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 V\iIvBpWg
function z = zernfun(n,m,r,theta,nflag) ~)!V8
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. xWC\954
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N WU+Jo@]y
% and angular frequency M, evaluated at positions (R,THETA) on the >K_$[qP3
% unit circle. N is a vector of positive integers (including 0), and XPc9z}/(e
% M is a vector with the same number of elements as N. Each element J[<D/WIH
% k of M must be a positive integer, with possible values M(k) = -N(k) O4b-A3:
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, F8|5_214'
% and THETA is a vector of angles. R and THETA must have the same vOvxQS}dBp
% length. The output Z is a matrix with one column for every (N,M) P+*rWJ8gQ
% pair, and one row for every (R,THETA) pair. ]X>QLD0W
% k$UzBxR
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike Xa? 6#
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), "6I-]:K-
% with delta(m,0) the Kronecker delta, is chosen so that the integral !T#8N7J>
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 9sfB+]}h
% and theta=0 to theta=2*pi) is unity. For the non-normalized '-nuH;r
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. giPhW>
% 4'}_qAT
% The Zernike functions are an orthogonal basis on the unit circle. UtW"U0A
% They are used in disciplines such as astronomy, optics, and Z3X&<Y5
% optometry to describe functions on a circular domain. u>3&.t@hU1
% NE=#5?6%g7
% The following table lists the first 15 Zernike functions. fwnYzd3
% M0;t%*1
% n m Zernike function Normalization Y1U"HqNl*
% -------------------------------------------------- V<~_OF
% 0 0 1 1 HdY3DdC%q
% 1 1 r * cos(theta) 2 bG?WB,1
% 1 -1 r * sin(theta) 2 OIXAjU*N
% 2 -2 r^2 * cos(2*theta) sqrt(6) ~kSnXJv
% 2 0 (2*r^2 - 1) sqrt(3) QigoRB!z#9
% 2 2 r^2 * sin(2*theta) sqrt(6) '1kj:Np
% 3 -3 r^3 * cos(3*theta) sqrt(8) XoH[MJC
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 0w'y#U)&8
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) {d?4;Kd
% 3 3 r^3 * sin(3*theta) sqrt(8) 6'No4[F
4n
% 4 -4 r^4 * cos(4*theta) sqrt(10) U!;aM*67
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 7=QC+XSO
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) RIVL 0Ig
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) f@F^W YQm
% 4 4 r^4 * sin(4*theta) sqrt(10) }]39
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% -------------------------------------------------- Vlp*'2VO
% [o[v"e\w
% Example 1: 7n\j"0z
% 0ez
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% % Display the Zernike function Z(n=5,m=1) ?,i#B'Z^
% x = -1:0.01:1; '^-4{Y^2E
% [X,Y] = meshgrid(x,x); x^='pEt{
% [theta,r] = cart2pol(X,Y); ~*cY& 9
% idx = r<=1; yqVaA 'w5
% z = nan(size(X)); Zjp5\+hHV
% z(idx) = zernfun(5,1,r(idx),theta(idx)); P/gb+V=g!
% figure @]ptY*
% pcolor(x,x,z), shading interp d4/`:?w
% axis square, colorbar gGiV1jN_
% title('Zernike function Z_5^1(r,\theta)') v_@#hf3
% YP\4XI
% Example 2: xXY)KI
N[
% xo)?XFM2
% % Display the first 10 Zernike functions S]K6qY
% x = -1:0.01:1; GdfKxSO
% [X,Y] = meshgrid(x,x); YnO1Lf@
% [theta,r] = cart2pol(X,Y); &6|^~(P?
% idx = r<=1; h@>rjeY@
% z = nan(size(X)); 9i2vWSga
% n = [0 1 1 2 2 2 3 3 3 3]; a9@l8{)RX
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; sNk>0 X[
% Nplot = [4 10 12 16 18 20 22 24 26 28]; Y(I*%=:$
% y = zernfun(n,m,r(idx),theta(idx)); H:{(CY?t
% figure('Units','normalized') : DX/r
% for k = 1:10 vu.S>2Wv
% z(idx) = y(:,k); ]N(zom_0d
% subplot(4,7,Nplot(k)) ">D(+ xr!)
% pcolor(x,x,z), shading interp %dk$K!5D0
% set(gca,'XTick',[],'YTick',[]) *l?%
o{
% axis square <>*''^
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) gH{\y5%rO
% end WfjUJw5x"s
% sm&rR=b
% See also ZERNPOL, ZERNFUN2. C O%O<_C
"w|k\1D
% Paul Fricker 11/13/2006 BE2\? q-
a+a%}76N
z`r4edk3
% Check and prepare the inputs: GLKN<2|2@y
% ----------------------------- (27F
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) aXK%m
error('zernfun:NMvectors','N and M must be vectors.') ,tR'0&=
end O*n%2Mam
Y`O}]*{>8R
if length(n)~=length(m) A_q3p\b
error('zernfun:NMlength','N and M must be the same length.') %k;FxUKi
end v!'@NW_
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n = n(:); z#E,96R
m = m(:); )RCqsFjK
if any(mod(n-m,2)) h9n<ped`A;
error('zernfun:NMmultiplesof2', ... \=G
Xe.}4d
'All N and M must differ by multiples of 2 (including 0).') )J6b:W
end eg~^wi
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if any(m>n) JK8@J9(#
error('zernfun:MlessthanN', ... MVL }[ J
'Each M must be less than or equal to its corresponding N.') xo_k"'f+
end "vRqtEBO@
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if any( r>1 | r<0 ) BiLreZ~"
error('zernfun:Rlessthan1','All R must be between 0 and 1.') .idl@%
end 4a \+o]
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Cg\)BHv~
error('zernfun:RTHvector','R and THETA must be vectors.') xY'YbHFz
end
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r = r(:); vAhO!5]>\
theta = theta(:); oJu4vGy0
length_r = length(r); %C][E^9
if length_r~=length(theta) x
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error('zernfun:RTHlength', ... wkpVX*DfRE
'The number of R- and THETA-values must be equal.') U)%u`C0
end ~u`! Gi
!<PTsk F
% Check normalization:
qmyZbo|8&
% -------------------- &E'>+6
if nargin==5 && ischar(nflag) `IRT w"
isnorm = strcmpi(nflag,'norm'); 9* Twx&
if ~isnorm 6)<o O(
error('zernfun:normalization','Unrecognized normalization flag.') o%>nu
end >)Z2bCe
else WIlS^?5I<
isnorm = false; ]G&\L~P
end 44{:UhJkx
vlyNQ7"%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% cCKda3v!O
% Compute the Zernike Polynomials <4HuV.K
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% G8-d%O p
daJ-H
% Determine the required powers of r: m/B9)JzY
% ----------------------------------- ';!UJWYl
m_abs = abs(m); J
2~B<=V
rpowers = []; I}0-
for j = 1:length(n) p
8Hv7*
rpowers = [rpowers m_abs(j):2:n(j)]; AG%es0D[H
end |-Klh
rpowers = unique(rpowers); )4~XZt1r
s/^=WV
% Pre-compute the values of r raised to the required powers, *<5lx[:4/x
% and compile them in a matrix: d}CMX$1
% ----------------------------- XxQ2g&USk
if rpowers(1)==0 'N/%SRk
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); n,%^R
rpowern = cat(2,rpowern{:}); 8(K~QvE~
rpowern = [ones(length_r,1) rpowern]; >Nqkz?67
else =n?@My?;
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); #!j&L6
rpowern = cat(2,rpowern{:}); 5d;K.O
end [beuDZA
g+{MvSj$
% Compute the values of the polynomials: r 24]2A
% -------------------------------------- ;b2>y>?[
y = zeros(length_r,length(n)); UM^hF%
for j = 1:length(n) l%w|f`B:
s = 0:(n(j)-m_abs(j))/2; ~%q e,
pows = n(j):-2:m_abs(j); u-cC}DP
for k = length(s):-1:1 kQcQi}e
p = (1-2*mod(s(k),2))* ... 2a}_|#*
prod(2:(n(j)-s(k)))/ ... .SFwjriZ
prod(2:s(k))/ ... 8u23@?
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... JsdEA
prod(2:((n(j)+m_abs(j))/2-s(k))); imuHSxcaV
idx = (pows(k)==rpowers); !LESRh?
y(:,j) = y(:,j) + p*rpowern(:,idx); :udZfA\sW
end _+7f+eB
@}}1xP4Sr
if isnorm y!Eh /KD
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 9$t@Gmn
end }Q*ec/^{f
end !2,.C+,
% END: Compute the Zernike Polynomials <m\TZQBD
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% &$
9bC't6
a#@opUn-
% Compute the Zernike functions: *tqeq y-X
% ------------------------------ {GY$J<5=
idx_pos = m>0; P|4a}SWU
idx_neg = m<0; Cq'r
'cBZ
_z<q9:
z = y; A-5%_M3\G
if any(idx_pos) HxAa,+k
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ijT^gsLL
end }\*|b@)]
if any(idx_neg) 8A=(,)`}9
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); f5eX%FR
end oWT0WS
!h;VdCCi#
% EOF zernfun