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 iKAusWj
function z = zernfun(n,m,r,theta,nflag) fzPZ|
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. bK*~ol
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N BJy;-(JP
% and angular frequency M, evaluated at positions (R,THETA) on the
3+U]?7t
% unit circle. N is a vector of positive integers (including 0), and Ll}yJ#3,
% M is a vector with the same number of elements as N. Each element BC7 7<R!E)
% k of M must be a positive integer, with possible values M(k) = -N(k) J=H)JH3
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, H=~9CJ+tc
% and THETA is a vector of angles. R and THETA must have the same /tj$luls5
% length. The output Z is a matrix with one column for every (N,M) Ia4)uV8
% pair, and one row for every (R,THETA) pair. 8ObeiVXf)
% tC)6
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike /.Q4~Hw%}
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), G%{0i20_
% with delta(m,0) the Kronecker delta, is chosen so that the integral D$q'FZH
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ~ap2m
% and theta=0 to theta=2*pi) is unity. For the non-normalized 4 b,N8
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. [Qj;/
% E^CiOTN
% The Zernike functions are an orthogonal basis on the unit circle. Tv$sqVe9
% They are used in disciplines such as astronomy, optics, and m;,xmEp
% optometry to describe functions on a circular domain. ^3~e/P KM
% /,tAoa~FA
% The following table lists the first 15 Zernike functions. !#N\b
% $B
.Qc!m
% n m Zernike function Normalization &c%Y<1e`%
% -------------------------------------------------- #b)e4vwCq
% 0 0 1 1 T@YGB]*Y
% 1 1 r * cos(theta) 2 C+N k"l9
% 1 -1 r * sin(theta) 2 m_7
nz!h
% 2 -2 r^2 * cos(2*theta) sqrt(6) 3z8C
% 2 0 (2*r^2 - 1) sqrt(3) ,o#kRWRG
% 2 2 r^2 * sin(2*theta) sqrt(6) ] d?x$>
% 3 -3 r^3 * cos(3*theta) sqrt(8) E>uVofhml
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) .\:J~(
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) X#p Wyo~
% 3 3 r^3 * sin(3*theta) sqrt(8) "484n/D
% 4 -4 r^4 * cos(4*theta) sqrt(10) N4!<Xj
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) E"PcrWB&
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) Lx[
,Z,kD
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) fiDl8=~@
% 4 4 r^4 * sin(4*theta) sqrt(10) !8Rw O%c(
% -------------------------------------------------- !0}\&<8/m
% <48<86TP
% Example 1: 0L-!!
c3
% ftbpqp'
% % Display the Zernike function Z(n=5,m=1) 6lFfS!ZFA
% x = -1:0.01:1; +OHGn;C
% [X,Y] = meshgrid(x,x); =xN= #
% [theta,r] = cart2pol(X,Y); n1v5Q2xw
% idx = r<=1; Ip
*g'
% z = nan(size(X)); L}k/9F.5
% z(idx) = zernfun(5,1,r(idx),theta(idx)); ;;U:Jtn2
% figure 1KE:[YQ1
% pcolor(x,x,z), shading interp m`A%
p
% axis square, colorbar n.}T1q|l
% title('Zernike function Z_5^1(r,\theta)') -ysn&d\rV
% A%bCMP
% Example 2: , H
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% ]uh3R{a/
% % Display the first 10 Zernike functions `BXS)xj
% x = -1:0.01:1; R9o- `Wz
% [X,Y] = meshgrid(x,x); Gh(
A%x)
% [theta,r] = cart2pol(X,Y); HIvZQQW|
% idx = r<=1; F5T3E?_
% z = nan(size(X)); ^+|De}`u
% n = [0 1 1 2 2 2 3 3 3 3]; uaPx"
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; uE5X~
% Nplot = [4 10 12 16 18 20 22 24 26 28]; H`QQG!
% y = zernfun(n,m,r(idx),theta(idx)); |NFZ(6vNh
% figure('Units','normalized') 9$*s8}|
% for k = 1:10 %&<LNEiUN
% z(idx) = y(:,k); A*yi"{FLi
% subplot(4,7,Nplot(k)) =d`5f@'rl
% pcolor(x,x,z), shading interp o^p
% set(gca,'XTick',[],'YTick',[]) 8At<Wic
% axis square E,[xUz"
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ]+Ixi o
% end [:EvTY
% _8?o'<!8?^
% See also ZERNPOL, ZERNFUN2. 2t#L:vY
eVh-_
% Paul Fricker 11/13/2006 $iw%(H
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`)$_YZq|SR
% Check and prepare the inputs: 5]Ajf;W\
% ----------------------------- 6sfwlT
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) }Fb!?['G5
error('zernfun:NMvectors','N and M must be vectors.') dFXc/VH')
end Q;/a F`
9WG{p[
if length(n)~=length(m) 4_?7&G0(
error('zernfun:NMlength','N and M must be the same length.') fPa9ofU/kr
end GIwh@4;
qCQ./"8
n = n(:); uKr1Z2
m = m(:); BRRj$)u
if any(mod(n-m,2)) j Ch=@<9
error('zernfun:NMmultiplesof2', ... .p`
pG3
'All N and M must differ by multiples of 2 (including 0).') ,El!fgL
end Q9F)
`TLzVB-j3
if any(m>n) u,.3
error('zernfun:MlessthanN', ... p<Z3tD;Z
'Each M must be less than or equal to its corresponding N.') ^C)n$L>C0
end ,L>
ar)B
= "ts`>
if any( r>1 | r<0 ) !RvRGRSyF
error('zernfun:Rlessthan1','All R must be between 0 and 1.') j{++6<tr
end +~zXDBS9
sN=6 gCau
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) F"+o@9]
error('zernfun:RTHvector','R and THETA must be vectors.') jdA
]2]
end =qVP] 9
Kb;dKQ
r = r(:); Dh|w^Q
theta = theta(:); C@\{ehG
length_r = length(r); &?,U_)x/
if length_r~=length(theta) p/6zEZ*
error('zernfun:RTHlength', ... \*vHB`.,ey
'The number of R- and THETA-values must be equal.') ?i\;:<e4
end m|tC24
f>j wN@(
% Check normalization: Wzq>JNny
% -------------------- }
l667N
if nargin==5 && ischar(nflag) kh$_!BT
isnorm = strcmpi(nflag,'norm'); {2d_"lHBt
if ~isnorm n
1b(\PA
error('zernfun:normalization','Unrecognized normalization flag.') IXLO>>`
end @exey
else ed 59B)?l
isnorm = false; b,H[I!. %
end %V!iQzL1
2.uA|~qH
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% B:TR2G9UT
% Compute the Zernike Polynomials }Nj97R
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% d;[u8t
l(W[_ D
% Determine the required powers of r: K]oM8H1
% ----------------------------------- ]w).8=I
m_abs = abs(m); zSTR^sgJ
rpowers = []; %hS|68pN6
for j = 1:length(n) B0}~G(t(
rpowers = [rpowers m_abs(j):2:n(j)]; D|bBu
end &Nl2sey
rpowers = unique(rpowers); yGBQ0o7E
QWnndI_4p
% Pre-compute the values of r raised to the required powers, G#`\(NW
% and compile them in a matrix: #^#Kcg
% ----------------------------- `|O yRU"EK
if rpowers(1)==0 >cMd\%^t
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); c~,23wP1
rpowern = cat(2,rpowern{:}); AnsjmR:Jv
rpowern = [ones(length_r,1) rpowern]; Fqq6^um
else km5~Gc}
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 8;P2A\X
rpowern = cat(2,rpowern{:}); =s97Z-
end 7Ey#u4Q
mdih-u(T|
% Compute the values of the polynomials: u^W2UE\
% -------------------------------------- .\3`2
y = zeros(length_r,length(n)); eJ8]g49mD6
for j = 1:length(n) * A|-KKo\
s = 0:(n(j)-m_abs(j))/2; 10[Jl5+t
pows = n(j):-2:m_abs(j); [s1pM1x
for k = length(s):-1:1 Z,7R;,qX
p = (1-2*mod(s(k),2))* ... Cr/`keR
prod(2:(n(j)-s(k)))/ ... DC+wD
Bp;
prod(2:s(k))/ ... 1nhtM
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... X&m'.PA
prod(2:((n(j)+m_abs(j))/2-s(k))); N^0uit
idx = (pows(k)==rpowers); GyI-)BlDC
y(:,j) = y(:,j) + p*rpowern(:,idx); %GEJnJ
end
4-Z()F
O09ke-lC
if isnorm ,{eUP0]
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); .0HZNWRtb
end :c[n\)U[aa
end C_fY %O
% END: Compute the Zernike Polynomials X<OSN&d
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% &O\(;mFc
B@v\eF;
% Compute the Zernike functions: `<" m%>
% ------------------------------ !G5a*8]
idx_pos = m>0; N[|Nxm0z/C
idx_neg = m<0; u'A#%}3
._:nw=Y0<}
z = y; (WlIwKP
if any(idx_pos) V:NI4dv/R
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); #%3rTU
end -ZOBAG*
if any(idx_neg) hv$yV%.`
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); YA(@5CZ
end cTZ.}eLh
E N^Uki`
% EOF zernfun