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 AN:RY/ %Wo
function z = zernfun(n,m,r,theta,nflag) ]rX?n
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. [Yahxw}
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N g ]PLW3
% and angular frequency M, evaluated at positions (R,THETA) on the $M3A+6["H
% unit circle. N is a vector of positive integers (including 0), and w]5f3CIm
% M is a vector with the same number of elements as N. Each element 39a]B`y
% k of M must be a positive integer, with possible values M(k) = -N(k) T~ q'y~9o
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, R82Zr@_
% and THETA is a vector of angles. R and THETA must have the same :+dWJNY:
% length. The output Z is a matrix with one column for every (N,M) 3PR7g
% pair, and one row for every (R,THETA) pair. w2C!>fJ]1
% z1@sEfk>
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike PuoJw~^h
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ZX5A%`<M
% with delta(m,0) the Kronecker delta, is chosen so that the integral }AH|~3|D
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, (!&O4C5
% and theta=0 to theta=2*pi) is unity. For the non-normalized a ~iEps
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. [sO<6?LY
% <+1w'-
% The Zernike functions are an orthogonal basis on the unit circle. d(B;vL@R2V
% They are used in disciplines such as astronomy, optics, and *,XJN_DKj
% optometry to describe functions on a circular domain. H1ui#5n2
% O@(.ei*HJ!
% The following table lists the first 15 Zernike functions. ~\s &]L
% #uw*8&%0
% n m Zernike function Normalization HgBEV
% -------------------------------------------------- )yH#*~X_
% 0 0 1 1 Y(!)G!CMc
% 1 1 r * cos(theta) 2 E_I6
% 1 -1 r * sin(theta) 2 \iLd6Qo_aq
% 2 -2 r^2 * cos(2*theta) sqrt(6) }lvP|6Y: y
% 2 0 (2*r^2 - 1) sqrt(3) E|A_|FS&%
% 2 2 r^2 * sin(2*theta) sqrt(6) "BNmpP
% 3 -3 r^3 * cos(3*theta) sqrt(8) :IKp7BS
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) {ZYCnS&?CL
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) h|>n3-k|p
% 3 3 r^3 * sin(3*theta) sqrt(8) `3s-%>
% 4 -4 r^4 * cos(4*theta) sqrt(10) Yiw^@T\H`
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) *l8vCa9Y
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 5lA 8e
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ] j?Fk$C
% 4 4 r^4 * sin(4*theta) sqrt(10) 3&d+U)E
% -------------------------------------------------- vlKKPS
% T-cVM>u\D
% Example 1: @3=<wz<
% }Mlz\'{
% % Display the Zernike function Z(n=5,m=1) gwjv&.T6^
% x = -1:0.01:1; G,*
uj0g
% [X,Y] = meshgrid(x,x); E0x$;CG!
% [theta,r] = cart2pol(X,Y); %_LHD|<
% idx = r<=1; J3JRWy@?P
% z = nan(size(X)); ]vyF&`phb
% z(idx) = zernfun(5,1,r(idx),theta(idx)); Oua/NF)
% figure {7s zo`U2
% pcolor(x,x,z), shading interp WW/m
/+
% axis square, colorbar O6 J<Lqgh
% title('Zernike function Z_5^1(r,\theta)')
NOr*+N\
% IHMyP~{
% Example 2: BTQC1;;N
% 1{glRY'
% % Display the first 10 Zernike functions yBjWPx?
% x = -1:0.01:1; !8M'ms>s=
% [X,Y] = meshgrid(x,x); s-DL=MD
% [theta,r] = cart2pol(X,Y); vPq\reKe
% idx = r<=1; t/BiZo|zl
% z = nan(size(X)); G7{:d
% n = [0 1 1 2 2 2 3 3 3 3]; Jg6[/7*m
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; ~PvzUT-^
% Nplot = [4 10 12 16 18 20 22 24 26 28]; R20GjWy=
% y = zernfun(n,m,r(idx),theta(idx)); kqB00
;
% figure('Units','normalized') IY6S\Gn
% for k = 1:10 /[T8/7;_l
% z(idx) = y(:,k); 9r*T3=u.S
% subplot(4,7,Nplot(k)) ]/naH#8G
% pcolor(x,x,z), shading interp No|{rYYKK
% set(gca,'XTick',[],'YTick',[]) } dlNMW
% axis square a2FIFWvW
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 74OM tLL$
% end O|m-k0n
% Nr+1N83S}
% See also ZERNPOL, ZERNFUN2. @Ec9Do>
LJ#P- `!{&
% Paul Fricker 11/13/2006 fJV VW
Q1B!W
(R,n`x2^
% Check and prepare the inputs: Om~C0
% ----------------------------- J#WPXE+Ds
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) \F3t&:
error('zernfun:NMvectors','N and M must be vectors.') pQ\ [F
end ]<= t
5ZxBmQ
if length(n)~=length(m) r!uAofIi_
error('zernfun:NMlength','N and M must be the same length.') S"z4jpqn3
end @vh>GiR){
@/iLC6QF
n = n(:); Uij$
eBN
m = m(:); |Ay#0uQ5Y
if any(mod(n-m,2)) 5xKR
]u
error('zernfun:NMmultiplesof2', ... > `M\xt
'All N and M must differ by multiples of 2 (including 0).') +[DVD
end bhYaG i0
\ed(<e>
if any(m>n) uI wyan-
error('zernfun:MlessthanN', ... OR{"9)I
'Each M must be less than or equal to its corresponding N.') $!@f{9+
end &YMj\KmlSg
56dl;Z)
if any( r>1 | r<0 ) ;0E4S
error('zernfun:Rlessthan1','All R must be between 0 and 1.') ~3 (>_r
end >6q@Tr
V5w^Le_^
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) P&;I]2#
error('zernfun:RTHvector','R and THETA must be vectors.') PGGJpD?
end q[ZYlF,Ho
VPbNLi
r = r(:); 'fsOKx4Z
theta = theta(:); E~Nr4vq
length_r = length(r); HC+R:Dz
if length_r~=length(theta) 'l;|t"R12
error('zernfun:RTHlength', ... Af~AE2b3"
'The number of R- and THETA-values must be equal.') v/dcb%
end oJy/PR3
<s>SnOD
% Check normalization: =t2epIr5
% -------------------- zx*f*L,6F
if nargin==5 && ischar(nflag) }Of^Y@{q.
isnorm = strcmpi(nflag,'norm'); k6\c^%x
if ~isnorm 40XI\yE_?
error('zernfun:normalization','Unrecognized normalization flag.') 3*<W`yed
end .v{ty
else XJ+sm^`vOf
isnorm = false; teb(\% ,
end
8:MYeE5
o~B=[
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% /~:ztv\$M"
% Compute the Zernike Polynomials c9@*
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% :&MiO3#+
paY%pU
% Determine the required powers of r: !O*n6}nPE
% ----------------------------------- r%4:,{HF
m_abs = abs(m); CAO$Zt
rpowers = []; ~7v^7;tT
for j = 1:length(n) .jU9{;[
rpowers = [rpowers m_abs(j):2:n(j)]; RA}PM?D/
end BKk*<WMD
rpowers = unique(rpowers); 9z#IdY$a
i2DR}%U
% Pre-compute the values of r raised to the required powers, "q8wEu,z[
% and compile them in a matrix: cQjJ9o7
% ----------------------------- ^]HwStn&=
if rpowers(1)==0 r\zK>GVm_
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); (@zn[Nq
rpowern = cat(2,rpowern{:}); O7W}Z1G
rpowern = [ones(length_r,1) rpowern]; 'CvZiW[_r
else S1."2AxO
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 9Bn
dbSi
rpowern = cat(2,rpowern{:}); Kmtr.]Nj
end QnqX/vnR
#AHIlUH"m
% Compute the values of the polynomials: Y+E@afsKs
% -------------------------------------- *T3"U|0_ y
y = zeros(length_r,length(n)); lWR
for j = 1:length(n) ;8!D8o(+
s = 0:(n(j)-m_abs(j))/2; .s+e
hZ
pows = n(j):-2:m_abs(j); ?~$y3<[
for k = length(s):-1:1 <]<50
p = (1-2*mod(s(k),2))* ... F~:5/-zs
prod(2:(n(j)-s(k)))/ ... <NUZPX29
prod(2:s(k))/ ... ZISR]xay
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 5HB4B <2
prod(2:((n(j)+m_abs(j))/2-s(k))); NJ~'`{3v
idx = (pows(k)==rpowers); x-"7{@lz
y(:,j) = y(:,j) + p*rpowern(:,idx); oq|K:<l
end C]k\GlhB
\%K6T)9
if isnorm L:31toGK
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); >[#4Pb7_Y
end :c\NBKHv*
end $]_=B Jyu
% END: Compute the Zernike Polynomials m+L:\mvA
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% )}EwEM
7M4iBk4I
% Compute the Zernike functions: 90q*V%cS
% ------------------------------ \"Np'$4eu
idx_pos = m>0; OSBE5
idx_neg = m<0; + 7Z%N9
hAY_dM
z = y; N7NK1<vw2
if any(idx_pos) vK$W)(Z
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); d"V^^I)yx&
end u`ZnxD>
if any(idx_neg) WA<~M)rb
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); %T&kK2d;
end H;v*/~zl
% $J^dF_0
% EOF zernfun