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, `0rRKlb j4
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, IkQe~;Y
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? tvGlp)?.
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? kutJd{68
-x{&an=
' Rc#^U*n
Lc%xc`n8B
0p `")/
function z = zernfun(n,m,r,theta,nflag) WFem#hq
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. r8,om^N6
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N TM-Fu([LMV
% and angular frequency M, evaluated at positions (R,THETA) on the kM;o0wi
% unit circle. N is a vector of positive integers (including 0), and Mb.4J2F ?
% M is a vector with the same number of elements as N. Each element `BjR.xMv
% k of M must be a positive integer, with possible values M(k) = -N(k) +?Ez}
BP
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, $ser+Jt=
% and THETA is a vector of angles. R and THETA must have the same r**f,PDZ
% length. The output Z is a matrix with one column for every (N,M) :3O5ET'1
% pair, and one row for every (R,THETA) pair. <h@]Ri
% vY_eDJ~'
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike %J!NL0x_
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ot }6D
% with delta(m,0) the Kronecker delta, is chosen so that the integral @Z q[e
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 0m
A(:"
% and theta=0 to theta=2*pi) is unity. For the non-normalized (hN?:q?'
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. *V DVC0R
% X3yS5whd(
% The Zernike functions are an orthogonal basis on the unit circle. r^5jh1
% They are used in disciplines such as astronomy, optics, and (;ADW+.`J
% optometry to describe functions on a circular domain. n}q$f|4!
% zN")elBi
% The following table lists the first 15 Zernike functions. V^sc1ak1Q
% i?-Y
% n m Zernike function Normalization n"Z |e tZ4
% -------------------------------------------------- ;A"\?i Q
% 0 0 1 1 *HeVACxo
% 1 1 r * cos(theta) 2 kP^*hO!%
% 1 -1 r * sin(theta) 2 `ET& VV
% 2 -2 r^2 * cos(2*theta) sqrt(6) #c:kCZt#
% 2 0 (2*r^2 - 1) sqrt(3) ``4?a7!!
% 2 2 r^2 * sin(2*theta) sqrt(6) i*CnoQH
% 3 -3 r^3 * cos(3*theta) sqrt(8) ^{[[Z.&R?
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) #U"1 9@|}
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) J@Yj\9U
% 3 3 r^3 * sin(3*theta) sqrt(8) J>h;_jA
% 4 -4 r^4 * cos(4*theta) sqrt(10) BIj
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) wE6A
7\k%
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) F0.z i>5
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Mk5RHDh
% 4 4 r^4 * sin(4*theta) sqrt(10) hKN6 y%
% -------------------------------------------------- ) rpq+~b
% b# ='^W3
% Example 1: %b?uW]j:
% 6$RpV'xz
% % Display the Zernike function Z(n=5,m=1) taDQ65
% x = -1:0.01:1; .iT4-
% [X,Y] = meshgrid(x,x); [K:29N9~4
% [theta,r] = cart2pol(X,Y); |,sMST%
% idx = r<=1; &*gbK6JB
% z = nan(size(X)); &,MFB
% z(idx) = zernfun(5,1,r(idx),theta(idx)); Ct!S Tk[2
% figure FYl3c
% pcolor(x,x,z), shading interp !\x?R6K
% axis square, colorbar {[^#h|U
% title('Zernike function Z_5^1(r,\theta)') Nfb`YU=
% PeNF+5s/K
% Example 2: :<utq|#s
% ir&.Z5=
% % Display the first 10 Zernike functions [r9d<Zi}{
% x = -1:0.01:1; B*79qq
% [X,Y] = meshgrid(x,x); zy>}L #
% [theta,r] = cart2pol(X,Y); wS$46M<
% idx = r<=1; u)~s4tP4
% z = nan(size(X)); vYnftJK&
% n = [0 1 1 2 2 2 3 3 3 3]; A*i_|]Q
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; .NnGVxc5*
% Nplot = [4 10 12 16 18 20 22 24 26 28]; rQj~[Y.c
% y = zernfun(n,m,r(idx),theta(idx)); BIfi:7I;Q
% figure('Units','normalized') vgThK9{m;
% for k = 1:10 9@y3IiZ"}
% z(idx) = y(:,k); XU9'Rfp
% subplot(4,7,Nplot(k)) %VJW@S>j/
% pcolor(x,x,z), shading interp Ue7 6py9
% set(gca,'XTick',[],'YTick',[]) %?=)!;[
% axis square RL&lKHA
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) OKPJuV`y6
% end %rcFT_
% {ERjeuDm]
% See also ZERNPOL, ZERNFUN2. m
=k%,J_
r/PKrw sC
.@k *p >K
% Paul Fricker 11/13/2006 &t_h'JX&
\ja `c)x
ny1 \4C
[hnK/4!
P4 6,o
% Check and prepare the inputs: jdlG#j-\
% ----------------------------- rBfg*r`)
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) j-32S!
error('zernfun:NMvectors','N and M must be vectors.') _9kIRmT{
end j@ v-|
D9o*8h2$
n(R_#,Hs
if length(n)~=length(m) o](.368+4
error('zernfun:NMlength','N and M must be the same length.') h=[-Er'B
end
~6d5zI4\
:hP58 }Q$
} yq
n = n(:); T 2|:nC)@
m = m(:); _}ele+
if any(mod(n-m,2)) ,sI35I J
error('zernfun:NMmultiplesof2', ... %6i=lyH-
'All N and M must differ by multiples of 2 (including 0).') fU
={a2
end oMc1:=EG
W~NYU
4B$bj`h
if any(m>n) P)1EA;
error('zernfun:MlessthanN', ... kl<g;3
'Each M must be less than or equal to its corresponding N.') 2AK}D%jfc
end (\&
62B1
!Uy>eji}
6~@5X}^<0
if any( r>1 | r<0 ) Z4@y?fv7s
error('zernfun:Rlessthan1','All R must be between 0 and 1.') 7K :FeW'N
end \ V?I+Gc
qZbHMTnT6
[YE?OQ7#
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) )X%oXc&C|
error('zernfun:RTHvector','R and THETA must be vectors.') 0jTMZ<&zZ
end a}yR p
K PggDKS
Cuv|6t75'
r = r(:); tJm{I)G
theta = theta(:); ^c'f<<z|7r
length_r = length(r); u){S$</
if length_r~=length(theta) 3:AU:
error('zernfun:RTHlength', ... 61,O%lV
'The number of R- and THETA-values must be equal.') kfK[u/<i
end E9R]sXf8
^A#x<J+
w4A#>;Qu*
% Check normalization: `^e*T'UPl
% -------------------- \(bj(any
if nargin==5 && ischar(nflag) yHOqzq56
isnorm = strcmpi(nflag,'norm'); dEET}s\
if ~isnorm 4if\5 P:j
error('zernfun:normalization','Unrecognized normalization flag.') UR,?! rJ^B
end Z@oKz:U
else ^O \q3HA_4
isnorm = false; )Ga8`t"
end u\3ZIb
UM\}aq=,
xT=ySa$|>
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% KBj@V6Q
% Compute the Zernike Polynomials 0%H24N
9.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |0]YA
hXTYTbTX
kQ[Jo%YT?E
% Determine the required powers of r: `u=oeM:
% ----------------------------------- #G~wE*VR$
m_abs = abs(m); tvCcyD%w
rpowers = []; X TM$a9)
for j = 1:length(n) t%HI1eO7h
rpowers = [rpowers m_abs(j):2:n(j)]; b=G4MZQ
end ogp{rY
rpowers = unique(rpowers); ]_\AHnJ
Hh\
4MNl
Iu%^*K%
% Pre-compute the values of r raised to the required powers, S*s:4uf
% and compile them in a matrix: l.uN$B
% ----------------------------- ->3uOF!q
if rpowers(1)==0 &t_A0z
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); yWmrdvL
rpowern = cat(2,rpowern{:}); $r):d
rpowern = [ones(length_r,1) rpowern]; ?(>k,[n
else HoL~j( {
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); z6 a,0&;-L
rpowern = cat(2,rpowern{:}); WV@X@]U
end i0b.AA
1y~L8!:L
7|{ B#
% Compute the values of the polynomials: VZTmzIk.Y
% -------------------------------------- "&Gw1.p
y = zeros(length_r,length(n)); #)FDl70S8
for j = 1:length(n) @Jm.HST#S8
s = 0:(n(j)-m_abs(j))/2; yYM_lobn
pows = n(j):-2:m_abs(j); hAlPl<BO#V
for k = length(s):-1:1 G LoiH#R
p = (1-2*mod(s(k),2))* ... S7Znz@
prod(2:(n(j)-s(k)))/ ... brj[c>ID
prod(2:s(k))/ ... OgQntj:%lN
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ovB=Zm
prod(2:((n(j)+m_abs(j))/2-s(k))); L,WkJe3
idx = (pows(k)==rpowers); w"BIv9N
y(:,j) = y(:,j) + p*rpowern(:,idx); D(!;V
KH
end ygMd$0:MN
b]"2VN
if isnorm 3Fgz)*Gu]
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); JV&Zwbu
end )=y.^@UT@
end ?THa5%8f
% END: Compute the Zernike Polynomials O/(3 87= U
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LNaeB(z"
dV)Y,Yx0${
z}iSq$
% Compute the Zernike functions: (X*'y*:
% ------------------------------ n%n'1AUP:
idx_pos = m>0; hN[X 1*
idx_neg = m<0; A0S8Dh$
(v]P<3%
b By'v/
z = y; PBCb0[\
if any(idx_pos) kp'b>&9r
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); )4@M`8
end q)NXyy4BT
if any(idx_neg) ,tau9>!
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); j,\tejl1
end Wa(W&]
bAN 10U
3'.!
+#
% EOF zernfun JIVo=5c}