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>JgBe0
function z = zernfun(n,m,r,theta,nflag) ANBuX6q
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. &nr{-][
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N TxF^zx\
% and angular frequency M, evaluated at positions (R,THETA) on the ynM~&]fk#k
% unit circle. N is a vector of positive integers (including 0), and jXf@JxQ
% M is a vector with the same number of elements as N. Each element B2]52Fg-"
% k of M must be a positive integer, with possible values M(k) = -N(k) 8,IF%Z+LI
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, +`Q]p "G
% and THETA is a vector of angles. R and THETA must have the same _h^.`Tz,
% length. The output Z is a matrix with one column for every (N,M) -~8PI2
% pair, and one row for every (R,THETA) pair. eEVB
% jnOnV1I"
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike =Mwuhk|*
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), %8u9:Cl):
% with delta(m,0) the Kronecker delta, is chosen so that the integral Nkj$6(N=zJ
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, KO8{eT9d
% and theta=0 to theta=2*pi) is unity. For the non-normalized MF'Z?M
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ?sdSi--
% lq_UCCnv5
% The Zernike functions are an orthogonal basis on the unit circle. Mo N/?VA
% They are used in disciplines such as astronomy, optics, and :tO4LEb
% optometry to describe functions on a circular domain. )-[$m%
% .qohHJ&
% The following table lists the first 15 Zernike functions. QObVJg,GD
% c]x-mj =
% n m Zernike function Normalization Z ;rM@x
% -------------------------------------------------- {0F/6GwUC
% 0 0 1 1 :n13v@q
% 1 1 r * cos(theta) 2 kZ@UQ{>`
% 1 -1 r * sin(theta) 2 D6@ c|O{Q
% 2 -2 r^2 * cos(2*theta) sqrt(6) Ey:?!
% 2 0 (2*r^2 - 1) sqrt(3) `=hCS0F
% 2 2 r^2 * sin(2*theta) sqrt(6) iYT?6Y|+
% 3 -3 r^3 * cos(3*theta) sqrt(8) i@rUZYF
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) rucw{)
_
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) ',`Qx{tQ)
% 3 3 r^3 * sin(3*theta) sqrt(8) J#Y0R"fo
% 4 -4 r^4 * cos(4*theta) sqrt(10) #A4WFZ
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) f9#srIx+
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) L3oL>r'|
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Ewkx4,`Ff
% 4 4 r^4 * sin(4*theta) sqrt(10)
{,Vvm*L/
% -------------------------------------------------- "ADI.
% ~{{S<S
v
% Example 1: u
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% 6c^?DLy9B
% % Display the Zernike function Z(n=5,m=1) o%j?}J7y
% x = -1:0.01:1; 7W SP0Xyz
% [X,Y] = meshgrid(x,x); p+?`ru
% [theta,r] = cart2pol(X,Y); x[TLlV:{
% idx = r<=1; 3s%DF,
% z = nan(size(X)); I$sXbM;z=
% z(idx) = zernfun(5,1,r(idx),theta(idx)); |X1axRO
% figure >%`SXB&9
% pcolor(x,x,z), shading interp RYvdfj.ij
% axis square, colorbar .zdaY,
U
% title('Zernike function Z_5^1(r,\theta)') ~:{ mKc
% O,
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% Example 2: L;*7p9
% w+')wyB
% % Display the first 10 Zernike functions Z>g&%3j
% x = -1:0.01:1; .9ZK@xM&?
% [X,Y] = meshgrid(x,x); ]XlBV-@b
% [theta,r] = cart2pol(X,Y); {9 |*au(K
% idx = r<=1;
d<xi/
% z = nan(size(X)); H~JgZ pw
% n = [0 1 1 2 2 2 3 3 3 3]; e}{#VB<
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; o<lmU8xB=
% Nplot = [4 10 12 16 18 20 22 24 26 28]; :+\sKEzL
% y = zernfun(n,m,r(idx),theta(idx)); QhRz57'
% figure('Units','normalized') {ly <%Q7j
% for k = 1:10 M __S)
% z(idx) = y(:,k); <L8FI78[*
% subplot(4,7,Nplot(k)) `"ks0@^U
% pcolor(x,x,z), shading interp ;lE=7[UJ3X
% set(gca,'XTick',[],'YTick',[]) b/oNQQM#Dk
% axis square NL|c5y<r
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) Pw]+6
% end -J6`
% a3M I+
% See also ZERNPOL, ZERNFUN2. .?APDr"QQH
(p#c p
% Paul Fricker 11/13/2006 0@{bpc rc
_\IA[-C+O
!jB}}&Ii
% Check and prepare the inputs: aUa+]H[
% ----------------------------- JT<JS6vw#
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 8*?H~q~
error('zernfun:NMvectors','N and M must be vectors.') U:7w8$_
end UzSDXhzObf
b-VQn5W
if length(n)~=length(m) X)j%v\#`U
error('zernfun:NMlength','N and M must be the same length.') on8$Kc
end )Z4iM;4]
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n = n(:); $tW E9_
m = m(:); 5G'2 Wby'#
if any(mod(n-m,2)) G2n.NW#d4
error('zernfun:NMmultiplesof2', ... '6\w4J(
'All N and M must differ by multiples of 2 (including 0).') 46
0/eW\
end +|GHbwvp
v h)CB8
if any(m>n) R86i2',
error('zernfun:MlessthanN', ... QYDI-<.(
'Each M must be less than or equal to its corresponding N.') #%$@[4"V
end qh}+b^Wi
.i )K#82
if any( r>1 | r<0 ) KMfIp:~
error('zernfun:Rlessthan1','All R must be between 0 and 1.') @JdeOL;
end l_04b];
,'YKL",
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 2\64~a^
error('zernfun:RTHvector','R and THETA must be vectors.') %sZ3Gpi
end elKp?YN
d7g$9&/q
r = r(:); +DefV,Ny
theta = theta(:); PQF
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length_r = length(r); ".AW
if length_r~=length(theta)
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error('zernfun:RTHlength', ... JB5%\
'The number of R- and THETA-values must be equal.') )2d1@]6#
end )9/iH(
753gcY#i
% Check normalization: lxD~l#)^ln
% -------------------- M`=\ijUwN
if nargin==5 && ischar(nflag) $b^ niL
isnorm = strcmpi(nflag,'norm'); YGyw^$.w
if ~isnorm GM^H
)8U
error('zernfun:normalization','Unrecognized normalization flag.') tycVcr\(
end 6 AY~>p
else pXQ$n:e
isnorm = false; d{WOO)j
end Y nTx)uW
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *fy aAv
% Compute the Zernike Polynomials 6PWw^Cd
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% .hf%L1N%F
"f3mi[
% Determine the required powers of r: /a}N6KUi
% ----------------------------------- D&N3LH
m_abs = abs(m); D7thLqA
rpowers = []; z+0#H39 &
for j = 1:length(n) &R<K>i
rpowers = [rpowers m_abs(j):2:n(j)]; "K|':3n|
end HmsXV_B8[Y
rpowers = unique(rpowers); N /2WUp
.[:WMCc\
% Pre-compute the values of r raised to the required powers, Qe9}%k6@E
% and compile them in a matrix: %6V=G5+W
% ----------------------------- a9 S&n5
if rpowers(1)==0 KeyHxU=?
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); U+D#
rpowern = cat(2,rpowern{:}); CRzLyiRvU&
rpowern = [ones(length_r,1) rpowern]; Ms%C:KG
else }Vm'0
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); :de4Fje/4y
rpowern = cat(2,rpowern{:}); }U%E-:
end =(,kjw88w
mxc^IRj
% Compute the values of the polynomials: JV2[jo}0N
% -------------------------------------- F
Zt;D
y = zeros(length_r,length(n)); @'J~(#}
for j = 1:length(n) & )-fC
s = 0:(n(j)-m_abs(j))/2; !;k
^
pows = n(j):-2:m_abs(j); 1iM(13jW
for k = length(s):-1:1 -)ri,v{:c
p = (1-2*mod(s(k),2))* ... 8l?@ o
prod(2:(n(j)-s(k)))/ ... >;xkiO>Y
prod(2:s(k))/ ... \w$e|[~
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 0V2~
prod(2:((n(j)+m_abs(j))/2-s(k))); 85FzIX-F%
idx = (pows(k)==rpowers); ej(w{vl
y(:,j) = y(:,j) + p*rpowern(:,idx); W3MH8z
end pqbKPpG
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if isnorm gi? wf
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); .+ic6
end THwq~c'
end ZmaW]3$
% END: Compute the Zernike Polynomials &b19s=Z,
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% BJZGQrsz
w- wJhc|
% Compute the Zernike functions: @]],H0
% ------------------------------ 0}Qd
idx_pos = m>0; U}-hV@y
idx_neg = m<0; ef:Zi_o
HhTD/
z = y; Y$ZDJNz
if any(idx_pos) o-AAx#@
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 'sjks sy.3
end D rouEm
if any(idx_neg) 4Rl~7|
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 4?x$O{D5?{
end **n109R
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% EOF zernfun