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 ^|(4j_.(e
function z = zernfun(n,m,r,theta,nflag) ;XQ lj?:
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. R9G)X]
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N vaJXX
% and angular frequency M, evaluated at positions (R,THETA) on the )0MshgM
% unit circle. N is a vector of positive integers (including 0), and chzR4"WZFt
% M is a vector with the same number of elements as N. Each element Vp"Ug,1
% k of M must be a positive integer, with possible values M(k) = -N(k) $50"3g!Y
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, w*}yw"gP*0
% and THETA is a vector of angles. R and THETA must have the same K(fLqXE%
% length. The output Z is a matrix with one column for every (N,M) UDtbfc7bk
% pair, and one row for every (R,THETA) pair. <>Ddxmw
% [c[MQA0
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike BG0Mj2
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), }_l
-'t
% with delta(m,0) the Kronecker delta, is chosen so that the integral /Py>HzRE:
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, i/~QJ1C
% and theta=0 to theta=2*pi) is unity. For the non-normalized HKN"$(Q
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. G2{ M#H
% @Qjl`SL%O^
% The Zernike functions are an orthogonal basis on the unit circle. *oX]=u&
% They are used in disciplines such as astronomy, optics, and L^{;jgd&T9
% optometry to describe functions on a circular domain. Mq lo:7
^F
% l~!fQ$~
% The following table lists the first 15 Zernike functions. ~.9o{?pbG
% EZumJ."
% n m Zernike function Normalization pQ^,. [[
% -------------------------------------------------- wW! r}I#
% 0 0 1 1 &W<>^C2v
% 1 1 r * cos(theta) 2 39aCwhh7v
% 1 -1 r * sin(theta) 2 Q>a7Ps@~
% 2 -2 r^2 * cos(2*theta) sqrt(6) n!eqzr{
% 2 0 (2*r^2 - 1) sqrt(3) zo7XmUI3P
% 2 2 r^2 * sin(2*theta) sqrt(6) 'BdmFKy1
% 3 -3 r^3 * cos(3*theta) sqrt(8) eGe[sv"k
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8)
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% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) Q] yT
% 3 3 r^3 * sin(3*theta) sqrt(8) lH@E %
% 4 -4 r^4 * cos(4*theta) sqrt(10) _Z66[T+M
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) kbp(
a+5
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 2]aZe4H.
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) io r [v
% 4 4 r^4 * sin(4*theta) sqrt(10) #+Yp^6zg
% -------------------------------------------------- .4C[D{4
% Lr?4Y
% Example 1: ncJFB,4
% J6(
RlHS;
% % Display the Zernike function Z(n=5,m=1) v;bP8)mI
% x = -1:0.01:1; kuj12
% [X,Y] = meshgrid(x,x); 7l#2,d4
% [theta,r] = cart2pol(X,Y); g
y e(/N+I
% idx = r<=1; *iRm`)zC(
% z = nan(size(X)); PVD ~W)0m*
% z(idx) = zernfun(5,1,r(idx),theta(idx)); _95}ifSVm
% figure qM1)3.)[:
% pcolor(x,x,z), shading interp Jm(&G
% axis square, colorbar !`
M;#
% title('Zernike function Z_5^1(r,\theta)') *)`kx
% 2^ ,H_PS
% Example 2: Y(
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% ,v}?{pc
% % Display the first 10 Zernike functions 0ve`
% x = -1:0.01:1; ,P@/=I5
% [X,Y] = meshgrid(x,x);
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% [theta,r] = cart2pol(X,Y); U!\2K~
% idx = r<=1; i2FD1*=/?
% z = nan(size(X)); ;]&~D
+XH
% n = [0 1 1 2 2 2 3 3 3 3]; u3*NO
)O
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; "0'*q<8
% Nplot = [4 10 12 16 18 20 22 24 26 28]; eN]>l
% y = zernfun(n,m,r(idx),theta(idx)); (,Ja
% figure('Units','normalized') lLkmcHu
% for k = 1:10 4P4 Fo1
% z(idx) = y(:,k); W%>i$:Qq
% subplot(4,7,Nplot(k)) {7=WU4$
% pcolor(x,x,z), shading interp G !1~i*P$u
% set(gca,'XTick',[],'YTick',[]) AvrL9D
% axis square
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% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) vcw>v={x
% end bCA2ik
% J+71FP`ZH
% See also ZERNPOL, ZERNFUN2. ]|,q|c ,
Z&dr0w8
% Paul Fricker 11/13/2006 a/QtJwIV
so!w !O@@
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\b8,
% Check and prepare the inputs: =sE2}/g
% ----------------------------- QY~<~<d+G
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) v@fe-T&0
error('zernfun:NMvectors','N and M must be vectors.') -t@y\vZF,
end 7b&JX'`Mb
\LdmGv@&
if length(n)~=length(m) &o*s !u
error('zernfun:NMlength','N and M must be the same length.') RIy5ww}3|
end {Ax)[<i
;-KAUgL2
n = n(:); _{LN{iqDv
m = m(:); %@}o'=[
if any(mod(n-m,2)) )-+\M_JK5
error('zernfun:NMmultiplesof2', ... rU=b?D)n!w
'All N and M must differ by multiples of 2 (including 0).') Mw"xm9(Q
end .M9d*qp`S
eg"=H50
if any(m>n) R^J.?>0
error('zernfun:MlessthanN', ... TL},Unq
'Each M must be less than or equal to its corresponding N.') RzA2*]%a
end pk-yj~F }
jWH{;V&ZV
if any( r>1 | r<0 ) A 1T<
error('zernfun:Rlessthan1','All R must be between 0 and 1.') #XTY7,@P
end E rop9T1
.FIt.XPzv
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 1t/dxB;
error('zernfun:RTHvector','R and THETA must be vectors.') 1~}m.ER
end =X-^YG3x
B{7Kzwh;
r = r(:); ]y3pE}R
theta = theta(:); kOs(?=
length_r = length(r); yicO!:bM
if length_r~=length(theta) )W&o?VRfO
error('zernfun:RTHlength', ... ^FP}
qW~;9
'The number of R- and THETA-values must be equal.') J DLTOLG
end $_Y/'IN`k
9[cp7 Rcb
% Check normalization: {S[I_\3
% -------------------- i 8l./Yt/
if nargin==5 && ischar(nflag)
-Y*VgoK%
isnorm = strcmpi(nflag,'norm'); &qJPwO
if ~isnorm ;% 2wGT
error('zernfun:normalization','Unrecognized normalization flag.') `J72+ RA
end ?h/xAl
else 8 YNu<
isnorm = false; >(hSW~i~
end Ne3R.g9;Z
r& vFikIz
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 7OB%A&
% Compute the Zernike Polynomials Q*]$)D3n
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Lj}>Xy(7<
C>.e+V+':
% Determine the required powers of r: <0CzB"Ap
% ----------------------------------- h }<0 /
m_abs = abs(m); 3pvYi<<D'
rpowers = []; e# t3u_
for j = 1:length(n) U1OFDXHG
rpowers = [rpowers m_abs(j):2:n(j)]; R)ERxz#
end 94\t1fE
rpowers = unique(rpowers); &~RR&MdZ2
BR+nL6sU
% Pre-compute the values of r raised to the required powers, z9[[C^C
% and compile them in a matrix: l
:/&E 6 9
% ----------------------------- ~A6 "sb=
if rpowers(1)==0 fX_#S|DlSG
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); [`d$X^<y;
rpowern = cat(2,rpowern{:}); Jlp<koy
rpowern = [ones(length_r,1) rpowern]; !<&m]K
else nSS>\$
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); !lAD
q|$
rpowern = cat(2,rpowern{:}); sONBQ9
end OA[&Za#w
z"lqrSJ:
% Compute the values of the polynomials: @}WNKS&m
% -------------------------------------- MU'@2c
y = zeros(length_r,length(n)); :p' VbQZ{
for j = 1:length(n) ^(ScgoXva
s = 0:(n(j)-m_abs(j))/2; P.djd$#
pows = n(j):-2:m_abs(j); ;imRh'-V6
for k = length(s):-1:1 $$hv`HE^l
p = (1-2*mod(s(k),2))* ... n"6;\
prod(2:(n(j)-s(k)))/ ... b.b@bq$1
prod(2:s(k))/ ... UfO7+_2
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... D==Mb~
prod(2:((n(j)+m_abs(j))/2-s(k))); 3o*FPO7?
idx = (pows(k)==rpowers); P-CB;\
y(:,j) = y(:,j) + p*rpowern(:,idx); 2edBQYWd
end rz%<AF Z
ZQ3_y $
if isnorm 6-B 9na
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); LvJGvj
end l?/Y
end c8{]]
% END: Compute the Zernike Polynomials JS2nXs1
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% C)Jn[/BD
+R6a}d/K
% Compute the Zernike functions: mf' ]O,
% ------------------------------ *#y;8
idx_pos = m>0; HRB[GP+
idx_neg = m<0; !g>.i`
aQ#qRkI
z = y; ?7[alV ~
if any(idx_pos) jTb-;4N'
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); {fV}gR2
end O oSb>Y/4
if any(idx_neg) r[_4Lo@G
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); e8}Ezy"^
end ~9=aT1S|
]JE TeZ^/
% EOF zernfun