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 'T qF}a7
function z = zernfun(n,m,r,theta,nflag) *""W`x
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. .tHc*Eh
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N }?6;;d#
% and angular frequency M, evaluated at positions (R,THETA) on the SfY9PNck\
% unit circle. N is a vector of positive integers (including 0), and {<}Hut:a
% M is a vector with the same number of elements as N. Each element }C/+zF6q
% k of M must be a positive integer, with possible values M(k) = -N(k) &~B8~U4%
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, ?uJX
% and THETA is a vector of angles. R and THETA must have the same GA[bo)"
% length. The output Z is a matrix with one column for every (N,M) Ijz*wq\s;
% pair, and one row for every (R,THETA) pair. >XiT[Ru
% ;:R2 P@6f
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike .YB/7-%M[
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ~5Mj:{B
% with delta(m,0) the Kronecker delta, is chosen so that the integral MwQt/Qv=
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, glROT@
% and theta=0 to theta=2*pi) is unity. For the non-normalized }F9#3W&`c
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. cCx{
")
% _.]mES|
% The Zernike functions are an orthogonal basis on the unit circle. {wz_ngQ
% They are used in disciplines such as astronomy, optics, and :.a184ax
% optometry to describe functions on a circular domain. f4d-eXGwx`
% (@^ySiU
% The following table lists the first 15 Zernike functions. XUUP#<,s
% Cv*K.T
% n m Zernike function Normalization :Zob"*T
% -------------------------------------------------- t7V7 TL!5'
% 0 0 1 1 B#5[PX
% 1 1 r * cos(theta) 2 [[N${ C
% 1 -1 r * sin(theta) 2 1Q9Hs(s
% 2 -2 r^2 * cos(2*theta) sqrt(6) +NvpYz
% 2 0 (2*r^2 - 1) sqrt(3) Nx*1m
BC
% 2 2 r^2 * sin(2*theta) sqrt(6) 9O Y ao
% 3 -3 r^3 * cos(3*theta) sqrt(8) OkT@ _U
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) {%y|A{}c
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) uT<<G)v)
% 3 3 r^3 * sin(3*theta) sqrt(8) D8Mq '$-
% 4 -4 r^4 * cos(4*theta) sqrt(10) ,PJC FQMR
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) YvP62c \
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) ^f"|<r
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Q uw|KL
% 4 4 r^4 * sin(4*theta) sqrt(10) =i;T?*@
% -------------------------------------------------- gnxD'1_
% 6zNWDUf
% Example 1: Pq(LW(
% ^~bdAO81
% % Display the Zernike function Z(n=5,m=1) $T7 qd
% x = -1:0.01:1; #&L7FBJ"*v
% [X,Y] = meshgrid(x,x); N{@~(>ee^
% [theta,r] = cart2pol(X,Y); @B(E&
% idx = r<=1; Q%J,:J
% z = nan(size(X)); kr
|k \
% z(idx) = zernfun(5,1,r(idx),theta(idx)); t6\--lk_
% figure 9zCuVUcd$.
% pcolor(x,x,z), shading interp 5gC>j(
% axis square, colorbar Lz:FR*
% title('Zernike function Z_5^1(r,\theta)') T:|p[Xbo
% ryA+Lli.
% Example 2: xpwy%uo
% e:.?T\
% % Display the first 10 Zernike functions Odh r=Hs
% x = -1:0.01:1; 2*Pk1vrI
% [X,Y] = meshgrid(x,x); lq,]E/<&
% [theta,r] = cart2pol(X,Y); ,7k1n{C)
% idx = r<=1; ~ kDJ-V
% z = nan(size(X)); ,]]IJ;:w
% n = [0 1 1 2 2 2 3 3 3 3]; QF*cdc<
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; A 2A_F|f
% Nplot = [4 10 12 16 18 20 22 24 26 28]; 'Yc^9;C(
% y = zernfun(n,m,r(idx),theta(idx)); zM<L_l&
% figure('Units','normalized') 5tLb
o
% for k = 1:10 \$ss
% z(idx) = y(:,k); oK4xRv8Hd
% subplot(4,7,Nplot(k)) b^ [ z'
% pcolor(x,x,z), shading interp 72*j6#zS
% set(gca,'XTick',[],'YTick',[]) {{gt>"D,
% axis square 5dD8s-;^T
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) .P?n<n#
% end #)[.Xz:U
% Vx>Q
% See also ZERNPOL, ZERNFUN2. [fo#){3K
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% Paul Fricker 11/13/2006 N#$]W"U
CQrP%}`r
ozl!vf# kv
% Check and prepare the inputs: NPM2qL9&J
% ----------------------------- yaWY>sB
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 7-}5
W
error('zernfun:NMvectors','N and M must be vectors.') Ld/6{w4ir
end S{f,EBE
k#8`996P
if length(n)~=length(m) |GsMLY:0
error('zernfun:NMlength','N and M must be the same length.') G#6Z@|kVw
end -!li,&,A1
IXR'JZ?fH
n = n(:); Em5,Zr_
m = m(:); ]+B.=mO_
if any(mod(n-m,2)) rX>b R/
error('zernfun:NMmultiplesof2', ... a)Pr&9I
'All N and M must differ by multiples of 2 (including 0).') oGl<i
end H=g%>W%3
,&o^}TFkg
if any(m>n) z1^fG)
error('zernfun:MlessthanN', ... niW"o-}
'Each M must be less than or equal to its corresponding N.') <hTHY E=
end ~kSOYvK$'
`NEi/jB
if any( r>1 | r<0 ) H270)Cwn+
error('zernfun:Rlessthan1','All R must be between 0 and 1.') o)7Ot\:E
end ^yq}>_
:M f8q!Q'
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) cs9h\]ZA
error('zernfun:RTHvector','R and THETA must be vectors.') .cw)Y#;IG
end ,R3TFVV!?
{&B_b|g*fW
r = r(:); HIvSpO
theta = theta(:); la!U
length_r = length(r); w%\{4T~
if length_r~=length(theta) ^~7Mv^A
error('zernfun:RTHlength', ... 8e,F{>N
'The number of R- and THETA-values must be equal.') mU?~s7
end S_OtY]gF
@F$}/
% Check normalization: llWY7u"
% -------------------- g7*Uuh#
if nargin==5 && ischar(nflag) ]j6K3
isnorm = strcmpi(nflag,'norm'); Tcc83_Iq
if ~isnorm k`|E&+og
error('zernfun:normalization','Unrecognized normalization flag.') vD?D]8.F~Q
end "Y&
else '-[hy>t
isnorm = false; H^@Hco>|
end U=69q]
qBh@^GxY),
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% =s]2?m
% Compute the Zernike Polynomials T6=|)UTe1
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 6DK).|@$r
vR2);ywX
% Determine the required powers of r: <*dcl2xS
% ----------------------------------- cg17e
m_abs = abs(m); y %61xA`#
rpowers = []; D M+MBK
for j = 1:length(n) e!gNd>b {
rpowers = [rpowers m_abs(j):2:n(j)]; r^<,f[yH
end ~_ZK93o(
rpowers = unique(rpowers); SOM? 0.
>3KlI
% Pre-compute the values of r raised to the required powers, l>pB\<LL
% and compile them in a matrix:
<HN+pi
% ----------------------------- @SiV3k
if rpowers(1)==0 rr1'|
k"
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 8]`s&d@GY
rpowern = cat(2,rpowern{:}); .
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rpowern = [ones(length_r,1) rpowern];
pV u[
else ?YZgH>7"
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 8Q<Nl=g>'
rpowern = cat(2,rpowern{:}); ly0L)L]\
end Ax;?~v4Z
Zy;jp*Q
% Compute the values of the polynomials: mI4GBp
% -------------------------------------- )j~{P
y = zeros(length_r,length(n)); iQ8{N:58DN
for j = 1:length(n) %7aJSuQN%
s = 0:(n(j)-m_abs(j))/2; knG:6tQ
pows = n(j):-2:m_abs(j); %aK[Yvo6
for k = length(s):-1:1 FZ+2{wIV^
p = (1-2*mod(s(k),2))* ... 8Nyz{T[
prod(2:(n(j)-s(k)))/ ... 'h'pM#D
prod(2:s(k))/ ... SM
RKEPwp&
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Y,Z$U| U
prod(2:((n(j)+m_abs(j))/2-s(k))); wzd(=*N
idx = (pows(k)==rpowers); 0|tyKP|J
y(:,j) = y(:,j) + p*rpowern(:,idx); IE996
end
~,&8)1
uj.$GAtO)
if isnorm (_@5V_U
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ?&eS }skL
end JU^Y27
end n/Fxjf0W
% END: Compute the Zernike Polynomials OEjX(F3=
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% U2<q dknB
6wwbH}*=?
% Compute the Zernike functions: Ji9o0Y R
% ------------------------------ H7&y79mB
idx_pos = m>0; E=,5%>C0#%
idx_neg = m<0; $poIWJM c
ciml:"nQ
z = y; R$
+RTG:E
if any(idx_pos) [|eIax xR,
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); zc;kNkV#1Y
end 36+/MvIT
if any(idx_neg) EHn!ZrQgh
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); __$ ;Z
end ^[m-PS(
E2w-b^,5
% EOF zernfun