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 eNIkiJ$uS
function z = zernfun(n,m,r,theta,nflag) skk-.9
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. EO4"Z@ji
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N >Sc$R0
% and angular frequency M, evaluated at positions (R,THETA) on the mtSNl|O&{
% unit circle. N is a vector of positive integers (including 0), and s,eld@
% M is a vector with the same number of elements as N. Each element xaGVu0q
% k of M must be a positive integer, with possible values M(k) = -N(k) r4;5b s6wm
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, MILIu;[{#r
% and THETA is a vector of angles. R and THETA must have the same ddUjs8VvJ
% length. The output Z is a matrix with one column for every (N,M) {toyQ)C7
% pair, and one row for every (R,THETA) pair. el <<D
% Fy}MXe"f
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike [<#<:h&\
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), B6tcKh9d,
% with delta(m,0) the Kronecker delta, is chosen so that the integral E[ )7tr
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, qT4I Y$h
% and theta=0 to theta=2*pi) is unity. For the non-normalized 8gVxiFjo
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. J{nyo1A
% s=H/b$v
% The Zernike functions are an orthogonal basis on the unit circle. , aRJ!AZ
% They are used in disciplines such as astronomy, optics, and l%sp[uqcg
% optometry to describe functions on a circular domain. p?dGZ2` [I
% 8\qCj.>S
% The following table lists the first 15 Zernike functions. ka?IX9t\
% w\"n!^ms
% n m Zernike function Normalization QOkE\ro
% -------------------------------------------------- ,W)IVc
% 0 0 1 1 ,cGwtt(
% 1 1 r * cos(theta) 2 &=s|
% 1 -1 r * sin(theta) 2 Vu|Br
% 2 -2 r^2 * cos(2*theta) sqrt(6) dO 1-c`
% 2 0 (2*r^2 - 1) sqrt(3) m
wRLzN
% 2 2 r^2 * sin(2*theta) sqrt(6) Pe+ 8~0o=R
% 3 -3 r^3 * cos(3*theta) sqrt(8) ^7ea6G"
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) ch5`fm
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) br34Eh
% 3 3 r^3 * sin(3*theta) sqrt(8) &xGfkCP.]
% 4 -4 r^4 * cos(4*theta) sqrt(10) T3u5al
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Y{Y;EY4
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 1jUhG2y
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ^*cMry
% 4 4 r^4 * sin(4*theta) sqrt(10) Q.pEUDq/
% -------------------------------------------------- P`Hd*xh".j
% y (c|5CQ
% Example 1: _SBp66
r
% Ie^Dn!0S
% % Display the Zernike function Z(n=5,m=1) s0XRL1kWr
% x = -1:0.01:1; +!L_E6pyXE
% [X,Y] = meshgrid(x,x); ADLa.{
% [theta,r] = cart2pol(X,Y); e6{[o@aM{
% idx = r<=1; p0[,$$pM
% z = nan(size(X)); )}k?r5g
% z(idx) = zernfun(5,1,r(idx),theta(idx)); =M/UHOY
% figure RB lOTQjv
% pcolor(x,x,z), shading interp Q
!RVD*(
% axis square, colorbar lJ2|jFY9
% title('Zernike function Z_5^1(r,\theta)') #FQm/Q<0
% I9:G9
% Example 2: )MD*)O
% ctc`^#q
% % Display the first 10 Zernike functions E1l\~%A
% x = -1:0.01:1; `L"p)5H
% [X,Y] = meshgrid(x,x); m]-v IUpb
% [theta,r] = cart2pol(X,Y); ;G4HMtL
% idx = r<=1; gq/ePSa
% z = nan(size(X)); AjL?Qh4
% n = [0 1 1 2 2 2 3 3 3 3]; aiR|.opIb
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; (:fE _H2z
% Nplot = [4 10 12 16 18 20 22 24 26 28]; Y6;0khp
% y = zernfun(n,m,r(idx),theta(idx)); A<YZBR_
% figure('Units','normalized') D)O6|DiO
% for k = 1:10 7/D9n9F
% z(idx) = y(:,k); l# !@{ <
% subplot(4,7,Nplot(k)) (.quX@w"m
% pcolor(x,x,z), shading interp uhw5O9
% set(gca,'XTick',[],'YTick',[]) {0)WS}&
% axis square qa0JQ_?o]
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) R@7GCj
% end H%01&u
% vHI"C %
% See also ZERNPOL, ZERNFUN2. d5sGkR`(
!0. 5
% Paul Fricker 11/13/2006 ?(,5eg
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j quSR=
% Check and prepare the inputs: VH7iH|eW
% -----------------------------
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if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) WfTdD.Xx
error('zernfun:NMvectors','N and M must be vectors.') a_pCjG89
end !7ZfT?&
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if length(n)~=length(m) j0pvLZjM
error('zernfun:NMlength','N and M must be the same length.') >+;
b>
end c> U{,z
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n = n(:); >Rki[SNb-b
m = m(:); MR)KLM0
if any(mod(n-m,2)) ,I2reG
error('zernfun:NMmultiplesof2', ... L>5!3b=b
'All N and M must differ by multiples of 2 (including 0).') M;p q2$
end :LIKp;
rt@-Pw!B
if any(m>n) y`B!6p
5j
error('zernfun:MlessthanN', ... "mP*}VF
'Each M must be less than or equal to its corresponding N.') e}Af"LI
end Pu%>j'A
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if any( r>1 | r<0 ) ^pQ;0[9Y0
error('zernfun:Rlessthan1','All R must be between 0 and 1.') "PX3%II
end SG|i/K|7
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) kkE)zF
error('zernfun:RTHvector','R and THETA must be vectors.') H`6Jq?\
end eVCkPv*
:7DVc&0
r = r(:); h$ETH1Ue
theta = theta(:); dVmAMQk.g
length_r = length(r); eR*
]<0=
if length_r~=length(theta) #g`cih=QL
error('zernfun:RTHlength', ... ]g-qWSKU
'The number of R- and THETA-values must be equal.') w7t"&=pF7
end W'2-3J
}rMpp[
% Check normalization: QRmQ>
% -------------------- a@=36gx)
if nargin==5 && ischar(nflag) 0[!gk]p
isnorm = strcmpi(nflag,'norm'); .vOpU4
if ~isnorm }Mb'tGW
error('zernfun:normalization','Unrecognized normalization flag.') @#--dOWYR
end C"` 'Re5)
else KlqJEtO_
isnorm = false; #<i><EG
end .Qi1I
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vzrD"
% Compute the Zernike Polynomials :qSi>KCGh
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ~
%YTJS
q;a*gqt
% Determine the required powers of r: X @jYQ.
% ----------------------------------- <,cIc]eX
m_abs = abs(m); ?nGf Wx^
rpowers = []; ]Y:
W[p
for j = 1:length(n) qT>&
v_<
rpowers = [rpowers m_abs(j):2:n(j)]; _:=OHURc
end dR, NC-*
rpowers = unique(rpowers); +i_f.Ipp
.6Lhy3x
% Pre-compute the values of r raised to the required powers, w4MMo
% and compile them in a matrix: ~CdseSo9
% ----------------------------- 6k=Wt7C
if rpowers(1)==0 }L7F
g%,
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); U*fj5
rpowern = cat(2,rpowern{:}); tG^ ?fc
rpowern = [ones(length_r,1) rpowern]; KsU&<eQ
else D*r Zaqy
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); [BR}4(7
rpowern = cat(2,rpowern{:}); 79B`w
#
end GxBPEIim
s1vYZ
% Compute the values of the polynomials: %b%<g%@i
% -------------------------------------- A8Z?[,Mq!
y = zeros(length_r,length(n)); E?h2e~ ,]
for j = 1:length(n) ,,#rv-*
s = 0:(n(j)-m_abs(j))/2; !2M[
pows = n(j):-2:m_abs(j); GKx,6E#JM
for k = length(s):-1:1 VJtTbt;>
p = (1-2*mod(s(k),2))* ... TN@JPoH
prod(2:(n(j)-s(k)))/ ... pW^ ?g|_}
prod(2:s(k))/ ... Q2pboZ86
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... QDT{Xg*I
prod(2:((n(j)+m_abs(j))/2-s(k))); \C2P{q/m
idx = (pows(k)==rpowers); x7kg_`\U
y(:,j) = y(:,j) + p*rpowern(:,idx); .,K?\WZ
end !#gE'(J;c
kt0{-\
p
if isnorm o-<_X&"a|5
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Bsk2&17z
end ;Owu:}
end ggsi`Z{j?
% END: Compute the Zernike Polynomials xI\s9_"Qy
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% TvG:T{jwy
!E#.WX
% Compute the Zernike functions: svRaU7<UDN
% ------------------------------ }vA
nP]!A5
idx_pos = m>0; A*U'SCg(G
idx_neg = m<0; V42*4hskL
eh/OCzWH
z = y; f4y;K>u7p
if any(idx_pos) z'D{:q
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); WHLKf
end Y[]+C8"O
if any(idx_neg) -2ij;pkIW$
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); x, G6`|Hl
end
7-g4S]r<
U7%pOpO!
% EOF zernfun