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 ^FK-e;J
function z = zernfun(n,m,r,theta,nflag) NO.5Vy
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. o@r~KFIe
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N cb _nlG!
% and angular frequency M, evaluated at positions (R,THETA) on the uBo~PiJ2"
% unit circle. N is a vector of positive integers (including 0), and oMF[<Xf
% M is a vector with the same number of elements as N. Each element j$khGR!
% k of M must be a positive integer, with possible values M(k) = -N(k) ljk,R
G
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, ]"U/3dL5
% and THETA is a vector of angles. R and THETA must have the same l gTw>r
% length. The output Z is a matrix with one column for every (N,M) uSNlI78D
% pair, and one row for every (R,THETA) pair. DbH'Qs?z
% Hr=?_Un"
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ZrDr/Q~
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), gPy}.g{tH$
% with delta(m,0) the Kronecker delta, is chosen so that the integral Qy|6A@
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, =b#,OXQ
% and theta=0 to theta=2*pi) is unity. For the non-normalized NE-c[|rq
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 4 _Idf
% ~> 5
% The Zernike functions are an orthogonal basis on the unit circle. 4Kn)5>
% They are used in disciplines such as astronomy, optics, and .\|}5J9W
% optometry to describe functions on a circular domain. `5t
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% {3\{aZ8)
% The following table lists the first 15 Zernike functions. _S6SCSFc
% z6bIv}
% n m Zernike function Normalization Z`{GjV3%wH
% -------------------------------------------------- Rj/ y.g
% 0 0 1 1 Hc-Ke1+
% 1 1 r * cos(theta) 2 Cg%}=
% 1 -1 r * sin(theta) 2 2M?L++i
% 2 -2 r^2 * cos(2*theta) sqrt(6) _SQ0`=+
% 2 0 (2*r^2 - 1) sqrt(3) LKu
,H
% 2 2 r^2 * sin(2*theta) sqrt(6) fBct%M 3
% 3 -3 r^3 * cos(3*theta) sqrt(8) p|'Rm]&jb
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) Ct9*T`Gl
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) ^1z)\p1
% 3 3 r^3 * sin(3*theta) sqrt(8) &,iPI2`O A
% 4 -4 r^4 * cos(4*theta) sqrt(10) D
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% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10)
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% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) qJEtB;J'
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 8jU6N*p/
% 4 4 r^4 * sin(4*theta) sqrt(10) ZTK)N
% -------------------------------------------------- r[ RO"Ej"
% ^uWj#
% Example 1: #i[V{J8.p
% H.[t&VO
% % Display the Zernike function Z(n=5,m=1) =1% <
% x = -1:0.01:1; 1Et{lrgh
f
% [X,Y] = meshgrid(x,x); u#v];6N
% [theta,r] = cart2pol(X,Y); , @dhJ8/
% idx = r<=1; >&uR=Yd
% z = nan(size(X)); $ D(q
% z(idx) = zernfun(5,1,r(idx),theta(idx)); zZ{(7Kfz
% figure 0*8uo
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% pcolor(x,x,z), shading interp GQ=Pkko
% axis square, colorbar qc@v"pIz'S
% title('Zernike function Z_5^1(r,\theta)') Zi ;7.P qL
% >Gxh=**F
% Example 2: 1F94e)M)"
% ;&]oV`Ib
% % Display the first 10 Zernike functions k= oCpXq^
% x = -1:0.01:1; =FXq=x%9+
% [X,Y] = meshgrid(x,x); P(Q}r7F~(
% [theta,r] = cart2pol(X,Y); =fy'w3m
% idx = r<=1; Z^ }4bR]
% z = nan(size(X)); hC...tk
% n = [0 1 1 2 2 2 3 3 3 3]; T6Ks]6m_
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; PW GNUNc
% Nplot = [4 10 12 16 18 20 22 24 26 28]; 3d*wZ9qz
% y = zernfun(n,m,r(idx),theta(idx)); nO .:f
% figure('Units','normalized') h9WyQl7
% for k = 1:10 S]}W+BF3
% z(idx) = y(:,k); H0Ck%5
% subplot(4,7,Nplot(k)) EF[I@voc
% pcolor(x,x,z), shading interp jinXK
% set(gca,'XTick',[],'YTick',[]) &Vmx<w
% axis square C?lZu\L
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) H(F9&6}
% end 2, r{zJ8
% m0+'BC{$u
% See also ZERNPOL, ZERNFUN2. @1iH4RE*
`& }C*i"
% Paul Fricker 11/13/2006 rZ^VKO`~I1
4#2iq@s
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% Check and prepare the inputs: %n:ymc
$}
% ----------------------------- uE:`Fo=y
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) yc3i> w`
error('zernfun:NMvectors','N and M must be vectors.') UWg+7RL
end ({kOgOeC
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if length(n)~=length(m) Q804_F
F#
error('zernfun:NMlength','N and M must be the same length.') m005*>IY
end `Fs- z
WTQd}f
n = n(:); o&U/e\zy
m = m(:); F@Cxjz
if any(mod(n-m,2)) 8c0ugM
error('zernfun:NMmultiplesof2', ... -q}I;
cH
'All N and M must differ by multiples of 2 (including 0).') NM&R\GI
end OZi4S3k
]8ob`F`m,
if any(m>n) Wc!.{2
error('zernfun:MlessthanN', ... Jqgo\r%`
'Each M must be less than or equal to its corresponding N.') UA}N
end EK<ly"S.
37nGFH`K2m
if any( r>1 | r<0 ) W"ldQ
error('zernfun:Rlessthan1','All R must be between 0 and 1.') }@Ou]o
end f`"@7-N
s.9_/cFWB
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ^9A,j}>o-
error('zernfun:RTHvector','R and THETA must be vectors.') mM)d`br
end ]O.Z4+6w
k#pNk7;MZ
r = r(:); FG6mh,C!
theta = theta(:); x|q|> dPB
length_r = length(r); [V _\SQV0
if length_r~=length(theta) -Gmg&yQ9
error('zernfun:RTHlength', ... Jyo(Etp
'The number of R- and THETA-values must be equal.') G>w+J'7
end TwLQ;Q
tA]Y=U+Q
% Check normalization: `CF.-Vl3J#
% -------------------- ^A' Bghy
if nargin==5 && ischar(nflag) hT?|:!ED.F
isnorm = strcmpi(nflag,'norm'); ?-D'xqc
if ~isnorm BhCOT+i;c
error('zernfun:normalization','Unrecognized normalization flag.') );oE^3]f
end U.p"JSH
L
else }D7} %P]
isnorm = false; ^muPjM+D
end r>3y87
KB6`OT^b{r
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% )ME'qA3K
% Compute the Zernike Polynomials u:GDM
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ua]?D2
C}8 3t~Q
% Determine the required powers of r: WDq~mi
% ----------------------------------- SWPb=[WEz
m_abs = abs(m); G+zIh}9
rpowers = []; +je{%,*
for j = 1:length(n) JPGEE1!B{b
rpowers = [rpowers m_abs(j):2:n(j)]; *#g[
jl4
end S^*ME*DDz
rpowers = unique(rpowers); [ %:%C]4
DZ5QC aA
% Pre-compute the values of r raised to the required powers, G*\U'w4w|*
% and compile them in a matrix: fe$O Pl~
% ----------------------------- gO,2:,
if rpowers(1)==0 #xBh62yIuP
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); b?deZ2"L#
rpowern = cat(2,rpowern{:}); r"\g6<RP
rpowern = [ones(length_r,1) rpowern]; p{S#>JTr
else P2>Y0"bY
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); .:V4>
rpowern = cat(2,rpowern{:}); V/W{d[86G
end 4VrL@c
@
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% Compute the values of the polynomials: E{W(5.kb;i
% -------------------------------------- +!Lz]@9K
y = zeros(length_r,length(n)); _yP02a^2
for j = 1:length(n) |+r5D4]e
s = 0:(n(j)-m_abs(j))/2; )W.Y{\D0
pows = n(j):-2:m_abs(j); TDR2){I
for k = length(s):-1:1 kQQhZ8Ch
p = (1-2*mod(s(k),2))* ... w6FVSU]sY
prod(2:(n(j)-s(k)))/ ... nMU[S+
prod(2:s(k))/ ... h(MS>=
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... L qdzqq
prod(2:((n(j)+m_abs(j))/2-s(k))); A
^U`c'$
idx = (pows(k)==rpowers); C3GI?|b
y(:,j) = y(:,j) + p*rpowern(:,idx); l_z@.</8P@
end TSHH=`cx
Jl|^
if isnorm JDj^7\`
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); \bzT=^Z;2
end &p6^
end fw+ VR.#2H
% END: Compute the Zernike Polynomials 9G"-~C"e3
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (043G[H'.
B#Z-kFn@
% Compute the Zernike functions: 2z615?2_U
% ------------------------------ 8@J5tFJ&%
idx_pos = m>0; to"[r
idx_neg = m<0; PHHX)xK
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z = y; QB|D_?]
if any(idx_pos) -e(,>9Q
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 8j<+ '
R
end KM jnY2
if any(idx_neg) ;|H(_J=6k
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); %eDJ]\*^X
end CKgbb4;<m[
vhj^R5=
% EOF zernfun