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 mW%?>Z1=>d
function z = zernfun(n,m,r,theta,nflag) .lhn;*Yi
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. |lH;Fq{\
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N w#i[_
% and angular frequency M, evaluated at positions (R,THETA) on the @5)
8L/[l
% unit circle. N is a vector of positive integers (including 0), and midsnG+jnf
% M is a vector with the same number of elements as N. Each element g/UaYCjM
% k of M must be a positive integer, with possible values M(k) = -N(k) hC_Vts[v/
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, fQ+VT|jzx
% and THETA is a vector of angles. R and THETA must have the same Cc?TSZ8[
% length. The output Z is a matrix with one column for every (N,M) *]J dHO
% pair, and one row for every (R,THETA) pair. UueD(T;p
% l!E7AKk8
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike AGA`fRVx
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), (SVWdgb
% with delta(m,0) the Kronecker delta, is chosen so that the integral (eCFWmO
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, SvvUkQ#1w
% and theta=0 to theta=2*pi) is unity. For the non-normalized a'\By?V]
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. n3MWs);5
% ;jK#[*y
% The Zernike functions are an orthogonal basis on the unit circle. 5W
=(+Q>C
% They are used in disciplines such as astronomy, optics, and @&1Wyp
% optometry to describe functions on a circular domain. 4\.V
% ,~zj=F
% The following table lists the first 15 Zernike functions. zm9TvoC%}
% HEqWoV]{d
% n m Zernike function Normalization zBf-8]"^
% -------------------------------------------------- xr(|*
% 0 0 1 1 +kdySWF
% 1 1 r * cos(theta) 2 Uh.Zi3X6}6
% 1 -1 r * sin(theta) 2 1gO2C$
% 2 -2 r^2 * cos(2*theta) sqrt(6) Q4s&E\}
% 2 0 (2*r^2 - 1) sqrt(3) "%8A:^1
% 2 2 r^2 * sin(2*theta) sqrt(6) v}J;ZIb
% 3 -3 r^3 * cos(3*theta) sqrt(8) 2}}?'PwwT
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) `P+(&taT
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) vjViX<#(V
% 3 3 r^3 * sin(3*theta) sqrt(8) !}3,B28
% 4 -4 r^4 * cos(4*theta) sqrt(10) (B>Zaro#
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 7dh1W@\
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) C-P06Q]
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) TY;U2.Ud
% 4 4 r^4 * sin(4*theta) sqrt(10) @D`zKYwX1
% -------------------------------------------------- VS?@y/\In
% &ntBU]<q
% Example 1: M/V(5IoP(
% ~!%0Z9>ap
% % Display the Zernike function Z(n=5,m=1) &A!KJ.
% x = -1:0.01:1; NnxM3*
% [X,Y] = meshgrid(x,x); UkR3}{i
% [theta,r] = cart2pol(X,Y); D1,O:+[;.
% idx = r<=1; aI#4H+/
% z = nan(size(X)); ^c9ThV.v
% z(idx) = zernfun(5,1,r(idx),theta(idx)); <2
% figure hQJWKAf,/
% pcolor(x,x,z), shading interp QF-)^`N
% axis square, colorbar }F`beoMAkM
% title('Zernike function Z_5^1(r,\theta)') |U[y_Y\a
% !^U6Z@&/R
% Example 2: 0/]_nd
% urY`^lX~
% % Display the first 10 Zernike functions 2xmk,&s
% x = -1:0.01:1; VlW9UF-W
% [X,Y] = meshgrid(x,x); b5ie <s
% [theta,r] = cart2pol(X,Y); ;np_%?is
% idx = r<=1; D#sf i,O
% z = nan(size(X)); m^!Sv?hV
% n = [0 1 1 2 2 2 3 3 3 3]; MM#cLw
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; ~
}KzJiL
% Nplot = [4 10 12 16 18 20 22 24 26 28]; eVnbRT2y&
% y = zernfun(n,m,r(idx),theta(idx)); {KaN,td9
% figure('Units','normalized') ]H 2R
% for k = 1:10 4E"d /
% z(idx) = y(:,k); 7#4%\f+'t
% subplot(4,7,Nplot(k)) R $b,h
% pcolor(x,x,z), shading interp I"!'AI-
% set(gca,'XTick',[],'YTick',[]) y~#\#w{
% axis square |paP<$
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) l[n@/%2
% end Rlg#z4m
% LZWS^77
% See also ZERNPOL, ZERNFUN2. {Qtq7q.
=Q?f96T
% Paul Fricker 11/13/2006 `!c,y~r[
@[r ={s\
?M&4pO&Y
% Check and prepare the inputs: $^vP<
% ----------------------------- H/i<_L P
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) DA <ynBQ
error('zernfun:NMvectors','N and M must be vectors.') Tx+ p8J|Yr
end QaMDGD
me. /o(!?
if length(n)~=length(m) 1k>naf~O
error('zernfun:NMlength','N and M must be the same length.') g37q/nEv
end ce5nG0@#
?:}Pa<D&K
n = n(:); 9y+[o
m = m(:); ltEF:{mLe#
if any(mod(n-m,2)) A^pW]r=Xtk
error('zernfun:NMmultiplesof2', ... N#Ag'i4HF
'All N and M must differ by multiples of 2 (including 0).') xURw,
end x YT}>#[
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if any(m>n) gB+
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error('zernfun:MlessthanN', ... PRpE$`WK
'Each M must be less than or equal to its corresponding N.') ;:_(7|
end 9--dRTG
5^F]tRz-
if any( r>1 | r<0 ) ??I:H
error('zernfun:Rlessthan1','All R must be between 0 and 1.') :`zV
[A:D
end ;f(n.i
{bTeAfbf]
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ,I39&;Iq
error('zernfun:RTHvector','R and THETA must be vectors.') R92R}=G!
end G;2[
%^')G+>i
r = r(:); e`ex]py<C
theta = theta(:); ?waebuj>
length_r = length(r); e?vj+ZlS$f
if length_r~=length(theta) \1{_lynD
error('zernfun:RTHlength', ... PSEWL6=]N
'The number of R- and THETA-values must be equal.') V2QW\2@$
end 86{ZFtv
sS'{QIRC'
% Check normalization: cKpQr7]ur
% -------------------- /#IH-2N
if nargin==5 && ischar(nflag) paYz[Xq
isnorm = strcmpi(nflag,'norm'); 82.HH5Z{
if ~isnorm iPkT*Cl8
error('zernfun:normalization','Unrecognized normalization flag.') +U=KXv
end \d5}5J]a&n
else 5*XH6g F
isnorm = false; }#|2z}!
end uH]
m]t
/1N)d?Pcl
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [)k2=67
% Compute the Zernike Polynomials r"[L0Cbb
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% "MTq{f2?
}
Ab_o#Zy
% Determine the required powers of r: ebD{ pc`&
% ----------------------------------- 'rh\CA/}D
m_abs = abs(m); DZ%8 |PmB
rpowers = []; Y)v%
for j = 1:length(n) aLHrl6"
rpowers = [rpowers m_abs(j):2:n(j)]; |QMT
A5
end `{WCrw6)
rpowers = unique(rpowers); -rRz@Cr
acy"ct*I
% Pre-compute the values of r raised to the required powers, XJ
_%!
% and compile them in a matrix: @M9_j{A
% ----------------------------- ? 9qAe
if rpowers(1)==0 |/t K-c6J
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); @@; 1%z
rpowern = cat(2,rpowern{:}); J:[3;Z
rpowern = [ones(length_r,1) rpowern]; hN}5u"pS
else Mi;Tn;3er
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); #-A5Z;TD.
rpowern = cat(2,rpowern{:}); }Uq/kei^P
end .$OjUlzr-H
%K`4k.gN
% Compute the values of the polynomials: {6DpPw^ "
% -------------------------------------- 7%X+O8
y = zeros(length_r,length(n)); ?SB5b ,
for j = 1:length(n) R,XD6' Q
s = 0:(n(j)-m_abs(j))/2; VgUvD1v?}
pows = n(j):-2:m_abs(j); y.%i
for k = length(s):-1:1 "^!j5fZ
p = (1-2*mod(s(k),2))* ... J511AoQ{R
prod(2:(n(j)-s(k)))/ ... 2Sv>C `FMU
prod(2:s(k))/ ... zabw!@]
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... x={kjym L
prod(2:((n(j)+m_abs(j))/2-s(k))); ;A`IYRzt
idx = (pows(k)==rpowers); Xk;Uk[
y(:,j) = y(:,j) + p*rpowern(:,idx); kK08W3@&t
end zv&ePq\#
O#A8t<f|M
if isnorm aS2a_!f
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 1fmSk$ y.9
end 5Gc_LI&v7
end iz,]%<_PE
% END: Compute the Zernike Polynomials #vnefIcBf
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 7O]J^H+7
Bi %Z2/
% Compute the Zernike functions: !>?4[|?n<
% ------------------------------ q|?`Gsr
idx_pos = m>0; ?=TL2"L
idx_neg = m<0; eUi> Mp
NU BpIx&
z = y; z&\Il#'\m+
if any(idx_pos) nYo&x'
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); xn0s`I[
end !k4 }v'=
if any(idx_neg) (K!M*d+
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); n U+pnkMj
end yIn/Y 0No
&Xj {:s#
% EOF zernfun