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 u2@:[:Ao
function z = zernfun(n,m,r,theta,nflag) xpRQ"6
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. #e{l:!uS\
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N F3;UH%L1
% and angular frequency M, evaluated at positions (R,THETA) on the vqJiMa j@Z
% unit circle. N is a vector of positive integers (including 0), and cQA;Y!Q#
% M is a vector with the same number of elements as N. Each element D)K/zh)
% k of M must be a positive integer, with possible values M(k) = -N(k) ikw_t?
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, :>\ i
% and THETA is a vector of angles. R and THETA must have the same I[c/)
N
% length. The output Z is a matrix with one column for every (N,M) P!0uAkt9C
% pair, and one row for every (R,THETA) pair. 6zaO$
% z|<6y~5,
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike Fnzv&
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), rMdOE&5G
% with delta(m,0) the Kronecker delta, is chosen so that the integral wHEt;rc(
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, Kj;Q;Ii
% and theta=0 to theta=2*pi) is unity. For the non-normalized #JWW ;M6F
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ~]O~a}]g(
% W{*U#:Jx1
% The Zernike functions are an orthogonal basis on the unit circle. qa.nm4"6+
% They are used in disciplines such as astronomy, optics, and T9}G:6
% optometry to describe functions on a circular domain. 4703\
HK
% |'aGj
% The following table lists the first 15 Zernike functions. [h
{zT)[
% .-awl1 W
% n m Zernike function Normalization N>/!e787OU
% -------------------------------------------------- =e$<["
% 0 0 1 1 K7 -AVMY
% 1 1 r * cos(theta) 2 6c$ so
% 1 -1 r * sin(theta) 2 8iGS=M
% 2 -2 r^2 * cos(2*theta) sqrt(6) RXxi7^ U
% 2 0 (2*r^2 - 1) sqrt(3) @@-n/9>vs
% 2 2 r^2 * sin(2*theta) sqrt(6) - 0R5g3^*/
% 3 -3 r^3 * cos(3*theta) sqrt(8) "v@Y[QI
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) z"Miy
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8)
W8z4<o[$
% 3 3 r^3 * sin(3*theta) sqrt(8) A<fKO <d
% 4 -4 r^4 * cos(4*theta) sqrt(10) Tty_P,
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Ti$G2dBO
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) NvW`x
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) r'/&{?Je/
% 4 4 r^4 * sin(4*theta) sqrt(10) YcOPqvQ
% -------------------------------------------------- =PYfk6j9
% )S@e&a|
% Example 1: \@&oK2f
% JZI)jIh
% % Display the Zernike function Z(n=5,m=1) DA]<30w
% x = -1:0.01:1; Q6)Wh6Cm
% [X,Y] = meshgrid(x,x); gB|>[6
% [theta,r] = cart2pol(X,Y); -@L7!,j
% idx = r<=1; >9dzl#
% z = nan(size(X)); ~tx|C3A`d
% z(idx) = zernfun(5,1,r(idx),theta(idx)); H1>~,zc>E
% figure K)b@,/ 5
% pcolor(x,x,z), shading interp A0X'|4I
% axis square, colorbar 7.)kG}q]
% title('Zernike function Z_5^1(r,\theta)') D+#OB|&Dn
% 3r^Ls[ey
% Example 2: C0C2]xx{
% QiH>!Ssw
% % Display the first 10 Zernike functions vT@*o=I
% x = -1:0.01:1; !ZNirvk
% [X,Y] = meshgrid(x,x); # dA9v7
% [theta,r] = cart2pol(X,Y); <<'%2q5
% idx = r<=1; `vjn,2S}
% z = nan(size(X)); E? lK(C
% n = [0 1 1 2 2 2 3 3 3 3]; {E=BFs
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; i4T=4q
% Nplot = [4 10 12 16 18 20 22 24 26 28]; `PY=B$?{4
% y = zernfun(n,m,r(idx),theta(idx)); |\.:h":!0~
% figure('Units','normalized') HuT4OGBFpC
% for k = 1:10 Cv[_N%3[
% z(idx) = y(:,k); AQ%B&Q(V1
% subplot(4,7,Nplot(k)) GFGW'}w-
% pcolor(x,x,z), shading interp hGU
m7
% set(gca,'XTick',[],'YTick',[]) 1;v,rs M
% axis square F8H4R7
8>;
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) G&Fe2&5!w
% end h;Hg/jv
% MO^Q 8v
% See also ZERNPOL, ZERNFUN2.
&x?m5%^l
p40;@gUug
% Paul Fricker 11/13/2006 >:Y"DX-
&]"Z x0t5%
[][ze2+b
% Check and prepare the inputs: Ec9%RAxl
% ----------------------------- >sjvE4s
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) !C(U9p. 0
error('zernfun:NMvectors','N and M must be vectors.')
F/SYmNp
end q2b>Z6!5
>,x&L[3
if length(n)~=length(m) j/t)=c
error('zernfun:NMlength','N and M must be the same length.') K
0e*K=UM
end ).)^\
qTrM*/m:]L
n = n(:); ToK=`0#LNK
m = m(:); 1B#iJZ}
if any(mod(n-m,2)) DHg)]FQ/
error('zernfun:NMmultiplesof2', ... Avww@$
'All N and M must differ by multiples of 2 (including 0).') wP7
E8'
end wpWZn[j
`_()|; !y
if any(m>n) q`VkA
\
error('zernfun:MlessthanN', ... I5*<J n
'Each M must be less than or equal to its corresponding N.') uZTbJ3$$
end :HM~!7e
KVevvy)W
if any( r>1 | r<0 ) 63(XCO
error('zernfun:Rlessthan1','All R must be between 0 and 1.') C|V5@O?;&
end P~#LbUP(
d\R "?Sg
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) K]1|#`n
error('zernfun:RTHvector','R and THETA must be vectors.') Z;z,dw
end JXjH}C
Q/@ pcU
r = r(:); O=vD6@QI
theta = theta(:); PMi.)%++
length_r = length(r); /2''EF';
if length_r~=length(theta) 'C=(?H)M
error('zernfun:RTHlength', ... @Gw.U>"!C
'The number of R- and THETA-values must be equal.') w`EC6ZN
end >;]S+^dXY
DR
@yd,
% Check normalization: D9H%jDv
% -------------------- ex#-,;T
if nargin==5 && ischar(nflag) ^ ;K"Y'f$
isnorm = strcmpi(nflag,'norm'); P1z:L
if ~isnorm &lID6{7 9Z
error('zernfun:normalization','Unrecognized normalization flag.') H?eG5
end V*r/0|vd
else L{GlDoFk
isnorm = false; (/^?$~m"
end ~$ Po3]{s
M;W&#Fz%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% y<*\D_J
% Compute the Zernike Polynomials [0 rH/{
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% #S]O|$&*
Xgl
%2'
% Determine the required powers of r: +vH#xc\'
% ----------------------------------- &> _aY #
m_abs = abs(m); 9ei<ou_s
rpowers = []; ;dtA-EfOZ
for j = 1:length(n) Lctp=X4
rpowers = [rpowers m_abs(j):2:n(j)]; mKE'l'9A_
end Unansk
rpowers = unique(rpowers); 's5H_ah
mI\[L2x
% Pre-compute the values of r raised to the required powers, rLY I\
% and compile them in a matrix: GY5JPl
% ----------------------------- 'R2*3<
if rpowers(1)==0 1H\5E~X
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); uhv_'Q
rpowern = cat(2,rpowern{:}); /cVZ/"
rpowern = [ones(length_r,1) rpowern]; gv&Hu$ca
else Y9
Bk$$#\
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false);
z).&0K
rpowern = cat(2,rpowern{:}); \F\xZ.r
end R&:Qy7"
z_#HJ}R=
% Compute the values of the polynomials: DjiI*HLNR
% -------------------------------------- >)Bv>HM
y = zeros(length_r,length(n)); ![eY%2;<
for j = 1:length(n) a<]vHC7
s = 0:(n(j)-m_abs(j))/2; wzmQRn;s
pows = n(j):-2:m_abs(j); E$A=*-u
for k = length(s):-1:1 Q'hs,t1<
p = (1-2*mod(s(k),2))* ... '*Tt$0#o
prod(2:(n(j)-s(k)))/ ... -G#m'W&
prod(2:s(k))/ ... {lUaN0O:
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... <u1`o`|-
prod(2:((n(j)+m_abs(j))/2-s(k))); t0?tXe.B
idx = (pows(k)==rpowers); RE-y5.kE^
y(:,j) = y(:,j) + p*rpowern(:,idx); {qU;>;(
end ^4p$@5zH
yn20*ix{
if isnorm cxFyN;7
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); d+5v[x~'
end ;#8xRLW
end YY$Z-u(
% END: Compute the Zernike Polynomials h2= wC.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ]US!3R^
+tG'
% Compute the Zernike functions: 7j(gW
% ------------------------------ W[e2J&G
idx_pos = m>0; h&!$ `)
idx_neg = m<0; ~fzuz'"^
pX$X8z%
z = y; ,% .)mf
if any(idx_pos) [A]
+Azc
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); jR+kx:+
end 0,8RA_Ca}
if any(idx_neg) 9%0^fhrJ
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); M~y}0Ik
end G
c,
;
0M"T[c
% EOF zernfun