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 CcTdLq
function z = zernfun(n,m,r,theta,nflag) kQ]4Bo
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. #<~oR5ddlb
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ;Fo7 -kK
% and angular frequency M, evaluated at positions (R,THETA) on the znB+RiV8
% unit circle. N is a vector of positive integers (including 0), and blLl1Ak
% M is a vector with the same number of elements as N. Each element <5E)6c_W)
% k of M must be a positive integer, with possible values M(k) = -N(k) xM=ydRu
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, Sp6==(:.
% and THETA is a vector of angles. R and THETA must have the same .]H/u
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% length. The output Z is a matrix with one column for every (N,M) 4{Q$^wD+.
% pair, and one row for every (R,THETA) pair. kbL7Xjk
% Ee_?aG
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% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike =0L%<@yA
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ScjeAC)
% with delta(m,0) the Kronecker delta, is chosen so that the integral yEMM@5W)8
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, F Uz1P
% and theta=0 to theta=2*pi) is unity. For the non-normalized >z~_s6#CP
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. u -)ED
% S}fQis
% The Zernike functions are an orthogonal basis on the unit circle. S\]9mHJI
% They are used in disciplines such as astronomy, optics, and KWxTN|>
% optometry to describe functions on a circular domain. qzNXz_#+u
% /0cm7[a ?
% The following table lists the first 15 Zernike functions. _M&n~ r
% 15VvZ![$V
% n m Zernike function Normalization mU(v9Jpf7
% -------------------------------------------------- z;?ztpa@
% 0 0 1 1 2}7 _Y6RS*
% 1 1 r * cos(theta) 2 $}IG+,L
% 1 -1 r * sin(theta) 2 ck%.D%=
% 2 -2 r^2 * cos(2*theta) sqrt(6) 'gXD?ARW
% 2 0 (2*r^2 - 1) sqrt(3) l-cBN^^
% 2 2 r^2 * sin(2*theta) sqrt(6) }9^'etD
% 3 -3 r^3 * cos(3*theta) sqrt(8) {\`y)k 7
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) @{UUB=}9
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) e|W;(@$<
% 3 3 r^3 * sin(3*theta) sqrt(8) -[J4nN &N
% 4 -4 r^4 * cos(4*theta) sqrt(10) RX%)@e/@
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) auB
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% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) #6
ni~d&0
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) O8A(OfX
% 4 4 r^4 * sin(4*theta) sqrt(10) &^K(9"
% -------------------------------------------------- #'},/Lm@
% HL]J=Gh
% Example 1: P3YM4&6XA
% l]~9BPsR
% % Display the Zernike function Z(n=5,m=1) x4PzP
% x = -1:0.01:1; }A]eC
% [X,Y] = meshgrid(x,x); Tt9cX}&&
% [theta,r] = cart2pol(X,Y); K2e68GU
% idx = r<=1; 8(&6*-7=
% z = nan(size(X)); "+4Jmf9
% z(idx) = zernfun(5,1,r(idx),theta(idx)); WO{7/h</
% figure :/%Y"0
% pcolor(x,x,z), shading interp Kxa1F,dZ
% axis square, colorbar l.]wBH#RS
% title('Zernike function Z_5^1(r,\theta)') Xn?.Od(
% #AP;GoIf"j
% Example 2: 5!S#}=f=
% D 5oYcGc
% % Display the first 10 Zernike functions 7QnWw0
% x = -1:0.01:1; JWaWOk(t=?
% [X,Y] = meshgrid(x,x); g\q4-
% [theta,r] = cart2pol(X,Y); si)>:e
% idx = r<=1; ?9O#b1f N
% z = nan(size(X)); b{,v?7^4
% n = [0 1 1 2 2 2 3 3 3 3]; A`JE(cIz3
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; pZK 1G
% Nplot = [4 10 12 16 18 20 22 24 26 28]; N
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% y = zernfun(n,m,r(idx),theta(idx)); X `EVjK
% figure('Units','normalized') k H<C9z2=
% for k = 1:10 ^|zag
% z(idx) = y(:,k); xo?'L&%
% subplot(4,7,Nplot(k)) +c!HXX
% pcolor(x,x,z), shading interp MRXw)NAw
% set(gca,'XTick',[],'YTick',[]) K?[Vz[-Fc
% axis square E3Y0@r
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 3uxf n=E
% end oJ*,a
% lwgwdB
% See also ZERNPOL, ZERNFUN2. $Zo|ta^
$M4Z_zle)
% Paul Fricker 11/13/2006 +TA~RCd
g 8uq6U
#]5KWXC'~
% Check and prepare the inputs: jIr\.i
% ----------------------------- +jZa A/
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) J5F@<vi
error('zernfun:NMvectors','N and M must be vectors.') 5@r6'Z
end j;b>~_ U%
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if length(n)~=length(m) ) 3"!Q+
error('zernfun:NMlength','N and M must be the same length.') LxGD=b
end Vt3*~Beb
<uS/8MP{
n = n(:); 52j3[in
m = m(:); 7g]mrI@
if any(mod(n-m,2)) Iox )-
error('zernfun:NMmultiplesof2', ... 6nE/8m
'All N and M must differ by multiples of 2 (including 0).') spm)X-[1
end %Vltc4QU
<QFayZ$
if any(m>n) T?f{.a)
error('zernfun:MlessthanN', ... &+@`Si=
'Each M must be less than or equal to its corresponding N.') 2)}*'_E9
end (0#$%US\
w'
J`$=
if any( r>1 | r<0 ) _0gdt4
error('zernfun:Rlessthan1','All R must be between 0 and 1.') q78OP}
end - E GZ
J
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) #BcUE?K*N
error('zernfun:RTHvector','R and THETA must be vectors.') g.di3GGi
end *S.FM.r
gCPH>8JwS0
r = r(:); [pp|*@1T
theta = theta(:); n{MTh_C4n
length_r = length(r); d7G@Z|R3p
if length_r~=length(theta) onRTX|#
error('zernfun:RTHlength', ... T:'JA
'The number of R- and THETA-values must be equal.') pO7OP"q1
end 'Ca;gi !U
c%hXj#;
% Check normalization: y;fF|t<y
% -------------------- $ .$nv~f
if nargin==5 && ischar(nflag) {V(~
isnorm = strcmpi(nflag,'norm'); W!\%v"
if ~isnorm a}f/<-L
error('zernfun:normalization','Unrecognized normalization flag.') 6@/k|t>OT
end Cj4Y, N
else ko[d axUB
isnorm = false; <yEApWd;
end 6Y\TVRR
_+aR|AEC
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% hGrX,.zj
% Compute the Zernike Polynomials v'?o#_La+
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% #"!ga)a%L
7bO>[RQB
% Determine the required powers of r: b:~#;$g
% ----------------------------------- w.J$(o(/
m_abs = abs(m); tF-l=ph}`
rpowers = []; ;qUB[Kw
for j = 1:length(n) j0~c2
rpowers = [rpowers m_abs(j):2:n(j)]; z7:*
,X
end H<fi,"X^
rpowers = unique(rpowers); Yl'8"
\HF
O%>*=h`P
% Pre-compute the values of r raised to the required powers, 0U*f"5F
% and compile them in a matrix: 8N"WKBj|_d
% ----------------------------- 9)S3{i6w
if rpowers(1)==0 $MQ<QP
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); NrXIaN
rpowern = cat(2,rpowern{:}); \ILNx^$EL
rpowern = [ones(length_r,1) rpowern]; c u";rnj
else Da8gOZ
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); wzxV)1jT
rpowern = cat(2,rpowern{:}); /({oN1X>i
end N;-%:nC
J
%A=
% Compute the values of the polynomials: @73kry v
% -------------------------------------- eXnSH$uI
y = zeros(length_r,length(n)); wN%lc3[/z2
for j = 1:length(n) -R]~kGa6m<
s = 0:(n(j)-m_abs(j))/2; H? z~V-8
pows = n(j):-2:m_abs(j); FCwE/ 2,
for k = length(s):-1:1 ']\SX*z?
p = (1-2*mod(s(k),2))* ... L;M@]
prod(2:(n(j)-s(k)))/ ... Z}vDP^rf
prod(2:s(k))/ ... cU ?F D
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... | Z7j
s"
prod(2:((n(j)+m_abs(j))/2-s(k))); F Xr\
idx = (pows(k)==rpowers); U<sGj~"#
y(:,j) = y(:,j) + p*rpowern(:,idx); JCBX?rM/
end O"o|8
l}M/
#*y.C[^5{
if isnorm 6m]?*k1HC
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); i4k [#x
end McS]aJfrk
end /E\04Bs
% END: Compute the Zernike Polynomials $n!5JS@40
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ^`SEmYb;
SYsO>`/ )
% Compute the Zernike functions: Hq<4G:#
% ------------------------------ pnp8`\cIH
idx_pos = m>0;
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idx_neg = m<0; a;IOL
FMF mn|
z = y; lo6upirZX
if any(idx_pos) Rsq EAdZw[
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); LQ%QFfC
end 9__Q-J
if any(idx_neg) TTa3DbFp%
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); [!3cWJCt
end +'93%/:
$iy!:Did
% EOF zernfun