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, HS.eK#:N
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, 3! KyO)8
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? t,~feW,
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? 7*+tG7I @
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function z = zernfun(n,m,r,theta,nflag) Vvn~G.&)
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. `j6O
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Z4k'c+
% and angular frequency M, evaluated at positions (R,THETA) on the uY&t9L8
% unit circle. N is a vector of positive integers (including 0), and w\JTMS$
% M is a vector with the same number of elements as N. Each element t4zKI~cO
% k of M must be a positive integer, with possible values M(k) = -N(k) Fp+fZU
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, pW<l9W
% and THETA is a vector of angles. R and THETA must have the same 9KL)5_6 M
% length. The output Z is a matrix with one column for every (N,M) 9*a"^
% pair, and one row for every (R,THETA) pair. {ZUgyGE{
% 2N&S__
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike Jk`0yJi$q
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), @ +>>TGC
% with delta(m,0) the Kronecker delta, is chosen so that the integral W
6R/{H
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, n}J!?zZc
% and theta=0 to theta=2*pi) is unity. For the non-normalized vf!lhV-UG+
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. O2~Q(q'
% D'Kiy
% The Zernike functions are an orthogonal basis on the unit circle. :<6gP(
% They are used in disciplines such as astronomy, optics, and ,u5iiR
% optometry to describe functions on a circular domain. 9+'*
% e1H2w?
s
% The following table lists the first 15 Zernike functions. 2Gc0pBqx
% _BND{MsX
% n m Zernike function Normalization 0[-@<w ^j
% -------------------------------------------------- a^)@}4
% 0 0 1 1 \k%j
% 1 1 r * cos(theta) 2 )5<c8lzp
% 1 -1 r * sin(theta) 2 0fw>/"v
% 2 -2 r^2 * cos(2*theta) sqrt(6) mN"g~o*
% 2 0 (2*r^2 - 1) sqrt(3) \lpvRZ\L&g
% 2 2 r^2 * sin(2*theta) sqrt(6) [58qC:
% 3 -3 r^3 * cos(3*theta) sqrt(8) P7qzZ
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) Tu=~iQ
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) iB*1Yy0DC
% 3 3 r^3 * sin(3*theta) sqrt(8) p=dM2>
% 4 -4 r^4 * cos(4*theta) sqrt(10) E>1%7"
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% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) nhB.>ReAi
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 97^)B4
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) R@[1a+}5
% 4 4 r^4 * sin(4*theta) sqrt(10) ?fvK<0S`
% -------------------------------------------------- o[k,{`M0
% 9t{Iv({6p
% Example 1: <)$JA
% cx+%lco!
% % Display the Zernike function Z(n=5,m=1) Y-P?t+l
% x = -1:0.01:1; QqB9I-_
% [X,Y] = meshgrid(x,x); x3=SMN|a
% [theta,r] = cart2pol(X,Y); "tU,.U
% idx = r<=1; Vdb X4^V
% z = nan(size(X)); kO'NT:
% z(idx) = zernfun(5,1,r(idx),theta(idx)); 4nD U-P#f
% figure tzG.)Uqs
% pcolor(x,x,z), shading interp aq]bF%7
% axis square, colorbar BA`K ,#Ft7
% title('Zernike function Z_5^1(r,\theta)') cD9axlJ
% $&FeR*$|g
% Example 2: `;3fnTI:1
% e`t-:~'
% % Display the first 10 Zernike functions fTV3lyk
% x = -1:0.01:1; @l&>C#K\
% [X,Y] = meshgrid(x,x); \`|OAC0a
% [theta,r] = cart2pol(X,Y); -h#9sl->
% idx = r<=1; f>ilk Q`
% z = nan(size(X)); 1y6{3AZm<
% n = [0 1 1 2 2 2 3 3 3 3]; ;c0z6E /
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; t|cTl/i
4
% Nplot = [4 10 12 16 18 20 22 24 26 28]; JrwR:_+|
% y = zernfun(n,m,r(idx),theta(idx)); =o,6iJ^?$m
% figure('Units','normalized') 9>[*y8[:0
% for k = 1:10 Tf.DFfV#y
% z(idx) = y(:,k); W< :7z
% subplot(4,7,Nplot(k)) 52z{
% pcolor(x,x,z), shading interp ~|=goHmm[
% set(gca,'XTick',[],'YTick',[]) PG'+vl
% axis square dW"=/UW
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) zr1A4%S"
% end )\fLS d
% ;Km74!.e7
% See also ZERNPOL, ZERNFUN2. {*t0WE&1t
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% Paul Fricker 11/13/2006 C?7I(b:
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% Check and prepare the inputs: H 5'Ke+4.e
% ----------------------------- 9 az{j1
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) i>=!6Hu2
error('zernfun:NMvectors','N and M must be vectors.') a X:,1^
end *BAR`+;U
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if length(n)~=length(m) +|TFxaVz
error('zernfun:NMlength','N and M must be the same length.') >sm<$'vZ/
end >):^Zs
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n = n(:); [vyi_0[
m = m(:); 5BB:.
if any(mod(n-m,2)) |Y]4PT#EE
error('zernfun:NMmultiplesof2', ... _!Ir|j.A
'All N and M must differ by multiples of 2 (including 0).') -5sKJt]+i
end b*W01ist
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if any(m>n) ])H[>.?K
error('zernfun:MlessthanN', ... Q,<V)
'Each M must be less than or equal to its corresponding N.') bz\-%$^k
end o=y0=,:a?9
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if any( r>1 | r<0 ) E7>D:BQ\2
error('zernfun:Rlessthan1','All R must be between 0 and 1.') /O&{fo
end k{-#2Qz
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) fsz:A"0H
error('zernfun:RTHvector','R and THETA must be vectors.') \S[I:fw#&
end b,):&M~p
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r = r(:); 5tQZf'pHfd
theta = theta(:); 5VhJ*^R`y
length_r = length(r); 8q_"aa,`
if length_r~=length(theta) 8\B]!
error('zernfun:RTHlength', ... c-q=Ct
'The number of R- and THETA-values must be equal.') %+0V0.
end \:D"#s%x
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% Check normalization: #[ ?E,
% -------------------- 1XPYI
if nargin==5 && ischar(nflag) l7vxTj@(-
isnorm = strcmpi(nflag,'norm'); Z|6,*XEc
if ~isnorm ^&Wa?
m.
error('zernfun:normalization','Unrecognized normalization flag.') "`Mowp*
end x_$`#m{hL5
else 1yV+~)by3
isnorm = false; g=L80$1
end
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% !@4 i:,p@
% Compute the Zernike Polynomials Z+g9!@'a
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% jN T+?2
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% Determine the required powers of r: 4c9a"v
% ----------------------------------- g#b9xTGJ^
m_abs = abs(m); s|\\"3
rpowers = []; X<mlaXwrA
for j = 1:length(n) x".!&5
rpowers = [rpowers m_abs(j):2:n(j)]; gnN"6r1
end xZ(ryE%
rpowers = unique(rpowers); )];Bo.QA
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% Pre-compute the values of r raised to the required powers, p-XO4Pc6
% and compile them in a matrix: Z~1uyr(
% ----------------------------- K7c[bhi_w
if rpowers(1)==0 hI 1or4V
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); PWk\#dJN&
rpowern = cat(2,rpowern{:}); oe<DP7e
rpowern = [ones(length_r,1) rpowern]; &>P<Zw-
else `lA_knS
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ,#U[)}im
rpowern = cat(2,rpowern{:}); zEk/15
end H*HL:o-[
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% Compute the values of the polynomials: qNi`OVh&
% -------------------------------------- [,56oMd~
y = zeros(length_r,length(n)); %U6A"?To
for j = 1:length(n) E<sd\~~A:
s = 0:(n(j)-m_abs(j))/2; WS//0
pows = n(j):-2:m_abs(j); 7#(0GZN9h%
for k = length(s):-1:1 aM+Am,n`@
p = (1-2*mod(s(k),2))* ... 3?e~J"WXC5
prod(2:(n(j)-s(k)))/ ... q~`dxq`}
prod(2:s(k))/ ... nzU;Bi^m
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 89Ir}bCr
prod(2:((n(j)+m_abs(j))/2-s(k))); }#h`1 uV
idx = (pows(k)==rpowers); |u]IOw&1
y(:,j) = y(:,j) + p*rpowern(:,idx); *vzEfmN:d
end '0w</g
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if isnorm 8]D0)
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 83J63Xa
end 1my1m
end $,zW0</P*l
% END: Compute the Zernike Polynomials 6aLRnH"Ud
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 9yz@hdG
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8([ MR
% Compute the Zernike functions: }N&?8s=
% ------------------------------ Z/czAr@4
idx_pos = m>0; G=]ox*BY
idx_neg = m<0; f,x;t-o+R
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mLx=Zes:.
z = y; 05:?5M4};
if any(idx_pos) k~F;G=P
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); U(Tl$#Bt
end ;;6$d{
if any(idx_neg) +NbiUCMX
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 67XUhnE
end F^Bk @
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% EOF zernfun Fv*Et-8tN5