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, Q;u pau
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, Vc2`b3"Br
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? nK,w]{<wG!
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? 3u;oQ5<(v
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function z = zernfun(n,m,r,theta,nflag) G?ZXWu.
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. J *yg&
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N (?c-iKGc
% and angular frequency M, evaluated at positions (R,THETA) on the ]3gSQ7
% unit circle. N is a vector of positive integers (including 0), and }}[2SH'nH
% M is a vector with the same number of elements as N. Each element Zh,71Umz
% k of M must be a positive integer, with possible values M(k) = -N(k) P%6~&woF
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, ]A"h&`Cvt
% and THETA is a vector of angles. R and THETA must have the same TO_e^A#
% length. The output Z is a matrix with one column for every (N,M) yZRzIb_
% pair, and one row for every (R,THETA) pair. q@&6#B
% xmX 4qtAL
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike /mMV{[
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), '7/)Ot(
% with delta(m,0) the Kronecker delta, is chosen so that the integral *fdTpXa
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, n ;Ei\\p!
% and theta=0 to theta=2*pi) is unity. For the non-normalized Gq6*SaTk
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. \8
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% ?CZd Ol
% The Zernike functions are an orthogonal basis on the unit circle. JLYi]nZ
% They are used in disciplines such as astronomy, optics, and U(Zq= M
% optometry to describe functions on a circular domain. ]yu:i-SfP
% j [a(#V{
% The following table lists the first 15 Zernike functions. VQs5"K"
% ;*N5Y}?j'
% n m Zernike function Normalization XuTD\g3)
% -------------------------------------------------- 5bIw?%dk(
% 0 0 1 1 u y+pP!<
% 1 1 r * cos(theta) 2 =vPj%oLp'a
% 1 -1 r * sin(theta) 2 So;<6~
% 2 -2 r^2 * cos(2*theta) sqrt(6) XG?8s
&
% 2 0 (2*r^2 - 1) sqrt(3) GVz6-T~\>
% 2 2 r^2 * sin(2*theta) sqrt(6) B[}6-2<>?C
% 3 -3 r^3 * cos(3*theta) sqrt(8) [m -bV$-d
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) *I+Q~4
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) LscGTs,
% 3 3 r^3 * sin(3*theta) sqrt(8) 4
:v=pZ
% 4 -4 r^4 * cos(4*theta) sqrt(10) fOHxtHM
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) jylD6IT
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) KXrjqqXs
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 5^cCY'I
% 4 4 r^4 * sin(4*theta) sqrt(10) K@2),(z
% -------------------------------------------------- Q/?$x*\>
% t7pFW^&
% Example 1: Fu~j8K
% df=f62
% % Display the Zernike function Z(n=5,m=1) TzZq(?V
% x = -1:0.01:1; ni<(K
0~
% [X,Y] = meshgrid(x,x); <%^&2UMg
% [theta,r] = cart2pol(X,Y); 7^285)UQA
% idx = r<=1; 6b,V;#Anj
% z = nan(size(X)); 7^Uv7<pw
% z(idx) = zernfun(5,1,r(idx),theta(idx)); y}
'@R$
% figure d5b%
W3
% pcolor(x,x,z), shading interp 2eogY#
% axis square, colorbar e.%nRhSs3
% title('Zernike function Z_5^1(r,\theta)') rOYx
b }1
% xo)P?-
% Example 2: ]|@^1we
% /QQ*8o8
% % Display the first 10 Zernike functions ^
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% x = -1:0.01:1; h;Kx!5)y
% [X,Y] = meshgrid(x,x); @WhHUd4s
% [theta,r] = cart2pol(X,Y); `V1]k_h
% idx = r<=1; Ef\-VKh
% z = nan(size(X)); LeQjvW9y
% n = [0 1 1 2 2 2 3 3 3 3]; x;S @bY
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; # _1`)VS
% Nplot = [4 10 12 16 18 20 22 24 26 28]; ~u{uZ(~
% y = zernfun(n,m,r(idx),theta(idx)); OI*H,Z"
% figure('Units','normalized') hp2t"t
% for k = 1:10 3$tdwe$S
% z(idx) = y(:,k); v19-./H^
j
% subplot(4,7,Nplot(k)) W^Yxny
% pcolor(x,x,z), shading interp 7$b1<.WX
% set(gca,'XTick',[],'YTick',[]) +vH4MwG$.&
% axis square H}!r|nG
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) h8P)%p
% end `uFdwO'DD
% pmM9,6P4@
% See also ZERNPOL, ZERNFUN2. >z03{=sAN
\bF{-" 7.
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% Paul Fricker 11/13/2006 Q8NX)R
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dG{A~Z z
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% Check and prepare the inputs: ;jXgAAz7
% ----------------------------- ixFi{_
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) @FeTz[
error('zernfun:NMvectors','N and M must be vectors.') eDMO]5}Ht
end 6<]lW
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if length(n)~=length(m) S!UaH>Rh
error('zernfun:NMlength','N and M must be the same length.') ^ c<Ve'-
end R5D1w+
)UR7i8]!0
%;_MGae
n = n(:); WY/}1X9.%
m = m(:); &HW9Jn
if any(mod(n-m,2)) fl(wV.Je|
error('zernfun:NMmultiplesof2', ... f?Lw)hMrA
'All N and M must differ by multiples of 2 (including 0).') *VcJ= b
2Y
end w}KkvP^
JI}'dU>*U:
}j%5t ~Qa
if any(m>n) Y|n"dMrL
error('zernfun:MlessthanN', ... UVP vOtZj
'Each M must be less than or equal to its corresponding N.') N['.BN
end yAt^;
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if any( r>1 | r<0 ) A&VG~r$
error('zernfun:Rlessthan1','All R must be between 0 and 1.') $mI Loy
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end QV!up^Zso
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) $Yq9P0Ya
error('zernfun:RTHvector','R and THETA must be vectors.') ueudRb
end ;TYBx24vD'
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r = r(:); /xhKd]Q
theta = theta(:); CTb%(<r
length_r = length(r); 5O%{{J
if length_r~=length(theta) qm}@!z^
error('zernfun:RTHlength', ... A"]YM'.
'The number of R- and THETA-values must be equal.') &Jj<h: *
end >6T8^Nt
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% Check normalization: ax`o>_)
% -------------------- jd:6:Fm
if nargin==5 && ischar(nflag) zPO9!?7|
isnorm = strcmpi(nflag,'norm'); (=0.in Z
if ~isnorm &~CI<\o P
error('zernfun:normalization','Unrecognized normalization flag.') ]kSG R
end .Mbz3;i0
else tw;}jh
isnorm = false; >Tgv11[
end =bOW~0Z1
dd;~K&_Q/i
fC`&g~yK'
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0kh6@y3
% Compute the Zernike Polynomials 4s-!7
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% la!~\wpa
9*gZ-#
P
pb\6|*
% Determine the required powers of r: FrS]|=LJhX
% ----------------------------------- ?,mmYW6TjB
m_abs = abs(m); o-5TC
rpowers = []; [,Gg^*umS
for j = 1:length(n) ,+k\p5P
rpowers = [rpowers m_abs(j):2:n(j)]; Y2AJ+
|
end [0!( xp^
rpowers = unique(rpowers); %b$>qW\*&
>:-$+I
B#A6v0Ta
% Pre-compute the values of r raised to the required powers, |Cv!,]9:r
% and compile them in a matrix: @d'j zs
% ----------------------------- p K*TE5]
if rpowers(1)==0 >MZ/|`[M
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); B,@i
rpowern = cat(2,rpowern{:}); ?uu*L6
rpowern = [ones(length_r,1) rpowern]; j2k"cmsKh
else ch]IzdD
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); kiEa<-]
rpowern = cat(2,rpowern{:}); HMXE$d=[
end -7ep{p-
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_BufO7`.
% Compute the values of the polynomials: )~>YH*g
% -------------------------------------- rq{$,/6.
y = zeros(length_r,length(n)); [Xkx_B
for j = 1:length(n) 6ujWNf
s = 0:(n(j)-m_abs(j))/2; vM={V$D&
pows = n(j):-2:m_abs(j); UQsN'r\tS
for k = length(s):-1:1 hrk r'3lv
p = (1-2*mod(s(k),2))* ... E.h*g8bXe
prod(2:(n(j)-s(k)))/ ... F ,kZU$
prod(2:s(k))/ ...
a?1Wq
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... !
nx{
X
prod(2:((n(j)+m_abs(j))/2-s(k))); w0.
u\
idx = (pows(k)==rpowers); tQVVhXQ7
y(:,j) = y(:,j) + p*rpowern(:,idx); P55fL-vo|}
end PCA4k.,T
*~`(RV
if isnorm :jf3HG
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ?6!LL5a.
end e-;}366}
end `[A];]
% END: Compute the Zernike Polynomials lE;!TQj:X
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ;uW FHc5@B
TeQV?ZQ#}
\U0Q<ot/7
% Compute the Zernike functions: ~*7]r`6\@
% ------------------------------ , gHDx
idx_pos = m>0; Om&Dw|xG8
idx_neg = m<0; \8tsDG(1 '
+ZYn? #IQ
]e3Ax(i)
z = y; =4!mAo}
if any(idx_pos) KvSG;
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); gw(z1L5
n
end 'w/hw'F6
if any(idx_neg) x-c"%Z|
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); M|-)GvR$J
end ,4rPg]r@
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% EOF zernfun ?2a $*(