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 J^
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function z = zernfun(n,m,r,theta,nflag) ;Gd~YGW^#
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. RUo9eQIPD
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N h-QLV[^
% and angular frequency M, evaluated at positions (R,THETA) on the OZ(dpV9.S
% unit circle. N is a vector of positive integers (including 0), and %!|O.xxRR
% M is a vector with the same number of elements as N. Each element +ts0^;QO2{
% k of M must be a positive integer, with possible values M(k) = -N(k) |.U)ll(c
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, s\3q!A?S3
% and THETA is a vector of angles. R and THETA must have the same w/m:{c Hk
% length. The output Z is a matrix with one column for every (N,M) (.23rVvnT@
% pair, and one row for every (R,THETA) pair. =.Tv)/ea
% n7! H:{L
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike XKU=oI0\j
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), Nneo{j
% with delta(m,0) the Kronecker delta, is chosen so that the integral A)NkT`<)
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, {C3Y7<
% and theta=0 to theta=2*pi) is unity. For the non-normalized T@YGB]*Y
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. C+N k"l9
% m_7
nz!h
% The Zernike functions are an orthogonal basis on the unit circle. +%0z`E\?M#
% They are used in disciplines such as astronomy, optics, and ]?LB?:6
% optometry to describe functions on a circular domain. r'4:)~]s
% 8e2?tmWM
% The following table lists the first 15 Zernike functions. A :e;k{J
% j*R,m1e8
% n m Zernike function Normalization A9:NKY{z
% -------------------------------------------------- D E/:['
% 0 0 1 1 CIC[1,
% 1 1 r * cos(theta) 2 .~D>5 JnEk
% 1 -1 r * sin(theta) 2 s0"e'
% 2 -2 r^2 * cos(2*theta) sqrt(6) )"<8K}%!
% 2 0 (2*r^2 - 1) sqrt(3) B80aw>M
% 2 2 r^2 * sin(2*theta) sqrt(6) >U!*y4
% 3 -3 r^3 * cos(3*theta) sqrt(8) cP>o+-)
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) md Gwh7/3
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) &^.57]
% 3 3 r^3 * sin(3*theta) sqrt(8) nk=$B(h
% 4 -4 r^4 * cos(4*theta) sqrt(10) N{Qxq>6 G
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) U5r}6D!)
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) K_&MoyJJ9f
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 9Kv|>#zff
% 4 4 r^4 * sin(4*theta) sqrt(10) <V&5P3)d9
% -------------------------------------------------- p( LZ)7/
% iCQ>@P]nE
% Example 1: L^`}J7r
% ,xi({{L*
% % Display the Zernike function Z(n=5,m=1) kLP0{A
% x = -1:0.01:1; b/("Y.r=
% [X,Y] = meshgrid(x,x); dJk9@u
% [theta,r] = cart2pol(X,Y); 6,b"
% idx = r<=1; dA~
3>f*b_
% z = nan(size(X)); 2I'~2o
% z(idx) = zernfun(5,1,r(idx),theta(idx)); YwDt.6(+,
% figure MgMD\
% pcolor(x,x,z), shading interp !36]ud&
% axis square, colorbar `ldz`yu6++
% title('Zernike function Z_5^1(r,\theta)') V"KS[>>f
% 8Cx^0
% Example 2: /n,a?Ft^N)
% j;~%lg=)
% % Display the first 10 Zernike functions b1?xeG#
% x = -1:0.01:1; ?&+9WJ<M
% [X,Y] = meshgrid(x,x); A;X=bj _&a
% [theta,r] = cart2pol(X,Y); ['qnn|
% idx = r<=1; :l u5Uu~
% z = nan(size(X)); TLa]O1=Bf.
% n = [0 1 1 2 2 2 3 3 3 3]; 0Q9T3X
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; -G |a*^
% Nplot = [4 10 12 16 18 20 22 24 26 28]; _GYMPq\%L#
% y = zernfun(n,m,r(idx),theta(idx)); _=XX~^I,
% figure('Units','normalized') QO;4}rq
% for k = 1:10 `)$_YZq|SR
% z(idx) = y(:,k); b7:0#l$
% subplot(4,7,Nplot(k)) N:5[,O<m_
% pcolor(x,x,z), shading interp 6sfwlT
% set(gca,'XTick',[],'YTick',[]) }Fb!?['G5
% axis square d l]#
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) n~IVNB*
% end ed!>)Cb
% [\z/Lbn
,.
% See also ZERNPOL, ZERNFUN2. e /K#>,
6QQfQ,
% Paul Fricker 11/13/2006 2'0K WYM
J>vMo@
*?p|F&J
% Check and prepare the inputs: 4Ft1@
% ----------------------------- bCv {1]RC2
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ?)4?V\$
error('zernfun:NMvectors','N and M must be vectors.') oA-:zz>wL
end !0VfbY9C
]2SI!Ai7
if length(n)~=length(m) S::=85[>z
error('zernfun:NMlength','N and M must be the same length.') KFRw67^
end g=@_Z"
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n = n(:); %|W.^q
m = m(:); a6xj\w
if any(mod(n-m,2)) uq3{hB#
error('zernfun:NMmultiplesof2', ... 7*o*6,/
'All N and M must differ by multiples of 2 (including 0).') &]6)LFm
end {}~: &.D
$^/0<i$
if any(m>n) 6aft$A}XnD
error('zernfun:MlessthanN', ... m!n/U-^
'Each M must be less than or equal to its corresponding N.') JAc_kl{4O
end El_Qk[X|A
c7uG9
if any( r>1 | r<0 ) QbFHfA2Ij
error('zernfun:Rlessthan1','All R must be between 0 and 1.') IIFMYl gF
end j V3)2C}
-Yi,_#3{
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) }=](p-] 5
error('zernfun:RTHvector','R and THETA must be vectors.')
g\fhp{gWB
end J97R0
Yf_6PGNzX
r = r(:); ,U,By~s
theta = theta(:); :fcM:w&
length_r = length(r); b,H[I!. %
if length_r~=length(theta) %V!iQzL1
error('zernfun:RTHlength', ... x+5k
<Xi}
'The number of R- and THETA-values must be equal.') gO?44^hMe
end {Bvj"mL]j
}Rvm &?~O
% Check normalization: H;ZHqcUX
% -------------------- W[bmzvJ_X
if nargin==5 && ischar(nflag) +>^7vq-\'
isnorm = strcmpi(nflag,'norm'); |iYg >
if ~isnorm % ~]xuP[
error('zernfun:normalization','Unrecognized normalization flag.') BcWcdr+}9
end F'PQqb {
else jjs&`Fy,
isnorm = false; ?WI3/>:<
end ;#+0L$<t
<~emx'F|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ZM#=`k9
% Compute the Zernike Polynomials FwAKP>6 *
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \kIMDg3}
t
!`Jse>
% Determine the required powers of r: CBT>"sYE1
% ----------------------------------- ^ZeJ[t&!#
m_abs = abs(m); km5~Gc}
rpowers = []; I+
l% Sn#\
for j = 1:length(n) =s97Z-
rpowers = [rpowers m_abs(j):2:n(j)]; 7Ey#u4Q
end t G.(flW,
rpowers = unique(rpowers);
,<,:8B
V3N0Og3
% Pre-compute the values of r raised to the required powers, `iKj
% and compile them in a matrix: ?9MVM~$
% ----------------------------- .lG5=Th!
if rpowers(1)==0 OKOu`Hz@
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); yqOuX>m 1c
rpowern = cat(2,rpowern{:}); j=+"Qz/hr_
rpowern = [ones(length_r,1) rpowern]; \uOdALZ
else Tpp &
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); `b5 @}',
rpowern = cat(2,rpowern{:}); A1Y7;-D
end CG]Sj*SA~
{i~8 :
% Compute the values of the polynomials: hjx)D
% -------------------------------------- #C*8X+._y
y = zeros(length_r,length(n)); /
jTT5
for j = 1:length(n) 4 {GU6v)f
s = 0:(n(j)-m_abs(j))/2; ygZ #y L
pows = n(j):-2:m_abs(j); O;Y:uHf
for k = length(s):-1:1 KLQTKMNv
p = (1-2*mod(s(k),2))* ... bF}V4"d,B3
prod(2:(n(j)-s(k)))/ ... q~K(]Ya/
prod(2:s(k))/ ... 9 t
n!t
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... i7[uLdQ
prod(2:((n(j)+m_abs(j))/2-s(k))); ]<uQ.~
idx = (pows(k)==rpowers); '(&%O8Yi
y(:,j) = y(:,j) + p*rpowern(:,idx); 2
+5e0/_V
end [&S}dQ"
U!w1AY|
if isnorm C.MoKa3
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ^}yg%+
end Ei>m0
~<\
end H(^bC5'
% END: Compute the Zernike Polynomials \[2lvft!
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% wmr-}Y!9u%
'Yy&G\S
% Compute the Zernike functions: @+,pN6}g
% ------------------------------ SU _SU".
idx_pos = m>0; w2(guL($
idx_neg = m<0; ^,Ydr~|T
s Wjy6;
z = y; cF T 9Lnz
if any(idx_pos) $WQq?1.9
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); !hxIlVd{
end %!Q`e79g8
if any(idx_neg) <msxHw
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); XkKC!
end g\oSG)
+0z 7KO%^^
% EOF zernfun