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, %dq|)r
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, %B04|Q
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? Df=Xbf>jt9
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? o=Ia{@
!Er)|YP
@'JA3V}
C ,[q#D4
EsjZ;D,c(
function z = zernfun(n,m,r,theta,nflag) n*A"}i`ix
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. ?pkGejcQ
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N H*h4D+Kxv
% and angular frequency M, evaluated at positions (R,THETA) on the '%KaAi$
% unit circle. N is a vector of positive integers (including 0), and @P6*4W
% M is a vector with the same number of elements as N. Each element I0} G,
q
% k of M must be a positive integer, with possible values M(k) = -N(k) j&Y{
CFuZ
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, Io]KlR@!T
% and THETA is a vector of angles. R and THETA must have the same mxmj
% length. The output Z is a matrix with one column for every (N,M) [ Ru( H
% pair, and one row for every (R,THETA) pair. NJPp6RZ%
% >JT^[i8[
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike "1ov<
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), eOs 4c`
% with delta(m,0) the Kronecker delta, is chosen so that the integral v6O5n(5,,
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, l#rr--];
% and theta=0 to theta=2*pi) is unity. For the non-normalized `W'S'?$
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. _TjRvILC
% T!QAcO
% The Zernike functions are an orthogonal basis on the unit circle. ,*g.?q@W2
% They are used in disciplines such as astronomy, optics, and 0EBHRY_F
% optometry to describe functions on a circular domain. :;N2hnHoG
% j&"GE':Y
% The following table lists the first 15 Zernike functions. Q=F^Y f
% f- ~]
% n m Zernike function Normalization .*nr3dY
% -------------------------------------------------- "hLmwz|a
% 0 0 1 1 UaM&/K9
% 1 1 r * cos(theta) 2 RW^e#z>m"E
% 1 -1 r * sin(theta) 2 |!*abc\`(`
% 2 -2 r^2 * cos(2*theta) sqrt(6) R|R3Ob.e
% 2 0 (2*r^2 - 1) sqrt(3) IZ9*
'0Z
% 2 2 r^2 * sin(2*theta) sqrt(6) `l+9g"q
% 3 -3 r^3 * cos(3*theta) sqrt(8) bipA{VU
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) =7Sw29u<
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) ew*;mQd
% 3 3 r^3 * sin(3*theta) sqrt(8) u^+
(5|
% 4 -4 r^4 * cos(4*theta) sqrt(10) #-K,,"
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 8QN/D\uq
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) o;'-^ LJ
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) )KcY<K
% 4 4 r^4 * sin(4*theta) sqrt(10) V*1-wg5>
% -------------------------------------------------- tS6r4d%~=
% A5%cgr% 6
% Example 1: Vl0Y'@{
% 7WEoyd
% % Display the Zernike function Z(n=5,m=1) CAbT9Wz&
% x = -1:0.01:1; Wo<kKkx2
% [X,Y] = meshgrid(x,x); ms/Q-
% [theta,r] = cart2pol(X,Y); ,Zb_Pu
% idx = r<=1; )C%S`d<%,
% z = nan(size(X)); \\$wg
% z(idx) = zernfun(5,1,r(idx),theta(idx)); @S?D}myD
% figure Z]=9=S|
.4
% pcolor(x,x,z), shading interp .oz(,$CS"
% axis square, colorbar 1L<X+,]@
% title('Zernike function Z_5^1(r,\theta)') q]OgT4ly
%
4B'-tV
% Example 2: }Fb966 $
% I_On0@%T5b
% % Display the first 10 Zernike functions !l~3K(&4
% x = -1:0.01:1; T*zy^we
% [X,Y] = meshgrid(x,x); J|N>}di
% [theta,r] = cart2pol(X,Y); -|`E'b81
% idx = r<=1; *sq+ Vc(
% z = nan(size(X)); P*9L3R*=N
% n = [0 1 1 2 2 2 3 3 3 3]; Pc=:j(
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; l#;o^H i
% Nplot = [4 10 12 16 18 20 22 24 26 28]; A?Gk8
% y = zernfun(n,m,r(idx),theta(idx)); @po|07
% figure('Units','normalized') &1ss
@-
% for k = 1:10 Gkz~xQy1T
% z(idx) = y(:,k); A3zO&4f
]
% subplot(4,7,Nplot(k)) U$T
(R2@
% pcolor(x,x,z), shading interp D]WU,a[$Bc
% set(gca,'XTick',[],'YTick',[]) eLyaTOZadu
% axis square o Np4> 7Lk
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ^li(q]g1!
% end [C( >e0r
% 02~GT_)$^
% See also ZERNPOL, ZERNFUN2. za[;d4<}k
o]m56
z)&GF$*
% Paul Fricker 11/13/2006 i0*6o3h
F=8gtk|U
;Ak 6*Sr
vAo|o*
]|)M /U *
% Check and prepare the inputs: C_
(s
% ----------------------------- )GF>]|CG
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) LOlj8T8Z
error('zernfun:NMvectors','N and M must be vectors.') eVujur$P
end ,: 4+hJ<q
%XK<[BF
`~ {0
if length(n)~=length(m) il>XV>
error('zernfun:NMlength','N and M must be the same length.') #;9n_)
end _33YgO
A_<1}8{L
HLp'^
n = n(:); tCirdwmg
m = m(:); rc)vVv
if any(mod(n-m,2)) vV 7L
:>
error('zernfun:NMmultiplesof2', ...
"xY]&
'All N and M must differ by multiples of 2 (including 0).') D-4\AzIb
end ro*$OLc/
p_Y U!j_VE
qW'5Zk
if any(m>n) ?ZlN$h^
error('zernfun:MlessthanN', ... 7T-}oNaJA\
'Each M must be less than or equal to its corresponding N.') )Qx&m}
end :h60
`]\:%+-
8n Oent0a
if any( r>1 | r<0 ) ctWH?b/ua
error('zernfun:Rlessthan1','All R must be between 0 and 1.') 5W~-|8m
end coFQu ;i
=}Xw}X+[WY
ejI nJ
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) o5|P5h
error('zernfun:RTHvector','R and THETA must be vectors.') 3QO*1P@q
end i>n)T
r-k,4Yz
3# r`e
r = r(:); Uv"O'Z
theta = theta(:); r2; )VS
length_r = length(r); VN!+r7w'
if length_r~=length(theta) T|FF&|Pk
error('zernfun:RTHlength', ... !$|h[ct
'The number of R- and THETA-values must be equal.') [_,Gk]F=
end 'Xw>?[BB
(jB_uMuS
qGPIKu
% Check normalization: R2!_)Rpf
% -------------------- A *_ |/o
if nargin==5 && ischar(nflag) 3a\.s9A"
isnorm = strcmpi(nflag,'norm'); li~#6$
if ~isnorm Q]oCzSi
error('zernfun:normalization','Unrecognized normalization flag.') `SGI
Qrb
end ww(.
else gm}[`GMU
isnorm = false; /~B\1
end 2or!v^^u
xfJ&11fG2
skRI\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% >[|Y$$
% Compute the Zernike Polynomials TB
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% YoEL|r|
x9{&rldC
R"
'=^
% Determine the required powers of r: C].w)B
% ----------------------------------- ,Xt!dT-
m_abs = abs(m); k%S;N{Qh@
rpowers = []; ZyQ+}rO
for j = 1:length(n) mrvPzoF,]
rpowers = [rpowers m_abs(j):2:n(j)]; KJ&~z? X
end jWL;ElM'
rpowers = unique(rpowers); uEPdL':}2
DeTD.)pS
4RXF.kJ3=
% Pre-compute the values of r raised to the required powers, v)AadtZ0d
% and compile them in a matrix: t9yjfyk9W
% ----------------------------- >u)DuZXj
if rpowers(1)==0 -<GSHckD
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); onOvE Y|R
rpowern = cat(2,rpowern{:}); Skn2-8;10
rpowern = [ones(length_r,1) rpowern]; !WD~zZ|
else CF?TW
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); jLLZZPBK
rpowern = cat(2,rpowern{:}); kbF+aS
end 3S_H hvB
5QoU&Hv
_g#v*7o2@
% Compute the values of the polynomials: qIIl,!&}A
% -------------------------------------- hz8Z)xjJ V
y = zeros(length_r,length(n)); lh?TEQ
for j = 1:length(n) oA1d8*i^E
s = 0:(n(j)-m_abs(j))/2; 9/nS?>11
pows = n(j):-2:m_abs(j); DKGZm<G>
for k = length(s):-1:1 7<ZCeM2x
p = (1-2*mod(s(k),2))* ... $sX X6K),
prod(2:(n(j)-s(k)))/ ... ;:)?@IuSy
prod(2:s(k))/ ... )(&WhZc Z
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... "uthFE
prod(2:((n(j)+m_abs(j))/2-s(k))); 0,x<@.pW
idx = (pows(k)==rpowers); )K+Tvx3(m
y(:,j) = y(:,j) + p*rpowern(:,idx); EhBYmc"&
end d^Jf(NE0Yo
AX= 4{b'
if isnorm `vijd(a?v
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); w[V71Iej
end TnvX&Y'
end ~YX!49XfHh
% END: Compute the Zernike Polynomials lN-[2vT<
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 8eVQnp*
]ZjydQjo)
c!{]Z_d\
% Compute the Zernike functions: =H\ig%%E@
% ------------------------------ u9 yXHf
idx_pos = m>0; 34$qV{Y%y
idx_neg = m<0; X!w&ib-
z^q ~|7
[MkXQwY
z = y; #
[0>wEq
if any(idx_pos) o|v_+<zD!
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); mJ3|UClPS
end {{\
d5CkX
if any(idx_neg) v_zVhEtY
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); Cy~Pfty
end F5#P{zk|
e I 6G
t*&O*T+fgy
% EOF zernfun *:*Kdt`'G