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^ vxlF
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， /o+,
=7hY
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？ ?ti7iBz?
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ /M v\~vg$1
m$pXe<
4x e:+sA.N
</:f-J%U/
>OZ+k(saL
function z = zernfun(n,m,r,theta,nflag) ,^:Zf| V
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. V4/P
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N G/2@ Mn-
% and angular frequency M, evaluated at positions (R,THETA) on the P}DrUND
% unit circle. N is a vector of positive integers (including 0), and Uu>YE0/)
% M is a vector with the same number of elements as N. Each element !ny; Y V
% k of M must be a positive integer, with possible values M(k) = -N(k) $-M 1<?5
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, XuoI19V[
% and THETA is a vector of angles. R and THETA must have the same kh^AH6{2
% length. The output Z is a matrix with one column for every (N,M) 6( D K\58
% pair, and one row for every (R,THETA) pair. s2b!Nib
% *z` {$hc
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike :}UWy?F
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 5(u7b
% with delta(m,0) the Kronecker delta, is chosen so that the integral Qb xjfW"/+
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ;9=9D{-4+
% and theta=0 to theta=2*pi) is unity. For the non-normalized ItC *[
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. P ,CJy|[L
% 4kxy7 ] W
% The Zernike functions are an orthogonal basis on the unit circle. f ,K1 a9.
% They are used in disciplines such as astronomy, optics, and Q%o
% optometry to describe functions on a circular domain. IC92lPM }
% tojJQ6;J
% The following table lists the first 15 Zernike functions. i ,4
% =
fuF]yL%
% n m Zernike function Normalization +qD4`aI
% -------------------------------------------------- g igDrf}
% 0 0 1 1 _o' jy^
% 1 1 r * cos(theta) 2 B/i,QBPF]
% 1 -1 r * sin(theta) 2 JEU?@J71O
% 2 -2 r^2 * cos(2*theta) sqrt(6) e>uV8!u
% 2 0 (2*r^2 - 1) sqrt(3) [^1;8Tbk
% 2 2 r^2 * sin(2*theta) sqrt(6) cV&(L]k >`
% 3 -3 r^3 * cos(3*theta) sqrt(8) 7bQ#M )}
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) xqmJP bA
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) *ZKfyn$+~
% 3 3 r^3 * sin(3*theta) sqrt(8) , $78\B^
% 4 -4 r^4 * cos(4*theta) sqrt(10) "aB]?4
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) =WJ*$j (
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) h9>~?1$lz
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) H ]}Iw5Z
% 4 4 r^4 * sin(4*theta) sqrt(10) ULjW589 zb
% -------------------------------------------------- W%Br%VQJ
% q NC.| R
% Example 1: e9k}n\t3
% ,yAvLY5 P
% % Display the Zernike function Z(n=5,m=1) L
a0 H
% x = -1:0.01:1; /<zBcpVNV
% [X,Y] = meshgrid(x,x); vRn^ n
% [theta,r] = cart2pol(X,Y); WTY{sq\'
o
% idx = r<=1; Ocx=)WKdW
% z = nan(size(X)); \hv*` ukF
% z(idx) = zernfun(5,1,r(idx),theta(idx)); 7 EQ
|p
% figure Lo7 R^>
% pcolor(x,x,z), shading interp `"A\8)6-
% axis square, colorbar @6h=O`X>
% title('Zernike function Z_5^1(r,\theta)')
p,]Hs{R
% e,`+6qP{
% Example 2: !_l W#feR
% <`H:Am `
% % Display the first 10 Zernike functions JgYaA*1X
% x = -1:0.01:1; hb_YdnG
% [X,Y] = meshgrid(x,x); 3AX /A+2
% [theta,r] = cart2pol(X,Y); @~QW~{y
% idx = r<=1; , Z&"@g
% z = nan(size(X)); PO<4rT+B
% n = [0 1 1 2 2 2 3 3 3 3]; JS!rZi
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; M2my>
% Nplot = [4 10 12 16 18 20 22 24 26 28]; 5<,}^4wWZ
% y = zernfun(n,m,r(idx),theta(idx)); .OXvv _?<
% figure('Units','normalized') C1)TEkc"C
% for k = 1:10 A;Xn#t ,(K
% z(idx) = y(:,k); ;gK+AU
% subplot(4,7,Nplot(k)) l4L&hY^
% pcolor(x,x,z), shading interp l_ >^LFOA
% set(gca,'XTick',[],'YTick',[]) t}_qtO7>
% axis square &" K 74
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) (!W:-|[K\
% end _4x X}Z;
% J@p[v3W
% See also ZERNPOL, ZERNFUN2. iNd 8M V
:T5l0h- eC
[=S@lURzm@
% Paul Fricker 11/13/2006 % 89f<F\V
x_2
[+Ol
)z2Tm4>iql
h1FM)n[E7
<M7@JgC &
% Check and prepare the inputs: h&NcN-["
% ----------------------------- FTtYzKX(bv
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) b kLm]n3
error('zernfun:NMvectors','N and M must be vectors.') F>96]71
2
end pWO,yxr:
T%
Kj >-
! Hdg
$,
if length(n)~=length(m) HGh`O\f8
error('zernfun:NMlength','N and M must be the same length.') 2/E3~X7
end 6EGh8H f
W*}q;ub;
_\"7
n = n(:); ~BD Vm Qa
m = m(:); Nt $/JBB[$
if any(mod(n-m,2)) Bei z*2-}a
error('zernfun:NMmultiplesof2', ... z )a8
^]`
'All N and M must differ by multiples of 2 (including 0).') %_K NAuM
end CmY'[ rI
`:}GE@]
Ip4CC '
if any(m>n) f,)[f M4
error('zernfun:MlessthanN', ... x\ *`i)su
'Each M must be less than or equal to its corresponding N.') LXJ"ct
end ^ :6v-
Yx
VkRvmKYl
UF|v=|*{#
if any( r>1 | r<0 ) eH(8T
error('zernfun:Rlessthan1','All R must be between 0 and 1.') iVFHr<zk
end O5{
>k
b U-Cd
zX{ [Z
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) .B6$U>>NS^
error('zernfun:RTHvector','R and THETA must be vectors.') g(;t,Vy,I
end )DI/y1
#T99p+O
A+iQH1C0h
r = r(:); .%M=dL>
theta = theta(:); j_o6+R k
length_r = length(r); L/" u,~[
if length_r~=length(theta) n^UrHHOL
error('zernfun:RTHlength', ... D""d-oI [
'The number of R- and THETA-values must be equal.') n-#?6`>a
end Y6?d
y\
Hh(_sewo
S5-}u)XnH
% Check normalization: A%"mySW
% -------------------- z%hB=V!~91
if nargin==5 && ischar(nflag) ]mn(lK
isnorm = strcmpi(nflag,'norm'); - 9UQs.Nv
if ~isnorm B=(m;A#G
error('zernfun:normalization','Unrecognized normalization flag.') s~6?p%
2]
end \(cu<{=rU
else ujXC#r&
isnorm = false; sG%Q?&-
end ']Nw{}eS`
";J1$a
Y@c! \0e$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% l=Jbu c
% Compute the Zernike Polynomials B ;SYO>.W
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Ja4O*C<
'%. lY9D
z F>|
9JU
% Determine the required powers of r: &\ F`M|c
% ----------------------------------- `jSxq66L p
m_abs = abs(m); CKNC"Y*X
rpowers = []; C o4QWyt:
for j = 1:length(n) $*N jvr7
rpowers = [rpowers m_abs(j):2:n(j)]; IR;lt 3
end #VgPg5k.<
rpowers = unique(rpowers); )Jz L
od"Oq?~/t
pUZbZ
U
% Pre-compute the values of r raised to the required powers, JpvE c!cli
% and compile them in a matrix: w6F4o;<PR
% ----------------------------- ;_@u@$=~
if rpowers(1)==0 1[
ME/r
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); * 8CI'UX
rpowern = cat(2,rpowern{:}); s_N?Y)lS+(
rpowern = [ones(length_r,1) rpowern]; y[UTuFv~Q
else k#_B^J&d
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); C&^"]-t
rpowern = cat(2,rpowern{:}); XkHO =
end : P>Wd3m
*[]7l]XK .
T$U,rOB"
% Compute the values of the polynomials: G'^Qi}o
% -------------------------------------- > )YaW cI
y = zeros(length_r,length(n)); zqh.U @
for j = 1:length(n) B<S uNbR
s = 0:(n(j)-m_abs(j))/2; ycg5S rg
pows = n(j):-2:m_abs(j); G1K5J`"*
for k = length(s):-1:1 Ms ;:+JI
p = (1-2*mod(s(k),2))* ... {9q~bt
prod(2:(n(j)-s(k)))/ ... y m<3
prod(2:s(k))/ ... )@F uw*
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Ai fnC4
prod(2:((n(j)+m_abs(j))/2-s(k))); y*0bHzJ
idx = (pows(k)==rpowers); ^31 X-}t v
y(:,j) = y(:,j) + p*rpowern(:,idx); (, Il>cR4
end nsQx\Tnhx
eGwrSF#a)
if isnorm R=yn4>I
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); HP}d`C5<R
end MDGD *Qn~
end & k*sxW'
% END: Compute the Zernike Polynomials DF|(C Qs9
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |_@ '_
bnt>j0E
N2[EdOJT_
% Compute the Zernike functions: n@<+D`[ .V
% ------------------------------ ~1jSz- s
idx_pos = m>0; . Xn w@\k'
idx_neg = m<0; DUUQz:?{J
*Hx{ eqC
4 8l!P(>?y
z = y; r)UtS4 7
if any(idx_pos) dY'/\dJ
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); P ~/Gla k
end 2{:bv~*I0F
if any(idx_neg) pT\>kqmj
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ;WxE0Q:!~
end ` 1aEV#;
4+qoq$F</
oh
c/{D2
% EOF zernfun = s^KZV
