下面这个函数大家都不会陌生，计算zernike函数值的，并根据此可以还原出图像来， @*6_Rp"@
我输入10阶的n、m，r,theta为38025*1向量，最后得到的z是29525*10阶的矩阵， u0? TMy.%
这个，跟我们用zygo干涉仪直接拟合出的36项zernike系数，有何关系呢？ SV95 g @
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ kMEXg zl
^ -~=U^2tC
Ha ZV 7
q:jv9eL.O
!](Mt?e
function z = zernfun(n,m,r,theta,nflag) D "fjk 1
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. dYwEVu6q
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N l)DcwkIG
% and angular frequency M, evaluated at positions (R,THETA) on the n@C#,v#^0
% unit circle. N is a vector of positive integers (including 0), and f D_3lbiL(
% M is a vector with the same number of elements as N. Each element u0[O /G
% k of M must be a positive integer, with possible values M(k) = -N(k) /K+;HAUTn
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, 4>Q] \\Lc
% and THETA is a vector of angles. R and THETA must have the same ]5ibg"{S
% length. The output Z is a matrix with one column for every (N,M) ~<W a $~oY
% pair, and one row for every (R,THETA) pair. @\-*aS_8>
% Rdd9JJsVd
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike -biw{
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), _ qQ
% with delta(m,0) the Kronecker delta, is chosen so that the integral 8) `
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, {JKG-0)z?
% and theta=0 to theta=2*pi) is unity. For the non-normalized <X1[j9Qtv0
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. \
sz ](X
% I;$tBgOWq
% The Zernike functions are an orthogonal basis on the unit circle. !HXsxNe
% They are used in disciplines such as astronomy, optics, and ! ([ v=O#
% optometry to describe functions on a circular domain. QqeF
% )J[Ady^5
% The following table lists the first 15 Zernike functions. K_N` My
% OWYY2&.h
% n m Zernike function Normalization b4ke'gx
% -------------------------------------------------- ecp0 hG`%
% 0 0 1 1 h=NXU9n%'
% 1 1 r * cos(theta) 2 - / 7@ A
% 1 -1 r * sin(theta) 2 $ 'a]lR
% 2 -2 r^2 * cos(2*theta) sqrt(6) `"iPJw14
% 2 0 (2*r^2 - 1) sqrt(3) j_zy"8Y{
% 2 2 r^2 * sin(2*theta) sqrt(6) QYB LU7
% 3 -3 r^3 * cos(3*theta) sqrt(8) D2:ShyYAS
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 0 R &7vn
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) OXoEA a
% 3 3 r^3 * sin(3*theta) sqrt(8) 4 T/ ~erc
% 4 -4 r^4 * cos(4*theta) sqrt(10) R3BK\kf&
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) D9r ;Ys%
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) r9-)+R
J
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) $7Lcn9 ?G
% 4 4 r^4 * sin(4*theta) sqrt(10) T?-K}PUcQ
% -------------------------------------------------- qNkX:|j
% L{ c\7
% Example 1: N,cj[6;T%
% MF::At[4
% % Display the Zernike function Z(n=5,m=1) 1< M~ #
% x = -1:0.01:1; ;/^O7KM -
% [X,Y] = meshgrid(x,x); + k
% [theta,r] = cart2pol(X,Y); f5n AD
% idx = r<=1; qMBEJ< o
% z = nan(size(X)); 6<n+p'+n
% z(idx) = zernfun(5,1,r(idx),theta(idx)); i} 5+\t[Q
% figure .Ag)/Xm(?
% pcolor(x,x,z), shading interp Yd ~Tzh
% axis square, colorbar 8O*O 5
% title('Zernike function Z_5^1(r,\theta)') \Fy HIs
% CT{ X$N
% Example 2: OadGwa\:s
% C2
!F
% % Display the first 10 Zernike functions mgEZiAV ?
% x = -1:0.01:1; Bq85g5Dc
% [X,Y] = meshgrid(x,x); 16N`xw+{
% [theta,r] = cart2pol(X,Y); OgyHX>} bH
% idx = r<=1; ! AL? bW
% z = nan(size(X)); dC">AW
% n = [0 1 1 2 2 2 3 3 3 3]; gHU0Pr9 '
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; LoS% FI
% Nplot = [4 10 12 16 18 20 22 24 26 28]; G9>
0w)r
% y = zernfun(n,m,r(idx),theta(idx)); E>}3MfL
% figure('Units','normalized') }/.b@`Dh;
% for k = 1:10 IAbH_+7O
% z(idx) = y(:,k); gO! :WD
% subplot(4,7,Nplot(k)) d!q)FRzi
% pcolor(x,x,z), shading interp Z9PG7h
% set(gca,'XTick',[],'YTick',[]) 5CM]-qbf@
% axis square M l,87fo
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) bd.t|A
% end 3ry0.
% ze Hs5P8}r
% See also ZERNPOL, ZERNFUN2. |Iq\ZX%q
zDA;FKZPp
WAh{*$Rpl
% Paul Fricker 11/13/2006 ljj}X JQ
lo cf6%2g~
p4wXsOQ}
0GiL(e|
@X0$X+]E*8
% Check and prepare the inputs: <UO'&?G
% ----------------------------- I!,FxOM|$
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Ha/-v?E
error('zernfun:NMvectors','N and M must be vectors.') T$9tO{
end q \\52 :\
25`6V>\
09rbu\h
if length(n)~=length(m) &r !* Y&
error('zernfun:NMlength','N and M must be the same length.') u+vUv~4A6
end l8Zz Kb-
S4 (lC%$|
1C\[n (9
n = n(:); 5i1Xumh 4
m = m(:); ukRbSJ5a5
if any(mod(n-m,2)) #a"gW,/K
error('zernfun:NMmultiplesof2', ... *H%Jgz,
'All N and M must differ by multiples of 2 (including 0).') th (<S
end *b]$lj
{%3sj"suB
[CJr8Qn
if any(m>n) M2e_)f:
error('zernfun:MlessthanN', ... _kT$/k
'Each M must be less than or equal to its corresponding N.') |\/Y<_)JD
end h48
jKL(
1-60gI1)
G4eY}3F7,4
if any( r>1 | r<0 ) }*I:0" WH
error('zernfun:Rlessthan1','All R must be between 0 and 1.') .#y.:Pb|e
end %B'*eBj~fw
I= 'S).
oh e0}~)V
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 9.qj Ee
error('zernfun:RTHvector','R and THETA must be vectors.') +\n8##oAI
end E)w^od wMU
H$i4OQ2
VdV18-ea
r = r(:); =tE7XC3X_
theta = theta(:); yb:Xjg7
length_r = length(r); B'Ll\<mq@
if length_r~=length(theta) 2-*zevPiG=
error('zernfun:RTHlength', ... )a%kAUNj
'The number of R- and THETA-values must be equal.') 8Yq_6
end w8df-]r
k-&fPE jG
%;|^*?!J0
% Check normalization: {m/h3hjFa
% -------------------- co$I htOv
if nargin==5 && ischar(nflag) 5&\%
isnorm = strcmpi(nflag,'norm'); b-rgiR$cg
if ~isnorm o%E^41M7E
error('zernfun:normalization','Unrecognized normalization flag.') HG/`5$L
+}
end 3 ;6Criq}
else D> |R.{
isnorm = false; -~-BQ!!(
end \.tnzP
D
5[_|+
vf+GC*f
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% VnB"0 "%w
% Compute the Zernike Polynomials `}YCUm[SI
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 8e 9ZgC|
&nk[gb
o\
}x^q?;7xW
% Determine the required powers of r: ;LM,<QJ
% ----------------------------------- VYb6#sl
m_abs = abs(m); 6ZCSCBW
rpowers = []; V~>
x \
for j = 1:length(n) :eIu<_,}
rpowers = [rpowers m_abs(j):2:n(j)]; k%5 o5Hx
end V9tG2m Lf>
rpowers = unique(rpowers); J~3+j6?%
D.hj9
4#o Lf1
% Pre-compute the values of r raised to the required powers, gxS*rzCG
% and compile them in a matrix: 7n,*3;I
% ----------------------------- O|o pNr
if rpowers(1)==0 [nO\Q3c|@$
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); *-gd k9
rpowern = cat(2,rpowern{:}); `J%iFm/5*
rpowern = [ones(length_r,1) rpowern]; &"(xd@V)]A
else tp-PE ?
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); Uk=-A
@q
rpowern = cat(2,rpowern{:}); -^i[
end Ps@a@d"83
)zzK\I6/EQ
u dhj$ :t
% Compute the values of the polynomials: N<lO!x1[H*
% -------------------------------------- dy^Zlu`
f
y = zeros(length_r,length(n)); DeTx7 i0
for j = 1:length(n) p_x@FA(
s = 0:(n(j)-m_abs(j))/2; Cx.GEY|0
pows = n(j):-2:m_abs(j); /T53"+7:0
for k = length(s):-1:1 Hy _ (
p = (1-2*mod(s(k),2))* ... `@$qy&AJ
prod(2:(n(j)-s(k)))/ ... Flrpk `4
prod(2:s(k))/ ... 7$8YBcZ6
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Cy'0O>v5
prod(2:((n(j)+m_abs(j))/2-s(k))); \^$ g%a
idx = (pows(k)==rpowers); afVl)2h
y(:,j) = y(:,j) + p*rpowern(:,idx); s}NE[Tw
end &R? \q*
}IM *V sk
if isnorm g]sc)4
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 1$&(ei]*:
end [Ybn pI
end owz 6j:
% END: Compute the Zernike Polynomials Ifg h yh<d
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ~(( '1+
jA&ZO>4
q97Z .o
% Compute the Zernike functions: R!mFMw"
% ------------------------------ 2$)xpET
idx_pos = m>0; Z2HH&3H A
idx_neg = m<0; k
E^%w?C
D"x; /I
bq mb|mD
z = y; o5NV4=
if any(idx_pos) Y8 c#"vm(
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); rHzwSR@}1
end f,Z* o
if any(idx_neg) : MfY8P)
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 8zDLX,M-
end ~N<zv( {lG
,4O|{Iu#n
_$g2;X >
% EOF zernfun 6:Fb>|]*PY
