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 vMB`TpZ
function z = zernfun(n,m,r,theta,nflag) lboi\GP|
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. @?r[
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% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 3.?kxac
% and angular frequency M, evaluated at positions (R,THETA) on the pZg}7F{$
% unit circle. N is a vector of positive integers (including 0), and HM(S}>
% M is a vector with the same number of elements as N. Each element r1)@ 7Nt
% k of M must be a positive integer, with possible values M(k) = -N(k) yMoV|U6
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, _rU%DL?
% and THETA is a vector of angles. R and THETA must have the same W dNOE;R
% length. The output Z is a matrix with one column for every (N,M) 3EN(Pz L
% pair, and one row for every (R,THETA) pair. o6[aP[~F
% <yHa[c`L
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike "bC1dl<
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), (7Q
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% with delta(m,0) the Kronecker delta, is chosen so that the integral
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% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, k Z>Xl- LV
% and theta=0 to theta=2*pi) is unity. For the non-normalized y:R!E *.L'
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. J>XMaI})U
% jJK`+J,i}X
% The Zernike functions are an orthogonal basis on the unit circle. TppuEC>
% They are used in disciplines such as astronomy, optics, and Dxlpo!
?#
% optometry to describe functions on a circular domain. JgmX=6N
% R@8pKCL.
% The following table lists the first 15 Zernike functions. ZcLW8L
% c?0.>^,B Q
% n m Zernike function Normalization aF41?.s
% -------------------------------------------------- ;0c
-+,
% 0 0 1 1 -FGQn
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% 1 1 r * cos(theta) 2 :K)7_]y
% 1 -1 r * sin(theta) 2 (Iz$_(
% 2 -2 r^2 * cos(2*theta) sqrt(6) 1\aJ[t
% 2 0 (2*r^2 - 1) sqrt(3) 74p=uQ
% 2 2 r^2 * sin(2*theta) sqrt(6) 4fyds< f
% 3 -3 r^3 * cos(3*theta) sqrt(8) ym=7EY?o
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) {%b*4x0?
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) tvlrUp
% 3 3 r^3 * sin(3*theta) sqrt(8) QU;bDNq,c
% 4 -4 r^4 * cos(4*theta) sqrt(10) ac%6eW0#
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) <n-}z[09
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) `UK'IN.il
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 8h%oJ4da
% 4 4 r^4 * sin(4*theta) sqrt(10) %Y:"5fH
% -------------------------------------------------- :+qd>;yf#
% &{uj3s&C
% Example 1: Bgvv6(i
% !JGe
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% % Display the Zernike function Z(n=5,m=1) -+ha4JOB
% x = -1:0.01:1; =+z +`ot
% [X,Y] = meshgrid(x,x); 8%ea(|Wjg
% [theta,r] = cart2pol(X,Y); ~EL3I
% idx = r<=1; x,% %^(
% z = nan(size(X)); EQTJ=\WFF
% z(idx) = zernfun(5,1,r(idx),theta(idx)); Z)^1~!w0
% figure "!w#E6gU
% pcolor(x,x,z), shading interp Rl/5eE8
% axis square, colorbar L GdM40
% title('Zernike function Z_5^1(r,\theta)')
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% {KGEv%
% Example 2: Kt#_Ln_6
% [`4
% % Display the first 10 Zernike functions rLpfybu
% x = -1:0.01:1; SIridZ*%
% [X,Y] = meshgrid(x,x); QHDR*tB:{
% [theta,r] = cart2pol(X,Y); !y\r.fm!A
% idx = r<=1; kfV}ta'^S
% z = nan(size(X)); n muzTFs=
% n = [0 1 1 2 2 2 3 3 3 3]; ,`
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% m = [0 -1 1 -2 0 2 -3 -1 1 3]; !*1$j7`tP
% Nplot = [4 10 12 16 18 20 22 24 26 28]; v8} vk]b
% y = zernfun(n,m,r(idx),theta(idx)); @u @~gEt
% figure('Units','normalized') [o"<DP6w
% for k = 1:10 ('k9X cTPP
% z(idx) = y(:,k); !sG#3sUe[
% subplot(4,7,Nplot(k)) Iz^vt#b
% pcolor(x,x,z), shading interp "P9(k>
% set(gca,'XTick',[],'YTick',[]) &"r /&7:
% axis square yiw4<]{IX
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 2Onp{,'}
% end ?Gl]O3@3
% 5MCnGg@
% See also ZERNPOL, ZERNFUN2. Lc#GBaJ
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% Paul Fricker 11/13/2006 x:K~?c3
jQrj3*V
Yu$QL@
% Check and prepare the inputs: er8T:.Py
% ----------------------------- V1&qgAy~
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ?o6X_UxW!
error('zernfun:NMvectors','N and M must be vectors.') $<QrV,T
end 8c\\-{
F)Iz:
if length(n)~=length(m) 9Vru,7g
error('zernfun:NMlength','N and M must be the same length.') R4y]<8}
end J= [D'h
} J[Z)u
n = n(:); @ry/zG#
m = m(:); Mbp7%^E"A
if any(mod(n-m,2)) L^jjf8_
error('zernfun:NMmultiplesof2', ... *4ido?
'All N and M must differ by multiples of 2 (including 0).') k2@]nW"S
end \%|Xf[AX
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if any(m>n) p0[+Zm{#l
error('zernfun:MlessthanN', ... /9e?uC6
'Each M must be less than or equal to its corresponding N.') Q5^ #:uZ
end
l*?_ @
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if any( r>1 | r<0 ) :sQ>oNnz
error('zernfun:Rlessthan1','All R must be between 0 and 1.') EE^x34&=
end P8(hHuO
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) o1vK2V
error('zernfun:RTHvector','R and THETA must be vectors.') c: r25
end
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D4b-Y[/"
r = r(:); &7i&"TNptP
theta = theta(:); Z5E; FGPb
length_r = length(r); P6&%`$
if length_r~=length(theta) 1uO2I&B
error('zernfun:RTHlength', ... !
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'The number of R- and THETA-values must be equal.') ftG3!}
end ;=7K*npT
&s(J:P$!
% Check normalization: |r*)U(c`
% -------------------- "M, 1ElQ
if nargin==5 && ischar(nflag) D#AqZS>B
isnorm = strcmpi(nflag,'norm'); S=0DQ19
if ~isnorm N+ak{3
error('zernfun:normalization','Unrecognized normalization flag.') W#%s0EN<_
end }jUsv8`}8R
else 9b&|'BBW
isnorm = false; XC5/$3'M&
end ESNI$[`
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% sO`
oapy
% Compute the Zernike Polynomials >{N}UNZ$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% FpiTQC7d
d=n@#|3
% Determine the required powers of r: @AF<Xp{
% ----------------------------------- "(3u)o9
m_abs = abs(m); P`ou:M{8
rpowers = []; 8Z0x*Ssk
for j = 1:length(n) <:V~_j6P0
rpowers = [rpowers m_abs(j):2:n(j)]; Bb:C^CHIQm
end L;*
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rpowers = unique(rpowers); +ID\u
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% Pre-compute the values of r raised to the required powers, .07`nIs"
% and compile them in a matrix: 'Y /0:)
% -----------------------------
p"#\E0GM
if rpowers(1)==0 00.x*v
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ."H;bfcL_
rpowern = cat(2,rpowern{:}); dYwkP^KB
rpowern = [ones(length_r,1) rpowern]; odSPl{. >d
else
v&|65[<
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 8ix_<$%
rpowern = cat(2,rpowern{:}); 1vxRhS&FY
end J\r\_P@;c
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% Compute the values of the polynomials: 6,A|9UX=`
% -------------------------------------- ~PI2G9
y = zeros(length_r,length(n)); 0eNdKE
for j = 1:length(n) W|7|XO
s = 0:(n(j)-m_abs(j))/2; bDM },(
pows = n(j):-2:m_abs(j); ts!tv6@
for k = length(s):-1:1 V6X )L>!xx
p = (1-2*mod(s(k),2))* ... RbX9PF"|+
prod(2:(n(j)-s(k)))/ ... 1>OlBp
prod(2:s(k))/ ... !1G
KpL
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... uYMn VE"
prod(2:((n(j)+m_abs(j))/2-s(k))); V9,<>
idx = (pows(k)==rpowers); ?1D!%jfi
y(:,j) = y(:,j) + p*rpowern(:,idx); u<Kowt<ci
end Tb$))O}
hO]F\0+
if isnorm Jv8:GgSg
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); rXi&8R[
end 1EXT^2!D
end &em~+83
% END: Compute the Zernike Polynomials n@8Y6+7i
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Cgx:6TRS
d ItfR'$
% Compute the Zernike functions: oFj_o
% ------------------------------ [ ,;e,ld
idx_pos = m>0; (dq_,LI
idx_neg = m<0; TP
rq:"K
,*J@ic7"
z = y; F:!6B b C
if any(idx_pos) Z*m^K%qJ
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); y>EW,%leC
end `(FjOd
K
if any(idx_neg) ]SCHni_
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); In1W/?
end WT'-.UX m
uu.X>agg
% EOF zernfun