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 7W+{U02O
function z = zernfun(n,m,r,theta,nflag) e&K7n@
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. -Vs;4-B{9
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N R*lq.7
% and angular frequency M, evaluated at positions (R,THETA) on the R:+?<U&
% unit circle. N is a vector of positive integers (including 0), and o#D'"Tn!
% M is a vector with the same number of elements as N. Each element ]ki) (Bb
% k of M must be a positive integer, with possible values M(k) = -N(k) 1\AcceJ|(w
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, 6BZi4:PDx
% and THETA is a vector of angles. R and THETA must have the same dKevhm)R"
% length. The output Z is a matrix with one column for every (N,M) P057]cAat<
% pair, and one row for every (R,THETA) pair. wzcv[C-x
% (Zej\lEN
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike |O' gT8
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), @PK
1
% with delta(m,0) the Kronecker delta, is chosen so that the integral iAeq%N1(0
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, {$7vd
% and theta=0 to theta=2*pi) is unity. For the non-normalized {cjp8W8hS
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. #WE
lL2&
% kX*.BZI}C
% The Zernike functions are an orthogonal basis on the unit circle. )EcfEym.>
% They are used in disciplines such as astronomy, optics, and =AF;3
% optometry to describe functions on a circular domain. WopA7J,
% }h|HT
% The following table lists the first 15 Zernike functions. 8M]QDgd.
% !,sQB_09C
% n m Zernike function Normalization @Y ?p-&
% -------------------------------------------------- qZlL6
% 0 0 1 1 	HV
% 1 1 r * cos(theta) 2 DD6K[\
% 1 -1 r * sin(theta) 2 /N")uuv
% 2 -2 r^2 * cos(2*theta) sqrt(6) \_)mWK,h
% 2 0 (2*r^2 - 1) sqrt(3) q AsTiT6r
% 2 2 r^2 * sin(2*theta) sqrt(6) n<eK\w
% 3 -3 r^3 * cos(3*theta) sqrt(8) T:!H^
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) er@.<Dc
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) r
jnf30
% 3 3 r^3 * sin(3*theta) sqrt(8) gEmsPk,
% 4 -4 r^4 * cos(4*theta) sqrt(10) 0&3zBL%Bo
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) %+(fdk-k+
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) +JB*1dz>8
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) I]Z"?T
% 4 4 r^4 * sin(4*theta) sqrt(10) }{[p<pU$C
% -------------------------------------------------- 51;Bc[)%
% 3g0v,7,Zv
% Example 1: nFefDdP
% LRdV_O1e6M
% % Display the Zernike function Z(n=5,m=1) Ng*O/g`%L
% x = -1:0.01:1; cA{,2CYc
% [X,Y] = meshgrid(x,x); n0uL^{B
% [theta,r] = cart2pol(X,Y); @y|JIBBRc
% idx = r<=1; " "CNw-^t
% z = nan(size(X)); >^v,,R8j
% z(idx) = zernfun(5,1,r(idx),theta(idx)); R78P](1\>
% figure _1jeaV9@
% pcolor(x,x,z), shading interp 1NAtg*`
% axis square, colorbar uK[gI6M
% title('Zernike function Z_5^1(r,\theta)') 461p 4)
% }9Q<<a
% Example 2: qIO)<5\[%d
% +@do<2l]
% % Display the first 10 Zernike functions BbgKaC q
% x = -1:0.01:1; "Fxw"I
<
% [X,Y] = meshgrid(x,x); 7V"Jfh4_
% [theta,r] = cart2pol(X,Y); vtq47i
% idx = r<=1; Mu_'C$zA
% z = nan(size(X)); 1Nz#,IdQ
% n = [0 1 1 2 2 2 3 3 3 3]; kP&Ekjt@
% m = [0 -1 1 -2 0 2 -3 -1 1 3];
G%%5lw!y'
% Nplot = [4 10 12 16 18 20 22 24 26 28]; rWp+kV[Ec>
% y = zernfun(n,m,r(idx),theta(idx)); zbDK$g6
% figure('Units','normalized') 4@@gC&:Y
% for k = 1:10 (V`ddP-
% z(idx) = y(:,k); OuB[[L
% subplot(4,7,Nplot(k)) raZ0B,;eFu
% pcolor(x,x,z), shading interp De49!{\a
% set(gca,'XTick',[],'YTick',[]) n&E/{o(
% axis square "g1Fg.o
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) sv#/ 78 ~|
% end Z}>+!Z
% WAVEwA`r
% See also ZERNPOL, ZERNFUN2. G+NTn\
K` <`l
% Paul Fricker 11/13/2006 _2xYDi
{InW%qSn_
i6k~j%0m
% Check and prepare the inputs: 'uLYah
% ----------------------------- Y_nlIcu
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) eS2VLVxu
error('zernfun:NMvectors','N and M must be vectors.') nyPW6VQ0n
end 9|>5;Ej
2VkA!o4nP
if length(n)~=length(m) >yr3C
error('zernfun:NMlength','N and M must be the same length.') QaAMiCZFR
end ?xo<Fv
lp5b&I_
n = n(:); ?MJ5GVeH
m = m(:); 0Pg@%>yb~
if any(mod(n-m,2)) dg;E,'e_
p
error('zernfun:NMmultiplesof2', ... liTAV9<
'All N and M must differ by multiples of 2 (including 0).') H&0S
end mz^[C7(q'(
mtNB09E(
if any(m>n) Le,+jm
error('zernfun:MlessthanN', ... ~h444Hp=
'Each M must be less than or equal to its corresponding N.') !)uXCg9U
end PML84*K -
2Zi&=Zj"
if any( r>1 | r<0 ) Y67i\U>?
error('zernfun:Rlessthan1','All R must be between 0 and 1.') [&{NgUgu"
end zfUkHL6
fq0[7Yb
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) s *<T5Z
error('zernfun:RTHvector','R and THETA must be vectors.') =L}$#Y8?
end .%mjE'
"C9.pdP\8
r = r(:); GI#TMFz3
theta = theta(:); z0 _/JwJn
length_r = length(r); v5`Odbc=w
if length_r~=length(theta) K#plSD^f=
error('zernfun:RTHlength', ... K?mly$
'The number of R- and THETA-values must be equal.') 8hvh
xp
end _ 4+=S)$
"RsH'`
% Check normalization: DT#Z6A
% -------------------- ZQrgYeQl"
if nargin==5 && ischar(nflag) ?a-}1A{
isnorm = strcmpi(nflag,'norm'); +4Lj}8,
if ~isnorm zy[|4Q(?
error('zernfun:normalization','Unrecognized normalization flag.') ktK/s!bgY
end =|bW >y
else pXHeUBY.
isnorm = false; ~7&O[
end .s<tQU
, MU9p*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% '#x<Fo~hT
% Compute the Zernike Polynomials vghn+P8
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% iMOf];O)
?8)$N
% Determine the required powers of r: @GE:<'_:{
% ----------------------------------- g3,F+
m_abs = abs(m); Q*AgFF%wn
rpowers = []; WnC0T5S?U
for j = 1:length(n) v4wXa:CJ
rpowers = [rpowers m_abs(j):2:n(j)]; +l_$}UN
end &0S/]E`_M
rpowers = unique(rpowers); M;qV%
k
\Rvsy;7
% Pre-compute the values of r raised to the required powers, b1qli5
% and compile them in a matrix: "Q<*H<e
% ----------------------------- ecy41y'~:
if rpowers(1)==0
) XHcrm&
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); T2MX_rt#D
rpowern = cat(2,rpowern{:}); t9
m],aH
rpowern = [ones(length_r,1) rpowern]; QYTwGThWR
else ^7~w yAr
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); %epK-q9[
rpowern = cat(2,rpowern{:}); ._z[T@!9
end :7Q,
`W9
"t"&6\
% Compute the values of the polynomials: q! U'DDEP
% -------------------------------------- '$n#~/#}
y = zeros(length_r,length(n)); uP[:P?,t
for j = 1:length(n) Yhd|1,m9f
s = 0:(n(j)-m_abs(j))/2; xF3H\`{4x
pows = n(j):-2:m_abs(j); 4\yKd8I
for k = length(s):-1:1 h8_~ OX
p = (1-2*mod(s(k),2))* ... _Uz}z#jt
prod(2:(n(j)-s(k)))/ ... f*SAbDE
prod(2:s(k))/ ... c F(]`49(
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... L)ry!BuHI
prod(2:((n(j)+m_abs(j))/2-s(k))); u +OfUBrf
idx = (pows(k)==rpowers); 0Ti>PR5M
y(:,j) = y(:,j) + p*rpowern(:,idx); +C !A@
end i@ avm7
;"/ "
if isnorm ".SQ*'Oc
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 9jwo f}OU
end +iPS=?S
end %lU$;cY
% END: Compute the Zernike Polynomials &j7l#Urq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% VgNt
XJ;JDch
% Compute the Zernike functions: ico(4KSk
% ------------------------------ V-w[\u
idx_pos = m>0; 2v<[XNX
idx_neg = m<0; ,uP1U@Cas
N7xkkAS{
z = y; ^MWfFpJV!]
if any(idx_pos) 7>m#Y'ppl@
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)');
qEpP%p
end P( W8XC
if any(idx_neg) G#! j`
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); `v)-v<
end E
2DTE
\0pJ+@\T9
% EOF zernfun