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 Ia&R/I
function z = zernfun(n,m,r,theta,nflag) a QH6akH
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. vDy&sgS$<
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N _;<!8e$C
% and angular frequency M, evaluated at positions (R,THETA) on the z\YIwrq3*
% unit circle. N is a vector of positive integers (including 0), and }\pI`;*O|
% M is a vector with the same number of elements as N. Each element jvT'N@
% k of M must be a positive integer, with possible values M(k) = -N(k) ;"3B,Yj
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, H'+7z-%G
% and THETA is a vector of angles. R and THETA must have the same #;!&8iH
% length. The output Z is a matrix with one column for every (N,M) =;}W)V|X)S
% pair, and one row for every (R,THETA) pair. 2[E wN!IZ
% ?b7\m":'
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ngY%T5-
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), /
)0hsQs
% with delta(m,0) the Kronecker delta, is chosen so that the integral k[=qx{Osx%
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 8~=*\
@^
% and theta=0 to theta=2*pi) is unity. For the non-normalized c:R?da
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. XtF
m5\U
% lame/B&nc
% The Zernike functions are an orthogonal basis on the unit circle. U"oNJ8&%|
% They are used in disciplines such as astronomy, optics, and @hLkU4S
% optometry to describe functions on a circular domain. YJi%vQ*]
% ]rcF/uQJ<n
% The following table lists the first 15 Zernike functions. qnm_#!&uHT
% JAbUK[:K
% n m Zernike function Normalization "kg`TJf=
% -------------------------------------------------- #-hO\
QdC
% 0 0 1 1 nHK(3Z4G
% 1 1 r * cos(theta) 2 Qm%F]nyy
% 1 -1 r * sin(theta) 2 H=dIZ
% 2 -2 r^2 * cos(2*theta) sqrt(6) @Z)|_
% 2 0 (2*r^2 - 1) sqrt(3) P
rt}
01$
% 2 2 r^2 * sin(2*theta) sqrt(6) Cu"Cpt[
% 3 -3 r^3 * cos(3*theta) sqrt(8) Bx\&7|,x
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 5*0zI\
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) ,'#TdLe
% 3 3 r^3 * sin(3*theta) sqrt(8) kmB!NxF>)F
% 4 -4 r^4 * cos(4*theta) sqrt(10) F_-Lu]*
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) f~IJ4T2#N
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) lU
WXXuO]
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 37AVk`a
% 4 4 r^4 * sin(4*theta) sqrt(10) i1iP'`r
% -------------------------------------------------- g40Hj Y
% %E?Srs}j
% Example 1: gGqrFh\
% ]5!3|UYS
% % Display the Zernike function Z(n=5,m=1) VK}4<u
% x = -1:0.01:1; $-4](br|
% [X,Y] = meshgrid(x,x); +X?ErQm
% [theta,r] = cart2pol(X,Y); igO>)XbsM
% idx = r<=1; XN<SKW(H3
% z = nan(size(X)); Q PH=`s
% z(idx) = zernfun(5,1,r(idx),theta(idx)); `y8pwWo-o
% figure :uL<UD,vu3
% pcolor(x,x,z), shading interp i,Ct AbMx
% axis square, colorbar
tm1=
% title('Zernike function Z_5^1(r,\theta)') +ikSa8)*i
% ?HEqv$n
% Example 2: F^ q{[Z
% HB07 n4 |
% % Display the first 10 Zernike functions
'g v0;L
% x = -1:0.01:1; *dBy<dIy
% [X,Y] = meshgrid(x,x); sqkWQ`Ur
% [theta,r] = cart2pol(X,Y); FaHOutP
% idx = r<=1; (f/(q-7VWt
% z = nan(size(X)); }~FX!F#oU
% n = [0 1 1 2 2 2 3 3 3 3]; X+vKY
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; U?[ (
% Nplot = [4 10 12 16 18 20 22 24 26 28]; W/'1ftn?D
% y = zernfun(n,m,r(idx),theta(idx)); q8 v iC|
% figure('Units','normalized') hCxg6e<[
% for k = 1:10 ]HKt7 %,
% z(idx) = y(:,k); ?d')#WnC
% subplot(4,7,Nplot(k)) ">V&{a-C4
% pcolor(x,x,z), shading interp Y&2FH/(M
% set(gca,'XTick',[],'YTick',[]) .#EU@Hc
% axis square yi7.9/;a
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) h*w9{[L
% end Y;'<u\^M"
% A;AQw
% See also ZERNPOL, ZERNFUN2. :X>Wd+lY:_
n,I3\l9
% Paul Fricker 11/13/2006 lyn%r
@@d_F<Ym[
Kda'N$|`
% Check and prepare the inputs: 6<&~R3dQ
% ----------------------------- *d._H1zT
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Hv6h7-
error('zernfun:NMvectors','N and M must be vectors.') dX(JV' 18A
end j^G=9r[,
g*k)ws
if length(n)~=length(m) Tigw+2
error('zernfun:NMlength','N and M must be the same length.') tE*BZXBlm
end J)nK9
VcjbRpTy&
n = n(:); !'f7;%7s
m = m(:); 5F kdGF
if any(mod(n-m,2)) Ek+R
error('zernfun:NMmultiplesof2', ... y)kxR
'All N and M must differ by multiples of 2 (including 0).') 6w.E Sm
end R hWQ:l]
9A|A@E#
if any(m>n) `_/bg(E
error('zernfun:MlessthanN', ... NuZ2,<~9
'Each M must be less than or equal to its corresponding N.') *'@Oo
end 3Z*r#d$nh:
JCWTB`EB>
if any( r>1 | r<0 ) I0XJ&P%
error('zernfun:Rlessthan1','All R must be between 0 and 1.') VL%. maj
end PD#,KqL:
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) i)#-VOhX)
error('zernfun:RTHvector','R and THETA must be vectors.') (\\;A?
end +\!.X_Ij
ki1(b]rf
r = r(:); \ `Hp/D1
theta = theta(:); c4z&HQd
length_r = length(r); O|Uz)Y94
if length_r~=length(theta) &ALnE:F
error('zernfun:RTHlength', ... MA
.;=T
'The number of R- and THETA-values must be equal.') U>tR :)
end #X Q/y} (
5lsslE+:J
% Check normalization: -K|1w'E
% -------------------- Ow0>qzTg
if nargin==5 && ischar(nflag) fPeS;
isnorm = strcmpi(nflag,'norm'); vF\>;pcT
if ~isnorm qbyYNlXqm
error('zernfun:normalization','Unrecognized normalization flag.') ^\}MG!l
end "FHJ_$!
else r1!1u7dr
t
isnorm = false; yr\ClIU
end B=A!hXNa
3.W[]zH/u
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% #]h
X."b2
% Compute the Zernike Polynomials %q5dV<X'c
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ]B>76?2W
#j2kT
% Determine the required powers of r: ?G `m;S
% ----------------------------------- BX_yC=S
m_abs = abs(m); RV~t%Sw^
rpowers = []; 8LV6E5Q
for j = 1:length(n) YsmRY=3
rpowers = [rpowers m_abs(j):2:n(j)]; @=kgK[t
9
end v3"6'.f;bY
rpowers = unique(rpowers); il^;2`]&
8AR8u!;8
% Pre-compute the values of r raised to the required powers, FJn-cR.n
% and compile them in a matrix: {
^o.f
% ----------------------------- ]>M\|,wh
if rpowers(1)==0 |WB-N g
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); &S4*x|-C&
rpowern = cat(2,rpowern{:}); .\_):j*
rpowern = [ones(length_r,1) rpowern]; |z)s9B;:#i
else E#!N8fQ
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); tW/k
rpowern = cat(2,rpowern{:}); Y`~B> J
end h,c*:
Kq[4I[+R
% Compute the values of the polynomials: "_/ih1z]
% -------------------------------------- W&5/1``u\
y = zeros(length_r,length(n)); kQkc+sGJf
for j = 1:length(n) [}szM^
s = 0:(n(j)-m_abs(j))/2; GDSV:]hL
pows = n(j):-2:m_abs(j); !hVbx#bXl
for k = length(s):-1:1 [IAUJ09>I
p = (1-2*mod(s(k),2))* ... 3(e_2v
prod(2:(n(j)-s(k)))/ ... !E$$FvL
prod(2:s(k))/ ... ^kfqw0!
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... t:2DB)
prod(2:((n(j)+m_abs(j))/2-s(k))); *2Pr1U
idx = (pows(k)==rpowers); j(%N.f6
y(:,j) = y(:,j) + p*rpowern(:,idx); &
/8Tth86
end i q`}c
|c
_(-jk4 L
if isnorm a&>NuMDI
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); {+9RJmZg
end zRna=h!
end d,GOP_N8I
% END: Compute the Zernike Polynomials y#'hOSR2
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 6f +aGz
;<Q%d~$xy}
% Compute the Zernike functions: hDxq9EF
% ------------------------------ `,]Bs*~
idx_pos = m>0; `X<B+:>v-
idx_neg = m<0; jn^X{R\
zT>!xGTu7~
z = y; }JFTe
g
if any(idx_pos) UDEGQ^)Xz|
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); X+!+&RAN*
end Z:9 Q~}x8
if any(idx_neg) b`X''6
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); oPi>]#X
end BwT[SI<Sg
>._d2.Q'
% EOF zernfun