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, [^"(%{H
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, E]e[Ty1
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? Fr Q-v]c
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? e]L3=R;
pC?1gc1G
PrYWha=c-
CI3_lWax%
+~v3D^L15
function z = zernfun(n,m,r,theta,nflag) 4s+J-l
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. xQ(KmP2hl
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ZkVvL4yIK
% and angular frequency M, evaluated at positions (R,THETA) on the )]e d;V
% unit circle. N is a vector of positive integers (including 0), and r^Rcjyc1
% M is a vector with the same number of elements as N. Each element 5)zj){wL
% k of M must be a positive integer, with possible values M(k) = -N(k) <45dy5!Tz
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, j2v[-N4 {J
% and THETA is a vector of angles. R and THETA must have the same G^eFS;
% length. The output Z is a matrix with one column for every (N,M) i|! 9o:
% pair, and one row for every (R,THETA) pair. k=q%FlE
% W0 ,"V'C
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike -<HvhW
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 5]O LV1Xt
% with delta(m,0) the Kronecker delta, is chosen so that the integral l`w|o
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ->#@rF:S
% and theta=0 to theta=2*pi) is unity. For the non-normalized Fn0LE~O}-8
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. o\V4qekk
% Vm%ux>}
% The Zernike functions are an orthogonal basis on the unit circle. sMpC4E
% They are used in disciplines such as astronomy, optics, and .$E~.6J %i
% optometry to describe functions on a circular domain. |$*9j""u
% p]IhQnj2
% The following table lists the first 15 Zernike functions. K&~#@I;
% 4lo}-@j
% n m Zernike function Normalization q,h.W JI
% -------------------------------------------------- KcyM2hE7
% 0 0 1 1 {xb%P!o`
% 1 1 r * cos(theta) 2 F#C 6.`B
% 1 -1 r * sin(theta) 2 U3iyuE
% 2 -2 r^2 * cos(2*theta) sqrt(6) ,%DAh
% 2 0 (2*r^2 - 1) sqrt(3) U~`^Y8UF
% 2 2 r^2 * sin(2*theta) sqrt(6) )k<~}wvQ0
% 3 -3 r^3 * cos(3*theta) sqrt(8) RBojT
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) j`-y"6)
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) (Y@|h%1W
% 3 3 r^3 * sin(3*theta) sqrt(8) G5@fqh6ws
% 4 -4 r^4 * cos(4*theta) sqrt(10) 4Fc1'
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) vWU4ZBT8G
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) N`GwL
aF
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) vT<wd#
% 4 4 r^4 * sin(4*theta) sqrt(10) ?ut juMdl
% -------------------------------------------------- rVW'KN
% MvwJ(3
% Example 1: [#h!3d|?B
% H
{Wpf9_
K
% % Display the Zernike function Z(n=5,m=1) Dve5m=
% x = -1:0.01:1;
l e/#J
% [X,Y] = meshgrid(x,x); &ZFAUE,[
% [theta,r] = cart2pol(X,Y); @V
CQ4X7T
% idx = r<=1; +OP:"Q_#
% z = nan(size(X)); D`@U[ `Sw
% z(idx) = zernfun(5,1,r(idx),theta(idx)); SPm5tU
% figure hl~F1"q)
% pcolor(x,x,z), shading interp YU"\Wd[
% axis square, colorbar |(8h:g
% title('Zernike function Z_5^1(r,\theta)') "TNUw&ih
% ':>*=&
% Example 2: S#z8H+'
% &l*dYzqq
% % Display the first 10 Zernike functions DG
FvRB
% x = -1:0.01:1; QX3![;0F
% [X,Y] = meshgrid(x,x); %QZ!Tb
% [theta,r] = cart2pol(X,Y); 1VsEic
% idx = r<=1; 9
3I9`!e
% z = nan(size(X)); H%%nB
% n = [0 1 1 2 2 2 3 3 3 3]; PP`n>v=n
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; u(a&x|WY
% Nplot = [4 10 12 16 18 20 22 24 26 28]; kO>{<$
% y = zernfun(n,m,r(idx),theta(idx)); dNt|"9~&
% figure('Units','normalized') -KiS6$-
% for k = 1:10 RN3D:b+
% z(idx) = y(:,k); W,J,h6{F
% subplot(4,7,Nplot(k)) 0'}?3/u-
% pcolor(x,x,z), shading interp .T
X& X
% set(gca,'XTick',[],'YTick',[]) 4V3
w$:,
% axis square !/Ps}.)A`
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) R?Q-@N>wE
% end 3k0%H]wt
% $yK!Q)e:
% See also ZERNPOL, ZERNFUN2. mR@Xt#
><7`$ 2Or
SX,zJ`"
% Paul Fricker 11/13/2006 VMXXBa&
ml2z
*s4!;2ZhsU
Br!&Y9
}w8AnaC
% Check and prepare the inputs: zPc;[uHT
% ----------------------------- !AHm+C_=Lg
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Z.(x|Q9
error('zernfun:NMvectors','N and M must be vectors.') 3%a37/|~y
end 7rg[5hP T
F'*&-l
0G 1o3[F
if length(n)~=length(m) PSE|4{'
error('zernfun:NMlength','N and M must be the same length.') 4BtdN-T}b
end #4u; `j"4=
8[ OiG9b
qZ}XjL
n = n(:); TZ2f-KI
m = m(:); N9<eU!4>
if any(mod(n-m,2)) bm.H0rHR4
error('zernfun:NMmultiplesof2', ... 0wcWDE
9
'All N and M must differ by multiples of 2 (including 0).') bb/MnhB
end r&DK> H
+rY0/T_0,
{`Z)'G\`
if any(m>n) lhTbg M
error('zernfun:MlessthanN', ... X>>rvlD N
'Each M must be less than or equal to its corresponding N.') M3kE91
end x6tY _lzJ
9@B+$~:}7
}VRo: sJb
if any( r>1 | r<0 ) e8.bH#
error('zernfun:Rlessthan1','All R must be between 0 and 1.') 2ZeL
end 8msDJ{,X
Nb1lawC
Akf9nT
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) E>D@#I>
error('zernfun:RTHvector','R and THETA must be vectors.') u{,^#I}
end
p]oo^
tPHiz%
ja2]VbB
r = r(:); 9g"H9)EZ^
theta = theta(:); TbuR?#
length_r = length(r); TW0^wSm
if length_r~=length(theta) 8hg(6 XUG
error('zernfun:RTHlength', ... -(G2@NG
'The number of R- and THETA-values must be equal.') oH0\6:S
end bJD"&h5
=Yk$Q\c
ez>@'yhK
% Check normalization: [PT_y3'%
% -------------------- O.dNhd$
if nargin==5 && ischar(nflag) KWo)}m*6
isnorm = strcmpi(nflag,'norm'); 4`F*] Ft
if ~isnorm =l+p nG
error('zernfun:normalization','Unrecognized normalization flag.') ^-_!:7TH]
end $;1~JOZh
else u4'Lm+&O
isnorm = false; d\f5\Y
end D 4wB
&~U
67eo~~nUtg
+!(hd
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 7d*<'k]{,
% Compute the Zernike Polynomials Yy}aQF#M
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% $j/F7.S
wSK?mS6
,3j*D+
% Determine the required powers of r: c#DTL/8"DO
% ----------------------------------- ORoraEK
m_abs = abs(m); {~"=6iyj
rpowers = []; Ow
I?(ruL'
for j = 1:length(n) JoYzC8/r
rpowers = [rpowers m_abs(j):2:n(j)]; fomkwN
end 9maw+ c!~
rpowers = unique(rpowers); )+G(4eIT
AN;?`AM;
QbWD&8T0O
% Pre-compute the values of r raised to the required powers, ){}#v&
% and compile them in a matrix: lV?SvXe
% ----------------------------- }bVyvH
if rpowers(1)==0 SUw{xGp
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); {e^llfj$#
rpowern = cat(2,rpowern{:}); )
l)5^7=W
rpowern = [ones(length_r,1) rpowern]; =
7?'S#
else 5c#L6 dA)
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false);
,Y!)V
rpowern = cat(2,rpowern{:}); 'e)t+
end ;H^!yj5H
~$$V=$&
#/:[ho{JQ
% Compute the values of the polynomials: yZ{YIy~
% -------------------------------------- O6pjuhMx
y = zeros(length_r,length(n)); vcmS]$}
for j = 1:length(n) rcK*",>
s = 0:(n(j)-m_abs(j))/2; + y^s
6j}
pows = n(j):-2:m_abs(j); [{ pc1U-
for k = length(s):-1:1 4u&doSXR
p = (1-2*mod(s(k),2))* ... P7o6B,9
prod(2:(n(j)-s(k)))/ ... ~(8A&!#,!
prod(2:s(k))/ ... c(jA"K[|b
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... cZYX[.oIB
prod(2:((n(j)+m_abs(j))/2-s(k))); Rq7ks To
idx = (pows(k)==rpowers); ubLLhf
y(:,j) = y(:,j) + p*rpowern(:,idx); ,rd+ dN
end DXUI/C f
i <bs{Cu_S
if isnorm _D:/?=y;e
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); |] YT6-?.
end sxqXR6p{
end >C i=H(8vN
% END: Compute the Zernike Polynomials &jJgAZ!
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% c"~TH.,d
"*0
szz'
1@y?OWC
% Compute the Zernike functions: y8L:nnSj
% ------------------------------ ZoUfQ!2*
idx_pos = m>0; #GF1MFkoS
idx_neg = m<0; qg O)@B+
@dXf_2Tv=
W1OGN4`C
z = y; T \A uL
if any(idx_pos) yH`xk%q_
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); L.R
end v/.2Z(sZ
if any(idx_neg) 8,R]R=
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); D >mLSh
end YPM>FDxDB
gO5;hd[l
}PXWRv.gW
% EOF zernfun Zy#r<j]T