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 *wml
4lh
function z = zernfun(n,m,r,theta,nflag) ]jUxL=]r
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. u vo2W!
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 0@R @L}m
% and angular frequency M, evaluated at positions (R,THETA) on the OzFA>FK0f;
% unit circle. N is a vector of positive integers (including 0), and rPV\ F
% M is a vector with the same number of elements as N. Each element M(,npW
% k of M must be a positive integer, with possible values M(k) = -N(k) *@'\4OO
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, O87Ptr8
% and THETA is a vector of angles. R and THETA must have the same * \@u,[,
% length. The output Z is a matrix with one column for every (N,M) Y4PB&pZ$O2
% pair, and one row for every (R,THETA) pair. D0?l$]aE
% W/O&(t
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ;c5Q"
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), tqY)
% with delta(m,0) the Kronecker delta, is chosen so that the integral V$Y5EX
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, G]fl33_}l
% and theta=0 to theta=2*pi) is unity. For the non-normalized 66"-Xf~u
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. I|=$.i
% utq*<,^
% The Zernike functions are an orthogonal basis on the unit circle. 2}P<}-?6
% They are used in disciplines such as astronomy, optics, and (,<ti):
% optometry to describe functions on a circular domain. A*ImruV
% i[b?W$]7
% The following table lists the first 15 Zernike functions. pU`4bT(w%
% T,h,)|:I^
% n m Zernike function Normalization xN\PQ,J
% -------------------------------------------------- DCfV
% 0 0 1 1 "M*Pt
% 1 1 r * cos(theta) 2 !m"(SJn"
% 1 -1 r * sin(theta) 2 CtwMMZXX3
% 2 -2 r^2 * cos(2*theta) sqrt(6) wVD-}n1"
% 2 0 (2*r^2 - 1) sqrt(3) 8phcekh+
% 2 2 r^2 * sin(2*theta) sqrt(6) p1p4t40<l
% 3 -3 r^3 * cos(3*theta) sqrt(8) 3\6UH
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) { U a19~'>
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) jyPY]r
% 3 3 r^3 * sin(3*theta) sqrt(8) j;fmmV@
% 4 -4 r^4 * cos(4*theta) sqrt(10) iulM8"P
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Au'[|Prr
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) STp}?Cb
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 4.bL>Y>c
% 4 4 r^4 * sin(4*theta) sqrt(10) GeN8_i[
% -------------------------------------------------- %k#Q)zWJ
% [#H$@g|CT
% Example 1: JZ&]"12]fR
% 9PEjV$0E2
% % Display the Zernike function Z(n=5,m=1) Nko;I?Fn
% x = -1:0.01:1; b9HE #*d,
% [X,Y] = meshgrid(x,x); Ftj3`Mu
% [theta,r] = cart2pol(X,Y); MId\dFu
% idx = r<=1; 0q"&AxNsP
% z = nan(size(X)); SE{$a3`UzP
% z(idx) = zernfun(5,1,r(idx),theta(idx)); Gk2\B]{
% figure <1]#E@
% pcolor(x,x,z), shading interp \}t(g}7T
% axis square, colorbar '6u;KIG
% title('Zernike function Z_5^1(r,\theta)') _I+#K M
% vBQ|h
% Example 2: |_ HH[s*U
% l"!.aIY"e
% % Display the first 10 Zernike functions d$g-u8
% x = -1:0.01:1; R@t?!`f!+
% [X,Y] = meshgrid(x,x); X:+;d8rCy
% [theta,r] = cart2pol(X,Y); T<~NB5&f
% idx = r<=1; Z6SM7?d
% z = nan(size(X)); K=VYRY
% n = [0 1 1 2 2 2 3 3 3 3]; ^3IO.`|
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; ~oz8B^7i;
% Nplot = [4 10 12 16 18 20 22 24 26 28]; py9(z`}
% y = zernfun(n,m,r(idx),theta(idx)); V[Fzh\2n
% figure('Units','normalized') B|gyr4]
% for k = 1:10 Gr&5 mniu
% z(idx) = y(:,k);
v! uD]}
% subplot(4,7,Nplot(k)) XKLkJZN
% pcolor(x,x,z), shading interp JadXd K=gE
% set(gca,'XTick',[],'YTick',[]) rgdDkWLXC
% axis square #-1 ;
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) T?:Vw laE
% end ~\<Fq \.x
% i}N'WV`!
% See also ZERNPOL, ZERNFUN2. y} AkF2:
]$!-%pNv
% Paul Fricker 11/13/2006 Xa#`VDh
*xA&t)z(i
4]u53`
% Check and prepare the inputs: 5Q,#Co
% ----------------------------- DzYi>
E:*
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ".onev^(
error('zernfun:NMvectors','N and M must be vectors.') c?"#x-<1s
end E$=!l{Ms
ie{9zO<d
if length(n)~=length(m) lhva|
error('zernfun:NMlength','N and M must be the same length.') rR&; 2
end Z\D!'FX
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n = n(:); ?<bByxa
m = m(:); Eb{Zm<TP
if any(mod(n-m,2)) b=horvs/!
error('zernfun:NMmultiplesof2', ... Hly2{hokq
'All N and M must differ by multiples of 2 (including 0).') ='a[(C&Y
end yt}Ve6 m
L,M=ogdb
if any(m>n) pca `nN!
error('zernfun:MlessthanN', ... &eKnLGKD
'Each M must be less than or equal to its corresponding N.') URdCV{@42
end =<MSM\Rb
4*< x0
if any( r>1 | r<0 ) ET;YAa*
error('zernfun:Rlessthan1','All R must be between 0 and 1.') O{SP4|0JV
end b1JXC=*@
nh"nSBRxk
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) \]dx;,T
error('zernfun:RTHvector','R and THETA must be vectors.') Z5/^pyc
end 8+5#FC7
rrbD0UzFA
r = r(:); @(M-ZO!D
theta = theta(:); @?lmho?
length_r = length(r); X Uc(7>k
if length_r~=length(theta) ;NQ9A &$)
error('zernfun:RTHlength', ... uMKO^D
'The number of R- and THETA-values must be equal.') b6Pi:!4
end e=&,jg?K
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% Check normalization: }vzP\
% -------------------- #{KYsDtvx
if nargin==5 && ischar(nflag) jLo(Uf
isnorm = strcmpi(nflag,'norm'); R?Zv
if ~isnorm X)^eaw]Q0
error('zernfun:normalization','Unrecognized normalization flag.') OV/H&fe
end uNSaw['0j
else >>/|Q:
isnorm = false; ?!TFoD2'
end [Z9
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2-=Ov@y2k!
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% rYe z$e^r
% Compute the Zernike Polynomials }#):ZPTs
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% kT)[<`p
3+vbA;R
% Determine the required powers of r: :0N}K}
% ----------------------------------- 1oU/gm$7\q
m_abs = abs(m); xe?!UCUb@
rpowers = []; Rr#Zcs!G
for j = 1:length(n) m#6RJbEz
rpowers = [rpowers m_abs(j):2:n(j)]; "i>?Tg^
end S;@nPzhc
rpowers = unique(rpowers); `R[cM; c2
v2eLH:6
% Pre-compute the values of r raised to the required powers, `|kW%L4
% and compile them in a matrix: E,IeW {6s
% ----------------------------- j=|cx+nb
if rpowers(1)==0 )&/ecx"2Q
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); }@6Tcn1
rpowern = cat(2,rpowern{:}); ]q^6az(Ud
rpowern = [ones(length_r,1) rpowern]; V'b$P2 ?^
else by}C;eN
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 9r2l~zE
rpowern = cat(2,rpowern{:}); JR6r3W
end 709/'#- ^
g{ ()
% Compute the values of the polynomials: hK]mnA[Y
% -------------------------------------- ,bTpD!
y = zeros(length_r,length(n)); _43'W{%
for j = 1:length(n) P^'TI[\L9
s = 0:(n(j)-m_abs(j))/2; 'Fq+\J#%
pows = n(j):-2:m_abs(j); T'6MAxEZUq
for k = length(s):-1:1 iYaS
p = (1-2*mod(s(k),2))* ... P{m(.EC_
prod(2:(n(j)-s(k)))/ ... vJ,r}$H3
prod(2:s(k))/ ... W kP`qD3
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 5aZbNV}-
prod(2:((n(j)+m_abs(j))/2-s(k))); @T.+:U@S
idx = (pows(k)==rpowers); ^(F@ #zN}
y(:,j) = y(:,j) + p*rpowern(:,idx); b-8}TTL>
end nh
XVc((
hs^K9Jt
if isnorm |hl:!j.t
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); E.N@qMn~
end L;GkG! g
end UaCfXTG
% END: Compute the Zernike Polynomials X 0vcBHh
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 6 2:FlW>
?3
S{>+'
% Compute the Zernike functions: 5Z@0XI
% ------------------------------ y5{Vx{V"Q
idx_pos = m>0; AZ.$g?3w
idx_neg = m<0; 2A=q{7s
3N[Rrxe2
z = y; *fCmZ$U:{
if any(idx_pos) c_4K
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); zq(4@S-TU
end r03%+:
if any(idx_neg) |:w)$i& *
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); S=<OS2W7+r
end 1*GL;W~ix*
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% EOF zernfun