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 Lo|NE[b:G
function z = zernfun(n,m,r,theta,nflag) |n|U;|'^
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. w&wA >q>&
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N gtaV6sD
% and angular frequency M, evaluated at positions (R,THETA) on the bxd3
% unit circle. N is a vector of positive integers (including 0), and TZ&4
% M is a vector with the same number of elements as N. Each element pW*{Mx
% k of M must be a positive integer, with possible values M(k) = -N(k)
Z;j/K
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, :"OZc7
~
% and THETA is a vector of angles. R and THETA must have the same mHK@(D7X
% length. The output Z is a matrix with one column for every (N,M) cW81
% pair, and one row for every (R,THETA) pair. *1|YLy
% ":UWowJO
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike msA' 5>
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), Ax5mP8S
% with delta(m,0) the Kronecker delta, is chosen so that the integral ^ [X|As2
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 'h!h!
% and theta=0 to theta=2*pi) is unity. For the non-normalized {f`lSu
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. olD@W
UB
% Y=P9:unG
% The Zernike functions are an orthogonal basis on the unit circle. Ph(]?MG\_
% They are used in disciplines such as astronomy, optics, and T7>48eH
% optometry to describe functions on a circular domain. .DgoOo%?"
% V;>9&'Z3
% The following table lists the first 15 Zernike functions. =&di4'`
% MuDFdbtR
% n m Zernike function Normalization :0
W6uFNOU
% -------------------------------------------------- |_l<JQvf`E
% 0 0 1 1 E/"YId `A
% 1 1 r * cos(theta) 2 ;jRL3gAe)
% 1 -1 r * sin(theta) 2 .+{nA}Bc
% 2 -2 r^2 * cos(2*theta) sqrt(6) a~8:rW^
% 2 0 (2*r^2 - 1) sqrt(3) QRsqPh&-
% 2 2 r^2 * sin(2*theta) sqrt(6) r+imn&FK8
% 3 -3 r^3 * cos(3*theta) sqrt(8) x2
w8zT6M
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) <MPeh&_3#
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) ,bB( 24LD
% 3 3 r^3 * sin(3*theta) sqrt(8) lTa1pp
Zw
% 4 -4 r^4 * cos(4*theta) sqrt(10) R(M}0JRm
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Pk$}%;@v
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 1U717u
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Hfh@<'NL]
% 4 4 r^4 * sin(4*theta) sqrt(10) .yZK.[x4
% -------------------------------------------------- o `b`*Z
% =jJ H^Y2
% Example 1: NY4!TOp
% 4fu'QZ(}
% % Display the Zernike function Z(n=5,m=1) Ty`-r5
% x = -1:0.01:1; JaH*
rDs-
% [X,Y] = meshgrid(x,x); 8# 6\+R
% [theta,r] = cart2pol(X,Y); L@7Qs6G2u
% idx = r<=1; ]WTf< W<
% z = nan(size(X)); Bj;\mUsk
% z(idx) = zernfun(5,1,r(idx),theta(idx)); Vh 2Bz
% figure /yLzDCKn
% pcolor(x,x,z), shading interp uQeqnGp
% axis square, colorbar }BA9Ka#%
% title('Zernike function Z_5^1(r,\theta)') *
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% W\HLal
% Example 2: A{4Dzm !
% q]F4Lq(
% % Display the first 10 Zernike functions l<u{6o
% x = -1:0.01:1; C>AcK#-x,{
% [X,Y] = meshgrid(x,x); A|2 <A
!
% [theta,r] = cart2pol(X,Y); WLE%d]'%M
% idx = r<=1; 6a7vlo
% z = nan(size(X)); #]?tY}~
% n = [0 1 1 2 2 2 3 3 3 3]; \|v `l{
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; .6\T`6H=a
% Nplot = [4 10 12 16 18 20 22 24 26 28]; J
cP~-cp
% y = zernfun(n,m,r(idx),theta(idx)); Kp8fh-4_
% figure('Units','normalized') AnRlH
% for k = 1:10 - oU@D
% z(idx) = y(:,k); Po1hq2-U8
% subplot(4,7,Nplot(k)) 9X!ET!
% pcolor(x,x,z), shading interp 9~=gwP
% set(gca,'XTick',[],'YTick',[]) zT$0xj8
% axis square dAL0.>|`0
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) lco~X DI
% end _B}9f
% a[q84[OQ
% See also ZERNPOL, ZERNFUN2. :*#rRQ>t
o1e4.-xI
% Paul Fricker 11/13/2006 a|U}Ammr
BlL|s=dlQV
:=y0'f
V(@
% Check and prepare the inputs: l`DtiJ?$$0
% ----------------------------- /CH(!\bQ
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) oE$hqd s
error('zernfun:NMvectors','N and M must be vectors.') it~Z|$
end itw{;j
i^R{Ul[
if length(n)~=length(m) tzPC/?
error('zernfun:NMlength','N and M must be the same length.') Rl1$?l6Rf
end e$HQuA~Q;
4b]_
#7Qm
n = n(:); }X.>4\B5
m = m(:); `N$!s7M
if any(mod(n-m,2)) k'g$2
error('zernfun:NMmultiplesof2', ... ?<!
nm&~
'All N and M must differ by multiples of 2 (including 0).') "@4ghot t
end u %'y_C3
_$8{;1$T?
if any(m>n) J,RDTXqn
error('zernfun:MlessthanN', ... l^ARW
E
'Each M must be less than or equal to its corresponding N.') vm|!{5l:=y
end Vd21,~^>g
cs
t&0
if any( r>1 | r<0 ) pL! a
error('zernfun:Rlessthan1','All R must be between 0 and 1.') mGO>""<:
end ALfiR(!
MA$Xv`6I\
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) "NKf0F
error('zernfun:RTHvector','R and THETA must be vectors.') @7fm1b
end Rnr#$C%
C-Ig_Nc
r = r(:); U,'EF[t
theta = theta(:); F;pQ \Y
length_r = length(r); Hng!'
if length_r~=length(theta) |:N>8%@6c
error('zernfun:RTHlength', ... p'g^Wh
'The number of R- and THETA-values must be equal.') 7lR<@$q
end JD ]OIh
2
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% Check normalization: \"'\MA
% -------------------- b~*i91)\
if nargin==5 && ischar(nflag) qi&D+~Gv!
isnorm = strcmpi(nflag,'norm'); (8G$(MK
if ~isnorm L%XXf3;c
error('zernfun:normalization','Unrecognized normalization flag.') -6`;},Yr
end W^k,Pmopy
else L7}i
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isnorm = false; ]-:1se
end N
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1[s0Lz
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 84^[/d;!
% Compute the Zernike Polynomials 3 z=\.R
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% j=9ze op
%
e #M iaX
% Determine the required powers of r: 4jGLAor|
% ----------------------------------- oNIFx5*Z
m_abs = abs(m); %'0&ElQ
rpowers = []; m2 O&2[g
for j = 1:length(n) P6YQK+
rpowers = [rpowers m_abs(j):2:n(j)]; (sCAR=5v\
end k;Hnu
rpowers = unique(rpowers); 4mJFvDZV`
,Kw5Ro`I:
% Pre-compute the values of r raised to the required powers, CW-A e
% and compile them in a matrix: `%=<R-/#7S
% ----------------------------- Y\(;!o0a
if rpowers(1)==0 \ha-"Aqze3
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); A=X-;N#
rpowern = cat(2,rpowern{:}); -zKxf@"
rpowern = [ones(length_r,1) rpowern]; =EpJZt
else 7$7n71o
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); jct./arK
rpowern = cat(2,rpowern{:}); H^
BYd%-
end ){ gAj
PsbG|~
% Compute the values of the polynomials: vq>l>as9O
% -------------------------------------- .S7:;%qL6
y = zeros(length_r,length(n)); 8+&JQ"UaB
for j = 1:length(n) opD-vDa h
s = 0:(n(j)-m_abs(j))/2; 5)M2r!\
pows = n(j):-2:m_abs(j); !re1EL
for k = length(s):-1:1 *
t!r@k
p = (1-2*mod(s(k),2))* ... r~>,$[|n})
prod(2:(n(j)-s(k)))/ ... WYszk ,E
prod(2:s(k))/ ... sV2iITFp
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... y@;%Uv&
prod(2:((n(j)+m_abs(j))/2-s(k))); wz=z?AZW
idx = (pows(k)==rpowers); [@G`Afaf
y(:,j) = y(:,j) + p*rpowern(:,idx); ;^ 3$kF
end IzUo0D*@
Im)EDTm$
if isnorm cp%ii'
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); d#>y }H9
end -5k2j^r;
end hO( RZ'{
% END: Compute the Zernike Polynomials ]tY:,Mfs
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% c1%rV`)]
y LM"+.?pL
% Compute the Zernike functions: :(p)1=I
% ------------------------------ KDTDJ8
idx_pos = m>0; o8ppMM8_R[
idx_neg = m<0; 8omC%a}9m
o~1 Kp!U
z = y; Phs-(3
if any(idx_pos) AIZBo@xg
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ?KP}#>Ba@
end BsLG^f
if any(idx_neg) _^\$"nw
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); v\%G|8+]
end Z
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`hrQw)5?r
% EOF zernfun