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， V0n8fez
b
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， ),$^h7[n
这个，跟我们用zygo干涉仪直接拟合出的36项zernike系数，有何关系呢？ (%G>TV
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ rS+ >oP}
X^i3(N
<SdOb#2
XW+-E^d
:*-O;Yw?S@
function z = zernfun(n,m,r,theta,nflag) c?6(mU\x
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. oG-Eac,
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N rddn"~lm1
% and angular frequency M, evaluated at positions (R,THETA) on the ?"kU+tCxg
% unit circle. N is a vector of positive integers (including 0), and Jg$ NYs.xZ
% M is a vector with the same number of elements as N. Each element D0L s~qr
% k of M must be a positive integer, with possible values M(k) = -N(k) [ C!m ,4
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, y^ D3}ds
% and THETA is a vector of angles. R and THETA must have the same C Y"i|s
% length. The output Z is a matrix with one column for every (N,M) &E@mCQ1
% pair, and one row for every (R,THETA) pair. IvI;Q0E-3
% `W7; -
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike #IeG/ t(
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), !:~C /B{
% with delta(m,0) the Kronecker delta, is chosen so that the integral )&-n -m@E
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, mS.!lkV
% and theta=0 to theta=2*pi) is unity. For the non-normalized nO;ox*Bk+8
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. '=d y
=
% `.wgRUhFH;
% The Zernike functions are an orthogonal basis on the unit circle. 24f N3
% They are used in disciplines such as astronomy, optics, and 8jiBLZkRf
% optometry to describe functions on a circular domain. xscR Bx
% (1's Bm7F
% The following table lists the first 15 Zernike functions. h}}7_I9
% iphdJZ/f
% n m Zernike function Normalization @?</8;%3W
% -------------------------------------------------- z; >O5a>z
% 0 0 1 1 3Q,p ,
% 1 1 r * cos(theta) 2 L l,nt
% 1 -1 r * sin(theta) 2 ]ed7Q3lq
% 2 -2 r^2 * cos(2*theta) sqrt(6)
F |_mCwA
% 2 0 (2*r^2 - 1) sqrt(3) v4\
m9Pu4
% 2 2 r^2 * sin(2*theta) sqrt(6) y }h2
% 3 -3 r^3 * cos(3*theta) sqrt(8) \;+ b1
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) lg@q}
]1
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) YT@N$kOg_
% 3 3 r^3 * sin(3*theta) sqrt(8) u@zT~\ h*
% 4 -4 r^4 * cos(4*theta) sqrt(10) MN>U jFA
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) j92+kq>Xd
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) vVo# nzeZ5
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Z
eWst w7
% 4 4 r^4 * sin(4*theta) sqrt(10) }~#qDrK
% -------------------------------------------------- ( e<p^T J]
% t 2q WB[r
% Example 1: 2 xi@5;!
% eqpnh^0}d
% % Display the Zernike function Z(n=5,m=1) 8
;=?Lw ?
% x = -1:0.01:1; x
o7 2JJ
% [X,Y] = meshgrid(x,x); b+BX >$
% [theta,r] = cart2pol(X,Y); U"Z % _[*
% idx = r<=1; ]`}EOS-Q
% z = nan(size(X)); | D8c=c%
% z(idx) = zernfun(5,1,r(idx),theta(idx)); 4 Q\~l(
% figure ^{g+HFTA@
% pcolor(x,x,z), shading interp Z3iX^
% axis square, colorbar *y W9-(
% title('Zernike function Z_5^1(r,\theta)') ?_/T$b ]
% fJY
b)sN
% Example 2: -AKbX kc~\
% @T sdgx8
% % Display the first 10 Zernike functions 6<UI%X
% x = -1:0.01:1; <%oT}K\;
% [X,Y] = meshgrid(x,x); M5S<N_+Pe
% [theta,r] = cart2pol(X,Y); fXkemB^)_
% idx = r<=1; %'dsb7n
% z = nan(size(X)); A OCiIPw
% n = [0 1 1 2 2 2 3 3 3 3]; Uq2 Qh@B
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; J-Fqw-<aFJ
% Nplot = [4 10 12 16 18 20 22 24 26 28]; P"c7h7
% y = zernfun(n,m,r(idx),theta(idx)); yMf["A vG
% figure('Units','normalized') uC1v^!D
% for k = 1:10 e#4 iue7U
% z(idx) = y(:,k); `Y7&}/OM
% subplot(4,7,Nplot(k)) 1;+(HB
% pcolor(x,x,z), shading interp {>#4{D00
% set(gca,'XTick',[],'YTick',[]) ;[-y>qU0
% axis square ! q*]_1
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) $h' >Zvf
% end =+wkjTO
% }-M% $ ~`
% See also ZERNPOL, ZERNFUN2. T <ekDhlr
+' ?axv6e
h}P""
% Paul Fricker 11/13/2006 L|w}#|-
O .P:~
K 7d] p0d'
<' b%
Hd=!
% Check and prepare the inputs: !rgdOlTR ^
% ----------------------------- `)eq TeW
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) O7T wM Yh
error('zernfun:NMvectors','N and M must be vectors.') -"3<L l
end @Tf5YZ*
^2um.`8
)`]} D[j
if length(n)~=length(m) !8 >tT
error('zernfun:NMlength','N and M must be the same length.') `=~d^wKYJ3
end |70L h +
q P>Gre
uEkUK|
n = n(:); _}wy|T&7k&
m = m(:); }))JzrqAe
if any(mod(n-m,2)) 68jq1Y
Pv
error('zernfun:NMmultiplesof2', ... tr\}lfK%
'All N and M must differ by multiples of 2 (including 0).') *HN0em
end Ot_ xeg;7
g4*]R>f
B^uQv|m
if any(m>n) bi[gyl#
error('zernfun:MlessthanN', ... hSD uByoi
'Each M must be less than or equal to its corresponding N.') QK%6Ncv
end p,+$7f1S
geu8$^
bI~(<-S~K
if any( r>1 | r<0 ) ByPzA\;e
error('zernfun:Rlessthan1','All R must be between 0 and 1.') KBo/GBD]|
end h $}&N
C/Dc1sj
hE>i~:~R
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 2@S{e$YK`
error('zernfun:RTHvector','R and THETA must be vectors.') `P< m `*
end u%=M4|7
zy9# *gGq
!&D &Gs
r = r(:); t`X-jr)g
theta = theta(:); }
. cP
length_r = length(r); YQ}bG{ V
if length_r~=length(theta) OQON~& ~
error('zernfun:RTHlength', ... wg[ D*a
'The number of R- and THETA-values must be equal.') dF%sD|<)
end 4X2/ n
3N,!y
ag FWye
% Check normalization: yA+:\%y$
% -------------------- | :i``gFj
if nargin==5 && ischar(nflag) p}N IZ)]$
isnorm = strcmpi(nflag,'norm'); :8bz+3p
if ~isnorm .^S#h
(A
error('zernfun:normalization','Unrecognized normalization flag.') b;O|-2AR
end cM;& $IjCt
else "[(I*
isnorm = false; t F<|Eja *
end .)>DFGb>H
'o2x7~C@
H`$s63
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% =kUN ^hb
% Compute the Zernike Polynomials t YmR<^
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 1wl8
.h2K $(/
:*"0o{
ie
% Determine the required powers of r: ozbu|9 +v
% ----------------------------------- gNO<`9q
m_abs = abs(m); UNJ]$x0
rpowers = []; fRe $}KX
for j = 1:length(n) 3Q`F x
rpowers = [rpowers m_abs(j):2:n(j)]; 6U k[_)1
end W,i SN}
rpowers = unique(rpowers); $*bd})y)I
1Ig@ gdmz
[ }|-% 4s
% Pre-compute the values of r raised to the required powers, z&o"K\y\
% and compile them in a matrix: ; 9pOtr
% ----------------------------- ?3"bu$@8
if rpowers(1)==0 `<h}Ygo>k/
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); xoD5z<<
rpowern = cat(2,rpowern{:}); '~ 3a(1@8
rpowern = [ones(length_r,1) rpowern]; ki#O ^vl
else gd6We)&
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); m Kwhd} V
rpowern = cat(2,rpowern{:}); x )wIGo
end ) w.cCDL c
7CzZHkTg
] }XK
% Compute the values of the polynomials: ;SF0}51
% -------------------------------------- Cyxt EzPp
y = zeros(length_r,length(n)); O&=?,zLO[
for j = 1:length(n) 'g8~539{&
s = 0:(n(j)-m_abs(j))/2; +.Kmpw4
pows = n(j):-2:m_abs(j); N0GID-W!/~
for k = length(s):-1:1 c xdhG "
p = (1-2*mod(s(k),2))* ... ysfR@ sH7
prod(2:(n(j)-s(k)))/ ... `wI<LTzXS
prod(2:s(k))/ ... nn"!x|c
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 2Av 3.u8%u
prod(2:((n(j)+m_abs(j))/2-s(k))); BYN<|=
idx = (pows(k)==rpowers); v6
DN:!&
y(:,j) = y(:,j) + p*rpowern(:,idx); h3D8eR.
end #F.;N<a
kDJ5x8Q#
if isnorm 4 w^B&e%
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 3ryIXC\v
end v9Oyboh(y
end KP7bU9odJ
% END: Compute the Zernike Polynomials EVM hc"L
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *plsZ*Q8
'8~7Ru\KyX
G8@({EY
% Compute the Zernike functions: iLn)Z0<\o
% ------------------------------ t<9oEjk["
idx_pos = m>0; M ]W'>g)G
idx_neg = m<0; I+w 3It
_/ZIDIn
|Sy | E
z = y; ?@ l9T)fF
if any(idx_pos) k/O|ia 6
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); _CP e
end D Y($
if any(idx_neg) l/`<iG%
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); a <FzHCw
end ZPn`.Qc
>fI<g8N D
db:b%1hk:
% EOF zernfun [ kgT"?w=
