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 Dq+rEt
function z = zernfun(n,m,r,theta,nflag) xmZ]mu,,$
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. s,ZJ?[/
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N elXY*nt8h
% and angular frequency M, evaluated at positions (R,THETA) on the Y;S+2])R2
% unit circle. N is a vector of positive integers (including 0), and >L?)f3_a
% M is a vector with the same number of elements as N. Each element \}t(g}7T
% k of M must be a positive integer, with possible values M(k) = -N(k) z7F~;IB*u
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, /kyuL]6
% and THETA is a vector of angles. R and THETA must have the same %"@KuqV
% length. The output Z is a matrix with one column for every (N,M) ciI;U/V
% pair, and one row for every (R,THETA) pair. z (r Q6
% =kohQ d.n
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike zLuej'
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), )DuOo83n["
% with delta(m,0) the Kronecker delta, is chosen so that the integral l"!.aIY"e
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, RH^8 "%\
% and theta=0 to theta=2*pi) is unity. For the non-normalized zzy%dc
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ro7\}O:I
% ,6,#Lc
% The Zernike functions are an orthogonal basis on the unit circle. T>d-f=(9KH
% They are used in disciplines such as astronomy, optics, and o
<8L,u(U
% optometry to describe functions on a circular domain. 1p>5ZkHb
% !0ySS {/
% The following table lists the first 15 Zernike functions. 31k.{dnm
% <9YRSE[Ed
% n m Zernike function Normalization K~AQ) ]pJI
% -------------------------------------------------- Q u2W
% 0 0 1 1 w<N[K>
% 1 1 r * cos(theta) 2 #Zk6
% 1 -1 r * sin(theta) 2 i;`rzsRb
% 2 -2 r^2 * cos(2*theta) sqrt(6) JC}y{R8
% 2 0 (2*r^2 - 1) sqrt(3) fZH:&EP
% 2 2 r^2 * sin(2*theta) sqrt(6) lM"@vNgK
% 3 -3 r^3 * cos(3*theta) sqrt(8) ?%(8RQ
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) \MQ|(
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) zCj]mH`es'
% 3 3 r^3 * sin(3*theta) sqrt(8) ZffK];D
% 4 -4 r^4 * cos(4*theta) sqrt(10) t.c XrX`k
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) h djv/
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) Hb=4k)-/]
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) *^ncb,1+i
% 4 4 r^4 * sin(4*theta) sqrt(10) GUE3|
% -------------------------------------------------- bNp
RGhlV
% |/[?]`
% Example 1: 2#ND(
% g5lf-}?
% % Display the Zernike function Z(n=5,m=1) `Q^G
k{9P
% x = -1:0.01:1; ]wWN~G)2lV
% [X,Y] = meshgrid(x,x); { :'#Ts<
% [theta,r] = cart2pol(X,Y); Wcl@H @
% idx = r<=1; `] ;*k2
% z = nan(size(X)); ^tIs57!
% z(idx) = zernfun(5,1,r(idx),theta(idx)); p ElF,Y
% figure 6:i{_YX(.S
% pcolor(x,x,z), shading interp J7/"8S_#N
% axis square, colorbar Q3u
P7j
% title('Zernike function Z_5^1(r,\theta)') \ $}^u5Y
% L+7L0LbNU
% Example 2: Zm!T4pL
% l4u_Z:<w
% % Display the first 10 Zernike functions kUUeyq
% x = -1:0.01:1; |E&a3TQW
% [X,Y] = meshgrid(x,x); .&=nP?ZPC6
% [theta,r] = cart2pol(X,Y); x6\EU=,
% idx = r<=1; Y` Oz\W
% z = nan(size(X)); w`&~m:R
% n = [0 1 1 2 2 2 3 3 3 3]; 8-3]Bm!
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; xCz(qR
% Nplot = [4 10 12 16 18 20 22 24 26 28]; }&Ngh4/
% y = zernfun(n,m,r(idx),theta(idx)); j[k&O)A{C
% figure('Units','normalized') lt`(R*B%
% for k = 1:10 gUwg\>UC
% z(idx) = y(:,k); wP8Wx~Q=
% subplot(4,7,Nplot(k)) !E8y!|7$
% pcolor(x,x,z), shading interp v8W .84e-
% set(gca,'XTick',[],'YTick',[]) pZUckQ
% axis square zBtlkBPu
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ?8X;F"Ba
% end <V0]~3
% _TY9!:&}q
% See also ZERNPOL, ZERNFUN2. MdmS
FJomUVR .
% Paul Fricker 11/13/2006 4qXO8T#~J=
?j9J6=2
#kjN!S*=
% Check and prepare the inputs: pyYm<dn
% ----------------------------- *UhYX)J
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) jU.z{(s
error('zernfun:NMvectors','N and M must be vectors.') w$8Su:g=
end ?-%Q[W
jI%v[]V
if length(n)~=length(m) }7&.FV"
error('zernfun:NMlength','N and M must be the same length.') E j`
end gpDH_!K
PKFjM~J
n = n(:); cu*8,*FU
m = m(:); >? >@&A/
if any(mod(n-m,2)) EK`}?>'
error('zernfun:NMmultiplesof2', ... E7X6Shng
'All N and M must differ by multiples of 2 (including 0).') x`~YTOfYk
end @a2n{
s)C5u;3!
if any(m>n) dJxdrs
error('zernfun:MlessthanN', ... _W]R|kYl$'
'Each M must be less than or equal to its corresponding N.') |`vwykhezO
end m1H|C 3u8
YbAa@Sq@
if any( r>1 | r<0 ) _#32hAI
error('zernfun:Rlessthan1','All R must be between 0 and 1.') 2q]y(kW+
end 35=kZXwG+4
(:Di/{i&r5
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) `t[b0; 'OH
error('zernfun:RTHvector','R and THETA must be vectors.') ~+DPq|-O
end X)7_@,7
EMy>X
r = r(:); #C^)W/dP
theta = theta(:); *{WhUHZF
length_r = length(r); d8D0 28d
if length_r~=length(theta) r{\c.\
error('zernfun:RTHlength', ... W
D 8
'The number of R- and THETA-values must be equal.') R|&jvG=|
end wO<.wPa`
xs#g
% Check normalization: |)~t^
% -------------------- zI-]K,!
if nargin==5 && ischar(nflag) n vzk P{
isnorm = strcmpi(nflag,'norm'); 3Ye{a<ckK
if ~isnorm PU8>.9x
error('zernfun:normalization','Unrecognized normalization flag.') NJ]AxFG
end zm>^!j
!
else 4# +i\H`
isnorm = false; o6bT.{8\
end $?P 5A E
7:/gO~gI
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |k.%e4
% Compute the Zernike Polynomials CcCcuxtR
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% tzIcR
#Z
tuK2D,6
% Determine the required powers of r: f4'WT
% ----------------------------------- ehTrjb3k
m_abs = abs(m); _c!$K#Yl{
rpowers = []; trx y3k;
for j = 1:length(n) N 2XL5<
rpowers = [rpowers m_abs(j):2:n(j)]; J2ZV\8t
end [?>\]
rpowers = unique(rpowers); W6c]a/
Rf4}((y7Y\
% Pre-compute the values of r raised to the required powers, .9NYa |+0
% and compile them in a matrix: 0RZ[]:(
% ----------------------------- L;GkG! g
if rpowers(1)==0 *Jwx,wF}4
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); -UB XWl
rpowern = cat(2,rpowern{:}); { )g
$
rpowern = [ones(length_r,1) rpowern]; 3xIelTf*
else /@ww"dmqU
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); %_@T'!]
rpowern = cat(2,rpowern{:}); \>$3'i=mQ
end 'fjouO
I+{2DY/}
% Compute the values of the polynomials: 'MgYSP<
% -------------------------------------- vSnGPLl
y = zeros(length_r,length(n)); do.AesdXaq
for j = 1:length(n) 4`e[gvh
s = 0:(n(j)-m_abs(j))/2; |:w)$i& *
pows = n(j):-2:m_abs(j); S=<OS2W7+r
for k = length(s):-1:1 1*GL;W~ix*
p = (1-2*mod(s(k),2))* ... vf5q8/a
prod(2:(n(j)-s(k)))/ ... 9?iA~r|+
prod(2:s(k))/ ... OKPNsN
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... `r'$l<(4WV
prod(2:((n(j)+m_abs(j))/2-s(k))); )b?$
4<X^
idx = (pows(k)==rpowers); lqTc6@:D
y(:,j) = y(:,j) + p*rpowern(:,idx); 9OYyR
end YP
Qix
hd*GDjmRQ/
if isnorm ^H0#2hFa
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); j1rR3)oP
end g=/!Ry=
end ,j6R/sg
% END: Compute the Zernike Polynomials u69UUkG
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ck< `kJ`b
7`j%5%q
% Compute the Zernike functions:
kRjNz~g
% ------------------------------ -UM|u_
idx_pos = m>0; 7gcR/HNeF
idx_neg = m<0; c@2a)S8Y]
D;&\)
z = y; YkFAu8b>
if any(idx_pos) )@ofczl6
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); {O:{F?
end w4fQ~rcUIc
if any(idx_neg) "F =NDF
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ! 1Hs;K
end OxPl0-]t
hF{gN3v5
% EOF zernfun