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, lg (>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, L<<v
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? 6NFLk+kqN
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? OH+2)X
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function z = zernfun(n,m,r,theta,nflag) Tc;BE
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. OJcI0(G
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N k|_
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% and angular frequency M, evaluated at positions (R,THETA) on the cz>`$Zz
% unit circle. N is a vector of positive integers (including 0), and !G~\9
% M is a vector with the same number of elements as N. Each element Me,AE^pgL'
% k of M must be a positive integer, with possible values M(k) = -N(k) #0qMYe>Y
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, oB}rd9
% and THETA is a vector of angles. R and THETA must have the same !.{{QwZ
% length. The output Z is a matrix with one column for every (N,M) f V/
% pair, and one row for every (R,THETA) pair. s.}:!fBk
% ?v@q&
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike '&xRb*
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), f7]C1!]
% with delta(m,0) the Kronecker delta, is chosen so that the integral ;}4e+`fF|
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, $J:~jY/J
% and theta=0 to theta=2*pi) is unity. For the non-normalized l>>,~
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. b WZX
% U
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% The Zernike functions are an orthogonal basis on the unit circle. }5;3c %
% They are used in disciplines such as astronomy, optics, and T^ah'WmNw
% optometry to describe functions on a circular domain. p7)b@,
% 0.t1p(x;
% The following table lists the first 15 Zernike functions. }JWk?
% b{JxTT}03
% n m Zernike function Normalization ?K?v64[
% -------------------------------------------------- w~9=6|_
% 0 0 1 1 &4l>_
% 1 1 r * cos(theta) 2
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% 1 -1 r * sin(theta) 2 |a Ht6F
% 2 -2 r^2 * cos(2*theta) sqrt(6) !g-19at
% 2 0 (2*r^2 - 1) sqrt(3) {~d8_%:b
% 2 2 r^2 * sin(2*theta) sqrt(6) o[eIwGxZ
% 3 -3 r^3 * cos(3*theta) sqrt(8) B,dKpz;kFg
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) O/Wc@Ln
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) ut^^,w{o>
% 3 3 r^3 * sin(3*theta) sqrt(8) =G2A Ufn
% 4 -4 r^4 * cos(4*theta) sqrt(10) "Q@ZS2;A
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) #
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% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) HPR*:t
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) =i)k@w_(x
% 4 4 r^4 * sin(4*theta) sqrt(10) NCysYmt
% -------------------------------------------------- ~v<,6BS<$Z
% \=/^H
% Example 1: DPsf]
% _-I 0f##.
% % Display the Zernike function Z(n=5,m=1) @=[SsS
% x = -1:0.01:1; ]LhNP}c
% [X,Y] = meshgrid(x,x); I806I@ix
% [theta,r] = cart2pol(X,Y); $.@)4Nu!_
% idx = r<=1; 0SziTM
% z = nan(size(X)); N^.!l_
% z(idx) = zernfun(5,1,r(idx),theta(idx)); xcYYo'U
% figure =w!14@W
% pcolor(x,x,z), shading interp i;>Hy|
% axis square, colorbar "i1~YE
% title('Zernike function Z_5^1(r,\theta)') =' cr@[~i
% #_bSWV4
% Example 2: Z*|qbu)
% ^dR5fAS
% % Display the first 10 Zernike functions o5FBqt
% x = -1:0.01:1; WV"{oED
% [X,Y] = meshgrid(x,x); LJMw-#61sj
% [theta,r] = cart2pol(X,Y); xe6 2gaT
% idx = r<=1; @GGPw9a
% z = nan(size(X)); Q
pY: L
% n = [0 1 1 2 2 2 3 3 3 3]; >p 7e6%
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; 8Mq]
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% Nplot = [4 10 12 16 18 20 22 24 26 28]; LPk85E
% y = zernfun(n,m,r(idx),theta(idx)); 3RP}lb
% figure('Units','normalized') n'JwT!
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% for k = 1:10 q<b;xx
% z(idx) = y(:,k); pFg9-xd%
% subplot(4,7,Nplot(k)) *qE[Y0Cd
% pcolor(x,x,z), shading interp xla9:*pPn
% set(gca,'XTick',[],'YTick',[]) )nS;]7pB@
% axis square bd2"k;H<o
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) k]"Rg2>%
% end v:<UbuJw
% zRJopcE<
% See also ZERNPOL, ZERNFUN2. s Hu~;)
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% Paul Fricker 11/13/2006 Ss&R!w9p
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$N=A, S
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% Check and prepare the inputs: ;ePmN|rq;
% ----------------------------- C``%<)WC
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) swnov[0
error('zernfun:NMvectors','N and M must be vectors.') CBTa9|57
end 2Fce| Tn
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if length(n)~=length(m) K{HRjNda#
error('zernfun:NMlength','N and M must be the same length.') vGC^1AM
end =1%3".
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n = n(:); ewk62{
m = m(:); UtiS?w6
if any(mod(n-m,2)) pscCXk(|A`
error('zernfun:NMmultiplesof2', ... fdN-Zq@'
'All N and M must differ by multiples of 2 (including 0).') t.>vLzrU
end hsljJvs
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if any(m>n) z1F[okLA
error('zernfun:MlessthanN', ... w'.ny<Pe
'Each M must be less than or equal to its corresponding N.') Y'Jb@l`$-
end d;(L@9HHD
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if any( r>1 | r<0 ) U,+kV?Z
error('zernfun:Rlessthan1','All R must be between 0 and 1.') TjlKy
end )D@1V=9,
z8= Gc$w!
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) dLal15Pb
error('zernfun:RTHvector','R and THETA must be vectors.') >NW
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end wI}5[m
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r = r(:); -";'l@D=
theta = theta(:); z(3mhMJY
length_r = length(r); EH] 5ZZ[Z
if length_r~=length(theta) W==HV0n
error('zernfun:RTHlength', ... =6q?XOM
'The number of R- and THETA-values must be equal.') ,$sq]_t
end * "ER8\
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=2)5_/9au
% Check normalization: OcMd'fwO
% -------------------- us4.-L
if nargin==5 && ischar(nflag) 5}~*,_J2Z
isnorm = strcmpi(nflag,'norm'); Y+V*$73`
if ~isnorm $ah, $B
error('zernfun:normalization','Unrecognized normalization flag.') 1U~AupHE
end Nj.(iBmr
else <{YP=WYW
isnorm = false; @| %t<{y^I
end djPr 4Nog
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% =q^o6{d0"
% Compute the Zernike Polynomials C1|e1
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 3>0/WbA:7E
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% Determine the required powers of r: q!""pr<n
% ----------------------------------- %zd1\We
m_abs = abs(m); //e.p6"8h
rpowers = []; H<%7aOwO2
for j = 1:length(n) BYVp~!u
rpowers = [rpowers m_abs(j):2:n(j)]; 7-w
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end }o=R7n%
rpowers = unique(rpowers); zScV 9,H1
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% Pre-compute the values of r raised to the required powers, T6Ue\Sp'
% and compile them in a matrix: QXq~e
% ----------------------------- "a5?cX;
if rpowers(1)==0 {.H}+ @0
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); .-`7Av+7
rpowern = cat(2,rpowern{:}); b\][ x6zJp
rpowern = [ones(length_r,1) rpowern]; .+ai
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else (~}yt .7K
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); qp
rpowern = cat(2,rpowern{:}); d~S.PRg=
end &>@nW!n
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%I(N
% Compute the values of the polynomials: uc=-+*D'I
% -------------------------------------- mV`Z]-$$i
y = zeros(length_r,length(n)); @XDU!<N
for j = 1:length(n) > g8;x#
s = 0:(n(j)-m_abs(j))/2; u~1[nH:
pows = n(j):-2:m_abs(j); }/(fe`7:
for k = length(s):-1:1 5U3="L
p = (1-2*mod(s(k),2))* ... Bu>srX9f
prod(2:(n(j)-s(k)))/ ... K^A\S
prod(2:s(k))/ ... Qgo0uuM
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... @ L% 3}
prod(2:((n(j)+m_abs(j))/2-s(k))); ub+>i
idx = (pows(k)==rpowers); k-Jj k3
y(:,j) = y(:,j) + p*rpowern(:,idx); M `Y~IG}
end D>?%p"e
UG&/0{j5XV
if isnorm Z\(+awv
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ut& RKr3
end H:,rNaz7D^
end T"in
% END: Compute the Zernike Polynomials V2i*PK
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% lY.FmF}k
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B\KvKT|\
% Compute the Zernike functions: 7AV !v`
% ------------------------------ -AD3Pd|Y[
idx_pos = m>0; Xy_+L_h^
idx_neg = m<0; NLoJmOi;L7
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z = y; r$=iM:kERC
if any(idx_pos) 8g(%6 ET
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); oSx]wZZ
end 5z5#_*)O
if any(idx_neg) |M)'@s:
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); :f 1*-y
end tP"C>#LO
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% EOF zernfun B3c
rms['