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 e"^* ~'mJ
function z = zernfun(n,m,r,theta,nflag) $-9m8}U(Y
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 8Z%C7
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% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N H)1< ;{:
% and angular frequency M, evaluated at positions (R,THETA) on the g9OO#C>
% unit circle. N is a vector of positive integers (including 0), and ;3NA,JA#Y
% M is a vector with the same number of elements as N. Each element #LEK?]y
% k of M must be a positive integer, with possible values M(k) = -N(k) A<.`HCv2
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, rJK3;d? E
% and THETA is a vector of angles. R and THETA must have the same weC$\st:D
% length. The output Z is a matrix with one column for every (N,M) :M(%sv</
% pair, and one row for every (R,THETA) pair. 31-%IkX+k
% T%K"^4k
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike uZ*;%y nQ
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), @%@uZqQ4
% with delta(m,0) the Kronecker delta, is chosen so that the integral #kT3Sx
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, +avu&2B
% and theta=0 to theta=2*pi) is unity. For the non-normalized /m%Y.:g
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 'l2'%@E>
% dC;@ Fn
% The Zernike functions are an orthogonal basis on the unit circle. W@jBX{k
% They are used in disciplines such as astronomy, optics, and z>+@pj
% optometry to describe functions on a circular domain. 01q5BQ7u
% t>><|~wp
% The following table lists the first 15 Zernike functions. ZZp6@@zyq'
% :a(er'A
% n m Zernike function Normalization 'cJHOd
% -------------------------------------------------- 1t/#ZT!X/
% 0 0 1 1 mjG-A8y
% 1 1 r * cos(theta) 2 >lxhXYp
% 1 -1 r * sin(theta) 2 \gy39xoW(
% 2 -2 r^2 * cos(2*theta) sqrt(6) QcG4~DEX4
% 2 0 (2*r^2 - 1) sqrt(3) he;;p ="!*
% 2 2 r^2 * sin(2*theta) sqrt(6) 7a 4G:
% 3 -3 r^3 * cos(3*theta) sqrt(8) "<x%kD
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) KOVGwEj
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) W#E-vi+l
% 3 3 r^3 * sin(3*theta) sqrt(8) AjB-&Z
% 4 -4 r^4 * cos(4*theta) sqrt(10) !Z2?dhS
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) sF}T9Ue
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 8@ck" LUzD
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) !T02@e/
% 4 4 r^4 * sin(4*theta) sqrt(10) Au08k}h<G
% -------------------------------------------------- !},_,J~(|
% m[,!
orq
% Example 1: U=MFNp+
% .<j\"X(
% % Display the Zernike function Z(n=5,m=1) v)>R)bzqe
% x = -1:0.01:1; B$"CoLC7+
% [X,Y] = meshgrid(x,x); j-@3jFu
% [theta,r] = cart2pol(X,Y); |13UJ
vR
% idx = r<=1; ~itrM3^"w
% z = nan(size(X)); u{maE ,
% z(idx) = zernfun(5,1,r(idx),theta(idx)); ]Ec\!,54u
% figure 6VpT*,2d~
% pcolor(x,x,z), shading interp 0q(}n v
% axis square, colorbar I|]~f[xI
% title('Zernike function Z_5^1(r,\theta)') 9mfqr$3
% >.N?y@
% Example 2: 4JSf t
t
% nE# p
Ry]
% % Display the first 10 Zernike functions JSCe86a7<E
% x = -1:0.01:1; >AI65g
% [X,Y] = meshgrid(x,x); oF[l<OY4
% [theta,r] = cart2pol(X,Y); uH S)
% idx = r<=1; ]P;Ng=a
% z = nan(size(X)); @w|'ip5@
% n = [0 1 1 2 2 2 3 3 3 3]; 6Pc3 ;X~
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; Q[J%
% Nplot = [4 10 12 16 18 20 22 24 26 28]; E!w%oTx{OR
% y = zernfun(n,m,r(idx),theta(idx)); ;:NW
% figure('Units','normalized') ;LM`B^Q]s
% for k = 1:10 v:kTZB
% z(idx) = y(:,k); qV2aa9p+
% subplot(4,7,Nplot(k)) /iFtW#K+
% pcolor(x,x,z), shading interp dUiv+K)ccQ
% set(gca,'XTick',[],'YTick',[]) 'N#,,d/G
% axis square
;L(2Ffk8
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ^3?]S{1/#
% end ^Rriu $\
% ]Z%9l(
% See also ZERNPOL, ZERNFUN2. U{Xg#UN
qELy'\
% Paul Fricker 11/13/2006 BMMWP
R:8\z0"L*
]Gm"U!h*
% Check and prepare the inputs: H.#<&5f
% ----------------------------- eCHT)35u
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) g9~>m JR
error('zernfun:NMvectors','N and M must be vectors.') (F9U`1~4
end w3oh8NRs_
d*;wHA,}F
if length(n)~=length(m) R+Q..9P
error('zernfun:NMlength','N and M must be the same length.') <RQ\nU
end Fy_D[g
uh#"4-v
n = n(:); SJ4[n.tPI
m = m(:); &0A^_Z .nA
if any(mod(n-m,2)) w+c%Y\:
error('zernfun:NMmultiplesof2', ... _qwKFC
'All N and M must differ by multiples of 2 (including 0).') n@IpO
i$Q
end _)AX/%^%
P:,@2el
if any(m>n) ^5n"L29V
error('zernfun:MlessthanN', ... @ov*Fh
'Each M must be less than or equal to its corresponding N.') ^i>Tm9vM
end t;g=@o9YA
? I7}4i7
if any( r>1 | r<0 ) VnqgN
error('zernfun:Rlessthan1','All R must be between 0 and 1.') imGg3'
end h8# 14?
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) rt- ^?2c?
error('zernfun:RTHvector','R and THETA must be vectors.') );-?~
end R0|dKKzS
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r = r(:); J${'?!N
theta = theta(:); zF'LbQz0[
length_r = length(r); t2V|moG
if length_r~=length(theta) w<}kY|A"=-
error('zernfun:RTHlength', ... VHwAO:+-
'The number of R- and THETA-values must be equal.') T \Zf`.mt
end n."vCP}O+
;Ih:$"$!
% Check normalization: Y|%s =0M
% -------------------- c;X8:Z=ja
if nargin==5 && ischar(nflag) J@$h'YUF
isnorm = strcmpi(nflag,'norm'); /Z':wu\
if ~isnorm "9Q @&C
error('zernfun:normalization','Unrecognized normalization flag.') 2/[J<c\G
end hsYS<]
else >iE/t$%1
isnorm = false; ]mO$Tg&s~
end ,mkXUW
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -B+Pl*
% Compute the Zernike Polynomials \53(D7+
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Tvdg:[V<
`XT8}9z!
% Determine the required powers of r: Y8$Y]2
% ----------------------------------- 6MsVV_/
m_abs = abs(m); u`K)dH,
rpowers = []; W|C>X=zTi
for j = 1:length(n) J3
Y-d7=|
rpowers = [rpowers m_abs(j):2:n(j)]; &A}@@d
end $q}zW%
rpowers = unique(rpowers); +OEheG8
x?5D>M/Y
% Pre-compute the values of r raised to the required powers, G3Z>,"w;=
% and compile them in a matrix: .X2fu/}
% ----------------------------- >"Tivc5
if rpowers(1)==0 _SVIY@K|/
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Vp"=8p#k
rpowern = cat(2,rpowern{:}); 3
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rpowern = [ones(length_r,1) rpowern]; ,q[aV 6kO
else [9}D+k F
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 3mmp5 d
rpowern = cat(2,rpowern{:}); idG}p+(;
end G&2UXr3
$-x@P9im
% Compute the values of the polynomials: NFYo@kX>
G
% -------------------------------------- {DP%=4
y = zeros(length_r,length(n)); .k_>
BD];
for j = 1:length(n) _BC%98:WP
s = 0:(n(j)-m_abs(j))/2; `B1r+uTP~
pows = n(j):-2:m_abs(j); B<V8:vOam
for k = length(s):-1:1 \:7G1_o
p = (1-2*mod(s(k),2))* ... 7IEG%FY
T
prod(2:(n(j)-s(k)))/ ... IF>dsAAI<
prod(2:s(k))/ ... Nj p?/r
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ,RA;X
prod(2:((n(j)+m_abs(j))/2-s(k))); (SH<]@s
idx = (pows(k)==rpowers); u;@~P
y(:,j) = y(:,j) + p*rpowern(:,idx); Ah_,5Z@&R
end H!H&<71-
pUp&eH
if isnorm 2cnyq$4k
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); bi:TX<K+
end obRYU|T
end 9Q*T'+V
% END: Compute the Zernike Polynomials +mgm39
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ) (4.7>
&"_5?7_N
% Compute the Zernike functions: \vKKq/f
% ------------------------------ ~4T:v_Q7g
idx_pos = m>0; CC,f*I
idx_neg = m<0; f+WN=-F\
r2h{#2
z = y; vV( ?A
if any(idx_pos) 2oO&8:`tv
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ktEdbALK
end t_Q\uo}
if any(idx_neg) !e<D2><^
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); REK(^1
h
end &/\Q 6$a
Kw/7X[|'G
% EOF zernfun