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， <Dm6CH
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， 8@h zw~>
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？ #0`"gR#+
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ dt`L}Yi
)sV#
b
R1Pnj
W$'pUhq\H
yFpHRfF}
function z = zernfun(n,m,r,theta,nflag) On'3K+(_
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. Z;>~<#!4
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ,x Tbt4J
% and angular frequency M, evaluated at positions (R,THETA) on the $--PA$H2 7
% unit circle. N is a vector of positive integers (including 0), and \^#1~Kx
% M is a vector with the same number of elements as N. Each element rM?D7a{q
% k of M must be a positive integer, with possible values M(k) = -N(k) fwq|8^S@
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, nZ bg
% and THETA is a vector of angles. R and THETA must have the same `% ENGB|
% length. The output Z is a matrix with one column for every (N,M) eGTK^p
% pair, and one row for every (R,THETA) pair. ~zm/n,Epb
% )K?GAj]Pq
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike > T-O3/KN
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), D{loX6
% with delta(m,0) the Kronecker delta, is chosen so that the integral W ^Rb~b^?
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, l ?R_ wu,Q
% and theta=0 to theta=2*pi) is unity. For the non-normalized I"!gzI`Sd
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ;
zv nDo x
% H9>&"=".
% The Zernike functions are an orthogonal basis on the unit circle. Bjo&
% They are used in disciplines such as astronomy, optics, and p|FX_4RjX
% optometry to describe functions on a circular domain. <Z__Q
% &*y ve}su
% The following table lists the first 15 Zernike functions. ('lnQD. Hd
% GawO>7w8
% n m Zernike function Normalization } O>Zu[8a
% -------------------------------------------------- 1(?J>{-lw
% 0 0 1 1 #NE^f2
% 1 1 r * cos(theta) 2 @T prS d
% 1 -1 r * sin(theta) 2 4bBxZY
% 2 -2 r^2 * cos(2*theta) sqrt(6) g.]'0)DMW
% 2 0 (2*r^2 - 1) sqrt(3) *nc4X9
% 2 2 r^2 * sin(2*theta) sqrt(6) >YfOR%mS4
% 3 -3 r^3 * cos(3*theta) sqrt(8) :L[6a>"neE
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) D1n2Z :9
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) L C7L O
% 3 3 r^3 * sin(3*theta) sqrt(8) )q^vitkjup
% 4 -4 r^4 * cos(4*theta) sqrt(10) #K l2K4
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) xS}H483h6W
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) %_[- [t3
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) u J` &hX
% 4 4 r^4 * sin(4*theta) sqrt(10) $ "8k|^Z3
% -------------------------------------------------- BQU5[8 l
% vxeT[/6i
% Example 1: gJ&!w8v.
% #$w#"Nr 9k
% % Display the Zernike function Z(n=5,m=1) ay_D.g xz
% x = -1:0.01:1; 95Qz1*TR
% [X,Y] = meshgrid(x,x); PNKT \y d
% [theta,r] = cart2pol(X,Y); +V@=G &Ou0
% idx = r<=1; $5b|@
% z = nan(size(X));
YBD {l
% z(idx) = zernfun(5,1,r(idx),theta(idx)); HD{`w1vcN
% figure tTGK 25&
% pcolor(x,x,z), shading interp \.c
) ^QQ
% axis square, colorbar H/k W
:k
% title('Zernike function Z_5^1(r,\theta)') |y% ].y)
% pK9^W T@
% Example 2: *Zi%Q[0Me
% ~(j'a!#Vvk
% % Display the first 10 Zernike functions ;a&:r7]=
% x = -1:0.01:1; EQ7n'Wqq
% [X,Y] = meshgrid(x,x); q|X4[E|{Q
% [theta,r] = cart2pol(X,Y); <j\;>3Q
% idx = r<=1; 66<\i ltUQ
% z = nan(size(X)); <aaDW
% n = [0 1 1 2 2 2 3 3 3 3]; .w_`d'}
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; I#X2 UQzP
% Nplot = [4 10 12 16 18 20 22 24 26 28]; G}#/`]o!K
% y = zernfun(n,m,r(idx),theta(idx)); /i y*3P,`
% figure('Units','normalized') 2 dD<]
% for k = 1:10 ZQ9oZHU m
% z(idx) = y(:,k); :6$4K"^1
% subplot(4,7,Nplot(k)) 0BB @ E(*
% pcolor(x,x,z), shading interp ; U qx&5P}
% set(gca,'XTick',[],'YTick',[]) cNo4UZvr
% axis square p/1 }>F|i
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) kbYg4t]FH
% end y)/$ge _U
% pOVg hllO
% See also ZERNPOL, ZERNFUN2. Bdq"6SK>
[du>ff
r\n
h.}s
% Paul Fricker 11/13/2006 2v9s@k/k)6
6G<Hi" I
i9\\evJs
:@>br+S
{#*? S>DA
% Check and prepare the inputs: 2b4pOM7W
% ----------------------------- ]EM)_ :tRf
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) -hVv
error('zernfun:NMvectors','N and M must be vectors.') F%.9f Uo
end {{]=zt|69
loC5o|Wh
K_&c5(-(_
if length(n)~=length(m) t$y&=v
error('zernfun:NMlength','N and M must be the same length.') g2aT`=&Z
end W\>^[c/
Vgqvv q<S
g8+4$2`ny
n = n(:); Tw} ?(\ya
m = m(:); Lov.E3S6;
if any(mod(n-m,2)) {V%%^Zhwy
error('zernfun:NMmultiplesof2', ... V/(`Ek-
'All N and M must differ by multiples of 2 (including 0).') #0?"J)
end +(z_"[l"
rysP)e
4T==A#Z
if any(m>n) -[zdX}x.:
error('zernfun:MlessthanN', ... 5w,lw
'Each M must be less than or equal to its corresponding N.') : $M9XZ~\
end vz;7} Zj]
<OR.q
V q3]7l
if any( r>1 | r<0 ) #MhNdH#
error('zernfun:Rlessthan1','All R must be between 0 and 1.') u [V4OU }%
end 2 ?Pt Z
<[Tq7cO0
<CZI7]PM7
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) LN@E\wRw{r
error('zernfun:RTHvector','R and THETA must be vectors.') ,] ~u:Y}
end !dfS|BA]
8%;}LK
Ci]'G>F@"
r = r(:); pT("2:)x
theta = theta(:); cm@jt\D
length_r = length(r); Zg)_cRR
if length_r~=length(theta) 'ox0o:
error('zernfun:RTHlength', ...
O D
'The number of R- and THETA-values must be equal.') 0n4g $JK7
end ^~ Ekg:`
~ A^E
wVx,JL5Jr
% Check normalization: w+t# Yb \7
% -------------------- AI&qU/}
if nargin==5 && ischar(nflag) *-?Wc z
isnorm = strcmpi(nflag,'norm'); Gx.P ]O 3
if ~isnorm @)o0GHNP
error('zernfun:normalization','Unrecognized normalization flag.') YSqv86
end |:./ hdcad
else ( V$Zc0
isnorm = false; qOW#Q:T
end r VUUH!
J5O.*&
`-4' /~G
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 8QT<M]N%
% Compute the Zernike Polynomials F;#zN
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \,2gTi,=
yB.G=90
+fM&su=wl
% Determine the required powers of r: x ZX`%f-
% ----------------------------------- #>= 8w9]
m_abs = abs(m); auRY|j
rpowers = []; _l<mu? "
for j = 1:length(n) jO=*:{#x
rpowers = [rpowers m_abs(j):2:n(j)]; %MN.O-Lc
end .l \r9I(
rpowers = unique(rpowers); IhE9snJ[
UgR : qjI
'@bJlJB9>
% Pre-compute the values of r raised to the required powers, L=RGL+f1 _
% and compile them in a matrix: 'G8 ?'u_)
% ----------------------------- sm
if rpowers(1)==0 h$pk<<
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); it)ZP H
rpowern = cat(2,rpowern{:}); 'd/*BjNp)
rpowern = [ones(length_r,1) rpowern]; =2%VZE7Vm
else } Gr&w-v
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 3rNc1\a;
rpowern = cat(2,rpowern{:}); o\4 CoeG
end (~&w-w3
&tj0M.-
2bLI%gg3
% Compute the values of the polynomials: `*aBRwvK~
% -------------------------------------- =u=Kw R
y = zeros(length_r,length(n)); 59 <hV?
for j = 1:length(n) $0oO
&)*
s = 0:(n(j)-m_abs(j))/2; x&Vm!,%:1
pows = n(j):-2:m_abs(j);
JCcZuwu[
for k = length(s):-1:1 =h6
sPJ
p = (1-2*mod(s(k),2))* ... 6Tw#^;q-
prod(2:(n(j)-s(k)))/ ... nMfFH[I4
prod(2:s(k))/ ... anw}w !@U
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... aJL^AG
prod(2:((n(j)+m_abs(j))/2-s(k))); 9Etz:?)b
idx = (pows(k)==rpowers); a {}|Bf<
y(:,j) = y(:,j) + p*rpowern(:,idx); {Sl57! U5
end (iJ1
;x
do-ahl,
if isnorm o*x*jn :hm
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); %<?0apO
end u~
~R9 .
end t kQH\5
% END: Compute the Zernike Polynomials &mj6 rIz
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% &
gJV{V5Ay
+')f6P;t>=
L%v^s 4@
% Compute the Zernike functions: uPYmHA} _/
% ------------------------------ B/5=]R
idx_pos = m>0; p ME{jD
idx_neg = m<0; ] sz3 ]"2
I/VxZ8T
,*4p?|A
z = y; )eUW5
tS
if any(idx_pos) @prG%vb"
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); {QBB^px
end y@j,a
if any(idx_neg) "dR |[a<#g
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); B!gGK|8
end 2^t#6XBk/
< B_Vc:Q
r1ws1 rr=
% EOF zernfun w ;d aC(:
