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， 2V@5: tf
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， 9} 6 _B|
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？ %k#+nad
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ q8$t4_ pF
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|1%% c
%
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function z = zernfun(n,m,r,theta,nflag) i?/Q7D<P
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 9&*
7+!
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Sl8+A+
% and angular frequency M, evaluated at positions (R,THETA) on the ]ltCJq
% unit circle. N is a vector of positive integers (including 0), and :Vxt2@p {
% M is a vector with the same number of elements as N. Each element h A ){>B<;
% k of M must be a positive integer, with possible values M(k) = -N(k) )3CM9P'0
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, E.*hY+kGZ
% and THETA is a vector of angles. R and THETA must have the same SPV + O{
% length. The output Z is a matrix with one column for every (N,M) edMCj
% pair, and one row for every (R,THETA) pair. d7kE}{,
% QKP
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% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike ,?yjsJd.
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ;((t|
% with delta(m,0) the Kronecker delta, is chosen so that the integral $}(Z]z}O ;
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, {LiJ=Ebt
% and theta=0 to theta=2*pi) is unity. For the non-normalized 1#x5
o2n
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. Hpix: To
% Wq3PN^
% The Zernike functions are an orthogonal basis on the unit circle. _9=87u0
% They are used in disciplines such as astronomy, optics, and (LK@w9)i;
% optometry to describe functions on a circular domain. (/u N+
% J~K O#`
% The following table lists the first 15 Zernike functions. OFr"RGW"
% 9C \}bT
% n m Zernike function Normalization $?F_Qsy{d
% -------------------------------------------------- } `L;. 9
% 0 0 1 1 C+/EPPi
% 1 1 r * cos(theta) 2 Lz1KDXr`)+
% 1 -1 r * sin(theta) 2 +}m`$B}mJ
% 2 -2 r^2 * cos(2*theta) sqrt(6) V<WWtu;3
% 2 0 (2*r^2 - 1) sqrt(3) g R!hN.I
% 2 2 r^2 * sin(2*theta) sqrt(6) -F/)-s6#!'
% 3 -3 r^3 * cos(3*theta) sqrt(8) 'ij+MU 1
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) nN& dtjoF
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) p8 S~`fjV
% 3 3 r^3 * sin(3*theta) sqrt(8) # fF5O2E'3
% 4 -4 r^4 * cos(4*theta) sqrt(10) Mc c%&j
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) dXDyY
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) #!_4Z X
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) f=91
Z_M
% 4 4 r^4 * sin(4*theta) sqrt(10) F7<M{h 5s
% --------------------------------------------------
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% $z OV*O2
% Example 1: p zRV X8
% NCg("n ,jx
% % Display the Zernike function Z(n=5,m=1) o Tvg%bX
% x = -1:0.01:1; / mJb$5 =1
% [X,Y] = meshgrid(x,x); IgJG,!>h
% [theta,r] = cart2pol(X,Y); \GHj_r
% idx = r<=1; n=b!c@f4
% z = nan(size(X)); Pj q9BK9p
% z(idx) = zernfun(5,1,r(idx),theta(idx)); B[mZQ&Gz`a
% figure 5q4wREh
% pcolor(x,x,z), shading interp .Od@i$E>&
% axis square, colorbar <>KQ8:
% title('Zernike function Z_5^1(r,\theta)') u L v
% L"0dB.
% Example 2: W/RB |TMT
% DBy%"/c
% % Display the first 10 Zernike functions ih("`//nP
% x = -1:0.01:1; !}|'1HIC
% [X,Y] = meshgrid(x,x); NfQ QJ@*
% [theta,r] = cart2pol(X,Y); vZQraY nJ
% idx = r<=1; -^_^ByJe
% z = nan(size(X)); R{H8@JLD
% n = [0 1 1 2 2 2 3 3 3 3]; Y, Lpv|
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; .`KzA]&#
% Nplot = [4 10 12 16 18 20 22 24 26 28]; KD\%B5Jy
% y = zernfun(n,m,r(idx),theta(idx)); &9gI?b8
% figure('Units','normalized') d?5oJ'JU
% for k = 1:10
= <A0;
% z(idx) = y(:,k); v#9i|
% subplot(4,7,Nplot(k)) l^ tRy_T:-
% pcolor(x,x,z), shading interp tHq a%
% set(gca,'XTick',[],'YTick',[]) E}zGY2Xx
% axis square NHU5J SlB
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ?!"pzDg
% end j7Zv"Vq@
% BQ ,749^S
% See also ZERNPOL, ZERNFUN2. uCt ?(E>
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q( EN] W],
% Paul Fricker 11/13/2006 KWYjN
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',Z]w;D!G
% Check and prepare the inputs: 6[F XgCb
% ----------------------------- 4QC_zyTE
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 3 %BI+1&T_
error('zernfun:NMvectors','N and M must be vectors.') $? Z}hU
end
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\ %xk u:
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if length(n)~=length(m) "J&WH~8+N
error('zernfun:NMlength','N and M must be the same length.') T#e|{ZCbq
end !mVq+_7]
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n = n(:); ]2&RN@
m = m(:); f6( 1 jx"
if any(mod(n-m,2)) <}xgp[O
error('zernfun:NMmultiplesof2', ... _/ 5
'All N and M must differ by multiples of 2 (including 0).') x!7!)]h
end x'G_z_<V
Y#P!<Q>}
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if any(m>n) lkp$rJ#6
error('zernfun:MlessthanN', ... >,Z n~8&Z
'Each M must be less than or equal to its corresponding N.') c<Ud[x.
end _9=cxw i<w
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if any( r>1 | r<0 ) WK0Iag Yw
error('zernfun:Rlessthan1','All R must be between 0 and 1.') 44 k8IYC*o
end :Ez*<;pF'
p?
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vXM {)
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 3 l
j^I
error('zernfun:RTHvector','R and THETA must be vectors.') ".pQM.T
end x*X {*?5@
; Ob^@OM
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r = r(:); W8-vF++R
theta = theta(:); 0=9$k
length_r = length(r); Ofb&W
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if length_r~=length(theta) oZL# *Z(h
error('zernfun:RTHlength', ...
fC}uIci
'The number of R- and THETA-values must be equal.') "2tKh!?Q
end D)[(
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R9K~b^`
% Check normalization: n b*`GE
% -------------------- LOwd mj
if nargin==5 && ischar(nflag) ]Ee$ulJ02
isnorm = strcmpi(nflag,'norm'); pz{ ]O_px
if ~isnorm bq8h ?Q
error('zernfun:normalization','Unrecognized normalization flag.') m,5?|J=
end ExFz@6@
else oe=1 [9T"
isnorm = false; puh-\Q/ P
end I,Jb_)H&t
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h >Z`&
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \nTV;@F
% Compute the Zernike Polynomials ^ME'D
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% * vqUOh
S`TQWWQo;
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% Determine the required powers of r: _1YC9}
% ----------------------------------- \IqCC h
m_abs = abs(m); YB:}L b
rpowers = []; ?O]RQXsZ2
for j = 1:length(n) $:A80(#+
rpowers = [rpowers m_abs(j):2:n(j)]; R$Qhu xT|
end \W\*'C8q\
rpowers = unique(rpowers); 3 m &
#\K"FE0PGz
N&$ ,uhmO
% Pre-compute the values of r raised to the required powers, +A$>F@ u
% and compile them in a matrix: 8WKY 4nkj
% ----------------------------- lO 0}
if rpowers(1)==0 E},zB*5TH
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); f)r6F JLU
rpowern = cat(2,rpowern{:}); L7. SH#m
rpowern = [ones(length_r,1) rpowern]; R.
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else xm=$D6O:
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); f'M([gn^_
rpowern = cat(2,rpowern{:}); rP!GS
_RG
end wAL}c(EHO
L
gy^ ^.
zXbA$c
% Compute the values of the polynomials: U: )Gc
% -------------------------------------- bUYjmb2g)
y = zeros(length_r,length(n)); vWa\8y f
for j = 1:length(n) )ac!@slb^7
s = 0:(n(j)-m_abs(j))/2; M23r/eg]
pows = n(j):-2:m_abs(j); J` { o`>
for k = length(s):-1:1 qmvQd8|XR
p = (1-2*mod(s(k),2))* ... >Ml5QO$*.q
prod(2:(n(j)-s(k)))/ ... d..JW{
prod(2:s(k))/ ... (S ?DKPnR
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... |WOc0 M[U
prod(2:((n(j)+m_abs(j))/2-s(k))); =([4pG
idx = (pows(k)==rpowers); ' d?6 L
y(:,j) = y(:,j) + p*rpowern(:,idx); <num!@ 2D
end }WBHuVcZG
>6)|># Wi
if isnorm q[/pE7FL
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); | :i d/
end <~:2~ r
end $2-_j) +
% END: Compute the Zernike Polynomials V\l@_%D[(v
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% sc6NON#
l/\D0\x2
:)&vf<JL
% Compute the Zernike functions: g= ,}j]tl
% ------------------------------ 9b@yDq3hQ
idx_pos = m>0; ;BKU
_}k=
idx_neg = m<0; B <a` o&?
>[~7fxjK-
y O*
z = y; < $otBC/%
if any(idx_pos) k1s5cg=n(
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); nb 6Y/`G
end ?ks.M'@
if any(idx_neg) n+i=Ff
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); &
d$X:
end *JQ* $$5
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["4Tn0g ;
% EOF zernfun 7?y 7fwER
