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 'j,Li(@}
function z = zernfun(n,m,r,theta,nflag) EkB6- nz
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. q:~`7I
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Cf1wM:K|8
% and angular frequency M, evaluated at positions (R,THETA) on the c^[1]'y
% unit circle. N is a vector of positive integers (including 0), and (HV~ '5D
% M is a vector with the same number of elements as N. Each element M5ySs\O4
% k of M must be a positive integer, with possible values M(k) = -N(k) Er)_[^)
HG
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, -RqAT 1
% and THETA is a vector of angles. R and THETA must have the same zQ6
-2 A
% length. The output Z is a matrix with one column for every (N,M) oN6*WNt J
% pair, and one row for every (R,THETA) pair. }Cq9{0by?a
% W|-N>,G
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 3EW f|6RI
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), A2O_pbQti
% with delta(m,0) the Kronecker delta, is chosen so that the integral Zxxy1Fl#.[
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, B 1ZHV^
% and theta=0 to theta=2*pi) is unity. For the non-normalized 8yo6v3JqC
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. |>o0d~s
% "/K&qj
% The Zernike functions are an orthogonal basis on the unit circle. <}Wy;!L
% They are used in disciplines such as astronomy, optics, and @tv];t
% optometry to describe functions on a circular domain. + x;ML
% g7}z
&S;_
% The following table lists the first 15 Zernike functions. vL=--#
% 2}#wdJ`
% n m Zernike function Normalization KutgW#+40
% -------------------------------------------------- 3_eml\CY
% 0 0 1 1 A7,$y!D
% 1 1 r * cos(theta) 2 `@.s!L(V
% 1 -1 r * sin(theta) 2 V8U`%/`N
% 2 -2 r^2 * cos(2*theta) sqrt(6) /%q9hI
% 2 0 (2*r^2 - 1) sqrt(3) !wb~A0m
% 2 2 r^2 * sin(2*theta) sqrt(6) ^(m6g &$(
% 3 -3 r^3 * cos(3*theta) sqrt(8) 1q233QSW)
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) LX?r=_\
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) N5an9r&z(1
% 3 3 r^3 * sin(3*theta) sqrt(8) .lF\b A|
% 4 -4 r^4 * cos(4*theta) sqrt(10) , ZP3F+XKb
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) GqD!W8+
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) r5qx! >
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) rs<&x(=Hv
% 4 4 r^4 * sin(4*theta) sqrt(10) .8PO7#
% -------------------------------------------------- y>cmKE
% [Fj#7VZK
% Example 1: B[_b J
*
% Z2j*%/
% % Display the Zernike function Z(n=5,m=1) 2=,Sz1`t
% x = -1:0.01:1; I/b8
% [X,Y] = meshgrid(x,x); [QqNsco)
% [theta,r] = cart2pol(X,Y); S{)n0/_
% idx = r<=1; Am?Hkh2
% z = nan(size(X)); >dm._*M
% z(idx) = zernfun(5,1,r(idx),theta(idx)); Z
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% figure O2/w:zOg'
% pcolor(x,x,z), shading interp #|_UA}Y
% axis square, colorbar 5eSTT#[+R
% title('Zernike function Z_5^1(r,\theta)') ._8cJf.ae
% ;pyJ O_R[
% Example 2: |mE+f]7$
% L(n~@gq
% % Display the first 10 Zernike functions R6$F<;nw
% x = -1:0.01:1; E!~2\qKT
% [X,Y] = meshgrid(x,x);
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% [theta,r] = cart2pol(X,Y); -GWzMBS S
% idx = r<=1; 8*PAgPj a
% z = nan(size(X)); MMr7,?,$
% n = [0 1 1 2 2 2 3 3 3 3]; HN~4-6[q
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; ec[[OIO
% Nplot = [4 10 12 16 18 20 22 24 26 28]; 3a)Q:#okD
% y = zernfun(n,m,r(idx),theta(idx)); c%Cae3;
% figure('Units','normalized') 4kF .
% for k = 1:10 _
*s
% z(idx) = y(:,k); m;+1;B
% subplot(4,7,Nplot(k)) nzJi)A./
% pcolor(x,x,z), shading interp K/d&c]
% set(gca,'XTick',[],'YTick',[]) xA'#JN<*
% axis square -qP[$Q
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) j=QR*8*
% end f=O>\
% aq \TO?
% See also ZERNPOL, ZERNFUN2. `&[:!U2]F
kCjI`=7$[
% Paul Fricker 11/13/2006 BOQV X&g%
~(L +4]
%c/"A8{ eb
% Check and prepare the inputs: y*Q-4_%,
% ----------------------------- 9.#R?YP$
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) >37}JUG
error('zernfun:NMvectors','N and M must be vectors.') (0m$W<
end zYF&Dv/u/
m9w
;a
if length(n)~=length(m) SA n=9MG
error('zernfun:NMlength','N and M must be the same length.') |A/_Qe|s2
end ZjW| qb
!,!tNs1 K
n = n(:); WM
)g(i~(
m = m(:); ;U3Vows
if any(mod(n-m,2)) n >PM_W
error('zernfun:NMmultiplesof2', ... Wc;D{p?Lb
'All N and M must differ by multiples of 2 (including 0).') Eq;frnw>q
end J3S+| x h~
&:f'{>3z
if any(m>n) /r@
error('zernfun:MlessthanN', ... 5nqdY*
'Each M must be less than or equal to its corresponding N.') +1fOW4!5
end vS__*}^
k#NMD4(%O
if any( r>1 | r<0 ) sZBO_](S
error('zernfun:Rlessthan1','All R must be between 0 and 1.') 6-}e-H
end J$*["y`+
L\CM);y
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Dx*oSP.qX
error('zernfun:RTHvector','R and THETA must be vectors.') vUx$[/<
end /M `y LI
E?D{/k,zZ
r = r(:); 7$P(1D4
theta = theta(:); ?Cfp=85ea!
length_r = length(r); :?6$}GcW
if length_r~=length(theta) vbh#[,lh
error('zernfun:RTHlength', ... zn$Ld,
'The number of R- and THETA-values must be equal.') W%Q>< 'c
end rWKLxK4oU
l<_mag/j9o
% Check normalization: _?LI0iIFx
% -------------------- I19F\
L`4
if nargin==5 && ischar(nflag) 1U9N8{xg9
isnorm = strcmpi(nflag,'norm'); zb,`K*Z{
if ~isnorm !O_^Rn+<2
error('zernfun:normalization','Unrecognized normalization flag.') >(KUYX?p
end "E!p1
else pR>QIZq<gT
isnorm = false; [N+ruc?)
end \ jdO,-(
2dW-WHaM
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% m|FONQ,@D
% Compute the Zernike Polynomials {\Y,UANZ
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% =H?5fT^
^qr[?ky]&
% Determine the required powers of r: Z i&X ,K~
% ----------------------------------- HV(*6b@
m_abs = abs(m); xl=|]8w
rpowers = []; q`zR 6
for j = 1:length(n) 9 NSYrIQ"
rpowers = [rpowers m_abs(j):2:n(j)]; }gaKO 5
end ~36XJ
rpowers = unique(rpowers); Z9bPj8d
|.nWy"L
% Pre-compute the values of r raised to the required powers, ,1h(k<-
% and compile them in a matrix: ?IO/zkeXg
% ----------------------------- tvCTC ey
if rpowers(1)==0 D"5~-9<
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 74wa
rpowern = cat(2,rpowern{:}); H}rP{`m
rpowern = [ones(length_r,1) rpowern]; P^+>QJ1
else ;%9ZL[-
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 5|z[%x~f
rpowern = cat(2,rpowern{:}); ueo3i1
end #R|4(HlL
Y:BrAa[
% Compute the values of the polynomials: 40/[uW"
% -------------------------------------- X)5O@"4 ?
y = zeros(length_r,length(n)); ^S$w,
for j = 1:length(n) v9kzMxs,
s = 0:(n(j)-m_abs(j))/2; w`:KexD+
pows = n(j):-2:m_abs(j); ^r$5];n
for k = length(s):-1:1 3E:<
p = (1-2*mod(s(k),2))* ... :D-vE7
prod(2:(n(j)-s(k)))/ ... wu'60po
prod(2:s(k))/ ... oWOZ0]H1
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Fd'L:A~
prod(2:((n(j)+m_abs(j))/2-s(k))); I!~Omr@P
idx = (pows(k)==rpowers); AP@d2{"m}
y(:,j) = y(:,j) + p*rpowern(:,idx); )~ kb7rfl
end L1K_|X
dq&d>f1
if isnorm Xu0*sQK
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); EQ-~e
end ),|bP`V
end ST.W{:X
% END: Compute the Zernike Polynomials ttrp|(
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% hw2Hn
j+seJg<_
% Compute the Zernike functions: p%'((!a2
% ------------------------------ g`8|jg0]`I
idx_pos = m>0; G&-h,"yo^
idx_neg = m<0; ['<rfK
`dhK$jYD
z = y; "w1jr 6"
if any(idx_pos) o,I642R~
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); yKJp37R
end O;"%z*g.
if any(idx_neg) I&0yUhn
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); =?hlgQ
end !h\3cs`QU
eS|p3jk;
% EOF zernfun