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 D7cOEL<
function z = zernfun(n,m,r,theta,nflag) Gs%IZo_
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. |1J=wp)#
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N d
(]t}
% and angular frequency M, evaluated at positions (R,THETA) on the vf(8*}'!Q
% unit circle. N is a vector of positive integers (including 0), and L'=2Uk#.D
% M is a vector with the same number of elements as N. Each element u38FY@U$
% k of M must be a positive integer, with possible values M(k) = -N(k) (x,w/1
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, QA7SQcd,
% and THETA is a vector of angles. R and THETA must have the same _KiaeVE
% length. The output Z is a matrix with one column for every (N,M) g/,fjM_
% pair, and one row for every (R,THETA) pair. oZ95 )'L,
% CK[2duf^~
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike Ao)hb4ex
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), FrD.{(/~
% with delta(m,0) the Kronecker delta, is chosen so that the integral 0L10GJ "(
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, fU^B
3S6X
% and theta=0 to theta=2*pi) is unity. For the non-normalized =
aSHb[hO
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. cC
w,b]
% {H s""/sb
% The Zernike functions are an orthogonal basis on the unit circle. ;hR!j!3}
% They are used in disciplines such as astronomy, optics, and :0>wm@qCQ
% optometry to describe functions on a circular domain. ])h={gI
% n
m(yFX?=
% The following table lists the first 15 Zernike functions. hH:7
% pgz3d{]ua
% n m Zernike function Normalization c/
%5IhX?
% -------------------------------------------------- ElAJR4'{*i
% 0 0 1 1 6'ye-}vD-
% 1 1 r * cos(theta) 2 ^zkTV_,cRp
% 1 -1 r * sin(theta) 2 fEc}c.!5
% 2 -2 r^2 * cos(2*theta) sqrt(6) on(P
% 2 0 (2*r^2 - 1) sqrt(3) SPW @TF1
% 2 2 r^2 * sin(2*theta) sqrt(6) j~c7nWfX
% 3 -3 r^3 * cos(3*theta) sqrt(8) >U~.I2sz
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) p%Ae"#_X%
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) 5P{dey!
% 3 3 r^3 * sin(3*theta) sqrt(8) LA$uD?YA
% 4 -4 r^4 * cos(4*theta) sqrt(10) 0K7]<\)
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) (u85$_C
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) ~!~VC)a*
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) G;615p1
% 4 4 r^4 * sin(4*theta) sqrt(10) 6HpSZa
% -------------------------------------------------- vIG8m@-!&;
% l)D18
% Example 1: N%6jZmKip
% ;+K:^*oJ
% % Display the Zernike function Z(n=5,m=1) @;_r`AT7
% x = -1:0.01:1; 1YR;dn
% [X,Y] = meshgrid(x,x); _6THyj$f
% [theta,r] = cart2pol(X,Y); ',8]vWsl
% idx = r<=1; Tz58@VY V
% z = nan(size(X)); liFNJd`|o+
% z(idx) = zernfun(5,1,r(idx),theta(idx)); `d6
{Tli
% figure z_!P0`
% pcolor(x,x,z), shading interp (Z.K3
% axis square, colorbar ttLChL
% title('Zernike function Z_5^1(r,\theta)') a}`4BMi3
% 0sVCTJ@
% Example 2: iKV;>gF,)v
% #!h:w
% % Display the first 10 Zernike functions ;3Fgy8T
% x = -1:0.01:1; <;#d*&]
% [X,Y] = meshgrid(x,x); R|{AIa{}
% [theta,r] = cart2pol(X,Y); `y0ZFh1>X
% idx = r<=1; /7|u2!#Ui
% z = nan(size(X)); 8gJ"7,}-'
% n = [0 1 1 2 2 2 3 3 3 3]; JO5~Vj_"
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; *La*j3|:
% Nplot = [4 10 12 16 18 20 22 24 26 28]; .Xo, BEjE/
% y = zernfun(n,m,r(idx),theta(idx)); A)040n
% figure('Units','normalized') N:0/8jmmO
% for k = 1:10 -x3QgDno
% z(idx) = y(:,k); ;M8N%
% subplot(4,7,Nplot(k)) j9%u&
% pcolor(x,x,z), shading interp HoymGU`w
% set(gca,'XTick',[],'YTick',[]) T_6,o[b8
% axis square ko
im@B
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) W2tIt&{
% end 9NaC7D$,
% b'Z#RIb
% See also ZERNPOL, ZERNFUN2. F0bmGDp@-
z|}Anc[\
% Paul Fricker 11/13/2006 P^v`5v
:~:(49l
^o !K0t*
% Check and prepare the inputs: h(d<':|
% ----------------------------- #g4X`AHB
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) "<3PyW?zt
error('zernfun:NMvectors','N and M must be vectors.') !rb)Y;WQt
end CeR4's7
[HtU-8:
if length(n)~=length(m) *ky5SM(NR
error('zernfun:NMlength','N and M must be the same length.') {#=q[jVi%1
end -#3B>VY
Mz40([{
n = n(:); A[XEbfDO
m = m(:); tAP~
if any(mod(n-m,2)) 4&K~EX"^T
error('zernfun:NMmultiplesof2', ... .pu]21m=
'All N and M must differ by multiples of 2 (including 0).') {qx}f^WV
end 93)&
@]WN|K
if any(m>n) @luv;X^%
error('zernfun:MlessthanN', ... p8[Z/]p
'Each M must be less than or equal to its corresponding N.') jFw?Ky2
end 0u
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+?*,J=/
if any( r>1 | r<0 ) kxWf1hIz0
error('zernfun:Rlessthan1','All R must be between 0 and 1.') ff?:_q+.N
end _R]la&^2F\
z^{VqC*o+
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 6T"[M
error('zernfun:RTHvector','R and THETA must be vectors.') AmRppbj/wO
end >\^:xxTf
z]=A3!H/Y
r = r(:); ^=pn!lK;^
theta = theta(:); ~(-B%Az
length_r = length(r); w80g)4V+
if length_r~=length(theta) |6"zIHvtc
error('zernfun:RTHlength', ... pUYa1 =
'The number of R- and THETA-values must be equal.') 8D)*~C'85E
end KxGK`'E'r
,;O+2TX
% Check normalization: tE9%;8;H
% -------------------- _yJd@
if nargin==5 && ischar(nflag) Q6RBZucv
isnorm = strcmpi(nflag,'norm'); j*q]-$ 2E
if ~isnorm \.9-:\'(
error('zernfun:normalization','Unrecognized normalization flag.') ;l &mA1+
end Kv{i_%j
else LC*@/((
isnorm = false; PD:"
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end #8%Lc3n
Pd%o6~_*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +<"sC+2
% Compute the Zernike Polynomials }a'8lwF%I
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 8D;>] >
z4&|~-m,
% Determine the required powers of r: tlCgW)<?
% ----------------------------------- (4>k+ H
m_abs = abs(m); 9%$4Ux*q
rpowers = []; y%cg
for j = 1:length(n) nr!kx)j
rpowers = [rpowers m_abs(j):2:n(j)]; (YGJw?]
end ]{0
2!
rpowers = unique(rpowers); J5mMx)t@
SE;Jl[PgcL
% Pre-compute the values of r raised to the required powers, pI( OI>~3
% and compile them in a matrix: mmu{K$9}I
% ----------------------------- |bO}|X
if rpowers(1)==0 ZxwI< T:&
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); cmZ39pjBJ
rpowern = cat(2,rpowern{:}); L/F!Y%=;[
rpowern = [ones(length_r,1) rpowern]; UCa(3p^V_
else k,0JW=Vh>|
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); hof:36 <
rpowern = cat(2,rpowern{:}); R}#?A%,*
end <I&X[Sqp
J3oH^
% Compute the values of the polynomials: *<i
{
Mb Q
% -------------------------------------- w=rh@S]
y = zeros(length_r,length(n)); 2Rc#{A
for j = 1:length(n) <omSK-
T-
s = 0:(n(j)-m_abs(j))/2; }(hx$G^M
pows = n(j):-2:m_abs(j); 0AZ Vc
for k = length(s):-1:1 dTB^6>H
p = (1-2*mod(s(k),2))* ... Cz+`C9#
prod(2:(n(j)-s(k)))/ ... \{\*h /m
prod(2:s(k))/ ... 0%<Fc9#
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... cD YKvrPY
prod(2:((n(j)+m_abs(j))/2-s(k))); <KoiZ{V
idx = (pows(k)==rpowers); Y#=0C*FS
y(:,j) = y(:,j) + p*rpowern(:,idx); .Qyq*6T3&
end V) a<)
[W,Ej
if isnorm jav7V"$
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ,_!pUal
end h
rW
end 5hr$tkkL
% END: Compute the Zernike Polynomials nVoL7ew+
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% `%ZM(9T
F
*=>=
% Compute the Zernike functions: i/6(~v
% ------------------------------ 9f\Lon4lX
idx_pos = m>0; `+CRUdr
idx_neg = m<0; DJdW$S7
}u5/
z = y; 1aP3oXLL
if any(idx_pos) D{x'k2=
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ,,sKPj[
end C*a>B,H
if any(idx_neg) tda#9i[pkH
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 9{RCh9
end 66(|3D X
_D1Uc|
% EOF zernfun