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 =KT7ZSTV
function z = zernfun(n,m,r,theta,nflag) :[(X!eP
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. y>8!qVX
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N \@OKB<ra
% and angular frequency M, evaluated at positions (R,THETA) on the SVXey?A;CJ
% unit circle. N is a vector of positive integers (including 0), and _a*Wk
% M is a vector with the same number of elements as N. Each element OY~5o&Oa
% k of M must be a positive integer, with possible values M(k) = -N(k) 7+T\
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, ?Pmj }f
% and THETA is a vector of angles. R and THETA must have the same
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% length. The output Z is a matrix with one column for every (N,M) lKIHBi
% pair, and one row for every (R,THETA) pair. |#5JI#,vX
% lW&glU(
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 3 ;.{
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% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), Q;r 0#"
% with delta(m,0) the Kronecker delta, is chosen so that the integral */\dH<
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, v-G(bw3
% and theta=0 to theta=2*pi) is unity. For the non-normalized 9FV#@uA}D
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. /q='~t
% aDza"Ln
% The Zernike functions are an orthogonal basis on the unit circle.
e%'9oAz
% They are used in disciplines such as astronomy, optics, and Bb:jy!jq_
% optometry to describe functions on a circular domain. ;5y4v
% -oF4mi8S
% The following table lists the first 15 Zernike functions. 0?,EteR
% `34[w=Zm
% n m Zernike function Normalization =#%e'\)a
% -------------------------------------------------- (a7IxW
% 0 0 1 1 L ?KEe>;r
% 1 1 r * cos(theta) 2 y
L&n)
% 1 -1 r * sin(theta) 2 8agd{bxU
% 2 -2 r^2 * cos(2*theta) sqrt(6) F w{8MQ2
% 2 0 (2*r^2 - 1) sqrt(3) {!oO>t
% 2 2 r^2 * sin(2*theta) sqrt(6) d:sUh
% 3 -3 r^3 * cos(3*theta) sqrt(8) BzWmV.5
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) wZrdr4j
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) (nda!^f_s
% 3 3 r^3 * sin(3*theta) sqrt(8) (2qo9j"j/Y
% 4 -4 r^4 * cos(4*theta) sqrt(10)
mH?^3T
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) o'Tqqrr
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) !2&h=;i~V
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ?wwY8e?S
% 4 4 r^4 * sin(4*theta) sqrt(10) ?Cu#(
% -------------------------------------------------- vgE5(fJh
% PVEEKKJP]J
% Example 1: >b*Pd
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% $ a5K
% % Display the Zernike function Z(n=5,m=1) )sNtwSl^
% x = -1:0.01:1; J)g(Nw,O
% [X,Y] = meshgrid(x,x); Ii|<:BW
% [theta,r] = cart2pol(X,Y); <j,7Z>Rk\x
% idx = r<=1; %8{' XJ!
% z = nan(size(X)); $g|g}>Sc
% z(idx) = zernfun(5,1,r(idx),theta(idx)); /h2`?~k+
% figure kt;X|`V{5z
% pcolor(x,x,z), shading interp )SDGj;j+
% axis square, colorbar )XO2DY1/&
% title('Zernike function Z_5^1(r,\theta)') $ h_ @`j
% g>f(5
% Example 2: VCc4nn#
% Mu:*(P/
% % Display the first 10 Zernike functions G0*$&G0nb
% x = -1:0.01:1; >)S
a#w;
% [X,Y] = meshgrid(x,x); D]oS R7h
% [theta,r] = cart2pol(X,Y); Y}f%/vus
% idx = r<=1; ]m}>/2oSs
% z = nan(size(X)); ^jCkM29eu
% n = [0 1 1 2 2 2 3 3 3 3]; l_f"}l
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; tU)+q?Mw
% Nplot = [4 10 12 16 18 20 22 24 26 28]; 80+"
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% y = zernfun(n,m,r(idx),theta(idx)); PiH#9XB
% figure('Units','normalized') 3rR(>}:[V
% for k = 1:10 *4(.=k
% z(idx) = y(:,k); =~HX/]zF
% subplot(4,7,Nplot(k)) VJ1`&
% pcolor(x,x,z), shading interp hR{Fn L
% set(gca,'XTick',[],'YTick',[]) LQ{4r1,u]
% axis square }l[t0C
t
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) g" M1HxlV
% end a<\m`
Es=
% Z)?"pBv'
% See also ZERNPOL, ZERNFUN2. 3d,|26I 7f
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w
% Paul Fricker 11/13/2006 2'@0|k,yC
%gf8'Q
mX2Qf8
% Check and prepare the inputs: {=R=\Y?r&
% ----------------------------- 8H{@0_M
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) LTa9'
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error('zernfun:NMvectors','N and M must be vectors.') %
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end ^rxXAc[
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if length(n)~=length(m) @7BH`b$)!
error('zernfun:NMlength','N and M must be the same length.') @P@t/
end K, 35*
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n = n(:); Crey}A/N
m = m(:); )T2Sw z/
if any(mod(n-m,2)) N:&Gv'`
error('zernfun:NMmultiplesof2', ... H ($=k-+5
'All N and M must differ by multiples of 2 (including 0).') n$~RgCf
end ?. ~@ lE
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if any(m>n) 8taaBM`:
error('zernfun:MlessthanN', ... mirMDJsl%
'Each M must be less than or equal to its corresponding N.') l5@k8tnz
end ?EtK/6dJZt
Y#rao:I
if any( r>1 | r<0 ) ;>YJ}:r"\
error('zernfun:Rlessthan1','All R must be between 0 and 1.') 61wGIN2,
end A).wjd(_,
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) <
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error('zernfun:RTHvector','R and THETA must be vectors.') O"{NHNG\oT
end 7,
O_'T &
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r = r(:); ,>e<mphM
theta = theta(:); &0N 3 p
length_r = length(r); t/y0gr tm6
if length_r~=length(theta) M'[J0*ip
error('zernfun:RTHlength', ... ThFI=K
'The number of R- and THETA-values must be equal.') Q+#, VuM
end 6rR}qV,+{
L-$GQGk{
% Check normalization: L]9*^al
% -------------------- <ZCjQkka>r
if nargin==5 && ischar(nflag) :x16N|z
isnorm = strcmpi(nflag,'norm'); M(5l Su
if ~isnorm H'2pmwk
error('zernfun:normalization','Unrecognized normalization flag.') *78TT\q<
end J/)Q{*`_
else [,lBY-Kz+
isnorm = false; zvSfW#
*
end Knn$<!>
H!7/U_AH
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 'S&5zwrH
% Compute the Zernike Polynomials c!6.D
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% UXe @c@3
QDLtilf :
% Determine the required powers of r: P PmE.%_
% ----------------------------------- S{&;
m_abs = abs(m); EK[~lIXg
rpowers = []; 7TlOF
for j = 1:length(n) a^|mF#
z
rpowers = [rpowers m_abs(j):2:n(j)]; 9D-PmSnv
end ALPZc:
rpowers = unique(rpowers); ~kF^0-JZY
j].XVn,
% Pre-compute the values of r raised to the required powers, Lw2EA 5
% and compile them in a matrix: 8BBuYY{
% ----------------------------- y1@{(CDp"
if rpowers(1)==0 _sx]`3/86
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 2gukK8R$
rpowern = cat(2,rpowern{:}); o5A@U0c_
rpowern = [ones(length_r,1) rpowern]; ,uK
}$l
else %nT!u!#
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ig jr=e
rpowern = cat(2,rpowern{:}); ?3"lI,!0
end A"d=,?yE
}eSaF@.
% Compute the values of the polynomials: #sN]6
% -------------------------------------- _-^a8F>/19
y = zeros(length_r,length(n)); -=@d2LY
for j = 1:length(n) tVFl`Xr
s = 0:(n(j)-m_abs(j))/2; g \&Z_
pows = n(j):-2:m_abs(j); K#tT \
for k = length(s):-1:1 0.=dOz r
p = (1-2*mod(s(k),2))* ... RMDzPda.
prod(2:(n(j)-s(k)))/ ... ={B%qq
prod(2:s(k))/ ... d3<7t
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 5{L~e>oS9
prod(2:((n(j)+m_abs(j))/2-s(k))); KZ>cfv-&a
idx = (pows(k)==rpowers); >-0Rq[)
y(:,j) = y(:,j) + p*rpowern(:,idx); 4*P#3 B'@V
end yxik`vmH
nD{o8;
if isnorm Jx!#y A;
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); W 2&o'(P\
end *%E4,(T
end _h6SW2:z!E
% END: Compute the Zernike Polynomials e
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% jEVDz
oIrO%v:'!
% Compute the Zernike functions: =;ClOy9
% ------------------------------ QV)>+6\
idx_pos = m>0; _Dr9 w&;<
idx_neg = m<0; u0zF::
O`K2mt\%
z = y; ,)@njC?J
if any(idx_pos) w;W# 'pE
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); kOdXbw9v
end %<8`(Uu5
if any(idx_neg) iO+,U} &
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); \2)D
end Swa0TiT(
%;_94!(hC
% EOF zernfun