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, (b//YyqN
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,
_$c o 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? qX5>[qf-
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? CU\gx*=E
UWC4PWL,>C
1g{}O^ul
$M,<=.oT
I<D7Jj
function z = zernfun(n,m,r,theta,nflag) G6zFQ\&f
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 6384$mT,S
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N {{Ox%Zm
% and angular frequency M, evaluated at positions (R,THETA) on the fEXFnQ#
% unit circle. N is a vector of positive integers (including 0), and jDb\4QyC
% M is a vector with the same number of elements as N. Each element zgEN2d
% k of M must be a positive integer, with possible values M(k) = -N(k) >"bW'
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, wrgB =o
% and THETA is a vector of angles. R and THETA must have the same )~=8Ssu
% length. The output Z is a matrix with one column for every (N,M) \^"Vqx
% pair, and one row for every (R,THETA) pair. G`O*AQ}[
% n]$rLm%^
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike s0;a j<J
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), !' ^l}K>
% with delta(m,0) the Kronecker delta, is chosen so that the integral ial{A6X
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 4bA^Gq
% and theta=0 to theta=2*pi) is unity. For the non-normalized oio{@#DX`
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ?SFQx\/
% SgewAng?@o
% The Zernike functions are an orthogonal basis on the unit circle. er7(Wph
% They are used in disciplines such as astronomy, optics, and GWuKDq
% optometry to describe functions on a circular domain. AJEbiP
% O)vGIp?f't
% The following table lists the first 15 Zernike functions. C}mhnU@
% !;|#=A9
% n m Zernike function Normalization hxMRmH[f:
% -------------------------------------------------- Eej
Lso#\
% 0 0 1 1 #W)m({}
% 1 1 r * cos(theta) 2 B;(U?gC
% 1 -1 r * sin(theta) 2 C_Q3^mLx
% 2 -2 r^2 * cos(2*theta) sqrt(6) S!u8JG1
% 2 0 (2*r^2 - 1) sqrt(3) a($7J6]M
% 2 2 r^2 * sin(2*theta) sqrt(6) r_$*euh@
% 3 -3 r^3 * cos(3*theta) sqrt(8) W%>T{}4
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) V9$T=[
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) u:|^L]{
% 3 3 r^3 * sin(3*theta) sqrt(8) _LwF:19Il
% 4 -4 r^4 * cos(4*theta) sqrt(10) P1rjF:x[*
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) R;Dj70g
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) fEL 9J{
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) \DujF>:
% 4 4 r^4 * sin(4*theta) sqrt(10) g@!U^mr*3
% -------------------------------------------------- /A,w{09G
% /g+-{+sx
% Example 1: Xrb7.Y0d
% 63l&
ihj
% % Display the Zernike function Z(n=5,m=1) 85G-`T
% x = -1:0.01:1; @z ",1^I
% [X,Y] = meshgrid(x,x); !hq*WtIk
% [theta,r] = cart2pol(X,Y); |E?r+]
% idx = r<=1; N!~]D[D
% z = nan(size(X)); SgxrU&::
% z(idx) = zernfun(5,1,r(idx),theta(idx)); dX/7n=
% figure I
m
I$~q'
% pcolor(x,x,z), shading interp ?H PAX
% axis square, colorbar z7IJSj1gQI
% title('Zernike function Z_5^1(r,\theta)') e&ysj:W5
"
% [yN+(^i
% Example 2: j8Z;}Ps
% @6~lZgXOV[
% % Display the first 10 Zernike functions ]P wS3:x
% x = -1:0.01:1; R&Nl!QTJj
% [X,Y] = meshgrid(x,x); ow9a^|@a
% [theta,r] = cart2pol(X,Y); y*^UGJC:
% idx = r<=1; Ph""[0n%o
% z = nan(size(X)); CBf[$[e
% n = [0 1 1 2 2 2 3 3 3 3]; _N|%i J5
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; ZS=H1
% Nplot = [4 10 12 16 18 20 22 24 26 28]; Hj
r'C?[
% y = zernfun(n,m,r(idx),theta(idx)); R]%"YQ V
% figure('Units','normalized') d*{Cv2A.
% for k = 1:10 ?&wrz
% z(idx) = y(:,k); oH6zlmqG"
% subplot(4,7,Nplot(k)) qI7KWUR
% pcolor(x,x,z), shading interp %dPk,Ylz
% set(gca,'XTick',[],'YTick',[]) %Ve@DF8G
% axis square o%~fJx:]y
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ?SgFD4<~P
% end &6OY^6<
% :a/rwZ[r
% See also ZERNPOL, ZERNFUN2. {Ia1H
E<+ G5j
^
34Ng
% Paul Fricker 11/13/2006 )-!)D
dlfjx
B,%6sa~I
p*lP9[7
8a8a:d
% Check and prepare the inputs: ^yB]_*WJ
% ----------------------------- !Q|a R
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ;6PU
error('zernfun:NMvectors','N and M must be vectors.') trrNu
end Jkj7ty.J
}:faHLYT
qX\85dPn@}
if length(n)~=length(m) 3>VL>;75[
error('zernfun:NMlength','N and M must be the same length.') ]*| hd/j
end {2:baoG-
M5:.\0_
n+sv2Wv:
n = n(:); (LTu=1
m = m(:); m]U
if any(mod(n-m,2)) _@>*]g
error('zernfun:NMmultiplesof2', ... </_QldL_
'All N and M must differ by multiples of 2 (including 0).') ]>)shH=Yx
end ^V; r
o`Z3}
`uPO+2
if any(m>n) wwdmz;0S
error('zernfun:MlessthanN', ... ib(|}7Je
'Each M must be less than or equal to its corresponding N.') rR@]`@9
end [VXQ&
m33&obSP
iSf%N>y'K
if any( r>1 | r<0 ) W gyRK2#!
error('zernfun:Rlessthan1','All R must be between 0 and 1.') d>F7i~W
end X~VI} dJ
axC{azo|
Ld_u Me?Z
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) QmSj6pB>
error('zernfun:RTHvector','R and THETA must be vectors.') ;q-c[TZC
end sT1OAK\^
4qDO(YWf
46T(1_Xt~
r = r(:); Zex~ $r
theta = theta(:); <#BK(W~$
length_r = length(r); a K6dy\
if length_r~=length(theta) BDfMFH[1
error('zernfun:RTHlength', ... K3:z5j.X
'The number of R- and THETA-values must be equal.') .&b^6$dC
end tBzE(vW
_"Y7}A\9
`/m]K~~
% Check normalization: -]KgLgJ
% -------------------- U*K4qJ6U
if nargin==5 && ischar(nflag) M)K!!Jqh
isnorm = strcmpi(nflag,'norm'); c(Y~5A{TXO
if ~isnorm )OQm,5F1
error('zernfun:normalization','Unrecognized normalization flag.') ][Tw^r&
end h/C{
else z<t2yh(DF
isnorm = false; DmgDhNXKq
end &0T7Uv-`
R $<{"b
+~F>:v?Rh
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 1^NC=IS9z
% Compute the Zernike Polynomials ?XVE{N
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ,O.iOT0=;
oAN,_1v)
.&]3wB~
% Determine the required powers of r: #QlxEs#%
% ----------------------------------- 2IHS)kkT|
m_abs = abs(m); _\dC<K *>
rpowers = []; [%LGiCU]
for j = 1:length(n) F',1R"/}
rpowers = [rpowers m_abs(j):2:n(j)]; cyd_xB5K
end Ye|gW=FUR
rpowers = unique(rpowers); +-t&li%F
#('R`~
BuM#&]s
% Pre-compute the values of r raised to the required powers, ~^Al#@
% and compile them in a matrix: -|#/KKF
% ----------------------------- \s8h.xjU
if rpowers(1)==0 kQ\l7xd
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); g; R
rpowern = cat(2,rpowern{:}); OFv-bb*YZ
rpowern = [ones(length_r,1) rpowern]; !N\_D
else r!{i2I|
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); p{qA%D
rpowern = cat(2,rpowern{:}); #Z]Cq0=
end #l)o<Z
NV{= tAR
R^@`]dX$
% Compute the values of the polynomials: XH0Vs.w
% -------------------------------------- uUBUUr
y = zeros(length_r,length(n)); XOS^&;
for j = 1:length(n) ]EN&EA"<
s = 0:(n(j)-m_abs(j))/2; RigS1A\2l
pows = n(j):-2:m_abs(j); "7(@I^'t6
for k = length(s):-1:1 B2BG*xa
p = (1-2*mod(s(k),2))* ... 'q/C: Yo
prod(2:(n(j)-s(k)))/ ... 4u2_xbT
prod(2:s(k))/ ... @/01MBs;
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... [D?xd/G
prod(2:((n(j)+m_abs(j))/2-s(k))); A&KY7[<AC{
idx = (pows(k)==rpowers); Bd>ATc+580
y(:,j) = y(:,j) + p*rpowern(:,idx); fe6Op
end #\ ="^z6
iRW5*-66f
if isnorm H-WNu+
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); G'HLnx}Yi
end 02^\np
end rP6k}
% END: Compute the Zernike Polynomials Cx) N;x
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% v+C D{Tc
,pZz`B#
>9g^-~X;v
% Compute the Zernike functions: 4Im}!q5;:<
% ------------------------------ )i-`AJK-'v
idx_pos = m>0; ;%>X+/.y0
idx_neg = m<0; 0icB2Jm:D}
DAN"&&
:w4 H$+j
z = y; "tK3h3/Xv
if any(idx_pos) f|!@H><
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); tJGPkeA
end k[1[Y{n.
if any(idx_neg) HqOnZ>D
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); -x/g+T-
end cwUor}<|
b]8\%=d
ws]d,]
% EOF zernfun 2NL|_W/