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 e9k}n\t3
function z = zernfun(n,m,r,theta,nflag) 0(@8
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. goIn7ei92
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N rZ w&[ G
% and angular frequency M, evaluated at positions (R,THETA) on the YpL{c* M
% unit circle. N is a vector of positive integers (including 0), and N%_-5Q)so
% M is a vector with the same number of elements as N. Each element o+/x8:
% k of M must be a positive integer, with possible values M(k) = -N(k) _S2QY7/
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, Pt";f
% and THETA is a vector of angles. R and THETA must have the same sBZKf8 @/
% length. The output Z is a matrix with one column for every (N,M) x4.-7%VV%
% pair, and one row for every (R,THETA) pair. A}H)ojG'v
% UKMrR9[x*
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ~WR6rc
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), i?4vdL8M
% with delta(m,0) the Kronecker delta, is chosen so that the integral t#6gjfIi
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, aR*z5p2-w
% and theta=0 to theta=2*pi) is unity. For the non-normalized ]*[S#Jk
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. G?'L1g[lc
% ,Z&"@g
% The Zernike functions are an orthogonal basis on the unit circle. PO<4rT+B
% They are used in disciplines such as astronomy, optics, and #x':qBv#
% optometry to describe functions on a circular domain. ~iEH?J%i1r
% _2 }i8q:
% The following table lists the first 15 Zernike functions. .OXvv _?<
% C1)TEkc"C
% n m Zernike function Normalization A;Xn#t ,(K
% -------------------------------------------------- ;gK+AU
% 0 0 1 1 l4L&hY^
% 1 1 r * cos(theta) 2 4SY]Q[
% 1 -1 r * sin(theta) 2 KosAc'/ M
% 2 -2 r^2 * cos(2*theta) sqrt(6) 6yv*AmFh
% 2 0 (2*r^2 - 1) sqrt(3) >[O
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% 2 2 r^2 * sin(2*theta) sqrt(6) MFiX8zwhx+
% 3 -3 r^3 * cos(3*theta) sqrt(8) {p
yo
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) Ol{)U;,`
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) _Bb/~^
% 3 3 r^3 * sin(3*theta) sqrt(8) oPo<F5M]d%
% 4 -4 r^4 * cos(4*theta) sqrt(10) &AZr(>
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) aoI{<,(
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 9_KUUA
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) C;G~_if4PR
% 4 4 r^4 * sin(4*theta) sqrt(10) 0Evmq3,9
% -------------------------------------------------- FL/@e$AK
% bn~=d@'
% Example 1: E`u=$~K
% d]0fgwwGC
% % Display the Zernike function Z(n=5,m=1) Rkw)IdB
% x = -1:0.01:1; 2}b1PMpZG
% [X,Y] = meshgrid(x,x); .v/s9'lB
% [theta,r] = cart2pol(X,Y); ~Pv4X2MO
% idx = r<=1; O}Fp\"
% z = nan(size(X)); kNd[M =%
% z(idx) = zernfun(5,1,r(idx),theta(idx)); ,Hch->?Og
% figure TPvS+_<oL{
% pcolor(x,x,z), shading interp azS"*#r6}
% axis square, colorbar R1 hb-
% title('Zernike function Z_5^1(r,\theta)') ZV,n-M =
% ncu
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% Example 2: 4F??9o8 }
% H}dsd=yO
% % Display the first 10 Zernike functions /V$[M
% x = -1:0.01:1; g$EjIHb
% [X,Y] = meshgrid(x,x); 9fzbR~s
% [theta,r] = cart2pol(X,Y); hz>&E,<8q
% idx = r<=1; s'tmak-}|
% z = nan(size(X)); r2M._}bF
% n = [0 1 1 2 2 2 3 3 3 3]; .NiPaUzc<
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; ,*bI0mFZ
% Nplot = [4 10 12 16 18 20 22 24 26 28]; [3]!*Cd
% y = zernfun(n,m,r(idx),theta(idx)); =V/$&96Q
% figure('Units','normalized') V\r5
% for k = 1:10 5owUQg,W
% z(idx) = y(:,k); HulN84
% subplot(4,7,Nplot(k)) [8^jwnAYS
% pcolor(x,x,z), shading interp Y"K7$+5#\
% set(gca,'XTick',[],'YTick',[]) iRPt0?$
% axis square L/"u,~[
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) n^UrHHOL
% end D""d-oI[
% n-#?6`>a
% See also ZERNPOL, ZERNFUN2. ;B:'8$j$
BBnj}XP*4
% Paul Fricker 11/13/2006 ZgcA[P
Yih^ZTf]O?
: N> 5{
% Check and prepare the inputs: +s
V$s]U
% ----------------------------- V2^(qpM!
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) d-#MRl$rtK
error('zernfun:NMvectors','N and M must be vectors.') `-hFk88
end xzyV|(
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if length(n)~=length(m) k =ru)
_$2
error('zernfun:NMlength','N and M must be the same length.') QukLsl]U
end v< xe(dC
7;dV]N
n = n(:); DQ?'f@I&*
m = m(:); &s_[~g<
if any(mod(n-m,2)) `|8)A)ZVT
error('zernfun:NMmultiplesof2', ... NFDi2L>Ba
'All N and M must differ by multiples of 2 (including 0).') b*n o.eB
end $"!"=v%B
%t([
if any(m>n) zbOEF
error('zernfun:MlessthanN', ... +w?RW^:Q=
'Each M must be less than or equal to its corresponding N.') &y;('w
end '&I.w p`^
J)6RXt*!
if any( r>1 | r<0 ) +`r;3kH ..
error('zernfun:Rlessthan1','All R must be between 0 and 1.') 'ZgrN14
end 7i`@`0
l`:M/z6"
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) W~$YKBW
error('zernfun:RTHvector','R and THETA must be vectors.') RCsQLKqF
end jQFAlO(E':
@{+c6.*}
r = r(:); C:"Al-
theta = theta(:); ;[R{oW
Nw
length_r = length(r); r{pTMcDS
if length_r~=length(theta) *r6+Vz
error('zernfun:RTHlength', ... ^%@(>:)0
'The number of R- and THETA-values must be equal.') "~:o#~F6
end VC:.ya|Z
[[}KCND
% Check normalization: EJ`JN|,M
% -------------------- +?5nkhH
if nargin==5 && ischar(nflag) i(Cd#1<
isnorm = strcmpi(nflag,'norm'); 6D_3Hwrs
if ~isnorm 3WZ]9v{k
error('zernfun:normalization','Unrecognized normalization flag.') ;f:}gMK
end x{`>Il
else 6J9^:gXW~
isnorm = false; K9\`Wu_qL
end 4eMNKIsvY$
]R~K-cN`
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
k_
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% Compute the Zernike Polynomials eGwrSF#a)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ujGvrYj
L=nyloz,0
% Determine the required powers of r: MDGD*Qn~
% ----------------------------------- ;i\m:8!;
m_abs = abs(m); "a
%5on
rpowers = []; )R.y>Ucb0
for j = 1:length(n) ^
ry
rpowers = [rpowers m_abs(j):2:n(j)]; |j($2.
end U6;,<-bL
rpowers = unique(rpowers); I&^B?"Y
8XZS BR(Z
% Pre-compute the values of r raised to the required powers, Hy`Ee7>
% and compile them in a matrix: -\O%f)R
% ----------------------------- 0Ah'G
if rpowers(1)==0 ^vPM\qP#g
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); r9G}[#DO
rpowern = cat(2,rpowern{:}); [LDsn]{
rpowern = [ones(length_r,1) rpowern]; &,/_"N"?D
else ~UA:_7#\M
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 8R<2I1xn2
rpowern = cat(2,rpowern{:}); 't'~p#$,F
end {XAm3's
FGY4 u4y
% Compute the values of the polynomials: kXK D>."E*
% -------------------------------------- b2]1Dfw
y = zeros(length_r,length(n)); FMMQO,BU
for j = 1:length(n) w7aC=B/{?i
s = 0:(n(j)-m_abs(j))/2; SC/|o
pows = n(j):-2:m_abs(j); y,e#e`
for k = length(s):-1:1 0IO#h{t
p = (1-2*mod(s(k),2))* ... u
hW@
Y+
prod(2:(n(j)-s(k)))/ ... jI:5[. Y
prod(2:s(k))/ ... VL4ErOoZ
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ]w ^9qS
prod(2:((n(j)+m_abs(j))/2-s(k))); s @\UZC
idx = (pows(k)==rpowers); "l0z?u
y(:,j) = y(:,j) + p*rpowern(:,idx); d;1%Ei3K
end (|H1zO
K'z|a{ru.{
if isnorm /sVy"48-
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); iv@ey-,<
end _
T ;+*
end Q v=F'
% END: Compute the Zernike Polynomials ], Xva`"
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 5R"My^G
e
lj] e
% Compute the Zernike functions: 9,8}4Y=GVI
% ------------------------------ X;`XkOjk
idx_pos = m>0; $]O;D~
idx_neg = m<0; 0G@sj7)]
x
xMV2&,Jq
z = y; ?VVtEmIN
if any(idx_pos) G1K72M}CW
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); t0t" =(d
end a?&{eMEe}
if any(idx_neg) .[YM0dt
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 5`] ;[M9
end lU6?p")F1
UOh%"h
% EOF zernfun