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, N-
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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, a]
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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? G)v
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那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? ?`zXLY9q7
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function z = zernfun(n,m,r,theta,nflag) W"^wnGa@a
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. D%6;^^WyUx
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N o*U]v
% and angular frequency M, evaluated at positions (R,THETA) on the B(xN Gs
% unit circle. N is a vector of positive integers (including 0), and $`R6=\|
% M is a vector with the same number of elements as N. Each element J]f3CU,<N
% k of M must be a positive integer, with possible values M(k) = -N(k) ;bHV
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, {:@tQdM:i8
% and THETA is a vector of angles. R and THETA must have the same ^P151*=D
% length. The output Z is a matrix with one column for every (N,M) Z87_ #5
% pair, and one row for every (R,THETA) pair. *HEuorl
% #Zrlp.M4
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike EdZ\1'&/9
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), g~(E>6Y
% with delta(m,0) the Kronecker delta, is chosen so that the integral oy<WsbnS
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ^&y$Wd]6
% and theta=0 to theta=2*pi) is unity. For the non-normalized 34\(7JO
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. }!IL]0q
% ,^#yo6-
% The Zernike functions are an orthogonal basis on the unit circle. pPd#N'\*
% They are used in disciplines such as astronomy, optics, and 5j~$Mj`
% optometry to describe functions on a circular domain. _6ay-u
% a!O0,y
% The following table lists the first 15 Zernike functions. >4t+:Ut:
% \=_{na_
% n m Zernike function Normalization AU2i%Q!
% -------------------------------------------------- J9~g|5
% 0 0 1 1 qucq,Yw
% 1 1 r * cos(theta) 2 yj^+G
% 1 -1 r * sin(theta) 2 \hCH>*x<
% 2 -2 r^2 * cos(2*theta) sqrt(6) [jmd
% 2 0 (2*r^2 - 1) sqrt(3) q$=#A7H>3)
% 2 2 r^2 * sin(2*theta) sqrt(6) 8#vc(04(
% 3 -3 r^3 * cos(3*theta) sqrt(8) -[-wkC8a
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) L|p
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% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) uu`G 2[t
% 3 3 r^3 * sin(3*theta) sqrt(8) g) -bW+]q
% 4 -4 r^4 * cos(4*theta) sqrt(10) }iuWAFZbGS
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) iX)%Q
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) cTG|fdgMW
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) o}ZdTf=
% 4 4 r^4 * sin(4*theta) sqrt(10) 1dK*y'rx
% -------------------------------------------------- >y,-v:Vy
% ti#7(^j
% Example 1: K5lmVF\$P
% Hw4%uS==V
% % Display the Zernike function Z(n=5,m=1) z*-2.}&U<
% x = -1:0.01:1; b9!FC$^J
% [X,Y] = meshgrid(x,x); 6fw(T.Pe
% [theta,r] = cart2pol(X,Y); 0\e IQp
% idx = r<=1; lv04g} W
% z = nan(size(X)); j:VbrR
% z(idx) = zernfun(5,1,r(idx),theta(idx)); !jTcsN%
% figure ^jx7@LgS=
% pcolor(x,x,z), shading interp jbAx;Xt'=M
% axis square, colorbar .X;3,D[w
% title('Zernike function Z_5^1(r,\theta)') 4T ~}
% 4M2j!Sw
% Example 2: .hifsB~
% &wV]"&-
% % Display the first 10 Zernike functions }9FSO9*&}
% x = -1:0.01:1; `G}TG(
% [X,Y] = meshgrid(x,x); f.9SB
% [theta,r] = cart2pol(X,Y); 7Ve1]) u
% idx = r<=1; sc}~8T
% z = nan(size(X)); 0.@&_XTPl
% n = [0 1 1 2 2 2 3 3 3 3]; V{!J-nO
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; 5;YMqUkw
% Nplot = [4 10 12 16 18 20 22 24 26 28]; jWrj?DV,2N
% y = zernfun(n,m,r(idx),theta(idx)); LA}Syt\F
% figure('Units','normalized') B\o Mn
% for k = 1:10 T:=lz:}I
% z(idx) = y(:,k); (^Y~/
% subplot(4,7,Nplot(k)) A|<jX}
% pcolor(x,x,z), shading interp s*-n^o-
% set(gca,'XTick',[],'YTick',[]) ?k(7 LX0j
% axis square {y_98N
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) vbyH<LPz5
% end Tu).K.p:
% 5?]hd*8
% See also ZERNPOL, ZERNFUN2. 24z< gO
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% Paul Fricker 11/13/2006 2h5nMI]'
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% Check and prepare the inputs: ww],y@da
% ----------------------------- ewctkI$,5
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) =A83W/4
error('zernfun:NMvectors','N and M must be vectors.') X:vghOt?
end z=q3Zo
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if length(n)~=length(m) [OC5l>
error('zernfun:NMlength','N and M must be the same length.') x|pg"v&[
end MkfBuW;)
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n = n(:); ;DFSzbF`
m = m(:); #h`
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if any(mod(n-m,2)) `p2+&&]S
error('zernfun:NMmultiplesof2', ... ;:\<gVi:
'All N and M must differ by multiples of 2 (including 0).') 8%A#`)fb
end /|C*
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if any(m>n) 0IBhb(X
error('zernfun:MlessthanN', ... D1zBsi94D
'Each M must be less than or equal to its corresponding N.') 5z7U1:
end C~2F9Pg
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if any( r>1 | r<0 ) }$SavB#SBP
error('zernfun:Rlessthan1','All R must be between 0 and 1.') mr*JJF0Z
end /Z'L^L%R
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) JtYP E?
error('zernfun:RTHvector','R and THETA must be vectors.') s4A43i'g!h
end 5m\<U`
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r = r(:); ,(d)Qg
theta = theta(:); [uC]*G]
length_r = length(r); &"f";
if length_r~=length(theta) TC!Yb_H}gN
error('zernfun:RTHlength', ... RYQ<Zr$!
'The number of R- and THETA-values must be equal.') Dz>^IMsY
end l? Udn0F
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% Check normalization: Qu?R8+"KS
% -------------------- =RA /
if nargin==5 && ischar(nflag) LClNxm2X
isnorm = strcmpi(nflag,'norm'); ]
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if ~isnorm 0m%|U'm|j
error('zernfun:normalization','Unrecognized normalization flag.') 5D\f8L
end i2E)P x
else Uzz'.K(Mv|
isnorm = false; *"?l ]d
end |=Eo?Q_
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% CR8/Ke
% Compute the Zernike Polynomials RDW8]=uM
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% /oR0+sH]
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% Determine the required powers of r: i6dHrx]:,
% ----------------------------------- GPkmf%FJ
m_abs = abs(m); |^: cG4e
rpowers = []; c`J.Tm[_u
for j = 1:length(n) QLXN*c
rpowers = [rpowers m_abs(j):2:n(j)]; t2/#&J]
end 7S '%
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rpowers = unique(rpowers); Wvbf"hq
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% Pre-compute the values of r raised to the required powers, P\yDa*m
% and compile them in a matrix: *W.C7=
% ----------------------------- >zw.GwN|
if rpowers(1)==0 U{7w#>V
.
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ]$ L|
rpowern = cat(2,rpowern{:}); _-q.Q^
rpowern = [ones(length_r,1) rpowern]; tjIl-IQ
else !nqUBa
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); /qMG=Z
rpowern = cat(2,rpowern{:}); .z]Wyx&/U
end g[1gF&
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%-)H^i~]%
% Compute the values of the polynomials: $;1#To
% -------------------------------------- pf1BN@
t
y = zeros(length_r,length(n)); wT;0w3.Z
for j = 1:length(n) wN@oYFoL
s = 0:(n(j)-m_abs(j))/2; 3kw,(-'1
pows = n(j):-2:m_abs(j); sF|5XjQ
for k = length(s):-1:1 0"kbrv2y
p = (1-2*mod(s(k),2))* ... kStnb?nk
prod(2:(n(j)-s(k)))/ ... sx7eC
prod(2:s(k))/ ... oC<.=2]
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... fKeT,U`W
prod(2:((n(j)+m_abs(j))/2-s(k))); Bzkoo J
idx = (pows(k)==rpowers); < vL,*.zd
y(:,j) = y(:,j) + p*rpowern(:,idx); `P@T$bC
end iIMd!Q.)@
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if isnorm p2ogn}`
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); T ?$:'XJ
end s%qF/70'
end tz5e"+Tz
% END: Compute the Zernike Polynomials fmQ_P.c
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% q1z"-~i)E
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% Compute the Zernike functions: @JtM5qB
% ------------------------------ u$>4F|=T
idx_pos = m>0; +1uF !G&l
idx_neg = m<0;
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z = y; C9~52+S
if any(idx_pos) :Pvzl1
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); \?Z{hmN
end 6hlc1?
if any(idx_neg) .LZwuJ^;
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 0O9Ni='Tn
end 9f2UgNqe9
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% EOF zernfun 1M}5>V{