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 zim]3%b*A;
function z = zernfun(n,m,r,theta,nflag) nQ2V
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. dmI,+hHtL
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ,6:ya8vB
% and angular frequency M, evaluated at positions (R,THETA) on the ,=whwl "tA
% unit circle. N is a vector of positive integers (including 0), and 6<jh0=$
% M is a vector with the same number of elements as N. Each element 1^ZQXUzl%i
% k of M must be a positive integer, with possible values M(k) = -N(k) S e/VOzzg
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, 3qU#Rg
;7
% and THETA is a vector of angles. R and THETA must have the same )X2=x^u*U
% length. The output Z is a matrix with one column for every (N,M) +U_> Bo
% pair, and one row for every (R,THETA) pair. 5m{!Rrb
% >fRI^Q,
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike }w .[ZeP
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), g BfYm
% with delta(m,0) the Kronecker delta, is chosen so that the integral VcKufV'
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, m-9{@kgAM?
% and theta=0 to theta=2*pi) is unity. For the non-normalized 8wz%e(
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. -02cI}e
% fQ c%a1'
% The Zernike functions are an orthogonal basis on the unit circle. Ht|No
% They are used in disciplines such as astronomy, optics, and I:l<t*
% optometry to describe functions on a circular domain. WtOpxAq
% Mqc"
% The following table lists the first 15 Zernike functions. S\=j; Uem
% b@j**O>[q)
% n m Zernike function Normalization O* `v1>
% -------------------------------------------------- 9[K".VeT]
% 0 0 1 1 S^0Po%d
% 1 1 r * cos(theta) 2 by; %k/
% 1 -1 r * sin(theta) 2 _V\rs{
5
% 2 -2 r^2 * cos(2*theta) sqrt(6) P @N7g`u3}
% 2 0 (2*r^2 - 1) sqrt(3) F0h`>{1%
% 2 2 r^2 * sin(2*theta) sqrt(6) o}H7;v8H
% 3 -3 r^3 * cos(3*theta) sqrt(8) &1)4B
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) a_Y*pOu
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) (#x<qi,T
% 3 3 r^3 * sin(3*theta) sqrt(8) mOji\qia
% 4 -4 r^4 * cos(4*theta) sqrt(10) EUH&"8
L
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) |hms'n0
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) ParOWs~W/
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Tbv", b
% 4 4 r^4 * sin(4*theta) sqrt(10) 1xN6V-qk
% -------------------------------------------------- 6\>S%S2:
% MzZYzz
% Example 1: kSx^Uu*
% pleLdGq
% % Display the Zernike function Z(n=5,m=1) OI0#@_L&
% x = -1:0.01:1; vf6_oX<Os
% [X,Y] = meshgrid(x,x); eX7dyM
% [theta,r] = cart2pol(X,Y); l_tr,3_w
% idx = r<=1; Sq^f}q
% z = nan(size(X)); .?{rd3[ec
% z(idx) = zernfun(5,1,r(idx),theta(idx)); y'\BpP
% figure qgREkb0
% pcolor(x,x,z), shading interp IB9[Lx
% axis square, colorbar tGHZU^B:}
% title('Zernike function Z_5^1(r,\theta)') zUX%$N+w}>
% (B|4wR\
% Example 2: JGQlx-qv
% S+(TRIjk
% % Display the first 10 Zernike functions tPu0r],`o
% x = -1:0.01:1; :pj00
% [X,Y] = meshgrid(x,x); lbM)U
% [theta,r] = cart2pol(X,Y); 48;6C g
% idx = r<=1; }
IJ
% z = nan(size(X)); {A2EGUmF2
% n = [0 1 1 2 2 2 3 3 3 3]; $|+q9o\
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; #ra"(/)
% Nplot = [4 10 12 16 18 20 22 24 26 28]; ]WlE9z7:8
% y = zernfun(n,m,r(idx),theta(idx)); HKu? J
% figure('Units','normalized') ]7<}EG
% for k = 1:10 _<tWy+.
% z(idx) = y(:,k); GJ YXCi
% subplot(4,7,Nplot(k)) n8W+q~sW%
% pcolor(x,x,z), shading interp Ln6\Iis
% set(gca,'XTick',[],'YTick',[]) :`('lrq
% axis square GIXxOea1
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) O ?`=<W/R
% end /{Ff)<Q.Z
% Yq~$Q4
% See also ZERNPOL, ZERNFUN2. ;',hwo_LBf
%`*`HU#X
% Paul Fricker 11/13/2006 6)<g%bH!
[O)(0
&'%b1CbE
% Check and prepare the inputs: kLc}a5;
% ----------------------------- |'@c ~yc
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) <1")JDW
error('zernfun:NMvectors','N and M must be vectors.') 5f5bhBZ<
end ,w>WuRN"
+; /]'
if length(n)~=length(m) 8MUY
error('zernfun:NMlength','N and M must be the same length.') "},0Cs
end 9A|deETa-
kmfz=q?
n = n(:); <ezv
m = m(:); 3FWl_d~uD
if any(mod(n-m,2)) 0
#*M'C#
error('zernfun:NMmultiplesof2', ... <'s_3AC
'All N and M must differ by multiples of 2 (including 0).') tE&@U$0>o
end P-B3<~*i!
21(8/F ~{
if any(m>n) &.dC%
error('zernfun:MlessthanN', ... "ecG\}R=
'Each M must be less than or equal to its corresponding N.') o}EipTL
end SePPI.n
j?!BHNs
if any( r>1 | r<0 ) LJ^n6 m|_
error('zernfun:Rlessthan1','All R must be between 0 and 1.') oW0A8_|9
end 6yDc4AX
lqD.epm
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) c?@WNv
error('zernfun:RTHvector','R and THETA must be vectors.') ;Y/{q B!
end g&z)y
_hM
#*?}v
r = r(:); 9\2<#,R1q
theta = theta(:); Cs2hi,s
length_r = length(r); >j5,Z]
if length_r~=length(theta) jg2UX
error('zernfun:RTHlength', ... (~C_zG
'The number of R- and THETA-values must be equal.') f?KHp|
end xZmO^F5KHj
!_zp'V]?
% Check normalization: rL{3O4O
% -------------------- q_0So}
if nargin==5 && ischar(nflag) !Q-h#']~L
isnorm = strcmpi(nflag,'norm'); _e2=BE`W)
if ~isnorm |r5e#3w
error('zernfun:normalization','Unrecognized normalization flag.') rE:"8d}z
end 5|T[:m
else y r4j
isnorm = false; +>zjTP7\e"
end 0D x,)C
dv?ael^
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% _(#HQd,i
% Compute the Zernike Polynomials {zTo[i
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% CV9o,rL
HR.^
y$IE
% Determine the required powers of r: :5Y
yI.T
% ----------------------------------- 7(ni_|$|
m_abs = abs(m); E5^P*6c(
rpowers = []; IJWUNKqo=
for j = 1:length(n) :v=^-&t
rpowers = [rpowers m_abs(j):2:n(j)]; ySfot`LQ
end 2kP0//
rpowers = unique(rpowers); %kS4v,I
pQQN8Y~^Y
% Pre-compute the values of r raised to the required powers, )K=%s%3h<
% and compile them in a matrix: bOEO2v'cQ
% ----------------------------- Yf=an`"
if rpowers(1)==0 VR8 kY&
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); vbo|q[z
rpowern = cat(2,rpowern{:}); 8R3x74fL
rpowern = [ones(length_r,1) rpowern]; x.5!F2$
else e8WuAI86
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); &.m.ruab
rpowern = cat(2,rpowern{:}); xz$-_NWW
end UN
FQ`L
l** gM
% Compute the values of the polynomials: q
<, b
% -------------------------------------- (D.B'V#>
y = zeros(length_r,length(n)); cO8':P5Q
for j = 1:length(n) e;|:W A
s = 0:(n(j)-m_abs(j))/2; {7'Evfn)
pows = n(j):-2:m_abs(j); _1c0pQ ^}3
for k = length(s):-1:1 W2$MH: j
p = (1-2*mod(s(k),2))* ... 65% WjO
prod(2:(n(j)-s(k)))/ ... 9\QeH'A
prod(2:s(k))/ ... Po)!vL"
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... mp!S<m
prod(2:((n(j)+m_abs(j))/2-s(k))); %>z4hH,
idx = (pows(k)==rpowers); |41NRGgY
y(:,j) = y(:,j) + p*rpowern(:,idx); p\'0m0*
end kFRl+,bi~
ifXGH>C
if isnorm pmWt7 }
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); O(R1D/A[
end ; ,vGw<|o
end Q!91uNL
% END: Compute the Zernike Polynomials c\Z.V*o
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% wV604eO(
X7bS{GT
% Compute the Zernike functions: & t.G4
% ------------------------------ bwC~
idx_pos = m>0; 483/ZgzT`
idx_neg = m<0; 3)6TnY/u6{
=O1py_m
z = y; y6hb-:
#1
if any(idx_pos) F3?PlH:Y
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); } SNZl`>
end !y$:}W?_
if any(idx_neg) nF)b4`Nd
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); |zkZF|-
end up@I,9C/
/q^\g4J
% EOF zernfun