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， =hlu,
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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， U]$3NIe
这个，跟我们用zygo干涉仪直接拟合出的36项zernike系数，有何关系呢？ M*uG`Eo&
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ GjG3aqP&!
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function z = zernfun(n,m,r,theta,nflag) '& :"/4@)
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. CB1u_E_
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 5w9<_W0d
% and angular frequency M, evaluated at positions (R,THETA) on the }5U f`pM8
% unit circle. N is a vector of positive integers (including 0), and JH#?}L/0Fe
% M is a vector with the same number of elements as N. Each element kMXl
{
% k of M must be a positive integer, with possible values M(k) = -N(k) Zv 93cv
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, j&5Xjl>4
% and THETA is a vector of angles. R and THETA must have the same l"8YI sir
% length. The output Z is a matrix with one column for every (N,M) Mr(3]EfgO
% pair, and one row for every (R,THETA) pair. 2T9Z{v
% 1L7,x @w
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike X; tk\Ixd
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), _{%H*PxTn=
% with delta(m,0) the Kronecker delta, is chosen so that the integral K(2s %
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, - %|I
% and theta=0 to theta=2*pi) is unity. For the non-normalized RwWQ$Eb_s
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. Qt 2hb
% kF .b)
% The Zernike functions are an orthogonal basis on the unit circle. ZxQP,Ys_Y
% They are used in disciplines such as astronomy, optics, and 7O#>N}|
% optometry to describe functions on a circular domain. %#~Wk|8} Q
% <5%We(3
% The following table lists the first 15 Zernike functions. uip]K{/A!e
% 9m{rQ P/
% n m Zernike function Normalization 6~L pBlb
% -------------------------------------------------- yM@cml6Ox
% 0 0 1 1 X*Zv,Wm
% 1 1 r * cos(theta) 2 75f.^4/%
% 1 -1 r * sin(theta) 2 AP%h!b5v
% 2 -2 r^2 * cos(2*theta) sqrt(6) clNP9{
% 2 0 (2*r^2 - 1) sqrt(3) ?|\Lm3%J
% 2 2 r^2 * sin(2*theta) sqrt(6) o m6R/K
% 3 -3 r^3 * cos(3*theta) sqrt(8) dQ]j
r.
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 7Z_iQ1
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) &3V4~L1aEg
% 3 3 r^3 * sin(3*theta) sqrt(8) +8M{y D9#
% 4 -4 r^4 * cos(4*theta) sqrt(10) ojri~erJE?
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) gN%R-e0
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) f&'md
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 65v'/m!ys
% 4 4 r^4 * sin(4*theta) sqrt(10) #A!0KN;GC2
% -------------------------------------------------- G)Y!aX
% 566EMy|
% Example 1: O9Aooe4 W=
% x&
S >Mr
% % Display the Zernike function Z(n=5,m=1) n^K]R}S
% x = -1:0.01:1; i{2KMa{K
% [X,Y] = meshgrid(x,x); _ sd?l
% [theta,r] = cart2pol(X,Y); xlAaIo)T
% idx = r<=1; }O/Nn0,
% z = nan(size(X)); #~b9H05D
% z(idx) = zernfun(5,1,r(idx),theta(idx)); ) =[Tgh
% figure ~$B ,K]
% pcolor(x,x,z), shading interp ryN-d%t?
% axis square, colorbar UWHC]V?
% title('Zernike function Z_5^1(r,\theta)') |@RO&F
% <OUApp H
% Example 2: 4/b#$o <I?
% 2 r,fF<WQ
% % Display the first 10 Zernike functions TR |; /yJ
% x = -1:0.01:1; e(Ve rd:c
% [X,Y] = meshgrid(x,x); #qWEyb2UZ
% [theta,r] = cart2pol(X,Y); qF ?S[Z;
% idx = r<=1; (_* a4xGF
% z = nan(size(X)); dx^3(#B
% n = [0 1 1 2 2 2 3 3 3 3]; ;1KhUf;&F
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; (w*$~ p
% Nplot = [4 10 12 16 18 20 22 24 26 28]; =" `y<J P
% y = zernfun(n,m,r(idx),theta(idx)); ]zO]*d=m
% figure('Units','normalized') ep5a BrN]"
% for k = 1:10 ,Gfnf%H\8>
% z(idx) = y(:,k); x {rt\OT
% subplot(4,7,Nplot(k)) 04s N 4C
% pcolor(x,x,z), shading interp \ys3&<;b
% set(gca,'XTick',[],'YTick',[]) MmX42;Pw
% axis square 2}NfR8
N
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}'])
#xmUND`@
% end
m ]\L1&
% bnlL-]]9z
% See also ZERNPOL, ZERNFUN2. ` F)Iv:;y,
IAfYlS#<yD
z, OMR`W
% Paul Fricker 11/13/2006 ZrTq)BZ
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% Check and prepare the inputs: |A2.W8`o
% ----------------------------- '@$?A>.cj
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) F?UL0Q|u v
error('zernfun:NMvectors','N and M must be vectors.') oR+Fn}mG
end p'H5yg3h
7 l?=$q>k"
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if length(n)~=length(m) .8"o&%$`V
error('zernfun:NMlength','N and M must be the same length.') (k[<>$hL*
end `p!.K9r7
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n = n(:); wtZe\ h
m = m(:); U<*dDE~z
if any(mod(n-m,2)) iB\d `NUf
error('zernfun:NMmultiplesof2', ... l!oU9
'All N and M must differ by multiples of 2 (including 0).') =%a.C(0&G
end w'UP#vT5&
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if any(m>n) 46m u,v
error('zernfun:MlessthanN', ... zP5H TEz
'Each M must be less than or equal to its corresponding N.') &=f%(,+
end UOa{J|k>h
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if any( r>1 | r<0 ) b7\nCRY
error('zernfun:Rlessthan1','All R must be between 0 and 1.') Sn a7r~ j
end d~.# K S
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) RlpW)\{j?
error('zernfun:RTHvector','R and THETA must be vectors.') %cBJ haR{(
end wt-)5f'{
r1,RloyZS
&e d. %:
r = r(:); {~Phc 2z
theta = theta(:); J H6\;G6
length_r = length(r); $[IuEdc/
if length_r~=length(theta) IuRKj8J)o
error('zernfun:RTHlength', ... e\\ I,
'The number of R- and THETA-values must be equal.') dD#A.C,Rz
end w@hm>6j
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qF> }"m
% Check normalization: Cf a?LgSz
% -------------------- ,;UVQwY
if nargin==5 && ischar(nflag) 1;SWfKU?.
isnorm = strcmpi(nflag,'norm'); N'TL &]
if ~isnorm d6wsT\S
error('zernfun:normalization','Unrecognized normalization flag.') d 'PjO-"g
end Zpg$:Rr
else uQrD}%GI
isnorm = false; xaejG/'iK
end e;,D!
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o ;.j_
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% CmM K\R.
% Compute the Zernike Polynomials )~rN{W<s`H
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% kAU[lPt*R
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% Determine the required powers of r: MH1??vW
% ----------------------------------- .#P'NF(5#
m_abs = abs(m); {73Z$w1%
rpowers = []; @MTm8E6au
for j = 1:length(n) #\Lt0
rpowers = [rpowers m_abs(j):2:n(j)];
"Qm
end GoZr[ =d
rpowers = unique(rpowers); B_nim[72
5^*I]5t8
"4|D"|wI)
% Pre-compute the values of r raised to the required powers, o |$l+TC
% and compile them in a matrix: j$siCsF
% ----------------------------- *JUP~/N r
if rpowers(1)==0 < OCy
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); b v~"_)C
rpowern = cat(2,rpowern{:}); cd#@"&r
rpowern = [ones(length_r,1) rpowern]; vEk
jd#
else DhYQ>Gv8U
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); UOi8>; k`
rpowern = cat(2,rpowern{:}); Z-.`JkKd8
end K#kU6/
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` ~F=
% Compute the values of the polynomials: +g>)Bur
% -------------------------------------- a)/!ifJ;
y = zeros(length_r,length(n)); 0ERA(=w5
for j = 1:length(n) =E,^ +`M
s = 0:(n(j)-m_abs(j))/2; 5L'X3g
pows = n(j):-2:m_abs(j); Z`Rrv$M!
for k = length(s):-1:1 p<: bP w
p = (1-2*mod(s(k),2))* ... 7X*$Fu<
prod(2:(n(j)-s(k)))/ ... :>c33X}
prod(2:s(k))/ ... \ym3YwP4/:
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... := C-P7
prod(2:((n(j)+m_abs(j))/2-s(k))); K1Snag
idx = (pows(k)==rpowers); _ ?]bd-E
y(:,j) = y(:,j) + p*rpowern(:,idx); 8XI G<Nc
end ;*Ldnj;B
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if isnorm EIPNR:6t
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ]yiwdQ
end |
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end w3WBgH
% END: Compute the Zernike Polynomials 7j| ^ZuI+
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ZxF RE#y~2
t2uX +1F
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% Compute the Zernike functions: )!z<q}i5
% ------------------------------ V{+'(<SV
idx_pos = m>0; V(3^ev/
idx_neg = m<0; F9IrbLS9c
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z = y; <[Oe.0SGu
if any(idx_pos) H]As2$[
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); !tJQ75Hwv
end 1fU g
if any(idx_neg) (?4m0Sn>#h
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 0}H7Xdkp
end WR,MqM20
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% EOF zernfun &y_Ya%Z3*e
