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 oJ0
#U
function z = zernfun(n,m,r,theta,nflag) t(- 5l
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. j&,%v+x
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N GYri\ <[
% and angular frequency M, evaluated at positions (R,THETA) on the )-LSn
% unit circle. N is a vector of positive integers (including 0), and _M[T8 "e(
% M is a vector with the same number of elements as N. Each element *3y:Wv T>
% k of M must be a positive integer, with possible values M(k) = -N(k) /gLi(Uw
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, p-%m/d?
% and THETA is a vector of angles. R and THETA must have the same }RkD7
% length. The output Z is a matrix with one column for every (N,M) "Ze<dB#,Y
% pair, and one row for every (R,THETA) pair. -$j|&l
% Io)@u~yz
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ]1KF3$n0
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), TSP#.QY
% with delta(m,0) the Kronecker delta, is chosen so that the integral z
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% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, (w,
Gv-S
% and theta=0 to theta=2*pi) is unity. For the non-normalized h&t9CpTfeJ
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ^:m7Qd?Z[
% N1z:9=(I
% The Zernike functions are an orthogonal basis on the unit circle. <o_(,,P%
% They are used in disciplines such as astronomy, optics, and f.u+({"ql
% optometry to describe functions on a circular domain. ^WIGd"^
% !\1Pu|
% The following table lists the first 15 Zernike functions. !bIhw}^C*
% -$kAWP8P4
% n m Zernike function Normalization 'sTMUPg`
% -------------------------------------------------- k/lDE
% 0 0 1 1 Z;GZ?NOlY
% 1 1 r * cos(theta) 2 5]@"f/
% 1 -1 r * sin(theta) 2 l=t$XWh!
% 2 -2 r^2 * cos(2*theta) sqrt(6) ]s:%joj%^
% 2 0 (2*r^2 - 1) sqrt(3) gLPgh%B4
% 2 2 r^2 * sin(2*theta) sqrt(6) {vAv ;m
% 3 -3 r^3 * cos(3*theta) sqrt(8) SH M@H93
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) R9lb<`
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) ioS(;2F
% 3 3 r^3 * sin(3*theta) sqrt(8) ;_=+h,n
% 4 -4 r^4 * cos(4*theta) sqrt(10) 8Ir
= @
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 0N>R!
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) %u02KmV.
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) <Yy|.=6 D
% 4 4 r^4 * sin(4*theta) sqrt(10) 4BAG GD2
% -------------------------------------------------- 0:4w@"Q
% VTvNn
% Example 1: 6.gk6
% 'nh^'i&0.
% % Display the Zernike function Z(n=5,m=1) 924a1
% x = -1:0.01:1; Q !G^CG
% [X,Y] = meshgrid(x,x); g\lEdxm6Sj
% [theta,r] = cart2pol(X,Y); %w3"B,k'9D
% idx = r<=1; |jE0H!j
% z = nan(size(X)); 0P_3%
% z(idx) = zernfun(5,1,r(idx),theta(idx)); :f5"w+
% figure a EmLf
% pcolor(x,x,z), shading interp Y|96K2BR
% axis square, colorbar jz72~+)T
% title('Zernike function Z_5^1(r,\theta)') +LsACSB
% OtFGo8
% Example 2: Z</.Ss 4
% -yP_S~\n
% % Display the first 10 Zernike functions Dk`(Wgk2
% x = -1:0.01:1; ct![eWsuB
% [X,Y] = meshgrid(x,x); wxSJ
% [theta,r] = cart2pol(X,Y); ,c9K]>8m`
% idx = r<=1; \t^h|<`
% z = nan(size(X)); $c<NEt_\
% n = [0 1 1 2 2 2 3 3 3 3]; w_]`)$9
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; >crFIkOJ
% Nplot = [4 10 12 16 18 20 22 24 26 28]; yRv4,{B}X>
% y = zernfun(n,m,r(idx),theta(idx)); /[RO>Z9
% figure('Units','normalized') #1oyRD-
% for k = 1:10 M"Q{lR
% z(idx) = y(:,k); DZE@C^0%
% subplot(4,7,Nplot(k)) -oR P ZtW
% pcolor(x,x,z), shading interp 5isqBu
% set(gca,'XTick',[],'YTick',[]) T.?}iz=ZEq
% axis square Ty;P`Uv]r
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) %{HeXe
% end Ek%mX"
% w=feXA3-S
% See also ZERNPOL, ZERNFUN2. &Y3r'"
'| rhm
% Paul Fricker 11/13/2006 f*46,`x
N>Q~WXvV#
I$HO[Z!
% Check and prepare the inputs: AD^Q`7K?uR
% ----------------------------- ATscP hk
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) {~cM 6W]f
error('zernfun:NMvectors','N and M must be vectors.') 3P2x%G p
end vA&MJD{
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if length(n)~=length(m) 'K*. ?M
error('zernfun:NMlength','N and M must be the same length.') ,A9_xdv5
end oo2CF!Xy
,,HoD~]rd
n = n(:); -fCR^`UOS
m = m(:); ]m<z
if any(mod(n-m,2)) {DWL 5V#M
error('zernfun:NMmultiplesof2', ... P}8cSX9
'All N and M must differ by multiples of 2 (including 0).') &Xh_`*]ox
end bAS/cuZs
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if any(m>n) <kOdd)X
error('zernfun:MlessthanN', ... 8$`$24Wx
'Each M must be less than or equal to its corresponding N.') n5>OZ3 E@
end 6%L#FSI
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if any( r>1 | r<0 )
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error('zernfun:Rlessthan1','All R must be between 0 and 1.') IC6r?
end oF L7dL
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) )
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error('zernfun:RTHvector','R and THETA must be vectors.')
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end "el}9OitC
~`X$bF
r = r(:); buKSZ
theta = theta(:); _?v&\j
length_r = length(r); W:8pmI
if length_r~=length(theta) <N{Y*,^z
error('zernfun:RTHlength', ... ,s`4k?y
'The number of R- and THETA-values must be equal.') 8h,=yAn5
end ToR@XL!%rP
sWv!ig_
% Check normalization: Z;~ 7L*|
% -------------------- !xvAy3
if nargin==5 && ischar(nflag) ~yiw{:\
isnorm = strcmpi(nflag,'norm'); YHzP/&0
if ~isnorm )|wC 1J!L
error('zernfun:normalization','Unrecognized normalization flag.') :hTmt{LjN
end kX%vTl7F
else Qo\?(EM
isnorm = false; O-&^;]ieJ
end @Nn'G{8OG
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% s"5wnp6pW
% Compute the Zernike Polynomials GB4^ 4Ajx
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% c2Wp 8l
tUi@'%>=5
% Determine the required powers of r: L$6W,D
% ----------------------------------- u0F{.fe
m_abs = abs(m); KAg-M#
rpowers = []; X`28?
for j = 1:length(n) *$Y_ %}
rpowers = [rpowers m_abs(j):2:n(j)]; Ug )eyu
end apjoIO-<
rpowers = unique(rpowers); W.BX6
<:4b4Nl
% Pre-compute the values of r raised to the required powers, 9Ed=`c
% and compile them in a matrix: bbT1p:RF
% ----------------------------- L~Y^O`c
if rpowers(1)==0 (_]D\g~
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); @MP ;/o+
rpowern = cat(2,rpowern{:}); gg/2R?O]
rpowern = [ones(length_r,1) rpowern]; q$PO.#
else Q^*4FH!W
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); .d6b?t
rpowern = cat(2,rpowern{:}); fJ=v?
end f2u4*X
E\
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% Compute the values of the polynomials: w=FU:q/
% -------------------------------------- KM?w{ ~9
y = zeros(length_r,length(n)); /ke[nr
for j = 1:length(n) TE:|w
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s = 0:(n(j)-m_abs(j))/2; m48Ab`
pows = n(j):-2:m_abs(j); YJ|U|[
for k = length(s):-1:1 "B>8on8O
p = (1-2*mod(s(k),2))* ... L+~XW'P?
prod(2:(n(j)-s(k)))/ ... @z^7*#vQv
prod(2:s(k))/ ... /U})mdFm
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... wg<t*6&'x
prod(2:((n(j)+m_abs(j))/2-s(k))); 2fg
P
idx = (pows(k)==rpowers); Z*Rgik
y(:,j) = y(:,j) + p*rpowern(:,idx); %C_c%3d
end h>F"GR?U_(
EQ.K+d*K][
if isnorm iBwM]Eyv.
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); hj}PL
end V|~o`(]
end Lp(i&A
% END: Compute the Zernike Polynomials ~E/=nv$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Shv$"x:W
tSe[*V4{'
% Compute the Zernike functions: $z`l{F4eMf
% ------------------------------ G[6i\Et
idx_pos = m>0; Lrmhr3
w5
idx_neg = m<0; \AIFIy
a?xZsR
z = y; zp5ZZcj_
if any(idx_pos) $+PyW(
r
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); I E{:{b\
end z,bK.KFSs
if any(idx_neg) -{q'Tmst
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); K>C@oE[W
end SSq4KFO1
[b_qC'K[
% EOF zernfun