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 eB2a1<S&@
function z = zernfun(n,m,r,theta,nflag) q'1
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%ZERNFUN Zernike functions of order N and frequency M on the unit circle. @T@lHc
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N i!u:]14>
% and angular frequency M, evaluated at positions (R,THETA) on the >1S39n5z.
% unit circle. N is a vector of positive integers (including 0), and }>$3B5}
% M is a vector with the same number of elements as N. Each element X-k$6}D
% k of M must be a positive integer, with possible values M(k) = -N(k) 'gv~M_
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, gEISnMH
% and THETA is a vector of angles. R and THETA must have the same bSgdVP-
% length. The output Z is a matrix with one column for every (N,M) ![Ll$Lr
% pair, and one row for every (R,THETA) pair. 'Hv=\p4$1
% AXT(D@sI=
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike fb|%)A=
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), L <W2a(
% with delta(m,0) the Kronecker delta, is chosen so that the integral ,0\Pr
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, V_"UiN"o
% and theta=0 to theta=2*pi) is unity. For the non-normalized hZwJ@ Vm#
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. aaRc?b'/
% Scd_tw.]|
% The Zernike functions are an orthogonal basis on the unit circle. MN<uIqG
% They are used in disciplines such as astronomy, optics, and *iiyU}x
% optometry to describe functions on a circular domain. K.r
"KxCm|
% v\3$$T)
% The following table lists the first 15 Zernike functions.
x=YV*
% \#7@"~<
% n m Zernike function Normalization G3${\'<
% -------------------------------------------------- [oDu3Qn
% 0 0 1 1 OKV/=]GS
% 1 1 r * cos(theta) 2 /vNHb_-
% 1 -1 r * sin(theta) 2 8Os: SC@Q
% 2 -2 r^2 * cos(2*theta) sqrt(6) Gy6PS{yY6t
% 2 0 (2*r^2 - 1) sqrt(3) t
.-%@,s
% 2 2 r^2 * sin(2*theta) sqrt(6) 5fY7[{2
% 3 -3 r^3 * cos(3*theta) sqrt(8) :R1F\FT*
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) yt[*4gF4
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) cH6<'W{*
% 3 3 r^3 * sin(3*theta) sqrt(8) 8fWk C<f}
% 4 -4 r^4 * cos(4*theta) sqrt(10) > JP}OS
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 1+v!)Y>Z&
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) D]'/5]~z<
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) U#g,XJ
% 4 4 r^4 * sin(4*theta) sqrt(10) Jk}Dj0o
% -------------------------------------------------- |3P dlIbO
% &`I 7aP|
% Example 1: ]=]fIKd
% U0@Qc}y
% % Display the Zernike function Z(n=5,m=1) R
"qt}4m
% x = -1:0.01:1; d^qTY?k.
% [X,Y] = meshgrid(x,x); Ft<B[bQ
% [theta,r] = cart2pol(X,Y); S+7u,%n/
% idx = r<=1; \\Te\l|L
% z = nan(size(X)); w)Z-, J
% z(idx) = zernfun(5,1,r(idx),theta(idx)); "'*Qq@!3?
% figure bsv!z\}
% pcolor(x,x,z), shading interp 71G\b|5
% axis square, colorbar 0mR^%+~
% title('Zernike function Z_5^1(r,\theta)') 2bAH)=
% JmF:8Q3H
% Example 2: 4,.[B7irR
% bj,cU)t0
% % Display the first 10 Zernike functions RC~ C}
% x = -1:0.01:1; 6Sz|3ms
% [X,Y] = meshgrid(x,x); qZYh^\
% [theta,r] = cart2pol(X,Y); L XHDX
% idx = r<=1; 8;$zD]{D1
% z = nan(size(X)); 1 Szv4
% n = [0 1 1 2 2 2 3 3 3 3]; @ n^2UJ
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; :vJ1Fo!
% Nplot = [4 10 12 16 18 20 22 24 26 28]; ZZrvl4h
% y = zernfun(n,m,r(idx),theta(idx)); Q?V'3ZZF!
% figure('Units','normalized') F*p@hl
% for k = 1:10 UTVqoCHA
% z(idx) = y(:,k); Kb~i9x&
% subplot(4,7,Nplot(k)) UId?a}J
% pcolor(x,x,z), shading interp Ma^}7D
/
% set(gca,'XTick',[],'YTick',[]) Jvr`9<`
% axis square TT^L)d
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) &0RKNpwg
% end Vc!'=&*
% 8fA8@O}
% See also ZERNPOL, ZERNFUN2. ?/}-&A"
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% Paul Fricker 11/13/2006 ?X@uR5?{
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% Check and prepare the inputs: 4P?R "Lk
% ----------------------------- <lP5}F87
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) l0lvca=;
error('zernfun:NMvectors','N and M must be vectors.') ;8g[y"I
end |Ge!;v
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if length(n)~=length(m) -u~AY#*
error('zernfun:NMlength','N and M must be the same length.') BHpj_LB-P
end &
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n = n(:); ;^ff35EE8
m = m(:); rO]2we/B,4
if any(mod(n-m,2)) qPn!.m$/
error('zernfun:NMmultiplesof2', ... :czUOZ_
'All N and M must differ by multiples of 2 (including 0).') Bpp(5
end /mwsF]Y
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if any(m>n) [<.dOe7|
error('zernfun:MlessthanN', ... $|VD+[jSV
'Each M must be less than or equal to its corresponding N.') jH[{V[<#X
end ;CDa*(e
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if any( r>1 | r<0 ) GpXU&A'r
error('zernfun:Rlessthan1','All R must be between 0 and 1.') ZJV;&[$[
end q OV$4[r
y$+_9VzYB
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Dv}VmC""
error('zernfun:RTHvector','R and THETA must be vectors.') tS[%C)
end z'}z4^35,
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r = r(:); V3$zlzSm,
theta = theta(:); ~vKDB$2
length_r = length(r); |`O210B@
if length_r~=length(theta) eKe[]/}e9
error('zernfun:RTHlength', ... gW^0A)5
'The number of R- and THETA-values must be equal.') v*^'|QyM7
end $.O(K4S
OQ+kOE&
% Check normalization: oT- Y
% -------------------- f<vZ4 IU
if nargin==5 && ischar(nflag) +oiuulA
isnorm = strcmpi(nflag,'norm'); K
Vnz{cx`
if ~isnorm 6OZn7:)Y
error('zernfun:normalization','Unrecognized normalization flag.') 4R&pb1eF
end mV|Z5 =f
else M<ba+Qn$
isnorm = false; Ur(< ]
end 7@Xi*Azd
@+Anp4%;Y
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% i}~U/.P
% Compute the Zernike Polynomials ><{Lh@{
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% c. uD%
Z@bKYfGM
% Determine the required powers of r: K%YR; )5A
% ----------------------------------- u XVs<im
m_abs = abs(m); y|(?>\jBl
rpowers = []; %)=c#H1
for j = 1:length(n) R2y~+tko?
rpowers = [rpowers m_abs(j):2:n(j)]; O7yIFqI=/
end yK w.69.
rpowers = unique(rpowers); ye`-U?7.
Z8o8>C\d9/
% Pre-compute the values of r raised to the required powers, B1o*phM
g
% and compile them in a matrix: G],W{<Pe
% ----------------------------- U?j[
8z
if rpowers(1)==0 )@6iQ
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); AizLzR$OG
rpowern = cat(2,rpowern{:}); [ N0"mE<
rpowern = [ones(length_r,1) rpowern]; dQI6.$?
else zRgl`zREr
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); du&9mOrr
rpowern = cat(2,rpowern{:}); 3e1^r_YI
end GE}>{x=^x
@JpkG%eK
% Compute the values of the polynomials: *[b22a4H(
% -------------------------------------- b1-'q^M
y = zeros(length_r,length(n)); nx@h
for j = 1:length(n) ?eri6D,86w
s = 0:(n(j)-m_abs(j))/2; &