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 xu9K\/{7
function z = zernfun(n,m,r,theta,nflag) v|Y:'5`V
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. |uT|(:i84,
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N _E0XUT!rA
% and angular frequency M, evaluated at positions (R,THETA) on the ^PDz"L<*
% unit circle. N is a vector of positive integers (including 0), and ?K]Cs&E4
% M is a vector with the same number of elements as N. Each element )U0`?kD
% k of M must be a positive integer, with possible values M(k) = -N(k) O ;,BzA-n
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, ]hY'A>4Uq
% and THETA is a vector of angles. R and THETA must have the same 4D(5WJ&
% length. The output Z is a matrix with one column for every (N,M) yn=BO`sgW
% pair, and one row for every (R,THETA) pair. LbX>@2(&
% @H%)!f]zWt
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike E`68Z/%
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), JL0>-kg
% with delta(m,0) the Kronecker delta, is chosen so that the integral >*/\Pg6^
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, A2 'W
% and theta=0 to theta=2*pi) is unity. For the non-normalized :u$nH9kwv
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ph*9,\c8
% Ni]V)wGE;
% The Zernike functions are an orthogonal basis on the unit circle. aH7i$U&
% They are used in disciplines such as astronomy, optics, and +o+e*B7Eh
% optometry to describe functions on a circular domain. rN0G|
% nT.i|(xd.
% The following table lists the first 15 Zernike functions. LLp/ SWe
% GZY8%.1{"a
% n m Zernike function Normalization cm`Jr#kl{
% -------------------------------------------------- epw*Px
% 0 0 1 1 o@SL0H-6|
% 1 1 r * cos(theta) 2 Q+L;k
R
% 1 -1 r * sin(theta) 2 CJ+/j=i;~c
% 2 -2 r^2 * cos(2*theta) sqrt(6) @Z9X^Y+u^h
% 2 0 (2*r^2 - 1) sqrt(3) B",5"'id
% 2 2 r^2 * sin(2*theta) sqrt(6) CG@3z@*?.
% 3 -3 r^3 * cos(3*theta) sqrt(8) GQ=Zp3[
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) iveJh2!#<
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) &sh5|5EC
% 3 3 r^3 * sin(3*theta) sqrt(8) 6&jW.G8/
% 4 -4 r^4 * cos(4*theta) sqrt(10) KVQ^-^
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) kn2s,%\`<p
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) ) -yJKmV
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) @\{L%y%a0
% 4 4 r^4 * sin(4*theta) sqrt(10) Da.eVU;
% -------------------------------------------------- fC6zDTis8A
% QH~;B[->
% Example 1: S$O+p&!X
% tOUpK20q.@
% % Display the Zernike function Z(n=5,m=1) QH z3
% x = -1:0.01:1; %H)^k${
% [X,Y] = meshgrid(x,x); Vf28R,~m
% [theta,r] = cart2pol(X,Y); 7 'T3Wc
% idx = r<=1; DxuT23.
(
% z = nan(size(X)); Uk@du7P1k
% z(idx) = zernfun(5,1,r(idx),theta(idx)); 4oxAC; L
% figure $7J9Yzp?L
% pcolor(x,x,z), shading interp
o[$~
% axis square, colorbar jh7-Fl`
% title('Zernike function Z_5^1(r,\theta)') AkMP)\Q
%
6z-ZJ|?
% Example 2: gX29c
% ,|5|aVfh
% % Display the first 10 Zernike functions @aQ};~
% x = -1:0.01:1; (!cG*FrN
% [X,Y] = meshgrid(x,x); =&%}p[
3g
% [theta,r] = cart2pol(X,Y); Ry47Fze
% idx = r<=1; &A/k{(.XP
% z = nan(size(X)); %XF>k)
% n = [0 1 1 2 2 2 3 3 3 3]; "2l$}G
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; H$D),s
gv
% Nplot = [4 10 12 16 18 20 22 24 26 28]; 2Dc2uU@`r
% y = zernfun(n,m,r(idx),theta(idx)); 38<Z=#S
% figure('Units','normalized') azK7kM~
% for k = 1:10 K_SURTys
% z(idx) = y(:,k); -hd@<+;E
% subplot(4,7,Nplot(k)) fBj-R~;0
% pcolor(x,x,z), shading interp *'i9
% set(gca,'XTick',[],'YTick',[]) RpmOg
% axis square e]9Z]a2
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) $O'IbA
% end 0|i3#G_~
% K*!qt(D&
% See also ZERNPOL, ZERNFUN2. b((>?=hh
I$0O4
% Paul Fricker 11/13/2006 nrEG4X9
=Ch^;Wyt
2gasH11M
% Check and prepare the inputs: @PL.7FM<v
% ----------------------------- `erKHZ]S
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) +nAbcBJAl
error('zernfun:NMvectors','N and M must be vectors.') f(Su
end @ajt
D-_2
VY#nSF`
if length(n)~=length(m)
;2y4^
error('zernfun:NMlength','N and M must be the same length.') luWr.<1
end 7oy}<9
TSKT6_IJw
n = n(:); {D$5M/$
m = m(:); @sdHB./
if any(mod(n-m,2)) dZWO6k9[H
error('zernfun:NMmultiplesof2', ... N^Hj%5
'All N and M must differ by multiples of 2 (including 0).') ''Y'ZsQ;
end v ^R:XdH
xqQLri}
if any(m>n) >vPv4e7&3
error('zernfun:MlessthanN', ... yM2}JsC
'Each M must be less than or equal to its corresponding N.') #3knKBH
end 2MU$OI0|
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if any( r>1 | r<0 ) 91#rP|88;
error('zernfun:Rlessthan1','All R must be between 0 and 1.') #E$*PAB
end 7 1+
bn
0-Ga2Go9
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) &cp
`? k
error('zernfun:RTHvector','R and THETA must be vectors.') n&%0G2m:
end ^wIg|Gc
fW
w+'xF!
r = r(:); Y|!m
theta = theta(:); ucYweXsO3
length_r = length(r); hiKyU!)Hv
if length_r~=length(theta) 5AbY 59
error('zernfun:RTHlength', ... nw-%!}Ot"
'The number of R- and THETA-values must be equal.') at+Nd K
end ^M)+2@6
`iNH`:[w
% Check normalization: 0N87G}Xu
% -------------------- .% 79(r^
if nargin==5 && ischar(nflag) {)n@Rq\=v
isnorm = strcmpi(nflag,'norm'); X #>:9
if ~isnorm M?_7*o]!
error('zernfun:normalization','Unrecognized normalization flag.') >{)\GK0i7
end U4NH9-U'
else r"9hpZH
isnorm = false; [XhG7Ly
end b]4\$ rW7
YU`}T<;bg
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% f>iDqC4
% Compute the Zernike Polynomials 7?;ZE:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% c'INmc
I|
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% Determine the required powers of r: `\r<3?
% ----------------------------------- fcTg/EXn
m_abs = abs(m); $|tk?Sps
rpowers = []; ,<BV5~T.|
for j = 1:length(n) Iw4[D#o
rpowers = [rpowers m_abs(j):2:n(j)]; A*~BkvPr
end 5\Rg%Ezl
rpowers = unique(rpowers); pr[V*C/
%O$=%"D6
% Pre-compute the values of r raised to the required powers, :*ZijN*{)$
% and compile them in a matrix: +|--}iE5n
% ----------------------------- P(UY}oU
if rpowers(1)==0 = q(?ALGc
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); H;seT XL
rpowern = cat(2,rpowern{:}); d`,z4_
rpowern = [ones(length_r,1) rpowern];
Q@!XVQx4
else )3WUyD*UZN
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); # w@FBFr@
rpowern = cat(2,rpowern{:}); }}Zg/(
end )KY4BBc
HB,?}S#TP
% Compute the values of the polynomials: EbeSl+iMx_
% -------------------------------------- v|KGzQx$.*
y = zeros(length_r,length(n)); ;H3~r^>c
for j = 1:length(n) rd;E /:`5
s = 0:(n(j)-m_abs(j))/2; f_Hh"Vh
pows = n(j):-2:m_abs(j); `oTV)J'~
for k = length(s):-1:1 P!SsMo6n
p = (1-2*mod(s(k),2))* ... "=ki_1/P
prod(2:(n(j)-s(k)))/ ... CkRilS<
prod(2:s(k))/ ... 1(pv3
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... +k
h
Tl:
prod(2:((n(j)+m_abs(j))/2-s(k))); 29l bOi
idx = (pows(k)==rpowers); ^E_chx-e}
y(:,j) = y(:,j) + p*rpowern(:,idx); _f~$iY
end JAM]neKiX
*&tTiv{^
if isnorm 3mHP=)
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); Vry*=X&Q
end njaKU?6%d2
end XSCcumde!
% END: Compute the Zernike Polynomials ^ZIs >.'
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% P'o]#Az
/'zXb_R,$
% Compute the Zernike functions: -Mf-8zw8G
% ------------------------------ =4sx(<
idx_pos = m>0; |S~$IFN4
idx_neg = m<0; 3ZN\F
d+vAm3.Dg
z = y; K%W;-W*'
if any(idx_pos) )H`V\H[0P
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); \=P(?!v
end i8KoJY"
if any(idx_neg) &^w"
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ,xR u74
end H )>3c1
t>OEzUd9
% EOF zernfun