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 1Eryw~,,9i
function z = zernfun(n,m,r,theta,nflag) q'X#F8v
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. j1ap,<\.k
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N *$mb~k^R
% and angular frequency M, evaluated at positions (R,THETA) on the Ie8K[ >
% unit circle. N is a vector of positive integers (including 0), and u =(.}
% M is a vector with the same number of elements as N. Each element f
uH3C~u7<
% k of M must be a positive integer, with possible values M(k) = -N(k) 9G6auk.m.O
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, 0zA:?}
% and THETA is a vector of angles. R and THETA must have the same wvr`~ e
% length. The output Z is a matrix with one column for every (N,M) .wtYostv
% pair, and one row for every (R,THETA) pair. Nvd(Tad
% PK_2
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ;b_<5S
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ;\T~Hc}&;
% with delta(m,0) the Kronecker delta, is chosen so that the integral J %E0Wd
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, F5w=tK
% and theta=0 to theta=2*pi) is unity. For the non-normalized A=*6|1w;
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. Ka"1gbJ|
% Yg1HvSw\
% The Zernike functions are an orthogonal basis on the unit circle. 8yuTT^
% They are used in disciplines such as astronomy, optics, and CY!H)6k
% optometry to describe functions on a circular domain. FGpV
]p
% =]<X6!0mR
% The following table lists the first 15 Zernike functions. .O{_^~w_q
% Y@b|/+
% n m Zernike function Normalization ~UsE"5
% -------------------------------------------------- M%Q_;\?]
% 0 0 1 1 ` ^z
l =
% 1 1 r * cos(theta) 2 _Vr}ipx-k
% 1 -1 r * sin(theta) 2 OoZv\"}!_
% 2 -2 r^2 * cos(2*theta) sqrt(6) a1v?{vu\E
% 2 0 (2*r^2 - 1) sqrt(3) "m}N
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% 2 2 r^2 * sin(2*theta) sqrt(6) %V-Hy ;V
% 3 -3 r^3 * cos(3*theta) sqrt(8) #Jfmt~ks'
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) sWP_fb1
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) ORfMp'uP=
% 3 3 r^3 * sin(3*theta) sqrt(8) YD5mJ[1t"2
% 4 -4 r^4 * cos(4*theta) sqrt(10) N,ZmGzNP)
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) b|Eo\l2
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) cs]3Rp^g
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) pq]>Ep
% 4 4 r^4 * sin(4*theta) sqrt(10) 2y9$ k\<xV
% -------------------------------------------------- AxCFZf 5
% X>MDX.Z
% Example 1: _wZr`E)
% : p7PiqQ
% % Display the Zernike function Z(n=5,m=1) &tlU.Whk+
% x = -1:0.01:1; m;u :_4
% [X,Y] = meshgrid(x,x); \YH*x`
% [theta,r] = cart2pol(X,Y); X@~R<
% idx = r<=1; ^pocbmg
% z = nan(size(X)); \Iz-<:gA'
% z(idx) = zernfun(5,1,r(idx),theta(idx)); ZVCa0Km
% figure Z%VgAV>>
% pcolor(x,x,z), shading interp NcIr;
}
% axis square, colorbar G-DOI
% title('Zernike function Z_5^1(r,\theta)') W!a'KI'
% iUf?MDE
% Example 2: #|
m*k
% ^O3p:X4u
% % Display the first 10 Zernike functions u4:6zU/{
% x = -1:0.01:1; .gw6W0\F
% [X,Y] = meshgrid(x,x); =K9-
% [theta,r] = cart2pol(X,Y); zY&/lWW._
% idx = r<=1; ^=w){]G
% z = nan(size(X)); 3MHByT%
% n = [0 1 1 2 2 2 3 3 3 3]; z s[zB#
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; !7Z?VEZ
% Nplot = [4 10 12 16 18 20 22 24 26 28]; ZV~9{E8
% y = zernfun(n,m,r(idx),theta(idx)); 12 bztlv
% figure('Units','normalized') .wcKG9u
% for k = 1:10 ezr'"1Ba}
% z(idx) = y(:,k); 6W N(Tw
% subplot(4,7,Nplot(k)) p@+D$
% pcolor(x,x,z), shading interp y~rtYI
% set(gca,'XTick',[],'YTick',[]) V}q=!zz
% axis square = /=?l
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) W_|7hwr
% end >]?!9@#IH
% OJ)XJL
% See also ZERNPOL, ZERNFUN2. x)e(g}n
/#e-x|L
% Paul Fricker 11/13/2006 !l1jQq_mK
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c.;<+dYsm*
% Check and prepare the inputs: PKt;]T0
% ----------------------------- 8?$XT
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) DbH'Qs?z
error('zernfun:NMvectors','N and M must be vectors.') mUwGr_)wj
end A55F *d
!F#^Peb
if length(n)~=length(m) #(r1b'jfP
error('zernfun:NMlength','N and M must be the same length.') [J43]
end pt9fOih[
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n = n(:); EZ)GW%Bm2
m = m(:); vOBXAF
if any(mod(n-m,2)) F ss@/-
error('zernfun:NMmultiplesof2', ... v'u}%FC
'All N and M must differ by multiples of 2 (including 0).') wWB^m@:4
end EdS7m,d
O|0} m
if any(m>n) *uvE`4V^Jg
error('zernfun:MlessthanN', ... MF4B 2d
'Each M must be less than or equal to its corresponding N.') Cg%}=
end 2M?L++i
_SQ0`=+
if any( r>1 | r<0 ) LKu
,H
error('zernfun:Rlessthan1','All R must be between 0 and 1.') (^LR9 CW
end ci{WyIh
Ct9*T`Gl
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ^1z)\p1
error('zernfun:RTHvector','R and THETA must be vectors.') &,iPI2`O A
end D
P+W*87J
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r = r(:); qJEtB;J'
theta = theta(:); 8jU6N*p/
length_r = length(r); ZTK)N
if length_r~=length(theta) r[ RO"Ej"
error('zernfun:RTHlength', ... ^uWj#
'The number of R- and THETA-values must be equal.') #i[V{J8.p
end ,HfdiGs}j
%1%@L7wP>
% Check normalization: M0"}>`1lJ
% -------------------- Xm[Cgt_?
if nargin==5 && ischar(nflag) q%8Ck)xz
isnorm = strcmpi(nflag,'norm'); #l-/!j
if ~isnorm 17B`
error('zernfun:normalization','Unrecognized normalization flag.') ;2iDa
end 'V(9ein^Q
else >Mk#19j[/
isnorm = false; -bQi4
end YEhPAQNj
5:X^Q.f;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% n_46;lD
% Compute the Zernike Polynomials c"^g*i2&0
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% khfWU
=FXq=x%9+
% Determine the required powers of r: ' |
bHu
% ----------------------------------- PgwNE wG
m_abs = abs(m); 55vI^SSA
rpowers = []; x_.}C%
for j = 1:length(n) y_N h5
rpowers = [rpowers m_abs(j):2:n(j)]; lyQNE3
end Z6_E/S
rpowers = unique(rpowers); x @uowx_&m
wTPHc:2
% Pre-compute the values of r raised to the required powers, r>x>aJ
% and compile them in a matrix: ~X%W2N2
% ----------------------------- =1Tn~)^O
if rpowers(1)==0 F`JW&r\
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); {xJ<)^fD8
rpowern = cat(2,rpowern{:}); u1_NC;
rpowern = [ones(length_r,1) rpowern]; &=hkB9
;
else Ai.^~#%X
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); '=]|"
rpowern = cat(2,rpowern{:}); W3-g]#\?
end yu@u0vlc
[rtMx8T
% Compute the values of the polynomials:
&L4>w.b"N
% -------------------------------------- f&L8<ASFo
y = zeros(length_r,length(n)); Ts
1
for j = 1:length(n) 53)*i\9&
s = 0:(n(j)-m_abs(j))/2; PBp+(o-
pows = n(j):-2:m_abs(j); C9"yu&l
for k = length(s):-1:1 \4roM1&[
p = (1-2*mod(s(k),2))* ... e[*%tx H
prod(2:(n(j)-s(k)))/ ... Xrd-/('2
prod(2:s(k))/ ... X(fT[A_2C
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... &U0Y#11Cx
prod(2:((n(j)+m_abs(j))/2-s(k))); :`20i*
idx = (pows(k)==rpowers); Ur2)];WZ
y(:,j) = y(:,j) + p*rpowern(:,idx); ,NoWAmv
end D|E,9|=v
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if isnorm
o
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y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); uEcK0>xp
end *d$r`.9j
end EawtT
% END: Compute the Zernike Polynomials b{hdEb
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +U*:WKdI?
j`ybz G^
% Compute the Zernike functions: p28=l5y+
% ------------------------------ >'|Wrz67Z
idx_pos = m>0; n`2LGc[rP
idx_neg = m<0; rWD*DmY@"
V"R ,omh
z = y; YKG}4{T
if any(idx_pos) kCZxv"Ts
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); *-.,QpgTX
end w>uo-88
if any(idx_neg) vK,.P:n
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); !=rJ~s
F/{
end (=/}i'
RqRyZ*n
% EOF zernfun