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 o[^% 0uVF
function z = zernfun(n,m,r,theta,nflag) ,U2
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%ZERNFUN Zernike functions of order N and frequency M on the unit circle. ;}3wT,=sN
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N pL>Q'{7s3
% and angular frequency M, evaluated at positions (R,THETA) on the xfqgK D>
% unit circle. N is a vector of positive integers (including 0), and r4jW=?|
% M is a vector with the same number of elements as N. Each element l%lkDh!$"
% k of M must be a positive integer, with possible values M(k) = -N(k) GAbX.9[V
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, Os9xZ
% and THETA is a vector of angles. R and THETA must have the same |UM':Ec
% length. The output Z is a matrix with one column for every (N,M) !l@IG C
% pair, and one row for every (R,THETA) pair. DqrS5!C
% NFPW#-TF
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike O'U0Y8HN
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), q~.\NKc
% with delta(m,0) the Kronecker delta, is chosen so that the integral R>[2}R30
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, +Tde#T&[
% and theta=0 to theta=2*pi) is unity. For the non-normalized URmx8=q
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. _S/bwPj|~y
% 4p&qH igG
% The Zernike functions are an orthogonal basis on the unit circle. }S3m
wp<Y
% They are used in disciplines such as astronomy, optics, and I-4csw<Qy
% optometry to describe functions on a circular domain. |vA3+kG
% gSK
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% The following table lists the first 15 Zernike functions. e{.2*>pH
% nX<!n\J T
% n m Zernike function Normalization ow]S 3[07
% -------------------------------------------------- l%.3hId-
% 0 0 1 1 cnC&=6=a<
% 1 1 r * cos(theta) 2 /K<Xr[z~y
% 1 -1 r * sin(theta) 2 m C_v!nL.
% 2 -2 r^2 * cos(2*theta) sqrt(6) 5 |{0|mP
% 2 0 (2*r^2 - 1) sqrt(3) ry3;60E\)
% 2 2 r^2 * sin(2*theta) sqrt(6) \TkBV?W
% 3 -3 r^3 * cos(3*theta) sqrt(8) f8_5.vlw
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) ,SuF1&4
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) ZU7e1VaZM
% 3 3 r^3 * sin(3*theta) sqrt(8) ~d?7\:n
% 4 -4 r^4 * cos(4*theta) sqrt(10) [oKc<o7)~"
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) jwyJ=W-
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) R*/%+
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) {%^q8l4j
% 4 4 r^4 * sin(4*theta) sqrt(10) y _>HQs,:
% -------------------------------------------------- SoS[yr
% .?CDWbzq
% Example 1: V'
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a
% KQaw*T[Q3w
% % Display the Zernike function Z(n=5,m=1) d%VGfSrKq
% x = -1:0.01:1; 8b!&TP~m1
% [X,Y] = meshgrid(x,x); 1$?O5.X:
% [theta,r] = cart2pol(X,Y); Erl"X}P
% idx = r<=1; jY$Bns&.w
% z = nan(size(X)); 1Jc-hrN-
% z(idx) = zernfun(5,1,r(idx),theta(idx)); Yq4_ss'nB
% figure BQ,]]}e43z
% pcolor(x,x,z), shading interp ;"
'`P[
% axis square, colorbar k]`I3>/L
% title('Zernike function Z_5^1(r,\theta)') +dSe"W9
% "]JE]n}Ulg
% Example 2: ]zmY]5
% \gki!!HQ
% % Display the first 10 Zernike functions QL"fC;xUn,
% x = -1:0.01:1; iW+ZI6@
% [X,Y] = meshgrid(x,x); ae{%*
\J
% [theta,r] = cart2pol(X,Y); $dFEC}1t
% idx = r<=1; '{QbjG%<P
% z = nan(size(X)); ^;$a_eR
% n = [0 1 1 2 2 2 3 3 3 3]; PbJn8o
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; K SDo)7`
% Nplot = [4 10 12 16 18 20 22 24 26 28]; {tk42}8k
% y = zernfun(n,m,r(idx),theta(idx)); Dsw(ti`@
% figure('Units','normalized') ]Hc`<P
% for k = 1:10 aN}yS=(Ff
% z(idx) = y(:,k); HZ5*PXg~
% subplot(4,7,Nplot(k)) &sh
%]o8
% pcolor(x,x,z), shading interp G?&0Z++
% set(gca,'XTick',[],'YTick',[]) tmDI2Z%7
% axis square M['8zN
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 29z+<?K{
% end =<y$5"|
% ce.'STm=
% See also ZERNPOL, ZERNFUN2. 8GN0487H
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% Paul Fricker 11/13/2006 pjvChl5
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5Z]`n
% Check and prepare the inputs: pi q%b]
% ----------------------------- Ry,_%j3
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 4gG&u33RrE
error('zernfun:NMvectors','N and M must be vectors.') }N#jA yp!
end NYM$0v`0YK
iSUn}%YFz!
if length(n)~=length(m) qtnLQl"M
error('zernfun:NMlength','N and M must be the same length.') ah>;wW!6/
end ;}#tm9S;
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n = n(:); eY"y[
m = m(:); XG@_Lcv*
if any(mod(n-m,2)) }at8b ^
error('zernfun:NMmultiplesof2', ... 7h<B:~(K
'All N and M must differ by multiples of 2 (including 0).') BLgmFE2
end f7)}A/$4+
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if any(m>n) Yn+/yz5k_
error('zernfun:MlessthanN', ... &Y\Vh}
'Each M must be less than or equal to its corresponding N.') [( BA:x1
end q{n~v>wU
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if any( r>1 | r<0 ) ?y>xC|kt
error('zernfun:Rlessthan1','All R must be between 0 and 1.') bqJL@!T
end 8wp)aGTcU
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) #*BcO-N
error('zernfun:RTHvector','R and THETA must be vectors.') %0YwaxXPn7
end Wy.2*+5FX0
HTao)`.
r = r(:); Q!7Er
theta = theta(:); gG1%.q
length_r = length(r); 2P`hdg
if length_r~=length(theta) ^2mmgN
error('zernfun:RTHlength', ... 5u'"m<4
'The number of R- and THETA-values must be equal.') pFXDo4eH
end J!3 X}@_N
{xi$'r
% Check normalization: sw6]Bc
% -------------------- )}\jbh>RH
if nargin==5 && ischar(nflag) G#ZU^%$M,
isnorm = strcmpi(nflag,'norm'); 3+u11'0=t
if ~isnorm tj;<Z.
error('zernfun:normalization','Unrecognized normalization flag.') =>-:o:Cu{
end cF_ Y}C
else {-2I^Ym 5i
isnorm = false; mg[=~&J^
end !R-M:|
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ~b+4rYNxU_
% Compute the Zernike Polynomials 4ZrX=e,
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% <%#M&9d)E
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% Determine the required powers of r: ~>3$Id:
% ----------------------------------- &s->,-,
m_abs = abs(m); *>h"}e41
rpowers = []; r@5_LD@f
for j = 1:length(n) 8G;
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rpowers = [rpowers m_abs(j):2:n(j)]; \\AufAkJ
end T~J6(,"
rpowers = unique(rpowers); r0379 _
}OZ%U2PU
% Pre-compute the values of r raised to the required powers, Ac0C,*|^
% and compile them in a matrix: 1q0DOf]!T
% ----------------------------- A6v02WG_1T
if rpowers(1)==0 }]$%aMxy T
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); xPWzm
hF
rpowern = cat(2,rpowern{:}); jq
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rpowern = [ones(length_r,1) rpowern]; \`}Rdr!p%
else W(Z_ac^e[
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 7dyGC:YuTL
rpowern = cat(2,rpowern{:}); i
2hP4<;h
end Eqc&iS~
S;sggeP7,
% Compute the values of the polynomials: |6'(yn
% -------------------------------------- 6+u}'mSj8
y = zeros(length_r,length(n)); N3
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for j = 1:length(n) .Qm"iOyM
s = 0:(n(j)-m_abs(j))/2; +kP)T(6
pows = n(j):-2:m_abs(j); e`
Z;}&
,
for k = length(s):-1:1 rCR?]1*Z
p = (1-2*mod(s(k),2))* ... _eb:"(m
prod(2:(n(j)-s(k)))/ ... :0|]cHm
prod(2:s(k))/ ... Tqz{{]%j~$
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... S1sNVW
prod(2:((n(j)+m_abs(j))/2-s(k))); U\a.'K50F
idx = (pows(k)==rpowers); #_0OYL`(mE
y(:,j) = y(:,j) + p*rpowern(:,idx); nd*9vxM
end {G&*\5W
`WQz_}TqB
if isnorm {XH!`\
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 1wP#?p)c
end =cI -<0QSn
end S&_Z,mT./
% END: Compute the Zernike Polynomials 2eo]D?}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Vp{! Ft8>
xS?[v&"2
% Compute the Zernike functions: j hf%ze
% ------------------------------ /?uA{/8
idx_pos = m>0; iU"jV*P]
idx_neg = m<0; KI)jP((
(8qD'(@
z = y; WP[h@#7<
if any(idx_pos) dZcRLLR
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); DjY&)oce(
end -x)Oo`
if any(idx_neg) xO?w8 *d
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); |YCGWJaci
end s\2t|d
`>KB8SY:qK
% EOF zernfun