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 ]..7t|^b&
function z = zernfun(n,m,r,theta,nflag) =SVb
k
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 6 U_P
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N jj6yf.r6c
% and angular frequency M, evaluated at positions (R,THETA) on the Hp5.jor(k
% unit circle. N is a vector of positive integers (including 0), and ?,^Aoy
% M is a vector with the same number of elements as N. Each element X}B]0z>
% k of M must be a positive integer, with possible values M(k) = -N(k) \@m^w"Ij
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, 5]~451
% and THETA is a vector of angles. R and THETA must have the same x4-_K%
% length. The output Z is a matrix with one column for every (N,M) {fa3"k_ke
% pair, and one row for every (R,THETA) pair.
]Gf`nJDV
% cUC!'+L
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ]-cSTtO
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), DhD^w;f]
% with delta(m,0) the Kronecker delta, is chosen so that the integral hO; XJyv
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, -mw`f)?Ev
% and theta=0 to theta=2*pi) is unity. For the non-normalized R'Uf#.
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. aKz:hG
% I`;SA~5
% The Zernike functions are an orthogonal basis on the unit circle. +F+M[ef<ws
% They are used in disciplines such as astronomy, optics, and odW K\e
% optometry to describe functions on a circular domain. Fs&r^ [/b
% xQkvK=~$
% The following table lists the first 15 Zernike functions. 9PdD =9HH
% vKBijmE
% n m Zernike function Normalization pD;fFLvN
% -------------------------------------------------- q5{h@}|M
% 0 0 1 1 Go(Td++HS
% 1 1 r * cos(theta) 2 i>e?$H,/
% 1 -1 r * sin(theta) 2 e%C_>
% 2 -2 r^2 * cos(2*theta) sqrt(6) gUY~
l= c
% 2 0 (2*r^2 - 1) sqrt(3) tmi)LRF
H
% 2 2 r^2 * sin(2*theta) sqrt(6) YO9;NA{sH
% 3 -3 r^3 * cos(3*theta) sqrt(8) oS^KC}X
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) Ug\$Ob5=q
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) LB`{35b-
% 3 3 r^3 * sin(3*theta) sqrt(8) 8p]9A,Uq&
% 4 -4 r^4 * cos(4*theta) sqrt(10) !RSJb
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) G`RQl@W>)(
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) bE?X?[K
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) iFnD`l6)
% 4 4 r^4 * sin(4*theta) sqrt(10) hkMVA
% -------------------------------------------------- >:HmIW0PLe
% K/K|[=bl
% Example 1: Ll.P>LH
% QD%!a{I
% % Display the Zernike function Z(n=5,m=1) N-W>tng_x
% x = -1:0.01:1; \rd%$hci
% [X,Y] = meshgrid(x,x); r1:CHIwK
% [theta,r] = cart2pol(X,Y); wf`A&P5tF
% idx = r<=1; ,B'fOJ.2
% z = nan(size(X)); ")<5VtV
% z(idx) = zernfun(5,1,r(idx),theta(idx)); i` Q&5KL
% figure {
&Vt]9
% pcolor(x,x,z), shading interp A9;,y'm^8
% axis square, colorbar R3%%;` c=
% title('Zernike function Z_5^1(r,\theta)') 8OiCldw:HN
% W/g_XQ
% Example 2: 4:5M,p
% m`}mbm^
% % Display the first 10 Zernike functions
1D_&n@
% x = -1:0.01:1; eph2&)D}Ep
% [X,Y] = meshgrid(x,x); hZlHY9[t?
% [theta,r] = cart2pol(X,Y); sUU[QP-
% idx = r<=1; [+Fajo;0
% z = nan(size(X)); t<o7 S:a"
% n = [0 1 1 2 2 2 3 3 3 3]; p=odyf1hK
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; \*[DR R0
% Nplot = [4 10 12 16 18 20 22 24 26 28]; qsQ{`E0
% y = zernfun(n,m,r(idx),theta(idx)); 7hTpjox2
% figure('Units','normalized') +abb[
% for k = 1:10 7Mk>`4D'c
% z(idx) = y(:,k); V~p01f"J
% subplot(4,7,Nplot(k)) YgdQC(ib
% pcolor(x,x,z), shading interp 2vh }:A_
% set(gca,'XTick',[],'YTick',[]) `!$6F:d_l
% axis square {xeJO:M3/
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) `So/G
% end AUu<@4R7
% 3!$+N\ #w
% See also ZERNPOL, ZERNFUN2. bv VkN
*@p"
% Paul Fricker 11/13/2006 %}e['d h
>lKu[nq;
`S0`3q}L3%
% Check and prepare the inputs: *CPp U|
% ----------------------------- n_Qua|R
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) YC')vv3o(
error('zernfun:NMvectors','N and M must be vectors.') $v #
end ~_Fx2T:X
JsNj!aeU%
if length(n)~=length(m) } C:i0Q
error('zernfun:NMlength','N and M must be the same length.') Il Qk W<
end heL`"Y2'y>
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n = n(:); [NAfy~X*
m = m(:); I;-Y2*
if any(mod(n-m,2)) GcDA0%i
error('zernfun:NMmultiplesof2', ... uAqiL>y
'All N and M must differ by multiples of 2 (including 0).') \Oq8kJ=
end q/@+.q
-fXQ62:S
if any(m>n) x"g)pGsT
error('zernfun:MlessthanN', ... "T{WOGU+
'Each M must be less than or equal to its corresponding N.') _cE_\Ay
end ('7$K
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if any( r>1 | r<0 ) ;N?(R\*8
error('zernfun:Rlessthan1','All R must be between 0 and 1.') &l3(+4Sh
end fLqjBG]<
!^&VZh
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) >~nr,V.q
error('zernfun:RTHvector','R and THETA must be vectors.') b>VV/j4!/
end g4b#U\D@)/
,h*N9}xYTi
r = r(:); ,dR.Sacv
theta = theta(:); %~x?C4L8
length_r = length(r); } 6!/Nb
if length_r~=length(theta) '~Cn+xf4]
error('zernfun:RTHlength', ... p]EugLEmG
'The number of R- and THETA-values must be equal.') nq HpYb6I0
end YI
?P@y
"3\y~<8%'
% Check normalization: ;cvMNU$fN
% -------------------- 8-NycG&)
if nargin==5 && ischar(nflag) hPSMPbI
isnorm = strcmpi(nflag,'norm'); &Ap9h#
dK
if ~isnorm ^!\AT!OT
error('zernfun:normalization','Unrecognized normalization flag.') E I(e3
end SMD*9&,
else :`zO%h
isnorm = false; xi(1H1KN5B
end Lv]%P.=[G
a`n)aXU l
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% QMz =e
% Compute the Zernike Polynomials l[c '%M |N
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% d}zh.O5P!
G@#lf@M]
% Determine the required powers of r: D\&S {
% ----------------------------------- wR@>U.XT@
m_abs = abs(m); Q&xH
rpowers = []; &H%/.4la
for j = 1:length(n) I51]+gEN
rpowers = [rpowers m_abs(j):2:n(j)]; F0p=|W
end cJaA*sg
rpowers = unique(rpowers); pT->qQ3;
;7qIm83
% Pre-compute the values of r raised to the required powers, !(F?`([A
% and compile them in a matrix: +4_, , I
% ----------------------------- m..ajYSQ
if rpowers(1)==0 (g@\QdH`|
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); k\.9iI'6
rpowern = cat(2,rpowern{:}); 3?a`@C&x
rpowern = [ones(length_r,1) rpowern]; BYX c
'K
else fV|uKs(W
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); x)Bbo9J
rpowern = cat(2,rpowern{:}); 0>Snps3*Z
end > v%.q]E6n
v,ZYh w
% Compute the values of the polynomials: @6yc^DAA
% -------------------------------------- ZI!:
y = zeros(length_r,length(n)); T,/rC{
for j = 1:length(n) @d0f +9d
s = 0:(n(j)-m_abs(j))/2; O*l,&5
pows = n(j):-2:m_abs(j); I U"
for k = length(s):-1:1 {\D&*
p = (1-2*mod(s(k),2))* ... h'-4nu;*
prod(2:(n(j)-s(k)))/ ... ?h&XIM(
prod(2:s(k))/ ... JkJ
@bh
Eu
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 8F8?1
prod(2:((n(j)+m_abs(j))/2-s(k))); B1)Eo2i#
idx = (pows(k)==rpowers); yO1
7C
y(:,j) = y(:,j) + p*rpowern(:,idx); dgpE3
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end 49Jnp>h
oYkd%N9P
if isnorm 6]b"n'G
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); XeI2<=@%
end c EYHB1*cT
end y<Q"]H.CkQ
% END: Compute the Zernike Polynomials H9(?yI@Zr#
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% /ovVS6Ai
Dhn7N8(LF!
% Compute the Zernike functions: J;>epM;*
% ------------------------------ "iK=
8
idx_pos = m>0; HXa[0VOx
idx_neg = m<0; dR]-R/1|
E)$>t}$
z = y; losqc *|
if any(idx_pos) BS##nS-[
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ,XO@ZBOM
end XG.[C>
if any(idx_neg) w li cuY?
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Jr!BDg
end ^ f! M"@
;nBf
% EOF zernfun