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 H#bu3*'
function z = zernfun(n,m,r,theta,nflag) Ej`G(
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. K%/g!t)
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N X`I=Z ysB
% and angular frequency M, evaluated at positions (R,THETA) on the HA0yX?f]
% unit circle. N is a vector of positive integers (including 0), and AgdU@&^
% M is a vector with the same number of elements as N. Each element y<y9'tx
% k of M must be a positive integer, with possible values M(k) = -N(k) Btc[
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, ;ZZmX]kz,M
% and THETA is a vector of angles. R and THETA must have the same S'sI[?\x
% length. The output Z is a matrix with one column for every (N,M) ;i 3C
% pair, and one row for every (R,THETA) pair. QmsS,Zljo
% _%aT3C}k
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike {|Fn<&G
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 4{"
v
% with delta(m,0) the Kronecker delta, is chosen so that the integral o^BX:\}
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, PC)V".W1
% and theta=0 to theta=2*pi) is unity. For the non-normalized q4u-mM7#7
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ' PmBNT
% *0 ;|
% The Zernike functions are an orthogonal basis on the unit circle. YMn=9EUp
% They are used in disciplines such as astronomy, optics, and Km0P)Z
% optometry to describe functions on a circular domain. r / L
% S,C/l1s
% The following table lists the first 15 Zernike functions. PO=A^ b
% m] @o1J
% n m Zernike function Normalization 7L!q{%}
% -------------------------------------------------- 'ExQG$t
% 0 0 1 1 vn96o]n
% 1 1 r * cos(theta) 2 Wt!NLlN8
% 1 -1 r * sin(theta) 2 &>hln<a>
% 2 -2 r^2 * cos(2*theta) sqrt(6) Qexv_:C
% 2 0 (2*r^2 - 1) sqrt(3) <U""CAE
% 2 2 r^2 * sin(2*theta) sqrt(6) ?w@KF%D
% 3 -3 r^3 * cos(3*theta) sqrt(8) L$f:D2Ei
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) )`m/vYKWL
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) P/dT;YhL
% 3 3 r^3 * sin(3*theta) sqrt(8) Za1VJ5-
% 4 -4 r^4 * cos(4*theta) sqrt(10) y~+U(-&.
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) luO4ap]*
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) h/#s\>)T
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ':T6m=yv
% 4 4 r^4 * sin(4*theta) sqrt(10) +*$@ K'VL
% -------------------------------------------------- {`[u XH?3d
% z%L\EP;o}
% Example 1: s|C4Jy_
% ww~gmz
% % Display the Zernike function Z(n=5,m=1) 1;[ZkRbzL
% x = -1:0.01:1; p87VJ}
% [X,Y] = meshgrid(x,x); @{8SC~ha
% [theta,r] = cart2pol(X,Y); I~7eu&QZ
% idx = r<=1; %|By ?i
% z = nan(size(X)); j;i7.B"[
% z(idx) = zernfun(5,1,r(idx),theta(idx)); n6
AP6PK7
% figure UmA'aq
% pcolor(x,x,z), shading interp a(eUdGJ
% axis square, colorbar 1V 2"sE
% title('Zernike function Z_5^1(r,\theta)') ;S^7Q5-
% sVT\e*4m}
% Example 2: \g\,
% %!Ak]|[7
% % Display the first 10 Zernike functions E3o J;E
% x = -1:0.01:1; n4Eqm33
% [X,Y] = meshgrid(x,x); -$_h]x*
W
% [theta,r] = cart2pol(X,Y); \Y}nehxG@
% idx = r<=1; Q
,)}t
% z = nan(size(X)); 5y|/}D>
% n = [0 1 1 2 2 2 3 3 3 3]; ;/.XAxkFL
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; wr;8o*~
% Nplot = [4 10 12 16 18 20 22 24 26 28]; 9WsGoZPn
% y = zernfun(n,m,r(idx),theta(idx)); EX^j^#N
% figure('Units','normalized') TZ%u;tBH:
% for k = 1:10 iKuSk~
% z(idx) = y(:,k); bcZ s+FOPd
% subplot(4,7,Nplot(k)) BF>3CW7
% pcolor(x,x,z), shading interp ^H
UNq[sQ
% set(gca,'XTick',[],'YTick',[]) B*j
AD2
% axis square l*C(FPw4
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) m>@ *-*8k
% end (E(kw="
% gsp|?)]x
% See also ZERNPOL, ZERNFUN2. fo30f=^Gi
hM @F|t3
% Paul Fricker 11/13/2006 4zM$I
.ahYjn
wWR9dsB.;
% Check and prepare the inputs: :TzHI
% ----------------------------- _4jRUsvjY
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) hZ@Wl6FG;
error('zernfun:NMvectors','N and M must be vectors.') 5%n
end DU/WB
(lY<\l
if length(n)~=length(m) bl;C=n
error('zernfun:NMlength','N and M must be the same length.') 7+vyN^XJ"5
end e8"?Qm7 J
REvY`
n = n(:); ?`%)3gx|
m = m(:); H%:~&_D
if any(mod(n-m,2)) sOBy)vq?\
error('zernfun:NMmultiplesof2', ... Z@I.socA
'All N and M must differ by multiples of 2 (including 0).') A<zSh}eh6
end OK}+:Y
;8
D31OT
if any(m>n) `_{^&W
WS
error('zernfun:MlessthanN', ... 3,cZ*4('d
'Each M must be less than or equal to its corresponding N.') c`(] j
w
end _pv<_
Sm
Htf|VpzMb
if any( r>1 | r<0 ) D|[~Py
error('zernfun:Rlessthan1','All R must be between 0 and 1.') Z?^~f}+
end BtN@P23>k.
D d$ SQ
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) A9[ELD>p
error('zernfun:RTHvector','R and THETA must be vectors.') =gb.%a{R
end U`es
n?m!
jHj*S9:`
r = r(:); \*0ow`|K
theta = theta(:); [p+6HF
length_r = length(r); =sk]/64h``
if length_r~=length(theta) N!}r(Dd*
error('zernfun:RTHlength', ... TrHz(no
'The number of R- and THETA-values must be equal.') n3t0Qc
end b[3K:ot+
pbe"
w=<
% Check normalization: x7=5 ;gf/X
% -------------------- T
_O|gU
if nargin==5 && ischar(nflag) 8%s_~Yc
isnorm = strcmpi(nflag,'norm'); -$#'
if ~isnorm u[_~ !y
error('zernfun:normalization','Unrecognized normalization flag.')
9I:H=5c
end {[
j+y
else 4
|E`
isnorm = false; 4%TY`
II
end 'mz
_JM
TixXA:Mf
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -o\r]24
% Compute the Zernike Polynomials 9WaKs d f
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% &n.7~C]R
piE9qXn
% Determine the required powers of r: tc%?{W\
% ----------------------------------- h[SuuW
m_abs = abs(m); |RBgJkS;8
rpowers = []; /XG4O
for j = 1:length(n) hVe@:1og#
rpowers = [rpowers m_abs(j):2:n(j)]; 5fK#*(x
end d`U{-?N>
rpowers = unique(rpowers); >W=
0N(
2-9'zN0u
% Pre-compute the values of r raised to the required powers, ,[rh7_
% and compile them in a matrix: ~G!>2 +L
% ----------------------------- $\xS~w
if rpowers(1)==0 ]~:9b[G2
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); D=U"L-rRs
rpowern = cat(2,rpowern{:}); cNzn2-qv
rpowern = [ones(length_r,1) rpowern]; jFBLElE
else )6# i>c-
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); @<5?q:9.8
rpowern = cat(2,rpowern{:}); Farcd!}
end $F!)S
rULrGoM
% Compute the values of the polynomials: io_4d2uBh
% -------------------------------------- K4Mv\! Q<8
y = zeros(length_r,length(n)); B1]dub9
for j = 1:length(n) Z[Gs/D
s = 0:(n(j)-m_abs(j))/2; zT[[WY4
pows = n(j):-2:m_abs(j); s7?Q[vN
for k = length(s):-1:1 fHek!Jv.
p = (1-2*mod(s(k),2))* ... Aen)r@Y:
prod(2:(n(j)-s(k)))/ ... zmH 8#
prod(2:s(k))/ ... H@$\SUc{
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... i?uJ<BdU[
prod(2:((n(j)+m_abs(j))/2-s(k))); Omkl|l9
idx = (pows(k)==rpowers); Z !Njfq5
y(:,j) = y(:,j) + p*rpowern(:,idx); ^lCys
end x4jn45]x@
"wi=aV9j
if isnorm Jrp{e("9
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); T!.6@g`x>
end (B@:0}>
end {FO>^~>l
% END: Compute the Zernike Polynomials iV5x-G`
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% _{ z.Tu
irSdqa/
% Compute the Zernike functions: [,s{ /OM
% ------------------------------ qkpnXQ
idx_pos = m>0; }~Z1C0t
idx_neg = m<0; *Z*4L|zT
[U_Su,
z = y; Wpo:'?!(M^
if any(idx_pos) ,/n<Qg"`
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); "G\OKt'Z
end 8<}f:9/
if any(idx_neg) ;h>s=D,r
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 5a1)`2V2M
end Ay'2!K,I
1?\ #hemL
% EOF zernfun