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 ?uE@C3 e
function z = zernfun(n,m,r,theta,nflag) @IBU{{
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. EMS$?"K
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 'n!Sco)C
% and angular frequency M, evaluated at positions (R,THETA) on the &PEw8: TX
% unit circle. N is a vector of positive integers (including 0), and onUF@3V
% M is a vector with the same number of elements as N. Each element |+Ub3<b[]
% k of M must be a positive integer, with possible values M(k) = -N(k) !r_2b! dy
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, r1xhplHH@
% and THETA is a vector of angles. R and THETA must have the same |uln<nM9
% length. The output Z is a matrix with one column for every (N,M) qH*Fv:qnM
% pair, and one row for every (R,THETA) pair. (wEaw|Zx
% =a./HCF
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike j1P#({z[
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), :]IYw!_-p
% with delta(m,0) the Kronecker delta, is chosen so that the integral pGSS
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, O<qo%fP
% and theta=0 to theta=2*pi) is unity. For the non-normalized ?{-y? %y
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. _WHGd&u
% J]4Uh_>)
% The Zernike functions are an orthogonal basis on the unit circle. UxVxnJ_
% They are used in disciplines such as astronomy, optics, and F%q}N,W
% optometry to describe functions on a circular domain. H5p&dNO
% q{oppali
% The following table lists the first 15 Zernike functions. #vvQ1ub
% s4{ >7`N2
% n m Zernike function Normalization THDyb9_g
% -------------------------------------------------- <bgFc[Z
% 0 0 1 1 Z\*jt B:
% 1 1 r * cos(theta) 2 RE75TqYW
% 1 -1 r * sin(theta) 2 *z\L
% 2 -2 r^2 * cos(2*theta) sqrt(6) [cf!%3>53
% 2 0 (2*r^2 - 1) sqrt(3) y8=H+Y
% 2 2 r^2 * sin(2*theta) sqrt(6) $2gZpO|
% 3 -3 r^3 * cos(3*theta) sqrt(8) W%^;:YQ9i
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) kG$U
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) iwT
PJGK|
% 3 3 r^3 * sin(3*theta) sqrt(8) XfH[:XG3
% 4 -4 r^4 * cos(4*theta) sqrt(10) $23dcC*hI
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) )*n2,n
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) _+2Jc}Yf
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Q !G^CG
% 4 4 r^4 * sin(4*theta) sqrt(10) g\lEdxm6Sj
% -------------------------------------------------- %w3"B,k'9D
% |jE0H!j
% Example 1: 0P_3%
% :f5"w+
% % Display the Zernike function Z(n=5,m=1) a EmLf
% x = -1:0.01:1; Y|96K2BR
% [X,Y] = meshgrid(x,x); jz72~+)T
% [theta,r] = cart2pol(X,Y); +LsACSB
% idx = r<=1; &i?>mt
% z = nan(size(X)); dw]jF=u
% z(idx) = zernfun(5,1,r(idx),theta(idx)); c.eA]m q
% figure R k@xv;t;
% pcolor(x,x,z), shading interp |KLCO'x
% axis square, colorbar j$Z:S~*
% title('Zernike function Z_5^1(r,\theta)') ]:r6
% ]KE"|}B
% Example 2: M|xs>+r*
% U[t/40W}P
% % Display the first 10 Zernike functions p? L*vcU
% x = -1:0.01:1; _/`H<@B_U
% [X,Y] = meshgrid(x,x); G2BB]] m3
% [theta,r] = cart2pol(X,Y); #[.aj2
% idx = r<=1; 5'zD}[2
% z = nan(size(X)); ];8S<KiS~
% n = [0 1 1 2 2 2 3 3 3 3]; 5>u,Qh
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; :M
_N
% Nplot = [4 10 12 16 18 20 22 24 26 28]; @X g5E
% y = zernfun(n,m,r(idx),theta(idx)); !{%BfZX<&
% figure('Units','normalized') qz6@'1
% for k = 1:10 p]erk
% z(idx) = y(:,k); ;dVYR=l
% subplot(4,7,Nplot(k)) bx8;`QMX
% pcolor(x,x,z), shading interp ni`uO<\U
% set(gca,'XTick',[],'YTick',[]) ::R5F4
% axis square T_/ n#e
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) JOFQyhY0>m
% end ~duF2m 72
% vkE a[7
% See also ZERNPOL, ZERNFUN2. ee\QK,QV
e> -fI_+b
% Paul Fricker 11/13/2006 "1HKD
?3=y]Vb+
N83c+vs%c
% Check and prepare the inputs: Hx#1TqC/
% ----------------------------- K|sk]2.
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 5~GH*!h%;
error('zernfun:NMvectors','N and M must be vectors.') eNc>^:&y*
end ALXie86a8
V18A|]k
if length(n)~=length(m) c%@<
h6
error('zernfun:NMlength','N and M must be the same length.') s_}q
end N/6!|F
v1}9i3Or#
n = n(:); F0x'^Z}Q;
m = m(:); 'B yB1NL
if any(mod(n-m,2)) A} v;uNS]
error('zernfun:NMmultiplesof2', ... _2
oZhJ
'All N and M must differ by multiples of 2 (including 0).') :Fh#"<A&&
end {j[a'Gb
#G!\MYfQt
if any(m>n) mr2fNA>kR
error('zernfun:MlessthanN', ... i#bcjH
'Each M must be less than or equal to its corresponding N.') b>]k=zd
end \zLKSJ]
"el}9OitC
if any( r>1 | r<0 ) ~`X$bF
error('zernfun:Rlessthan1','All R must be between 0 and 1.') )0?u_Z]w9
end Tnoy#w}Ve
.oH)eD
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) g1v=a
error('zernfun:RTHvector','R and THETA must be vectors.') IN7Cpg~9%
end K( r@JW
ToR@XL!%rP
r = r(:); sWv!ig_
theta = theta(:); Z;~ 7L*|
length_r = length(r); !xvAy3
if length_r~=length(theta) ~yiw{:\
error('zernfun:RTHlength', ... YHzP/&0
'The number of R- and THETA-values must be equal.') :hTmt{LjN
end 1+9!W
21[=xboU
% Check normalization: Y^tUcBm\
% -------------------- {PKf]m
if nargin==5 && ischar(nflag)
*I.eCMDa
isnorm = strcmpi(nflag,'norm'); Q6;bORN
if ~isnorm [JYy
error('zernfun:normalization','Unrecognized normalization flag.') 4^T_" W}
end W:>XXUU
else {t!Pv2y<
isnorm = false; moRo>bvN~
end ^h!}jvqE
9#E)H?`g
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% K57u87=*X?
% Compute the Zernike Polynomials `Wd4d2aLG
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
~S\8 '
lYT_Y.%I
% Determine the required powers of r: zZ 94_8b
% ----------------------------------- I,W`s
m_abs = abs(m); qSt\ 6~
rpowers = []; M|fC2[]v B
for j = 1:length(n) @,m 7%,
rpowers = [rpowers m_abs(j):2:n(j)]; XhUVDmeUMb
end 9[R+m3V/`
rpowers = unique(rpowers); rvuasr~
-"rANP-UI
% Pre-compute the values of r raised to the required powers, nK}-^Ur
% and compile them in a matrix: j'`-3<k
% ----------------------------- UCj{
&
if rpowers(1)==0 Jl<pWjkZZ
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); P9W?sPnC5
rpowern = cat(2,rpowern{:}); 5mX^{V&^
rpowern = [ones(length_r,1) rpowern]; WO6R04+WV
else Qb|@DMq%
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); .}Eckqkp
rpowern = cat(2,rpowern{:}); + w'q5/`
end \5}*;O@
_nM 7SK
% Compute the values of the polynomials: !v8](UI8-
% -------------------------------------- tz5\O}
y = zeros(length_r,length(n)); (8~D^N6Z
for j = 1:length(n) zkquXzlgB
s = 0:(n(j)-m_abs(j))/2; Yv.7-DHNl
pows = n(j):-2:m_abs(j); g7{:F\S
for k = length(s):-1:1 tUt_Q;%yC
p = (1-2*mod(s(k),2))* ... ~C>clkZ
prod(2:(n(j)-s(k)))/ ... l#~pK6@W
prod(2:s(k))/ ... bFSs{\zE
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... "'C5B>qO
prod(2:((n(j)+m_abs(j))/2-s(k))); 51tZ:-1!
idx = (pows(k)==rpowers); NFF!g]QN
y(:,j) = y(:,j) + p*rpowern(:,idx); ^7a@?|,q8
end Ww"]3
yb,X
}"Et
if isnorm N>CNgUyP
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); T;]Ob3(BpW
end p[&b@U#
end a?xZsR
% END: Compute the Zernike Polynomials &*745,e
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% $+PyW(
r
I E{:{b\
% Compute the Zernike functions: z,bK.KFSs
% ------------------------------ -{q'Tmst
idx_pos = m>0; K>C@oE[W
idx_neg = m<0; SSq4KFO1
[b_qC'K[
z = y; Fy0sn|
if any(idx_pos) W23Q>x&S
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); |>OBpb
end t fD7!N{
if any(idx_neg) =dsEt\
j
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); yZN~A:
end e)N<r
4j8$&~/
% EOF zernfun