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， c yz3,3\e
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， x U`p|(SS-
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？ #QMz<P/Gl6
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ _.8S&
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function z = zernfun(n,m,r,theta,nflag) /9fR'EO{x
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. vx5Zl& 6r
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N [d ]9Oa4
% and angular frequency M, evaluated at positions (R,THETA) on the /mzlH
% unit circle. N is a vector of positive integers (including 0), and 0LJ v '
% M is a vector with the same number of elements as N. Each element {I't]Qj_e
% k of M must be a positive integer, with possible values M(k) = -N(k) e$ rZ 5X
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, IjnU?Bf
% and THETA is a vector of angles. R and THETA must have the same g[4WzDF*
% length. The output Z is a matrix with one column for every (N,M) }@d @3
% pair, and one row for every (R,THETA) pair. 13x p_j
% >fQ MXfoY
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike h <<v^+m
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ^^ixa1H<
% with delta(m,0) the Kronecker delta, is chosen so that the integral "3Y0`&:D
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, IJcsmNW m
% and theta=0 to theta=2*pi) is unity. For the non-normalized x7 , 5
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. }Jj}%XxKs
% s!$a \ k
% The Zernike functions are an orthogonal basis on the unit circle. ;
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% They are used in disciplines such as astronomy, optics, and a9Zq{Ysj
% optometry to describe functions on a circular domain.
rjnrju+
% ^} >w<'0
% The following table lists the first 15 Zernike functions. 5 \VWC I
% DZtsy!xA
% n m Zernike function Normalization a0)QH
% -------------------------------------------------- DkDmE
% 0 0 1 1 7WzxA=*#
% 1 1 r * cos(theta) 2 5]:U9ts#
% 1 -1 r * sin(theta) 2 Nu)NqFG,
% 2 -2 r^2 * cos(2*theta) sqrt(6) X|]A T9W
% 2 0 (2*r^2 - 1) sqrt(3) (KZ{^X?a
% 2 2 r^2 * sin(2*theta) sqrt(6) _7_Y={4=`
% 3 -3 r^3 * cos(3*theta) sqrt(8) PXNuL&
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 5wU]!bxr
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) *. w 9c
% 3 3 r^3 * sin(3*theta) sqrt(8) #&e-|81H
% 4 -4 r^4 * cos(4*theta) sqrt(10) Dk5 1z@
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) yyTnL 2Y9
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) S)" Jf?
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) z},# ~L6$q
% 4 4 r^4 * sin(4*theta) sqrt(10) g2Z`zQA7
% -------------------------------------------------- XfIJ 4ZM5
% ]JQULE)
% Example 1: +^F Zq$NP
% 6[AL|d
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% % Display the Zernike function Z(n=5,m=1) 4 s9LB
% x = -1:0.01:1; &m;*<}X
% [X,Y] = meshgrid(x,x); :e+jU5;]3
% [theta,r] = cart2pol(X,Y); ] 7c=PC
% idx = r<=1; aw&,S"A@
% z = nan(size(X)); $M:*T.3
% z(idx) = zernfun(5,1,r(idx),theta(idx)); A?OQE9'
% figure (A.C]hD
% pcolor(x,x,z), shading interp -M Bxl`JU
% axis square, colorbar a(ZcmYzXU
% title('Zernike function Z_5^1(r,\theta)') j3ls3H&
% ?:eV%`7
% Example 2: HTTC TR
% gI|~|-'
% % Display the first 10 Zernike functions _+3::j~;m
% x = -1:0.01:1; Qn2&nD%zi
% [X,Y] = meshgrid(x,x); &mM0AA'\?H
% [theta,r] = cart2pol(X,Y); S$-7SEkO+
% idx = r<=1; <9b &<K:
% z = nan(size(X)); */S_Icf
% n = [0 1 1 2 2 2 3 3 3 3]; [{/jI\?v
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; ChQx a
% Nplot = [4 10 12 16 18 20 22 24 26 28]; )D%~` ,#pQ
% y = zernfun(n,m,r(idx),theta(idx)); J] r^W)O
% figure('Units','normalized') 5 SQ 8}Or3
% for k = 1:10 j![\& z
% z(idx) = y(:,k); z\4.Gm-
% subplot(4,7,Nplot(k)) 7 _[L o4_
% pcolor(x,x,z), shading interp F_P~x(X
% set(gca,'XTick',[],'YTick',[]) }Ou}+ ^Bc
% axis square dqcL]e
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ZWm6eD
% end GTxk%
% &BSn?
% See also ZERNPOL, ZERNFUN2. RT8 ?7xFc
bcz:q/f}@
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% Paul Fricker 11/13/2006 AQ^u
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% Check and prepare the inputs: =-Ck4 e *T
% ----------------------------- tO&^>&;5
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) X5w$4Kj&4l
error('zernfun:NMvectors','N and M must be vectors.') q1ma%eiN
end #lO Mm9
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\
B% +fw
if length(n)~=length(m) TkF[x%o
error('zernfun:NMlength','N and M must be the same length.') Pc ] H P
end 1xx}~|F?|
5~S5F3
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n = n(:); BFt> 9x]T
m = m(:); NX&_p!_V
if any(mod(n-m,2)) wdoR%b{M
error('zernfun:NMmultiplesof2', ... Eh BKj |y
'All N and M must differ by multiples of 2 (including 0).') gI`m.EH}}N
end *=xr-!MEk
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if any(m>n) og>uj>H &
error('zernfun:MlessthanN', ... x|29L7 i
'Each M must be less than or equal to its corresponding N.') Gp\
kU:}&
end [Pb OfxxgA
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if any( r>1 | r<0 ) 5E
< kwi
error('zernfun:Rlessthan1','All R must be between 0 and 1.') J,6yYIq
end ;9'OOz|+1
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= WJ NWt>
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) :2)/FPL6
error('zernfun:RTHvector','R and THETA must be vectors.') bQ5\ ]5M
end 4`=m u}Y2
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r = r(:); <\S:'g"(
theta = theta(:); R/a*LSe@&
length_r = length(r); \.}c9*)
if length_r~=length(theta) |gY^)9ei
error('zernfun:RTHlength', ... BD7N i^qI$
'The number of R- and THETA-values must be equal.') "J3x_~,[4m
end k== h|\|
ij U*|8n{>
h@wgd~X9
% Check normalization: 2b8L\$1q
% -------------------- /_ajaz%
if nargin==5 && ischar(nflag) rQ snhv
isnorm = strcmpi(nflag,'norm'); f|oh.z_R
if ~isnorm h
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error('zernfun:normalization','Unrecognized normalization flag.') .5{ab\_af
end p{dj~ &v
else Qe(:|q _
isnorm = false; mB)bcuPv
end 1yY0dOoLG)
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% *DhiN
% Compute the Zernike Polynomials |
VDV<g5h
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% oe~b}:
B#1;r-^P<
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% Determine the required powers of r: sW8dPw
O
% ----------------------------------- Rbv;?'O$L
m_abs = abs(m); eb$#A _m
rpowers = []; Eu04e N
for j = 1:length(n) eh#(eua0/
rpowers = [rpowers m_abs(j):2:n(j)]; IMONgFBS
end 0+b1vhQ
rpowers = unique(rpowers); b5n'=doR/I
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7
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% Pre-compute the values of r raised to the required powers, 8DaL,bi*.
% and compile them in a matrix: Od)C&N=y
% ----------------------------- Pg7Yp2)Oli
if rpowers(1)==0 d m%8K6|
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); <1M-Ro?5k
rpowern = cat(2,rpowern{:}); y4fdq7i~}9
rpowern = [ones(length_r,1) rpowern]; uf T`"i
else r" ,GC]
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); qJUK_6|3
rpowern = cat(2,rpowern{:}); ` sU/& P
end Pk)1WK7E
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VVOd]2{
% Compute the values of the polynomials: K", N!koj
% --------------------------------------
M \Kx'N
y = zeros(length_r,length(n)); G,w(d@
for j = 1:length(n) JqiP>4Uwm^
s = 0:(n(j)-m_abs(j))/2; v| 2T%y_
u
pows = n(j):-2:m_abs(j); R{T$[ $6S
for k = length(s):-1:1 Mf``_=K
p = (1-2*mod(s(k),2))* ... bA->{OPkT
prod(2:(n(j)-s(k)))/ ... x-3 \Ls[I
prod(2:s(k))/ ... lnR{jtWP
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... sD wqH.L
prod(2:((n(j)+m_abs(j))/2-s(k))); :9 ^*
^T
idx = (pows(k)==rpowers); Y:a]00&)#Y
y(:,j) = y(:,j) + p*rpowern(:,idx); pz>>)c`
end VW4r{&rS
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if isnorm "'\$
g[k
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); y'*K|a TG
end !C: $?o U
end 0lR5<^B
% END: Compute the Zernike Polynomials ~q Oa\#x_
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% XpJ7o=?W3
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[ibu/ W$
% Compute the Zernike functions: |
%Vh`HT
% ------------------------------ b SU~XGPB
idx_pos = m>0; 'b{]:Y
idx_neg = m<0; <UQbt N-B\
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@sC`!Rmy'-
z = y; n 7-6-
#
if any(idx_pos) E~oOK Q5W
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ^DwYOo 2B
end X}\:_/
if any(idx_neg) d-dEQKI?;
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ,\%c^,HLJ
end )P |),S,;Z
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0KOgw*>_
% EOF zernfun }U"&8%PZr
