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, \_6OC Vil
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, HfNDD|Zz
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? LJlZ^kh
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? ]2SI!Ai7
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function z = zernfun(n,m,r,theta,nflag) uRV<?y%
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. Pt,ebL~
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N y2L#:[8
% and angular frequency M, evaluated at positions (R,THETA) on the %r{3wH#D@
% unit circle. N is a vector of positive integers (including 0), and )(M7lq.e7
% M is a vector with the same number of elements as N. Each element /u<nLj 1
% k of M must be a positive integer, with possible values M(k) = -N(k) OW;tT=ql
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, gk0.zz([
% and THETA is a vector of angles. R and THETA must have the same $rB3m~c|
% length. The output Z is a matrix with one column for every (N,M) 9=l.T/?sf
% pair, and one row for every (R,THETA) pair. (t^n'V
% S^I,Iz+`S'
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike >H][.@LyR
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), jyS=!ydn+
% with delta(m,0) the Kronecker delta, is chosen so that the integral )=pD%$iq
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, E$s/]wnr[
% and theta=0 to theta=2*pi) is unity. For the non-normalized KxGX\
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. . RVVWqW
% SuBeNA[&
% The Zernike functions are an orthogonal basis on the unit circle. +xv!$gJEj
% They are used in disciplines such as astronomy, optics, and w&h2y4
% optometry to describe functions on a circular domain. ;Y9=!.Ak0y
% Pn.bVV:
% The following table lists the first 15 Zernike functions. 6c4&VW
% 6aO2:|:yP
% n m Zernike function Normalization '_s}o<
% -------------------------------------------------- uLeRZSC
% 0 0 1 1 X?r48l??
% 1 1 r * cos(theta) 2 gbBy/_b
% 1 -1 r * sin(theta) 2 yY{kG2b,
% 2 -2 r^2 * cos(2*theta) sqrt(6) {16<^
% 2 0 (2*r^2 - 1) sqrt(3) 5X.ebd;PT
% 2 2 r^2 * sin(2*theta) sqrt(6)
%V G/
% 3 -3 r^3 * cos(3*theta) sqrt(8) Ji'(`9F&a
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) Y
qdWctUY
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) F4#g?R::U
% 3 3 r^3 * sin(3*theta) sqrt(8) 6SM:x]`##,
% 4 -4 r^4 * cos(4*theta) sqrt(10) B/f0P(7
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) fN%jJ-[d
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) >>Ar$
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) I`RBj `IF
% 4 4 r^4 * sin(4*theta) sqrt(10) 3k$[r$+"
% -------------------------------------------------- P\m7 -
% U'( sn
% Example 1: _;9!
% nt1CTWKM8^
% % Display the Zernike function Z(n=5,m=1) )+y G+
% x = -1:0.01:1; gT+Bhr
% [X,Y] = meshgrid(x,x); A?!I/|E^;
% [theta,r] = cart2pol(X,Y); Wl"0m1G
% idx = r<=1; 4Cb9%Q0
% z = nan(size(X)); XE3aXK'R
% z(idx) = zernfun(5,1,r(idx),theta(idx)); k_|^ kdWJ
% figure NW9n
% pcolor(x,x,z), shading interp 7k%T<;V
% axis square, colorbar [U
=Uo*
% title('Zernike function Z_5^1(r,\theta)') FyL_xu\e
% yqOuX>m 1c
% Example 2: j=+"Qz/hr_
% mg:!4O$K
% % Display the first 10 Zernike functions Tpp &
% x = -1:0.01:1; G* b2,9&F
% [X,Y] = meshgrid(x,x); A~(l{g
% [theta,r] = cart2pol(X,Y); u`:hMFTID
% idx = r<=1; =1;=
% z = nan(size(X)); 9%)=`W
% n = [0 1 1 2 2 2 3 3 3 3]; "VxWj}+]
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; !LM<:kf.|
% Nplot = [4 10 12 16 18 20 22 24 26 28]; :6kj EI
% y = zernfun(n,m,r(idx),theta(idx)); 4\5uY
% figure('Units','normalized') eLD?jTi'
% for k = 1:10 .ae O}^
% z(idx) = y(:,k); (n{wg(R
% subplot(4,7,Nplot(k)) *!e(A ]&
% pcolor(x,x,z), shading interp q~K(]Ya/
% set(gca,'XTick',[],'YTick',[]) 9 t
n!t
% axis square iX{G]< n
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ]<uQ.~
% end AN:@fZ
% )QiQn=Ce
% See also ZERNPOL, ZERNFUN2. K!AAGj`
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th5g\h%j*
% Paul Fricker 11/13/2006 ^t"iX9
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% Check and prepare the inputs: N6oq90G
% ----------------------------- G28O%jD?
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 'WyTI^K9
error('zernfun:NMvectors','N and M must be vectors.') `Kl`VP=c
end h( QYxI,|
}1 vT)
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if length(n)~=length(m) nx":"LFI
error('zernfun:NMlength','N and M must be the same length.') vm23U^VJ
end -]G(ms;}/Y
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n = n(:); }*U[>Z-eO
m = m(:); eEc4bVQa
if any(mod(n-m,2)) u8zbYd3
error('zernfun:NMmultiplesof2', ... uUR~&8ERX
'All N and M must differ by multiples of 2 (including 0).') 7XrfuG*L$
end "R
#k~R
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Zc&&[g
if any(m>n) 1m<RwI3s
error('zernfun:MlessthanN', ... l?E a#
'Each M must be less than or equal to its corresponding N.') q!'rz
end c/W=$3
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if any( r>1 | r<0 ) e59dVFug.U
error('zernfun:Rlessthan1','All R must be between 0 and 1.') Si}HX!s
end Mc sTe|X
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ri~<~oB2:
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) r4_eTrC,
error('zernfun:RTHvector','R and THETA must be vectors.') g8;D/
end -#`c5y}P
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r = r(:); ]bCq=6ZKR
theta = theta(:); o(A|)c4k
length_r = length(r); .?C%1a&_l
if length_r~=length(theta) G*[P<<je_
error('zernfun:RTHlength', ... }b3/b
'The number of R- and THETA-values must be equal.') lw%?z/HDf
end [}mA`5
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Z*{]
,
% Check normalization: beY=g7|
% -------------------- \@a$'
if nargin==5 && ischar(nflag) nHFrG
=o,
isnorm = strcmpi(nflag,'norm'); RH)EB<PV
if ~isnorm VUU]Pu &
error('zernfun:normalization','Unrecognized normalization flag.') pI`?(5iK6|
end JD>d\z2QC
else `\>.h
isnorm = false; ,n,RFa
end `XTh1Z\
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% B=L&bx
% Compute the Zernike Polynomials .uo.N
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ]T!
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_gNz9$S
% Determine the required powers of r: ;|%dY{L-
% ----------------------------------- vEM(bT=H
m_abs = abs(m); wJb#g0
rpowers = []; ewNz%_2
for j = 1:length(n) bte~c
rpowers = [rpowers m_abs(j):2:n(j)]; .@ C{3$,VG
end l2%bF8]z
rpowers = unique(rpowers); qr\!*\9
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FX<b:#
% Pre-compute the values of r raised to the required powers, _GQz!YA
% and compile them in a matrix: NMO-u3<6.
% ----------------------------- @\_x'!R
if rpowers(1)==0 _:n b&B
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); fBtm%f
rpowern = cat(2,rpowern{:}); - "*r
rpowern = [ones(length_r,1) rpowern]; !33#. @[
else hlZ@Dq%f
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); {Ee>n^1
rpowern = cat(2,rpowern{:}); [36,eK
end tqPx$s
b<I9 MR
&}mw'_ I
% Compute the values of the polynomials: 3
vP(SIF
% -------------------------------------- PALl sGlf
y = zeros(length_r,length(n)); eg"Gjp-4=
for j = 1:length(n) y@bcYOh3
s = 0:(n(j)-m_abs(j))/2; _?7#MWe&
pows = n(j):-2:m_abs(j); g_*T?;!.U
for k = length(s):-1:1 ^ OJyN,A
p = (1-2*mod(s(k),2))* ... "bg'@:4F
prod(2:(n(j)-s(k)))/ ... *MNHT`Y^o
prod(2:s(k))/ ... "i.r@<)S
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 1xNVdI
prod(2:((n(j)+m_abs(j))/2-s(k))); BIaDY<j90
idx = (pows(k)==rpowers); %,@vWmn
y(:,j) = y(:,j) + p*rpowern(:,idx); D*5hrkV9
end fqz28aHh
ub0zJTFJ#
if isnorm Mkp/0|Q*
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 1RLY $M
end <O?y-$~
end sH,kW|D
% END: Compute the Zernike Polynomials ;wiao(t>4N
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 1PaUI#X"2F
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HC {XX>F^
% Compute the Zernike functions: mAgF73,3
% ------------------------------ O40+M)e]
idx_pos = m>0; wmNHT _
idx_neg = m<0; 4Ph0:^i_
+`mGK:>
zHWSE7!
z = y; 80}+MWdo
if any(idx_pos) 75!9FqMZ}
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); 'PZ|:9FX!
end ] U@o0
if any(idx_neg) C<^YVeG
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); GJi~y
end vq*Q.0 M+
rr`;W}3
C#rc@r,F
% EOF zernfun %OR|^M