 r^4 * cos(4*theta) sqrt(10) 4}j}8y2)H
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) .<hv&t
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) :g_ +{4
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) W0hLh<Go
% 4 4 r^4 * sin(4*theta) sqrt(10) a)b@en;v
% -------------------------------------------------- |V]E8Qt
% 2V 'Tt3
% Example 1: |3@]5f&
% "5bk82."
% % Display the Zernike function Z(n=5,m=1) (>23[;.0
% x = -1:0.01:1; ktb.fhO
% [X,Y] = meshgrid(x,x); '(*D3ysU
% [theta,r] = cart2pol(X,Y); 6 , ~aV
% idx = r<=1; cMAfW3j: ;
% z = nan(size(X)); K*[wr@)u
% z(idx) = zernfun(5,1,r(idx),theta(idx)); oQO3:2a
% figure Atw^C+"vW&
% pcolor(x,x,z), shading interp c:5BQr
'
% axis square, colorbar QB>e(j%
% title('Zernike function Z_5^1(r,\theta)') S/aPYrk>6
% 9X~^w_cdk
% Example 2: cj)~7 WF
% T@.CwV
% % Display the first 10 Zernike functions wAYc)u#
% x = -1:0.01:1; >LSA?dy!?
% [X,Y] = meshgrid(x,x); -TWo-iu^
% [theta,r] = cart2pol(X,Y); 5`Z#m:+u
% idx = r<=1; ;MD{p1w
% z = nan(size(X)); #.RI9B
% n = [0 1 1 2 2 2 3 3 3 3]; *lSIT]1
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; 8wd2\J,]
% Nplot = [4 10 12 16 18 20 22 24 26 28]; s+11) ~
% y = zernfun(n,m,r(idx),theta(idx)); U_?RN)>j
% figure('Units','normalized') \I=:,cz*,
% for k = 1:10 &0`L; 1R
% z(idx) = y(:,k); `,O^=HBM
% subplot(4,7,Nplot(k)) M
5h U.3.L
% pcolor(x,x,z), shading interp ORTM[cL
% set(gca,'XTick',[],'YTick',[]) OZ&aTm :
% axis square ADDp m-]
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) :H{8j}"
% end E {MSi"
% <LE>WfmC
% See also ZERNPOL, ZERNFUN2. bH&H\ Mx_k
\l~h#1|%;s
% Paul Fricker 11/13/2006 &nYmVwi?"Q
&wfM:a/c
STMcMm3
% Check and prepare the inputs: {+MMqJCa
% ----------------------------- :?TV6M
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ~zx-'sc?
error('zernfun:NMvectors','N and M must be vectors.') C-7.Sa
end 2iu;7/
-?[:Zn~$a
if length(n)~=length(m) aSj$62G"
error('zernfun:NMlength','N and M must be the same length.') S@_GjCpn
end mP-+];gg
=
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n = n(:); s`{O-
m = m(:); LQe<mZ<
if any(mod(n-m,2)) Y9u2:y!LdL
error('zernfun:NMmultiplesof2', ... J_,y?}.e3
'All N and M must differ by multiples of 2 (including 0).') 4%pvw;r
end cg4,PI%hz
8PQ& 7o
if any(m>n) laAG%lq/'
error('zernfun:MlessthanN', ... YG% Zw
'Each M must be less than or equal to its corresponding N.') C5m*pGImG
end g7F>o76M
QwiC2}/
if any( r>1 | r<0 ) Uhf
-}Jdw
error('zernfun:Rlessthan1','All R must be between 0 and 1.') 3,GSBiK3}
end k~H-:@
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) QeK~A@|F&
error('zernfun:RTHvector','R and THETA must be vectors.') JS4pJe\q
end yF*JzE 7,
l4; LV7Ji
r = r(:); jE{z4en
theta = theta(:); A;kB"Tx
length_r = length(r); kAqk~.
if length_r~=length(theta) 5<u+2x8|
error('zernfun:RTHlength', ... PW}Yts7p
'The number of R- and THETA-values must be equal.') L%"&_v#a^
end `VHm,g2
'=oV
% Check normalization: Ws=J)2q
% -------------------- h"[
][
if nargin==5 && ischar(nflag) 4m~\S)ad
isnorm = strcmpi(nflag,'norm'); "k+QDQ3=
if ~isnorm JO
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error('zernfun:normalization','Unrecognized normalization flag.') E*kS{2NAq
end 1vobfZ-w9
else X/@Gx 4
isnorm = false; hM;E UWv
end wc;5tb#
<4Ak$E%"
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% XVY^m}pMe
% Compute the Zernike Polynomials A/'G.H
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -wY6da*.W
'0[l'Dt'
% Determine the required powers of r: 4kx#=MLt
% ----------------------------------- R^D~ic
N
m_abs = abs(m); k(s3~S2h
rpowers = []; p.zU9rID
for j = 1:length(n) [}FP_Su$6
rpowers = [rpowers m_abs(j):2:n(j)]; 7m1*Q@D
end r8@:Ko= a
rpowers = unique(rpowers); }=wSfr9g
Nz2}Ma 2
% Pre-compute the values of r raised to the required powers, 0^hz 1\g
% and compile them in a matrix: 8R)*8bb
% -----------------------------
}UX >O
if rpowers(1)==0 2f4 *r^
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 'I;pS)sb
rpowern = cat(2,rpowern{:}); b+hZ<U/
rpowern = [ones(length_r,1) rpowern]; ~fr1O`8
else bvAO(`
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); .sCo,
rpowern = cat(2,rpowern{:}); 64[j:t=N
end WWD\EDnS
iHTxD1D+H
% Compute the values of the polynomials: <>p\9rVp*^
% -------------------------------------- Q5baY\"9^
y = zeros(length_r,length(n)); No j6Ina
for j = 1:length(n) 8^+Qn/b_%
s = 0:(n(j)-m_abs(j))/2; 7kleBDDT
pows = n(j):-2:m_abs(j); .0Cpqn,[
for k = length(s):-1:1 ;5oY)1
p = (1-2*mod(s(k),2))* ... 89~) nV)
prod(2:(n(j)-s(k)))/ ... cJL>,Z<|%
prod(2:s(k))/ ... b>G!K)MS3
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... AM\`v'I*6
prod(2:((n(j)+m_abs(j))/2-s(k))); [S'ngQ"f`
idx = (pows(k)==rpowers); }(ot IqE
y(:,j) = y(:,j) + p*rpowern(:,idx); d[jxU/.p;
end C#;}U51:t
GN(PH/fO9
if isnorm z;1yZ4[G
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); I 8e{%PK
end z9E*Mh(NE
end ZCV&v47\p_
% END: Compute the Zernike Polynomials mR?OSeeB
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ."cC^og
km.xy_v
% Compute the Zernike functions: _epi[zf@
% ------------------------------ =f?| f
idx_pos = m>0; *S`&
XPj
idx_neg = m<0; >|mmJ4T
J$@3,=L6V
z = y; fk;39$[
if any(idx_pos) BPtU]Bv-
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); vxY7/ _]
end HSq&'V
if any(idx_neg) L~CwL
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); r C$ckug
end B!yAam#^
>4b-NS/}0
% EOF zernfun