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 ^-7{{/
function z = zernfun(n,m,r,theta,nflag) g|l|)T.s
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. &($Zs'X
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N h!>NS ?X7
% and angular frequency M, evaluated at positions (R,THETA) on the (G6N@>V(`
% unit circle. N is a vector of positive integers (including 0), and p}swJ;S
% M is a vector with the same number of elements as N. Each element U^X8{,8O
% k of M must be a positive integer, with possible values M(k) = -N(k) }u7&SU
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, 3# T_(
% and THETA is a vector of angles. R and THETA must have the same /%GMbO_
% length. The output Z is a matrix with one column for every (N,M) 4.mbW
% pair, and one row for every (R,THETA) pair. ui6B
% V/-~L]G
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike }tT*Ch?u
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), *:A)j?(
% with delta(m,0) the Kronecker delta, is chosen so that the integral QWGFXy,=1
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, eDSBs3k7H
% and theta=0 to theta=2*pi) is unity. For the non-normalized *8CE0;p'k
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. k||DcwO
% 0Z{(,GU
% The Zernike functions are an orthogonal basis on the unit circle. }t #Hq
% They are used in disciplines such as astronomy, optics, and t|zLR
% optometry to describe functions on a circular domain. ,/>~J]:\;
% H{T)?J~
% The following table lists the first 15 Zernike functions. HCifO
% *ha9Vq@X
% n m Zernike function Normalization D r $N{d
% -------------------------------------------------- pf`li]j'V
% 0 0 1 1 [0e]zyB+
% 1 1 r * cos(theta) 2 Lsozl<@
% 1 -1 r * sin(theta) 2 3,B[%!3d
% 2 -2 r^2 * cos(2*theta) sqrt(6) i=<(fq
% 2 0 (2*r^2 - 1) sqrt(3)
*H
RxC
% 2 2 r^2 * sin(2*theta) sqrt(6) :PaFC{O)*
% 3 -3 r^3 * cos(3*theta) sqrt(8) P5P<-T{-c
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) jWW2&cBm\
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) 0,;FiOp
% 3 3 r^3 * sin(3*theta) sqrt(8) HnqZ7%jeN
% 4 -4 r^4 * cos(4*theta) sqrt(10) kB]|4CG{
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) OkO"t
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5)
Z{n7z$s*
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) HF\L`dJX?
% 4 4 r^4 * sin(4*theta) sqrt(10) EH$wWl^
% -------------------------------------------------- {UYqRfgbZ
% 3r{'@Y
=)Y
% Example 1: (<.1o_Q-LU
% UrxgKTry
% % Display the Zernike function Z(n=5,m=1) "v3u$-xN1
% x = -1:0.01:1; (|5g`JDG
% [X,Y] = meshgrid(x,x); sEvJ!$Tt?I
% [theta,r] = cart2pol(X,Y); <STjB,_s
% idx = r<=1; xI~\15PhG
% z = nan(size(X)); }wkBa]
% z(idx) = zernfun(5,1,r(idx),theta(idx)); 7F'61}qL
% figure R/O_*XY
% pcolor(x,x,z), shading interp 73.o{V
% axis square, colorbar r% '2a+}D
% title('Zernike function Z_5^1(r,\theta)') Gz@%UIv
% nhCB])u8l
% Example 2: I"JT3[*s
% "rjJ"u1
% % Display the first 10 Zernike functions n(f&uV_):
% x = -1:0.01:1; 1=(i{D~
% [X,Y] = meshgrid(x,x); XLbrE|0A?
% [theta,r] = cart2pol(X,Y); #G{T(0<F
% idx = r<=1; 9Jk(ID'c
% z = nan(size(X)); y~S[0]y>
% n = [0 1 1 2 2 2 3 3 3 3]; *}w.xt
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; {@,
L
% Nplot = [4 10 12 16 18 20 22 24 26 28]; $~~=SOd0
% y = zernfun(n,m,r(idx),theta(idx)); Y*Q(v
% figure('Units','normalized') kb7\qH!n
% for k = 1:10 nQ(#'9
% z(idx) = y(:,k); dF.T6b
% subplot(4,7,Nplot(k)) VBCj.dw
% pcolor(x,x,z), shading interp 4GHIRH
C%[
% set(gca,'XTick',[],'YTick',[]) q-8 GD7
% axis square ga~vQ7I_
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 'b^:"\t'Rh
% end ^3yjE/Wi"
% .D>lv_kp
% See also ZERNPOL, ZERNFUN2. _RmE+ Xg2
>Ia(g0
% Paul Fricker 11/13/2006 %mYIXsuH
7R2)Klt
d,)F #;^5
% Check and prepare the inputs: l9L;Tjj
% ----------------------------- v S+~4Q41
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) .$OInh
error('zernfun:NMvectors','N and M must be vectors.') #U_u~7?H$
end IkZ_N #m
~fUSmc
if length(n)~=length(m) P`%ppkzV6
error('zernfun:NMlength','N and M must be the same length.') BA>0
+
end
Qom@-A
S2s-TpjB<
n = n(:); jN<]yhqf
m = m(:); E8dp
if any(mod(n-m,2)) N7jRdT2k%
error('zernfun:NMmultiplesof2', ... s,29_z7
'All N and M must differ by multiples of 2 (including 0).') OLR1/t`V
end ( gFA? aD<
V_1# 7
if any(m>n) qlxW@|
error('zernfun:MlessthanN', ... uHIWbF<0oo
'Each M must be less than or equal to its corresponding N.') -$kJERvy
end =!c+|X`
Kk(ucO
if any( r>1 | r<0 ) 7 r<>^j'
error('zernfun:Rlessthan1','All R must be between 0 and 1.') *Fc&DQT(
end .0-m=3mp2
/t^lI%&
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) k$ M4NF~$
error('zernfun:RTHvector','R and THETA must be vectors.') 4a |Fx
end >y~_Hh(TSL
eEh0T%9K
r = r(:); !U!E_D.O
theta = theta(:); <`*P/V
length_r = length(r); q{ 1U
if length_r~=length(theta) ;$E[u)l
error('zernfun:RTHlength', ... #dt2'V- ,
'The number of R- and THETA-values must be equal.') o5@ jMU;
end Ft rw3OxN
8'[wa
% Check normalization: M!l5,ycF
% -------------------- r97[!y1gt
if nargin==5 && ischar(nflag) D5b_m|7%
isnorm = strcmpi(nflag,'norm'); v`w?QIB]
if ~isnorm NXNon*"
error('zernfun:normalization','Unrecognized normalization flag.') 15:@pq\
end S:uEK
else a0.3$
isnorm = false; +"cyOC
end {wXN kq
K@~#Gdnl
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% EM/+1
_u
% Compute the Zernike Polynomials q$rA-`jw
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +|<b0Xd
(NC>[
% Determine the required powers of r: ;T+U&U0d|
% ----------------------------------- -b}S3<15@
m_abs = abs(m); 3/=QZ8HA&-
rpowers = []; D*gVS
for j = 1:length(n) pe%)G6@G
rpowers = [rpowers m_abs(j):2:n(j)]; g VJ#LJ
end mRY6[*u
rpowers = unique(rpowers); UeMe4$m
15 11<,
% Pre-compute the values of r raised to the required powers, gtGKV
% and compile them in a matrix: N:[;E3?O
% ----------------------------- 5 hadA>d
if rpowers(1)==0 si_HN{
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); s)8M? |[`I
rpowern = cat(2,rpowern{:}); C'2 =0oou
rpowern = [ones(length_r,1) rpowern]; ]q7 LoH'S
else yN<fmi};c
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); hr6e 1Er
rpowern = cat(2,rpowern{:}); =DTOI
end KBqaI((
cu?(P;mQi
% Compute the values of the polynomials: {4aY}=
-Q*
% -------------------------------------- ]"g >> N
y = zeros(length_r,length(n)); vW-`=30
for j = 1:length(n) sg"D;b:X
s = 0:(n(j)-m_abs(j))/2; `$SEkYdt
pows = n(j):-2:m_abs(j); uEGPgYY (
for k = length(s):-1:1 lO:{tV
p = (1-2*mod(s(k),2))* ... *F*jA$aY
prod(2:(n(j)-s(k)))/ ... K[gWXBP
prod(2:s(k))/ ... 3@`H<tP'6o
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... `N.$LY;8
prod(2:((n(j)+m_abs(j))/2-s(k))); rL
sK-qQ
idx = (pows(k)==rpowers); nWF4[<t
y(:,j) = y(:,j) + p*rpowern(:,idx); zHOE.V2Qo
end y*b.eO
`-EH0'w~"
if isnorm }R&5qpl
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Qb't*2c%
end i;hc]fYb=K
end n`z+ w*
% END: Compute the Zernike Polynomials _6UAeZ*M
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Wejwj/EU%
e_c;D2'F
% Compute the Zernike functions: G68Nv:
% ------------------------------ .e2A*9,
idx_pos = m>0; {I-a;XBX
idx_neg = m<0; DGZY~(]
%^5 @z1d,
z = y; <j
9Mt=8M
if any(idx_pos) 51M^yG&M
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 1:x nD
end +Sd,l>8\
if any(idx_neg) \}x'>6zr2
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ]AA%J@
end ZutB_uW
/uE^H%9h
% EOF zernfun