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, gBF2.{"^
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, $:{r#mM
这个,跟我们用zygo干涉仪直接拟合出的36项zernike系数,有何关系呢? jm|x=s3}h
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? b^SQCX+P
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function z = zernfun(n,m,r,theta,nflag) xJ/<G$LNJ0
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. '}\#bMeObg
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Z*9Qeu-N:
% and angular frequency M, evaluated at positions (R,THETA) on the "OIra2O
% unit circle. N is a vector of positive integers (including 0), and 3LxhQVx2
% M is a vector with the same number of elements as N. Each element X/=*o;":
% k of M must be a positive integer, with possible values M(k) = -N(k) yuTSzl25,/
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, M
Y2=lT
% and THETA is a vector of angles. R and THETA must have the same k0%*{IVPN
% length. The output Z is a matrix with one column for every (N,M) `k ^d)9
% pair, and one row for every (R,THETA) pair. )#^5$5
% qDMVZb-(#
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike )<fa1Gz#^
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), f!3$xu5
% with delta(m,0) the Kronecker delta, is chosen so that the integral 3WOm`<
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, \!+sL JP
% and theta=0 to theta=2*pi) is unity. For the non-normalized sZ-A~X@g
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. [?dsS$Y3
% O{4G'CgN(
% The Zernike functions are an orthogonal basis on the unit circle. L7\rx w
% They are used in disciplines such as astronomy, optics, and 3Pj#k|(f[0
% optometry to describe functions on a circular domain. Ukf4Q\@w
% b7thu5
% The following table lists the first 15 Zernike functions. w=dTa5
% I}?+>cf
% n m Zernike function Normalization ,'7 X|z/_>
% -------------------------------------------------- \Zpg,KOT
% 0 0 1 1 B)q 5m
y
% 1 1 r * cos(theta) 2 z5V~m_RO
% 1 -1 r * sin(theta) 2 Yqpe2II7
% 2 -2 r^2 * cos(2*theta) sqrt(6) 91|0{1
% 2 0 (2*r^2 - 1) sqrt(3) #@quuiYq
% 2 2 r^2 * sin(2*theta) sqrt(6) B) 5QI
% 3 -3 r^3 * cos(3*theta) sqrt(8) vz\^Aa
#fv
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) hd~3I4D
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) 5,i0QT"
% 3 3 r^3 * sin(3*theta) sqrt(8) &d!Q%
% 4 -4 r^4 * cos(4*theta) sqrt(10) |a>W9Y m
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) )~u<u:N
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) qs9q{n-Aj
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) jcC"SqL
% 4 4 r^4 * sin(4*theta) sqrt(10) %%7~<=rk
% -------------------------------------------------- _LYI#D
% E`M, n,
% Example 1: <k41j=d
% t08E
2sI
% % Display the Zernike function Z(n=5,m=1) p3Ey[kURp
% x = -1:0.01:1; h$[tEmD%
% [X,Y] = meshgrid(x,x); aMLtZ7i>
% [theta,r] = cart2pol(X,Y); vVA)x~^
% idx = r<=1; /=+y[y3`
% z = nan(size(X)); w&b?ze{
% z(idx) = zernfun(5,1,r(idx),theta(idx)); 3 P)N,
% figure =!?4$vW
% pcolor(x,x,z), shading interp y3^>a5z!x
% axis square, colorbar u{Rgk:bn
% title('Zernike function Z_5^1(r,\theta)') qm!&(8NfK
% MBjo9P(
% Example 2: :iKk"r,2P[
% K6..N\7
% % Display the first 10 Zernike functions A9 D vU)1
% x = -1:0.01:1; lD K<gd
% [X,Y] = meshgrid(x,x); 9i8 ~
% [theta,r] = cart2pol(X,Y); *(w#*,lv
% idx = r<=1; bvR0?xnq
% z = nan(size(X)); Z(~v{c %<
% n = [0 1 1 2 2 2 3 3 3 3]; [k<w'n*
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; q]^Q?r<g::
% Nplot = [4 10 12 16 18 20 22 24 26 28]; f@)GiLC'"
% y = zernfun(n,m,r(idx),theta(idx)); ]:K[{3iM
% figure('Units','normalized') +|iJQF
% for k = 1:10 <$:Hf@tpMo
% z(idx) = y(:,k); V1d{E 0lM
% subplot(4,7,Nplot(k)) YXFUZ9a#e
% pcolor(x,x,z), shading interp 5nQxVwY
% set(gca,'XTick',[],'YTick',[]) Ok*aP+Wq
% axis square u A=x~-I
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) HOY@<'
% end vgyv~Px]AW
% :JI&ngWK
% See also ZERNPOL, ZERNFUN2. MODi:jsl
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% Paul Fricker 11/13/2006 F?3zw4Vt~
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T`Jj$Lue{
% Check and prepare the inputs: `%5~>vPS
% ----------------------------- l`V^d
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) eGEeWJ}[$
error('zernfun:NMvectors','N and M must be vectors.') BQ
/0z^A
end wq6.:8Or-]
%s(Ri6R&
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if length(n)~=length(m) Q"Ur*/-U
error('zernfun:NMlength','N and M must be the same length.') J;mvD^`g
end ]y52%RAKI
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n = n(:); 4w)aAXK
m = m(:); C3#mmiL-
if any(mod(n-m,2)) 1#OM~v6B
error('zernfun:NMmultiplesof2', ... !#' y#
'All N and M must differ by multiples of 2 (including 0).') )RZ:\:c
end :}[RDF?
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if any(m>n) V+y yy-/
error('zernfun:MlessthanN', ... @x4IxGlUs
'Each M must be less than or equal to its corresponding N.') uLK4tQ
end -$0w-M8'
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if any( r>1 | r<0 ) FsXqF&{
error('zernfun:Rlessthan1','All R must be between 0 and 1.') 8`4Z%;1
end ~ 6`Ha@
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) F/(z3Kf
error('zernfun:RTHvector','R and THETA must be vectors.') `~S; UG
end /#]4lFk:h
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r = r(:); >qT4'1S*g
theta = theta(:); 9bVPMq7}i
length_r = length(r); v_s(
if length_r~=length(theta) Hb:@]!r>
error('zernfun:RTHlength', ... !U?Z<zh
'The number of R- and THETA-values must be equal.') <6(&w9WY
end hiM nU
N-Jp; D
D$OUy}[2`.
% Check normalization: rcx'`CIJ
% -------------------- 9}_ccq
if nargin==5 && ischar(nflag) tI-u@
g
isnorm = strcmpi(nflag,'norm'); <`/22S"
if ~isnorm e>a4v8
error('zernfun:normalization','Unrecognized normalization flag.') *>%tx k:)
end S.$/uDwo
else q8_8rp-@
isnorm = false; qx+ .v2G
end S7fX1y[
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ow!NH,'Hy
% Compute the Zernike Polynomials x_K8Gr#Z 0
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 6$k"B/k
u#&ZD|
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% Determine the required powers of r: BA
9c-Ay
% ----------------------------------- / ~\ I
m_abs = abs(m); ),u)#`.l
G
rpowers = []; Munal=wL
for j = 1:length(n) F=qG+T
rpowers = [rpowers m_abs(j):2:n(j)]; 4sCzUvI~Y1
end /eI]!a
rpowers = unique(rpowers); m[t4XK
)^^Eh=Kbj
*Mg. *N
% Pre-compute the values of r raised to the required powers, Pgp`g.$<
% and compile them in a matrix: \F
}s"#
% ----------------------------- |8:IH@K*
if rpowers(1)==0 sPod)w?e
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); GqT0SP
rpowern = cat(2,rpowern{:}); #Xa TUT
rpowern = [ones(length_r,1) rpowern]; MS~|F^g
else g=gWkN
<
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); X-$\DXRIo
rpowern = cat(2,rpowern{:}); lNQ8$b
end N;A#K7A[@
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% Compute the values of the polynomials: :qQpBr$
% -------------------------------------- NPFrn[M$
y = zeros(length_r,length(n)); f L}3I(VK
for j = 1:length(n) 1;Dug
s = 0:(n(j)-m_abs(j))/2; \~O}V~wE
pows = n(j):-2:m_abs(j); ,8vqzI
for k = length(s):-1:1 -x)zyq6
p = (1-2*mod(s(k),2))* ... ;<9 dND
prod(2:(n(j)-s(k)))/ ... =%\y E0#
prod(2:s(k))/ ... >>nt3q
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... sr*3uI-)L
prod(2:((n(j)+m_abs(j))/2-s(k))); '0juZ~>}
idx = (pows(k)==rpowers); 4 )U,A~!
y(:,j) = y(:,j) + p*rpowern(:,idx); r z
end !|1GraiS
k^vsQ'TD
if isnorm iLyJ7zby
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 1syI%I1
end QS*!3?%
end ]0+5@c
% END: Compute the Zernike Polynomials Y5Ub[o
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% fF\s5f#:
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\U0p?wdr:
% Compute the Zernike functions: (1|_Nr
% ------------------------------ b/I_iJ8t
idx_pos = m>0; 6]/LrM, 23
idx_neg = m<0; 9AxeA2/X
/;[Zw8K7
te 0a6
z = y; ^zv,VD
if any(idx_pos) OjUZ-_J
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); n&`=.[+A
end S"/M+m+ ]
if any(idx_neg) is2OJ,
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ZE!dg^-L
end :+G1=TuXw~
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% EOF zernfun 7LQLeQvB