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 YR70BOxK
function z = zernfun(n,m,r,theta,nflag) Om<a<q
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. @CoIaUVP
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N V+\Wb[zDJ
% and angular frequency M, evaluated at positions (R,THETA) on the TvM~y\s
% unit circle. N is a vector of positive integers (including 0), and WA qINLdX
% M is a vector with the same number of elements as N. Each element K:M8h{Ua
% k of M must be a positive integer, with possible values M(k) = -N(k) +t.b` U`-
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, RFGffA&
% and THETA is a vector of angles. R and THETA must have the same l] vm=7:
% length. The output Z is a matrix with one column for every (N,M) +_!QSU,@
% pair, and one row for every (R,THETA) pair. @W<m4fi
% VUc%4U{Cti
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike RCrCs
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), =M1I>
% with delta(m,0) the Kronecker delta, is chosen so that the integral #Z #-Ht
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, #mT"gs
% and theta=0 to theta=2*pi) is unity. For the non-normalized A,]h),b
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. hPh-+Hb
%
9sP0D
% The Zernike functions are an orthogonal basis on the unit circle. `L
zPotz
% They are used in disciplines such as astronomy, optics, and =I<R! ZSN
% optometry to describe functions on a circular domain. ,uvRi)O>a
% bcyzhK=
% The following table lists the first 15 Zernike functions. .}t
e>]A*
% VVZ'i.*_3?
% n m Zernike function Normalization GyIV
Hby
% -------------------------------------------------- @~e5<:|5#
% 0 0 1 1 hxx.9x>ow
% 1 1 r * cos(theta) 2 6863xOv{T
% 1 -1 r * sin(theta) 2 mw!F{pw
% 2 -2 r^2 * cos(2*theta) sqrt(6) _t$sgz&
% 2 0 (2*r^2 - 1) sqrt(3) ?[AD=rUC
% 2 2 r^2 * sin(2*theta) sqrt(6) wJ]d&::@h
% 3 -3 r^3 * cos(3*theta) sqrt(8) SBpL6~NW
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) sK{e*[I>W
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) x:;kSh
% 3 3 r^3 * sin(3*theta) sqrt(8) 8}[).d160
% 4 -4 r^4 * cos(4*theta) sqrt(10) XSDpRo
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Y73C5.dNcE
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) IPk4
;,
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) )4OxY[2J
% 4 4 r^4 * sin(4*theta) sqrt(10) ixFi{_
% -------------------------------------------------- +0&/g&a\R
% `A >@]d
% Example 1: AdEMa}u6
% xAr\gu
% % Display the Zernike function Z(n=5,m=1) -~0^P,yQ
% x = -1:0.01:1; S!UaH>Rh
% [X,Y] = meshgrid(x,x); ^ c<Ve'-
% [theta,r] = cart2pol(X,Y); R5D1w+
% idx = r<=1; )UR7i8]!0
% z = nan(size(X)); %;_MGae
% z(idx) = zernfun(5,1,r(idx),theta(idx)); ZH8,KY"
% figure &HW9Jn
% pcolor(x,x,z), shading interp CY1Z'
% axis square, colorbar t!XwW$@
% title('Zernike function Z_5^1(r,\theta)') WLT"ji0w2
% (e~N q
% Example 2: +2{Lh7Ks
% Oz95
% % Display the first 10 Zernike functions 6N4~~O
% x = -1:0.01:1; L_T5nD^D
% [X,Y] = meshgrid(x,x); p'%s=TGwv
% [theta,r] = cart2pol(X,Y); N['.BN
% idx = r<=1; yAt^;
% z = nan(size(X)); [~HN<>L@C
% n = [0 1 1 2 2 2 3 3 3 3]; siI;"?
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; bw7@5=?;
% Nplot = [4 10 12 16 18 20 22 24 26 28]; DUS6SO
% y = zernfun(n,m,r(idx),theta(idx)); QV!up^Zso
% figure('Units','normalized') ,F|f. 7;
% for k = 1:10 (HVGlw'`
% z(idx) = y(:,k); EwN}l
% subplot(4,7,Nplot(k)) zfU{Kd
% pcolor(x,x,z), shading interp ;I}fBZ3
% set(gca,'XTick',[],'YTick',[]) K-4PI+qQ\
% axis square dH!*!r>
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) HfVZ~PP
% end &ncvGDGi
% L,\Iasv
% See also ZERNPOL, ZERNFUN2. }7Uoh(d
r@V!,k#S
% Paul Fricker 11/13/2006 ^W^OfY
>6T8^Nt
>7|VR:U?B
% Check and prepare the inputs: eFgA 8kY)
% ----------------------------- 3BI1fXT4=j
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) K0~rN.C!0
error('zernfun:NMvectors','N and M must be vectors.') It(_v
end 4 KiY6)
dN q$}
if length(n)~=length(m) K1KreYlF
error('zernfun:NMlength','N and M must be the same length.') By|4m
end Xvu(vA
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n = n(:); *@5 @,=d
m = m(:); =bOW~0Z1
if any(mod(n-m,2)) dd;~K&_Q/i
error('zernfun:NMmultiplesof2', ... fC`&g~yK'
'All N and M must differ by multiples of 2 (including 0).') 4x34u}l
end 4s-!7
e6*8K@LHB
if any(m>n) dPlV>IM$z
error('zernfun:MlessthanN', ... @JMiO^
'Each M must be less than or equal to its corresponding N.') .#gzP2 [q
end Ui~>SN>s
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=
if any( r>1 | r<0 ) !L(^(;$Kgr
error('zernfun:Rlessthan1','All R must be between 0 and 1.') (QEG4&9
end QRUz`|U
^qs $v06
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) SUiOJ[5,
error('zernfun:RTHvector','R and THETA must be vectors.') D*jM1w_`
end )9g2D`a4
X?O[r3<
r = r(:); .v
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theta = theta(:); /^ts9:
length_r = length(r); I7onX,U+
if length_r~=length(theta) ytImB`'\
error('zernfun:RTHlength', ... Txu/{M,
'The number of R- and THETA-values must be equal.') $Sq:q0
end !$JT e
kiEa<-]
% Check normalization: HMXE$d=[
% -------------------- -7ep{p-
if nargin==5 && ischar(nflag) 5pX6t
isnorm = strcmpi(nflag,'norm'); {}9a6.V;}
if ~isnorm YK_7ip.a[
error('zernfun:normalization','Unrecognized normalization flag.') =_CzH(=f#
end %9"H
else /ZX}Nc g
isnorm = false; hN_]6,<\
end OUnA;_
4W75T2q#
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% F9^S"qv$
% Compute the Zernike Polynomials E.h*g8bXe
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% F ,kZU$
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% Determine the required powers of r: KNl$3nX
% ----------------------------------- >*bvw~y,
m_abs = abs(m); + {]j]OP
rpowers = []; ^iA9%zp
for j = 1:length(n) }>\C{ClI
rpowers = [rpowers m_abs(j):2:n(j)]; [),ige
end q.vIc
?a
rpowers = unique(rpowers); kJU2C=m@e2
P}iE+Z3
% Pre-compute the values of r raised to the required powers, G@0&8
% and compile them in a matrix: lE;!TQj:X
% ----------------------------- ;uW FHc5@B
if rpowers(1)==0 gYj'(jB
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); rv;3~'V
rpowern = cat(2,rpowern{:}); y =@N|f!
rpowern = [ones(length_r,1) rpowern]; GgU/!@
else _1^'(5f$
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ~DWl s.
rpowern = cat(2,rpowern{:}); *8q.YuZ
end )7@0[>
UiWg<_<t
% Compute the values of the polynomials: 2wn2.\v M
% -------------------------------------- 9WHddDA
y = zeros(length_r,length(n)); iU-j"&L5
for j = 1:length(n) %O<BfIZ
s = 0:(n(j)-m_abs(j))/2; 1C.VnzRnJ
pows = n(j):-2:m_abs(j); jIyQ]:* p
for k = length(s):-1:1 _F{C\}
p = (1-2*mod(s(k),2))* ... 2%1hdA<
prod(2:(n(j)-s(k)))/ ... [QTV9
prod(2:s(k))/ ... ?2a $*(
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... V&i;\ 9
prod(2:((n(j)+m_abs(j))/2-s(k))); GbyJ:
idx = (pows(k)==rpowers); Efe 7gE'
y(:,j) = y(:,j) + p*rpowern(:,idx); 5;?yCWc
end y(Td/rY.
^Cmyx3O^
if isnorm 0:+E-^X
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); zDp 2g)
end J,G
lIv.A
end 8t`?#8D}
% END: Compute the Zernike Polynomials z#N@ 0R
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -&f$GUTJ
`/g
UV
% Compute the Zernike functions: ^aQ"E9
% ------------------------------ K,]=6Rj
idx_pos = m>0; j pOp.
idx_neg = m<0; +p^u^a
<#.g=ay
z = y; =sFTxd_"iQ
if any(idx_pos) !wNO8;(
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); <VcQ{F
end d _
e WcI
if any(idx_neg) iE{&*.q_}>
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); @;kSx":b
end BY*Q_Et
>p/`;Kq@
% EOF zernfun