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 U@DIO/C,m`
function z = zernfun(n,m,r,theta,nflag) %I?uO(
@
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. `Fnt#F}
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N iku) otUc
% and angular frequency M, evaluated at positions (R,THETA) on the R{u/r%
% unit circle. N is a vector of positive integers (including 0), and r;SA1n#
% M is a vector with the same number of elements as N. Each element
'f]\@&Np
% k of M must be a positive integer, with possible values M(k) = -N(k) D&$%JT'3
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, QF
Vy2 q
% and THETA is a vector of angles. R and THETA must have the same {|a=
% length. The output Z is a matrix with one column for every (N,M) Wu?4oF
% pair, and one row for every (R,THETA) pair. 6o!+E@V
b
% 8Y_wS&eB
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike LL4yafh
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), J1KV?aR
% with delta(m,0) the Kronecker delta, is chosen so that the integral [O7:<co
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, +<7`Gn(n3
% and theta=0 to theta=2*pi) is unity. For the non-normalized ;(5b5PA
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ~{/"fTif
% oYI7 .w
% The Zernike functions are an orthogonal basis on the unit circle. rK7m(
% They are used in disciplines such as astronomy, optics, and 5Z@OgR
% optometry to describe functions on a circular domain. AQ7w5}g+V
% V]&0"HX2r!
% The following table lists the first 15 Zernike functions. -YPUrU[)
% EPkmBru
^
% n m Zernike function Normalization ef *Vs
% -------------------------------------------------- o)GLh^g_I'
% 0 0 1 1 PS7ta?V
QC
% 1 1 r * cos(theta) 2 W^v3pH-y#
% 1 -1 r * sin(theta) 2 "Y-_83
% 2 -2 r^2 * cos(2*theta) sqrt(6) Y|stxeOC
% 2 0 (2*r^2 - 1) sqrt(3) #0GvL=}k
% 2 2 r^2 * sin(2*theta) sqrt(6) Rf9;jwU
% 3 -3 r^3 * cos(3*theta) sqrt(8) dn!#c=
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) sba+J:#w
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) @|BaZq,g
% 3 3 r^3 * sin(3*theta) sqrt(8) u?,M`w0'
% 4 -4 r^4 * cos(4*theta) sqrt(10) $q%r}Cdg
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) VB=$D|Ll
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) z3>ldT
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) _!2bZ:emG
% 4 4 r^4 * sin(4*theta) sqrt(10) W:VRLT>w>
% -------------------------------------------------- Vz[tgb]-
% c%tb6@C
% Example 1: Okxuhzn>"
% X"lPXoCN
% % Display the Zernike function Z(n=5,m=1) U|yXJ.Z3
% x = -1:0.01:1; ~?E.U,R
% [X,Y] = meshgrid(x,x); 9
M>.9~
% [theta,r] = cart2pol(X,Y); dPvRbwH<
% idx = r<=1; O1xK\ogv
% z = nan(size(X)); v{tw ;Z#
% z(idx) = zernfun(5,1,r(idx),theta(idx)); g4z*6L,u
% figure 7\.{O$Q
% pcolor(x,x,z), shading interp ^6g^ Q*"
% axis square, colorbar J;8M._
% title('Zernike function Z_5^1(r,\theta)') :Q]P=-Y8
% pg0Sq9qCN
% Example 2: dA03,s
% IPHZ~'M
% % Display the first 10 Zernike functions xNAX)v3Z
% x = -1:0.01:1; Q^trKw~XNy
% [X,Y] = meshgrid(x,x); '/O >#1
% [theta,r] = cart2pol(X,Y); L/*D5k%J
% idx = r<=1; /hF@Xh%hY
% z = nan(size(X)); w&F.LiX^
% n = [0 1 1 2 2 2 3 3 3 3]; ;8Qx~:c
% m = [0 -1 1 -2 0 2 -3 -1 1 3];
}%)]b*3
% Nplot = [4 10 12 16 18 20 22 24 26 28]; [8%R*}
% y = zernfun(n,m,r(idx),theta(idx)); <b>g^ `}?D
% figure('Units','normalized') HAKB@h)
% for k = 1:10 8@rddk
% z(idx) = y(:,k); t nvCtuaR
% subplot(4,7,Nplot(k)) !a9`]c
% pcolor(x,x,z), shading interp >a%C'H.A9
% set(gca,'XTick',[],'YTick',[]) ag02=}Q'r
% axis square tXXnHEz
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) nY M2Vxi0+
% end ka=EOiX.
% yor6h@F1
% See also ZERNPOL, ZERNFUN2.
Q h~
9Ib#A
% Paul Fricker 11/13/2006 dQljG.PiK
i U"2uLgb
v{r,Wy3
% Check and prepare the inputs: 0]k-0#JM
% ----------------------------- 2e?a"Vss
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) M4}b lh#
error('zernfun:NMvectors','N and M must be vectors.') r}nz )=\Cj
end Ci9]#)"c
8{4SaT.-Rm
if length(n)~=length(m) )`5=6i
error('zernfun:NMlength','N and M must be the same length.') GtLnh~)
end !-AK@`i.
F<0GX!p4u
n = n(:); ^!A@:}t>
m = m(:); nq%GLUH
if any(mod(n-m,2)) Q@(tyW+8U@
error('zernfun:NMmultiplesof2', ... sD=iHO
Am
'All N and M must differ by multiples of 2 (including 0).') 5c
($~EFr
end $97EeE:{M
9M;k(B!
if any(m>n) :meq4!g{1
error('zernfun:MlessthanN', ... S; Fj9\2)I
'Each M must be less than or equal to its corresponding N.') S;tv4JY
end rO[ Zx'a
wl5+VC*l0
if any( r>1 | r<0 ) l1UN.l'p
error('zernfun:Rlessthan1','All R must be between 0 and 1.') <wTD}.n
end sjj,q?
#-7w|
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Y^ 2]*e%
error('zernfun:RTHvector','R and THETA must be vectors.') Z/oP?2/Afh
end w%?6s 3
dV7~C@k6k8
r = r(:); $:IEp V{
theta = theta(:); !n3J6%b9y/
length_r = length(r); ,V`[;~49
if length_r~=length(theta) St|B9V?eEB
error('zernfun:RTHlength', ... M32Z3<
'The number of R- and THETA-values must be equal.') |Ye%HpTTv
end >5M Hn@
2p;N|V
% Check normalization: w$$vR
% -------------------- ^3lEfI<pBm
if nargin==5 && ischar(nflag) |PutTcjQ
isnorm = strcmpi(nflag,'norm'); N
VBWF
if ~isnorm s#>``E!
error('zernfun:normalization','Unrecognized normalization flag.') aX}:O
end V9/P kuT
else ;%mYsQ
isnorm = false; {GhM,-%e
end q3e^vMK"
ICm/9Onh&
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% !g7bkA
% Compute the Zernike Polynomials J_N`D+m
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% XAb-K?)
|m>{< :
% Determine the required powers of r: l~'NqmXe
% ----------------------------------- ~9JLqN"
m_abs = abs(m); Rdl^-\BV
rpowers = []; &pN/+,0E
for j = 1:length(n) ~@ML>z7
rpowers = [rpowers m_abs(j):2:n(j)]; (4"Azo*~![
end hx:"'m5
rpowers = unique(rpowers); hWAZP=H
Q|Go7MQZ@k
% Pre-compute the values of r raised to the required powers, [fIElH<
% and compile them in a matrix: ;To][J
% ----------------------------- J`[He$7)
if rpowers(1)==0 2>h.K/pC
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); R6E.C!EI
rpowern = cat(2,rpowern{:}); dZ{yNh.]
rpowern = [ones(length_r,1) rpowern]; j7v?NY
else G21cJi*
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 7#9yAS+x(
rpowern = cat(2,rpowern{:}); 69JC!du
end H-'~c\)
.!yw@kg
% Compute the values of the polynomials: 0})mCVBY
% -------------------------------------- #9u2LK
y = zeros(length_r,length(n)); 3}V-'!
for j = 1:length(n) Uv%?z0F<C
s = 0:(n(j)-m_abs(j))/2; xyPz_9
pows = n(j):-2:m_abs(j); HV\l86}
for k = length(s):-1:1 65AG#O5R
p = (1-2*mod(s(k),2))* ... D>m!R[!o
prod(2:(n(j)-s(k)))/ ... N3?@CM^hHw
prod(2:s(k))/ ... +5oK91o[y
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... oa:30@HSb
prod(2:((n(j)+m_abs(j))/2-s(k))); Qv/Kb w
N{
idx = (pows(k)==rpowers); \zv?r:1t
y(:,j) = y(:,j) + p*rpowern(:,idx); @ !m+s~~]h
end p}9bZKyf
\%$z!]S>
if isnorm HRF;qR9v
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); +"F 9yb
end WJF#+)P:Y
end qgk6 \&K[
% END: Compute the Zernike Polynomials L>{p>
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% WbH#@]+DN
mrId`<L5l{
% Compute the Zernike functions: OM 4,Sevk
% ------------------------------ :8 jaW?~
idx_pos = m>0; 7FvtWE*
idx_neg = m<0; FCPiU3
x/^,{RrPk
z = y; ?JI:>3e
if any(idx_pos) gbL!8Z1h
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); j/PNi@
end 3PgiV%]
if any(idx_neg) 0
V3`rK
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); =#K$b *#
end g1B[RSWv
5&N55?G6
% EOF zernfun