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 x@,B))WlGr
function z = zernfun(n,m,r,theta,nflag) NAEAvXj
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. xayd_RB 9
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N - f%J_`
% and angular frequency M, evaluated at positions (R,THETA) on the Vg8c}>7
% unit circle. N is a vector of positive integers (including 0), and tD3v`Ke
% M is a vector with the same number of elements as N. Each element 690;\O '
% k of M must be a positive integer, with possible values M(k) = -N(k) "5$2b>_UE
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, N/eFwv.Er
% and THETA is a vector of angles. R and THETA must have the same e4Jx%v?_P
% length. The output Z is a matrix with one column for every (N,M) #w]@yL]|is
% pair, and one row for every (R,THETA) pair. FK`M+ j
% ?8@EBPpC
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike *d,Z?S/
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), PRyzUG&
% with delta(m,0) the Kronecker delta, is chosen so that the integral a3E.rr;b
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, vI+X9C?
% and theta=0 to theta=2*pi) is unity. For the non-normalized U:O&FE
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 2)+ddel<Z
% |C.[eHe&D
% The Zernike functions are an orthogonal basis on the unit circle. sWX\/Iyy2p
% They are used in disciplines such as astronomy, optics, and WRfhxl
% optometry to describe functions on a circular domain. +p_>fO
% g7<u eF
% The following table lists the first 15 Zernike functions. C;oT0(
% v
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% n m Zernike function Normalization cR/z; *wr7
% -------------------------------------------------- dp#'~[ j
% 0 0 1 1 ev%}\^Vl[
% 1 1 r * cos(theta) 2 y,vrMWDy
% 1 -1 r * sin(theta) 2 .
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% 2 -2 r^2 * cos(2*theta) sqrt(6) PitDk
1T
% 2 0 (2*r^2 - 1) sqrt(3) hYU4%"X
% 2 2 r^2 * sin(2*theta) sqrt(6) Bq#B+JwX
% 3 -3 r^3 * cos(3*theta) sqrt(8) IRB BLXv7\
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) Tn(c%ytN
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) nM6/c
% 3 3 r^3 * sin(3*theta) sqrt(8) /WJ+e
% 4 -4 r^4 * cos(4*theta) sqrt(10) A&($X)t
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) #tQ__V
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) vHxLn/
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) "o>gX'm*
% 4 4 r^4 * sin(4*theta) sqrt(10) Q[.HoqWK
% -------------------------------------------------- KPMId`kf
% qr_:zXsob_
% Example 1: EiWsVic[
% c:sk1I,d~^
% % Display the Zernike function Z(n=5,m=1) a<mM
)[U
% x = -1:0.01:1;
n>:|K0u"
% [X,Y] = meshgrid(x,x); a) 5;Od
% [theta,r] = cart2pol(X,Y); QPT%CW61M
% idx = r<=1; Z2hIoCT
% z = nan(size(X)); |sklY0?l(
% z(idx) = zernfun(5,1,r(idx),theta(idx)); ?_Y2'O
% figure 6=i@ttAK
% pcolor(x,x,z), shading interp a C<
% axis square, colorbar 2C_/T8
% title('Zernike function Z_5^1(r,\theta)') "`8~qZ7k
% JN:EcVuy
% Example 2: h!h<!xaclW
% ;Vh5nO
% % Display the first 10 Zernike functions -iJ @K
% x = -1:0.01:1; PcK;L(
% [X,Y] = meshgrid(x,x); _vgFcE~E@
% [theta,r] = cart2pol(X,Y); t~@~XI5
% idx = r<=1; O[/l';i
% z = nan(size(X)); ;E]^7T
% n = [0 1 1 2 2 2 3 3 3 3]; [Uw/;Kyh
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; EoD[,:*
% Nplot = [4 10 12 16 18 20 22 24 26 28]; etkKVr;Kv
% y = zernfun(n,m,r(idx),theta(idx)); [[;vZ
% figure('Units','normalized') dyMj=e
% for k = 1:10 l/F'W}
% z(idx) = y(:,k); {Wp5Ane
% subplot(4,7,Nplot(k)) nFY6K%[
% pcolor(x,x,z), shading interp ^J{tOxO=l
% set(gca,'XTick',[],'YTick',[]) X9oxni#
% axis square v<c@bDZ>
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 8*t8F\U#
% end NT}r6V(Aju
% 9XSZD93L
% See also ZERNPOL, ZERNFUN2. [>N`)]fP
u#uT|a.
% Paul Fricker 11/13/2006 &mJ
+#vT
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}"CX`
% Check and prepare the inputs: BqA
% ----------------------------- :`w'}h7m
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) slWO\AYiO
error('zernfun:NMvectors','N and M must be vectors.') 4W$t28)
end ="*:H)
;)nV
if length(n)~=length(m) /9..hEq^
error('zernfun:NMlength','N and M must be the same length.') mqFo`Ee
end G^Q8B^Lg
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n = n(:); gZLzE*NZ
m = m(:); @CJ`T&
if any(mod(n-m,2)) ]&mN~$+C
error('zernfun:NMmultiplesof2', ... 1>"[b8a/
'All N and M must differ by multiples of 2 (including 0).') m5/d=k0l
end D#I^;Xg0h
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if any(m>n) ySEhi_)9^
error('zernfun:MlessthanN', ... ~&
@UH
'Each M must be less than or equal to its corresponding N.') _'"whZ)2
end WFTXSHcG
-4!9cE
if any( r>1 | r<0 ) 8UahoNrSt
error('zernfun:Rlessthan1','All R must be between 0 and 1.') =KctAR;
end l9eCsVQ~V
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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.') h/a|-V}m&
end --}5%6
:4V8Iz 71
r = r(:); <HC5YA)4
theta = theta(:); S);SfNh%CL
length_r = length(r); yD-L:)@"
if length_r~=length(theta) F^/1 u
error('zernfun:RTHlength', ... %gb4(~E+N
'The number of R- and THETA-values must be equal.') sOY+X
end v3ky;~ke
..5rW0lr
% Check normalization: &Oih#I
% -------------------- =5v=<, ]
if nargin==5 && ischar(nflag) LW$(;-rY
isnorm = strcmpi(nflag,'norm'); 1YrIcovi-
if ~isnorm }CCTz0[D"
error('zernfun:normalization','Unrecognized normalization flag.') k+D"LA%J
end ip`oL_c
else 7l~d_<h
isnorm = false; b,tf]Z-
end P,}cH;w6Ck
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% !)1gGXRY
% Compute the Zernike Polynomials '[z529HN
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ]"2;x
D\ ;(BB
% Determine the required powers of r: U_8 Z&
% ----------------------------------- 5!b+^UR;z
m_abs = abs(m); X^td`}F/=V
rpowers = []; C;UqLMrOI
for j = 1:length(n) 6VsgZ"Il
rpowers = [rpowers m_abs(j):2:n(j)]; KqD]GS#(
end j+9;Cp]N V
rpowers = unique(rpowers); S /kM#
]+
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% Pre-compute the values of r raised to the required powers, cOq'MDr
% and compile them in a matrix: L2,.af6+
% ----------------------------- )43\q Iu\
if rpowers(1)==0 v/m} {&K
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); w1&\heSQ
rpowern = cat(2,rpowern{:}); +&*D7A>~p
rpowern = [ones(length_r,1) rpowern]; g5OKhL0u
else AVnH|31dC~
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 9Ev<t\B
rpowern = cat(2,rpowern{:}); v><c@a=[
end 2I|`j^
l+vD`aJ 3
% Compute the values of the polynomials: t4P`#,:8
% -------------------------------------- 7'~Oai~r
y = zeros(length_r,length(n)); nr{#Krkb
for j = 1:length(n) i!a.6Gq
s = 0:(n(j)-m_abs(j))/2; )-s9CWJv
pows = n(j):-2:m_abs(j);
L* 0$x
for k = length(s):-1:1 x@)G@'vV|
p = (1-2*mod(s(k),2))* ... u^4$<fd
prod(2:(n(j)-s(k)))/ ... lM|}K-2
prod(2:s(k))/ ... \2c3Nsra
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ]<xzCPB
prod(2:((n(j)+m_abs(j))/2-s(k))); CQANex4&\
idx = (pows(k)==rpowers); Hh1]\4D,4
y(:,j) = y(:,j) + p*rpowern(:,idx); x<'<E@jpU;
end m}$7d5
j%`%
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if isnorm kdP*{
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); cp)BPg
end z%Eok
end ~z kzuh
% END: Compute the Zernike Polynomials @"G+kLv0
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% !\}X?Gf
1VR|z
% Compute the Zernike functions: Pxvf"SXX
% ------------------------------ >lV'}0u)
idx_pos = m>0; rHa*WA;TE
idx_neg = m<0; DP8%/CV!*
_qO'(DKylC
z = y; t+ vz=`
if any(idx_pos) ! }>CEE
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 0sA+5*mdM
end S0'
ACt`
if any(idx_neg) rQD^O4j R
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); PWBcK_4i%
end S?[@/35)
<5 }
% EOF zernfun