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 FVKW9"AyW
function z = zernfun(n,m,r,theta,nflag) T<I=%P)
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. jM&r{^(
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N .)+hH y
% and angular frequency M, evaluated at positions (R,THETA) on the 5o/&T"]@
% unit circle. N is a vector of positive integers (including 0), and gh>>Ibf
% M is a vector with the same number of elements as N. Each element iL=
m{
% k of M must be a positive integer, with possible values M(k) = -N(k) zSE<"(a
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, /1 RAAa
% and THETA is a vector of angles. R and THETA must have the same 1RKW2RCaW_
% length. The output Z is a matrix with one column for every (N,M) TyKWy0x-3
% pair, and one row for every (R,THETA) pair. ^T.E+2=>z
% g!i45-n3gt
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike =0-
$W5E
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), i%{3W:!4t
% with delta(m,0) the Kronecker delta, is chosen so that the integral 0A:n0[V:]
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 5VO;s1
% and theta=0 to theta=2*pi) is unity. For the non-normalized @Eb2k!T
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. $o +5/c?|
% !6G?zipB
% The Zernike functions are an orthogonal basis on the unit circle. J>^\oAgpE
% They are used in disciplines such as astronomy, optics, and TM8=U-A
% optometry to describe functions on a circular domain. }dxDtqb
% ^ZM0c>ev=l
% The following table lists the first 15 Zernike functions. {T'GQz+R"
% JxjI]SF02
% n m Zernike function Normalization dDDGM:]
% -------------------------------------------------- @R m-CWa
% 0 0 1 1 \*\R1_+
% 1 1 r * cos(theta) 2 -B$~`2-
% 1 -1 r * sin(theta) 2 @h?shW=^
% 2 -2 r^2 * cos(2*theta) sqrt(6) 3M0+"l(X
% 2 0 (2*r^2 - 1) sqrt(3) ~Z ~v
% 2 2 r^2 * sin(2*theta) sqrt(6) j$da8] !
% 3 -3 r^3 * cos(3*theta) sqrt(8) ,&Wn [G<2
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) Kd3?I5t
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) :y2p@#l#
% 3 3 r^3 * sin(3*theta) sqrt(8) T<-=nX
% 4 -4 r^4 * cos(4*theta) sqrt(10) |BZDhd9<{
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) D^U:
ih
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) z^nvMTC
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Gq#~vr
% 4 4 r^4 * sin(4*theta) sqrt(10) !'=15&5@
% -------------------------------------------------- |KY EK|
% LwuF0\
% Example 1: K={qU[_O
% g`k?AM\
% % Display the Zernike function Z(n=5,m=1) (!5LW'3B
% x = -1:0.01:1; >\<*4J$PZ
% [X,Y] = meshgrid(x,x); O;HY%
% [theta,r] = cart2pol(X,Y); qW_u
% idx = r<=1; S<nq8Ebmw
% z = nan(size(X)); ^")F7`PF
% z(idx) = zernfun(5,1,r(idx),theta(idx)); r$wZt
% figure 2 }vg U$a
% pcolor(x,x,z), shading interp 1x~U*vbhQ
% axis square, colorbar "tS'b+SJ-S
% title('Zernike function Z_5^1(r,\theta)') ftk%EYT;
% M!M!Ni
% Example 2: KyP)Qzp
% $!A:5jech
% % Display the first 10 Zernike functions uk`8X`'
% x = -1:0.01:1; s|bM%!$1
% [X,Y] = meshgrid(x,x); I-NN29Sk
% [theta,r] = cart2pol(X,Y); "()sb? &
% idx = r<=1; P
,eH5w"
% z = nan(size(X)); !SHj$Jwa'
% n = [0 1 1 2 2 2 3 3 3 3]; ']ood!
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; qu6DQ@
~YC
% Nplot = [4 10 12 16 18 20 22 24 26 28]; vOI[Z0Lq9h
% y = zernfun(n,m,r(idx),theta(idx)); %qsvtc`
% figure('Units','normalized') 9O,,m~B
% for k = 1:10 ALd;$fd qf
% z(idx) = y(:,k); smAC,-6]~
% subplot(4,7,Nplot(k)) qBk``!|s]
% pcolor(x,x,z), shading interp fvo<(c#Y#
% set(gca,'XTick',[],'YTick',[]) +:jT=V"X
% axis square P}3}ek1Ax
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 1D([@)^
% end dpN@#w
% a?cn9i)#
% See also ZERNPOL, ZERNFUN2. Y^ve:Z
vC/[^
% Paul Fricker 11/13/2006 X}4}&
\6j^kY=
://U^sFL
% Check and prepare the inputs: iy5R5L2
% ----------------------------- @u4=e4eF`
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) )
6DSH`-;
error('zernfun:NMvectors','N and M must be vectors.') eQX`,9:5
end YwT-T,oD
W,hWOO
if length(n)~=length(m) Z&yaSB
error('zernfun:NMlength','N and M must be the same length.') wod/&!)]A
end s'a= _cN
T>]sQPg
n = n(:); ,qFA\cO*
m = m(:); f!GHEhQ9
if any(mod(n-m,2)) J0<p4%Cf
error('zernfun:NMmultiplesof2', ... jPu5nwvUV>
'All N and M must differ by multiples of 2 (including 0).') :pKG\A
end q7]>i!A
(RF>s.B<
if any(m>n) Zy]s`aa
error('zernfun:MlessthanN', ... ij)Cm]4(2
'Each M must be less than or equal to its corresponding N.') m^Lj+=Z"
end 7|D|4!i2Y
o$bUY7_
if any( r>1 | r<0 ) 99ASIC!
error('zernfun:Rlessthan1','All R must be between 0 and 1.') D,W\ gP/h%
end mb\t/p
0'ZYO.y
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) m3
IP7h'
error('zernfun:RTHvector','R and THETA must be vectors.') Z^6#4Q]YC
end .;Y
x*]
|+ 7f2C
r = r(:); !;}2F-
theta = theta(:); J1 tDO?
length_r = length(r); {/UhUG
if length_r~=length(theta) ,w\ wQn>]K
error('zernfun:RTHlength', ... 03E3cp"
'The number of R- and THETA-values must be equal.') wL
eHQ]
end :I/
[yd6gH
% Check normalization: lCFU1 GHH
% -------------------- T^)plWw
if nargin==5 && ischar(nflag) IRB& j%LA
isnorm = strcmpi(nflag,'norm'); F3 f@9@b
if ~isnorm "a(1s},
error('zernfun:normalization','Unrecognized normalization flag.') $N\+,?
end dq8+m(7k
else ~zMKVM1Q.,
isnorm = false; O)5#Fcp(
end 5#u.pu
eY3=|RR
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {})y^L
% Compute the Zernike Polynomials f'_S1\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 8eww7k^R
1o#vhk/"+
% Determine the required powers of r: V4?Oc2mS
% ----------------------------------- (5(fd.m+_
m_abs = abs(m); C={mi#G[/
rpowers = []; C"No5r'K3
for j = 1:length(n) @zs1>\J7
rpowers = [rpowers m_abs(j):2:n(j)]; q%.bnF/Yd
end 8nu> gA
rpowers = unique(rpowers); |uQ[W17^N
RUc \u93n
% Pre-compute the values of r raised to the required powers, 2fBYT4*P;
% and compile them in a matrix: ?z"YC&Tp
% ----------------------------- '?k' 6R$'\
if rpowers(1)==0 <,-,?
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); =+(Q.LmhC
rpowern = cat(2,rpowern{:}); 65"uD7;
rpowern = [ones(length_r,1) rpowern]; -7L
else '_E c_F
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 0%;MVMH
rpowern = cat(2,rpowern{:}); C,='3^Nc
end M\jB)@)
$P_x v
% Compute the values of the polynomials: LO}z)j~W
% -------------------------------------- 1w) fu
y = zeros(length_r,length(n)); r$?Vx_f`Q
for j = 1:length(n) u7~mnl
s = 0:(n(j)-m_abs(j))/2; QB9A-U<J
pows = n(j):-2:m_abs(j); .J:;_4x
for k = length(s):-1:1 |Ib.)
p = (1-2*mod(s(k),2))* ... ,N;v~D$Y
prod(2:(n(j)-s(k)))/ ... U_}hfLILi
prod(2:s(k))/ ... -PXoMZx%
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 5])8qb/F
prod(2:((n(j)+m_abs(j))/2-s(k))); ze$Y=<S
idx = (pows(k)==rpowers); mc~`
y(:,j) = y(:,j) + p*rpowern(:,idx);
"$Y(NFb
end q@w"yz>
6*V8k%H
if isnorm u:eW0Ows"
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); -Fa98nV.WB
end *CT.G'bQX
end )ZeLaa P
% END: Compute the Zernike Polynomials ac3_L$X[
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% `_0)kdu
`+Xe'ey
% Compute the Zernike functions: :=Nb=&lst
% ------------------------------ CJ:uYXJJ:z
idx_pos = m>0; [}@n*D$
idx_neg = m<0; wU.'_SBfB
k|l5 "&K~.
z = y; 9G+y.^/6
if any(idx_pos) m.Twgin
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); bbO+%-(X
end uGM>C"
if any(idx_neg) `{%-*f^
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 3 ^pYCK%
end (A2U~j?Ry}
6G$/NW=L
% EOF zernfun