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?wFEADn
function z = zernfun(n,m,r,theta,nflag) s;$
eq);
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. u*H2kn[DU
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N k%c ?$n"
% and angular frequency M, evaluated at positions (R,THETA) on the c*LnLK/m
% unit circle. N is a vector of positive integers (including 0), and qB"y'UW8
% M is a vector with the same number of elements as N. Each element ] _#[oS
% k of M must be a positive integer, with possible values M(k) = -N(k) bx`(d@
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, >N44&W
% and THETA is a vector of angles. R and THETA must have the same M*@MkN*u&
% length. The output Z is a matrix with one column for every (N,M) X/'B*y'=U
% pair, and one row for every (R,THETA) pair. #Etz}:%W
% r`6XF
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike QULrE+@
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), W5sVQ`S-
% with delta(m,0) the Kronecker delta, is chosen so that the integral E-n!3RQ(w
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, b/WVWDyob/
% and theta=0 to theta=2*pi) is unity. For the non-normalized `\#Qr|GC
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 9KCnitU
% ]+,Z()
% The Zernike functions are an orthogonal basis on the unit circle. >e8t
% They are used in disciplines such as astronomy, optics, and _U'edK]R
% optometry to describe functions on a circular domain. vR#A7y @!
% 5wr0+Xo
% The following table lists the first 15 Zernike functions. '(I"54W
% (9'MdH
% n m Zernike function Normalization lD\lFN(:
% -------------------------------------------------- <XGOcekG
% 0 0 1 1 @$Z5Ag!
% 1 1 r * cos(theta) 2 Hk$|.TjzI
% 1 -1 r * sin(theta) 2 P0UMMn\-#
% 2 -2 r^2 * cos(2*theta) sqrt(6) YjLPW@
% 2 0 (2*r^2 - 1) sqrt(3) Cl i k
% 2 2 r^2 * sin(2*theta) sqrt(6) nL@P{,J
% 3 -3 r^3 * cos(3*theta) sqrt(8) oM QH-\(}
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) "RZ)pav?
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) l&5| =
% 3 3 r^3 * sin(3*theta) sqrt(8) Mm|HA@W^
% 4 -4 r^4 * cos(4*theta) sqrt(10) oa47TqFt
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) >0B[
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) dzggl(
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) @v@'8E Q
% 4 4 r^4 * sin(4*theta) sqrt(10) $ 'HiNP
{c
% -------------------------------------------------- &)<]AG.vd!
% S ^2'O7uj
% Example 1:
PDM>6U
% ;/>~|@
% % Display the Zernike function Z(n=5,m=1) AaKILIIQZ
% x = -1:0.01:1; :cIE8<\%
% [X,Y] = meshgrid(x,x); ,T"(97"
% [theta,r] = cart2pol(X,Y); cb|`)"<HN
% idx = r<=1; Pbd#Fu;
% z = nan(size(X)); Iu%/~FgPj{
% z(idx) = zernfun(5,1,r(idx),theta(idx)); ?Q:se
% figure %|r@q
% pcolor(x,x,z), shading interp (47jop0RDQ
% axis square, colorbar nr-VzF7zu
% title('Zernike function Z_5^1(r,\theta)') <P$b$fh/
% 29x
"E$e
% Example 2: />.&
% mpK|I|-
% % Display the first 10 Zernike functions &> }MoB
% x = -1:0.01:1; =@w};e#D
% [X,Y] = meshgrid(x,x); a5]~%xdK
% [theta,r] = cart2pol(X,Y); CDoZv""
% idx = r<=1; ]:m*7p\uk
% z = nan(size(X)); *!'00fv
% n = [0 1 1 2 2 2 3 3 3 3]; +~8/7V22
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; m6+2rD
% Nplot = [4 10 12 16 18 20 22 24 26 28]; tJ2l_M^
% y = zernfun(n,m,r(idx),theta(idx)); KDg!Y(m{
% figure('Units','normalized') z8vFQO\I"
% for k = 1:10 \`|,wLgH
% z(idx) = y(:,k); 7o0ej#
% subplot(4,7,Nplot(k)) *l_1T4]S
% pcolor(x,x,z), shading interp bZ )3{
% set(gca,'XTick',[],'YTick',[]) 6Q>:g"_
% axis square .:l78>f
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) <J[*~v%(
% end t~,!a? S7
% Hagj^8
% See also ZERNPOL, ZERNFUN2. [ivJ&'vB
)1lYfJ
% Paul Fricker 11/13/2006 |VaXOdD`&
b>Vs5nY!
gaTI:SKzc
% Check and prepare the inputs: q+|Dm<Ug
% ----------------------------- :%!=Ej.J
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) vE6/B"b
error('zernfun:NMvectors','N and M must be vectors.') $o{f)'.>n
end Lr40rLx;u
C0KP,JS&
if length(n)~=length(m) tdZ: w
error('zernfun:NMlength','N and M must be the same length.') eEezd[p
end cg$7`/U
%+>I1G
n = n(:); X
B65,l
m = m(:); EC?!%iO`
if any(mod(n-m,2)) -%%2Pz0I
error('zernfun:NMmultiplesof2', ... f<0-'fGJd
'All N and M must differ by multiples of 2 (including 0).') +!.=M8[
end e?RHf_d3T-
?6tuo:gP
if any(m>n) 1fEV^5I
error('zernfun:MlessthanN', ... lq1pgM ?Kf
'Each M must be less than or equal to its corresponding N.') "1h|1'S50?
end 3u+~!yz
|CStw"Fog
if any( r>1 | r<0 ) HO&#Lv
error('zernfun:Rlessthan1','All R must be between 0 and 1.') W#-M|
end [$-y8`~(
7&l
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) _oe2pL&
error('zernfun:RTHvector','R and THETA must be vectors.') !oM1
end *gVRMSrx4
3 T&m
r = r(:); DQKhR sC
theta = theta(:); )CihqsA2
length_r = length(r); a"#5JcR3
if length_r~=length(theta) tw\/1wa.
error('zernfun:RTHlength', ... "d%":F(
'The number of R- and THETA-values must be equal.') o`h F1*yp
end %UgyGQeo
YadyRUE
% Check normalization: OW1[Y-o[
% -------------------- #}e)*(
if nargin==5 && ischar(nflag) `')3}
isnorm = strcmpi(nflag,'norm'); 70*Y4'u}A
if ~isnorm /d8PDc "
error('zernfun:normalization','Unrecognized normalization flag.') A5Y z|
end 8Qek![3^
else '0/t |V<
isnorm = false; M2vYOg`t:c
end v:s~Y
o D:?fs]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 8(5}Jo+
% Compute the Zernike Polynomials sq-[<ryk
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% /u>")f
ndW??wiM
% Determine the required powers of r: Ol D]*=.cO
% ----------------------------------- u=
!?<Q
m_abs = abs(m); vezX/x D?
rpowers = []; F|%[s|s
for j = 1:length(n) Pou`PNvH
rpowers = [rpowers m_abs(j):2:n(j)]; Z?CmD;W
end v#nYH?+~mJ
rpowers = unique(rpowers); I tp7X
G W|~sE +
% Pre-compute the values of r raised to the required powers, wUW+S5"K
% and compile them in a matrix: N1+%[Uh9)
% ----------------------------- 9.D'!
if rpowers(1)==0 K7U`
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); vX/~34o]\
rpowern = cat(2,rpowern{:}); *siS4RX2
rpowern = [ones(length_r,1) rpowern]; :74)nbS
else kImS'i{A
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); N[z7<$$
rpowern = cat(2,rpowern{:}); UIovv%7zZ
end V!a\:%#^Y
#3\F<AJ<VB
% Compute the values of the polynomials: WFsa8qv
% -------------------------------------- d%u|)
=7
y = zeros(length_r,length(n)); ~t.*B& A
for j = 1:length(n) G>d@lt
s = 0:(n(j)-m_abs(j))/2; W6
f *>
pows = n(j):-2:m_abs(j); +8v^J8q0
for k = length(s):-1:1 AQQeLdTq
p = (1-2*mod(s(k),2))* ... +tES:3Pi
prod(2:(n(j)-s(k)))/ ... jf~/x>Q
prod(2:s(k))/ ... ^ejU=0+cN
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 3a"4Fn
prod(2:((n(j)+m_abs(j))/2-s(k))); 7rbl+:y2
idx = (pows(k)==rpowers); E[)`+:G]
y(:,j) = y(:,j) + p*rpowern(:,idx); q}U^H
end BXnSkT7
aS-rRL|\L
if isnorm gH(,>}{^K
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); t+|c)"\5h
end [wj&.I{^s
end B9&"/tT
% END: Compute the Zernike Polynomials #t>w)`bA-
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LIT{rR#8
B|/=E470G
% Compute the Zernike functions: r**u=q%p
% ------------------------------ N3!x7J7A
idx_pos = m>0; h%8[];*DpN
idx_neg = m<0; OjCTTz
j[.R|I|
z = y; V{HP8f91
if any(idx_pos) 2$V]XSe
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); n?
e&I>1W
end WSz#g2a
if any(idx_neg) Cb%?s
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); BlF>TI%2
end 'j 'bhG
GKTrf\"c
% EOF zernfun