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 qeaA&(|5
function z = zernfun(n,m,r,theta,nflag) O|v
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%ZERNFUN Zernike functions of order N and frequency M on the unit circle. VRS 2cc
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N cfoYnM
% and angular frequency M, evaluated at positions (R,THETA) on the }++5_Z_
% unit circle. N is a vector of positive integers (including 0), and [{F%LRCo-
% M is a vector with the same number of elements as N. Each element 6Dm+'y]l
% k of M must be a positive integer, with possible values M(k) = -N(k) l+
T,2sd
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, {^&@gkYY
% and THETA is a vector of angles. R and THETA must have the same bY#;E;'7
% length. The output Z is a matrix with one column for every (N,M) 2eok@1
% pair, and one row for every (R,THETA) pair. [}""@?
% q#1X[A()
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike D6=HYqdj
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), & 5
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% with delta(m,0) the Kronecker delta, is chosen so that the integral %"7WXOv&z
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, {y );vHf$
% and theta=0 to theta=2*pi) is unity. For the non-normalized `
%' z
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. R "E<8w
% ^o%_W0_r
% The Zernike functions are an orthogonal basis on the unit circle. (zah890//
% They are used in disciplines such as astronomy, optics, and ]G1R0 Q
% optometry to describe functions on a circular domain. jmW^`%;7
%
\ sf!
% The following table lists the first 15 Zernike functions. ~%aJFs
% irFc}.dI
% n m Zernike function Normalization rycJyiw<-
% -------------------------------------------------- _:,.yRez
% 0 0 1 1 ag]*DsBt
% 1 1 r * cos(theta) 2 Pc4R!Tc
% 1 -1 r * sin(theta) 2 nGZ\<-
% 2 -2 r^2 * cos(2*theta) sqrt(6) =49o U
% 2 0 (2*r^2 - 1) sqrt(3) Ve:&'~F2 s
% 2 2 r^2 * sin(2*theta) sqrt(6) ib50LCm
% 3 -3 r^3 * cos(3*theta) sqrt(8) $y6rvQ
2>S
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) Rkv
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) j6X LyeG7
% 3 3 r^3 * sin(3*theta) sqrt(8) Qg>L,ZO
% 4 -4 r^4 * cos(4*theta) sqrt(10) ]IXAucI]
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) X\G)81Q.S
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) wG:$6
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) -><QFJ
% 4 4 r^4 * sin(4*theta) sqrt(10) Dh8(HiXf:
% -------------------------------------------------- -R@JIe_28f
% jlRS:$|R0
% Example 1: oQBiPN+v.3
% !d|8'^gc
% % Display the Zernike function Z(n=5,m=1) 9L=;KtE1
% x = -1:0.01:1; nh. b/\o
% [X,Y] = meshgrid(x,x); ho|8U
% [theta,r] = cart2pol(X,Y); (+$ol'i
% idx = r<=1; qnTi_c
% z = nan(size(X)); tBTJmih"
% z(idx) = zernfun(5,1,r(idx),theta(idx)); j/`Up
% figure b.6ZfB,+G
% pcolor(x,x,z), shading interp o~}1oN
% axis square, colorbar oYg/*k7EDX
% title('Zernike function Z_5^1(r,\theta)') 45r|1<R o
% YZ{jP?x
% Example 2: vu>YH)N_h
% |?|K\UF(Y
% % Display the first 10 Zernike functions wV
%8v\
% x = -1:0.01:1; :D^Y?
% [X,Y] = meshgrid(x,x); johmJLC
% [theta,r] = cart2pol(X,Y); Ku&*`dME
% idx = r<=1; Ahd\TH
% z = nan(size(X)); xLLC)~
% n = [0 1 1 2 2 2 3 3 3 3]; xtu]F
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; Kd
TE{].d
% Nplot = [4 10 12 16 18 20 22 24 26 28]; B{N=0 cSi
% y = zernfun(n,m,r(idx),theta(idx)); w+3>DEfz
% figure('Units','normalized') sMN>wbHwh[
% for k = 1:10 Y"s
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% z(idx) = y(:,k); &:C{/QnA
% subplot(4,7,Nplot(k))
B[Ix?V4yy
% pcolor(x,x,z), shading interp zv|M*Wu
% set(gca,'XTick',[],'YTick',[]) Bd.Z+#%l"
% axis square `J]<_0kX}%
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) t{[gKV-b
% end ^$ 8Vh=D
% 1riBvBT
% See also ZERNPOL, ZERNFUN2. g8rp|MOH
KWtu,~O_u
% Paul Fricker 11/13/2006 ;*"!:GR%h
olHH9R9:
$]Rl__;
% Check and prepare the inputs: L;4[ k;5
% ----------------------------- tu7+LwF7
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ?L8&(&1@VD
error('zernfun:NMvectors','N and M must be vectors.') $:PF9pY(
end %f>X-*}NI-
8Yo-~,Gb
if length(n)~=length(m) DXt]b,
error('zernfun:NMlength','N and M must be the same length.') )#)nBM2\
end 1mY+0
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n = n(:); JgxE|#*7U
m = m(:); Y>(ZsHu
if any(mod(n-m,2)) HDa~7wE
error('zernfun:NMmultiplesof2', ... R Co eJ|
'All N and M must differ by multiples of 2 (including 0).') :QxL 9&"
end 0,;E.Py?.
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if any(m>n) 0~(\lkh*!9
error('zernfun:MlessthanN', ... H-;&xzAI
'Each M must be less than or equal to its corresponding N.') ev)rOcOU
end ',L{CQA?c
kQqBHA
if any( r>1 | r<0 ) "sz.v<F0:s
error('zernfun:Rlessthan1','All R must be between 0 and 1.') 6#OL
;Y]_
end $'WapxF
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) W?SP .-I
error('zernfun:RTHvector','R and THETA must be vectors.') =#
k<Kw#
end BUcaj.S
R>/QARX
r = r(:); M"k3zK,
theta = theta(:); fF8a 1XV
length_r = length(r); :;" aUHU'
if length_r~=length(theta) Eqz4{\
error('zernfun:RTHlength', ... .Z(S4wV
'The number of R- and THETA-values must be equal.') Xtu:
end KK&<Vw|O\
EX+={U|ua$
% Check normalization: Vy?R/
Uu
% -------------------- q[PD
if nargin==5 && ischar(nflag) s_S<gR
isnorm = strcmpi(nflag,'norm'); < fojX\}3
if ~isnorm BFzcoBu-
error('zernfun:normalization','Unrecognized normalization flag.') v9j4|w
end "N?%mCPI
else +YGw4{\EL
isnorm = false; VEFwqB1l
end $|`t9-EA/
z5|e\Z
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3i@ "D
% Compute the Zernike Polynomials 7yq7a[Ra
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% >N+bU{s
]Ssw32yn
% Determine the required powers of r: 0U>t>&,"
% ----------------------------------- 3p?<iVE
m_abs = abs(m); F20wf1^
rpowers = []; kg/+vJ
for j = 1:length(n) (>!]A6^L~
rpowers = [rpowers m_abs(j):2:n(j)]; 0)6i~Mg lY
end fD3jwPL
rpowers = unique(rpowers); fg>B
pmow[e
% Pre-compute the values of r raised to the required powers, ~$?y1Yv
% and compile them in a matrix: []2$rJZD9
% ----------------------------- 73^T*
if rpowers(1)==0 m>Yo9/XpZ
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); =sUl`L+w,L
rpowern = cat(2,rpowern{:}); ';;p8bv+
rpowern = [ones(length_r,1) rpowern]; i-:8TfI,
else L&!g33J&
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ,w9#%=xE
rpowern = cat(2,rpowern{:}); 7\\~xSXh
end A-Q{*{^#
`uM0,Z
% Compute the values of the polynomials: ] dm1Qm
% -------------------------------------- ]<\;d
B
y = zeros(length_r,length(n)); |dB1R%
for j = 1:length(n) )JY_eG&2Dx
s = 0:(n(j)-m_abs(j))/2; i&}zcGC
pows = n(j):-2:m_abs(j); 1Rb XM n
for k = length(s):-1:1 ^.Ih,@N6
p = (1-2*mod(s(k),2))* ... ,E/Y@sajn+
prod(2:(n(j)-s(k)))/ ... @^y?Bh9jQ
prod(2:s(k))/ ... =,>TpE
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... zDvP7hl
prod(2:((n(j)+m_abs(j))/2-s(k))); 7 BnenHD
idx = (pows(k)==rpowers); [6&CloY3
y(:,j) = y(:,j) + p*rpowern(:,idx); xnRp/I
end AihL>a%
P- `~]]
if isnorm 3gV&`>@
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); YjX!q]56
end r:WgjjA%
end IQk#
% END: Compute the Zernike Polynomials t=E|RYC(k
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% c:@OX[##
>^a"Z[s[
% Compute the Zernike functions: R+kZLOE
% ------------------------------ 8}pcanPg
idx_pos = m>0; >XXMIz:
idx_neg = m<0; k8x&aH
O yH!V&w
z = y; FVC2 XxP
if any(idx_pos) mSk :7ozZ
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); o
{XwLi
end |U#w?eE=
if any(idx_neg) &JXHDpd$a^
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); C#**)
end eUKl
Co
_;J9q}X
% EOF zernfun