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, ~g6`Cp`
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, 2!7)7wlj0
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? Zs$Qo->F
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? W$I^Ej}>$
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function z = zernfun(n,m,r,theta,nflag) )37 .H^7
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. <hS %I
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N -F-RWs{yS
% and angular frequency M, evaluated at positions (R,THETA) on the =}$YZuzmU
% unit circle. N is a vector of positive integers (including 0), and G>>`j2:y
% M is a vector with the same number of elements as N. Each element |3a1hCxt
% k of M must be a positive integer, with possible values M(k) = -N(k) 74h[YyVi
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, lU}y%J@
% and THETA is a vector of angles. R and THETA must have the same 4Z&i\#Q
% length. The output Z is a matrix with one column for every (N,M) 5Dhpcgq<<
% pair, and one row for every (R,THETA) pair. !'
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% s%{8$>8V.
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike e1EFZ,EcaO
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 4GiHp7Y&A
% with delta(m,0) the Kronecker delta, is chosen so that the integral ;j#(%U]Vp
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, &7}\mnhB
% and theta=0 to theta=2*pi) is unity. For the non-normalized P?zPb'UVqa
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. :skNEY].
% iPD5
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% The Zernike functions are an orthogonal basis on the unit circle. 9L"Z
~CUL
% They are used in disciplines such as astronomy, optics, and s y ]k
% optometry to describe functions on a circular domain. N`G*
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% L,m'/}$
% The following table lists the first 15 Zernike functions. +5zLQ>]z
% XMR$I&;G8
% n m Zernike function Normalization `YK2hr
% -------------------------------------------------- =&5^[:ksB
% 0 0 1 1 THQd`Lj
% 1 1 r * cos(theta) 2 DR d|m<Z
% 1 -1 r * sin(theta) 2 9i&(VzY[=
% 2 -2 r^2 * cos(2*theta) sqrt(6) fku\O<1
% 2 0 (2*r^2 - 1) sqrt(3) j[^(<R8
% 2 2 r^2 * sin(2*theta) sqrt(6) /|kR=
~
% 3 -3 r^3 * cos(3*theta) sqrt(8) ="k9
y
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) ]YO &_#
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) gJ;
*?Uq(
% 3 3 r^3 * sin(3*theta) sqrt(8) xbN)z
% 4 -4 r^4 * cos(4*theta) sqrt(10) -w[j`}([P9
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) !mM`+XH
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 8RA]h?$$J
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) vxey$Ir
% 4 4 r^4 * sin(4*theta) sqrt(10) MHuQGc"e+4
% -------------------------------------------------- a5)<roWQ
% B8f BX!u/
% Example 1: 4*)a3jI?
% #:~MtV
% % Display the Zernike function Z(n=5,m=1) :RxWHh3O
% x = -1:0.01:1; jHU5>Gt-}
% [X,Y] = meshgrid(x,x); E8Rk
b}
% [theta,r] = cart2pol(X,Y); GG9YAu
% idx = r<=1; !XJvhsKX y
% z = nan(size(X)); y1oQ4|KSI
% z(idx) = zernfun(5,1,r(idx),theta(idx)); C1x"q9|\`
% figure &n}eF-
% pcolor(x,x,z), shading interp 4
8}\
% axis square, colorbar pX\Y:hCug
% title('Zernike function Z_5^1(r,\theta)') DX*eN"z[
% &B3[:nS2
% Example 2: 3pV^Oe^9
% cE|Z=}4I7
% % Display the first 10 Zernike functions 75^U<Hz-3{
% x = -1:0.01:1; !xIK<H{*
% [X,Y] = meshgrid(x,x);
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% [theta,r] = cart2pol(X,Y); + YjK#
% idx = r<=1; RF#S=X6
% z = nan(size(X)); fMRv:kNAt
% n = [0 1 1 2 2 2 3 3 3 3]; qwERy{]Sp;
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; <$V!y
dO
% Nplot = [4 10 12 16 18 20 22 24 26 28]; @`IMR$'
% y = zernfun(n,m,r(idx),theta(idx)); #Yqj27&
% figure('Units','normalized') oB$P6
% for k = 1:10 |5;:3K+
% z(idx) = y(:,k); &f;<[_QI=
% subplot(4,7,Nplot(k)) d'x'hp%
% pcolor(x,x,z), shading interp Xf"B\%,(`
% set(gca,'XTick',[],'YTick',[]) bg =<) s
% axis square ++m^z` D
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) w@jC#E\
% end LGau!\
% pZ IDGy=~
% See also ZERNPOL, ZERNFUN2. " iz'x-wy
]ZbZ]
b W/^2B
% Paul Fricker 11/13/2006 qubyZ8hx
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%VSjMZ
~+HZQv3Y
% Check and prepare the inputs: ) ]y^RrD
% ----------------------------- d:_3V rRZ
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Z
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error('zernfun:NMvectors','N and M must be vectors.') gdx2&~
end a%IJ8t+mn
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/`2t$71)
if length(n)~=length(m) ` 465
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error('zernfun:NMlength','N and M must be the same length.') T2%{pcdV/
end vhEXtjL
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n = n(:); "Q[rM1R
m = m(:); v)!C
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if any(mod(n-m,2)) ;;Y>7Kn!u
error('zernfun:NMmultiplesof2', ... z5UY0>+VdS
'All N and M must differ by multiples of 2 (including 0).') m,qMRcDF
end *=KX0%3
c:@lR/oe"
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if any(m>n) E[nJ'h<h
error('zernfun:MlessthanN', ... "h84D&V
'Each M must be less than or equal to its corresponding N.') Ln4zy*v{
end "A>/m"c]*
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if any( r>1 | r<0 ) T 8.
to
error('zernfun:Rlessthan1','All R must be between 0 and 1.') .Jvy0B} B
end 5TB==Fj ?
-!s?d5k")
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) DEFh&n
error('zernfun:RTHvector','R and THETA must be vectors.') y?}R,5k
end Tg-HR8}X
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r = r(:); W3h{5\d!
theta = theta(:); Z4ZR]eD
length_r = length(r); #n5DK{e
if length_r~=length(theta) sZ7RiH+I
error('zernfun:RTHlength', ... 4Up3x+bg
'The number of R- and THETA-values must be equal.') Wb7z&vj
end "+BNas^rF
+'!4kwT R
f:K3 P[|
% Check normalization: ;/-X;!a>
% -------------------- 8va&*J?
2
if nargin==5 && ischar(nflag) _ITA $#
isnorm = strcmpi(nflag,'norm'); q_gsYb
if ~isnorm c9<&+
error('zernfun:normalization','Unrecognized normalization flag.') b- FJMY
end Zi4Ektj2
else |Ox!tvyr
isnorm = false; l4LowV7
end x#0B
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% M .J
% Compute the Zernike Polynomials km[PbC
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Do\YPo_Mr
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% Determine the required powers of r: 'aAay*1
% ----------------------------------- iJsa;|2/
m_abs = abs(m); noLb
rpowers = []; +'{@Xe}
for j = 1:length(n) y~jYGN
rpowers = [rpowers m_abs(j):2:n(j)]; s(3iGuT
end xn`<g|"#
rpowers = unique(rpowers); :
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9>1
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% Pre-compute the values of r raised to the required powers, Z"u|-RoBV
% and compile them in a matrix: yS2[V,vS7
% ----------------------------- w*3DIVlxL
if rpowers(1)==0 1qgzb
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Dn9AOi!
rpowern = cat(2,rpowern{:}); ap%
Y}
rpowern = [ones(length_r,1) rpowern]; 7lJs{$
P
else u}L;/1,B
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); _hy<11S;
rpowern = cat(2,rpowern{:}); 4t<l9Ilp
end (q|EC;
n!>#o1Qr
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% Compute the values of the polynomials: oO8opS7F
% -------------------------------------- 4CdST3
y = zeros(length_r,length(n)); faJ>,^V#
for j = 1:length(n) _);;@T
s = 0:(n(j)-m_abs(j))/2; #VA8a=t
pows = n(j):-2:m_abs(j); /cN. -lEo%
for k = length(s):-1:1 ~l=Jx*
p = (1-2*mod(s(k),2))* ... >FRJvZ6
prod(2:(n(j)-s(k)))/ ... Z%uDz3I\Q"
prod(2:s(k))/ ... 8pQ:B/3=
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... UVIR
P#
prod(2:((n(j)+m_abs(j))/2-s(k))); dAZh# i[
idx = (pows(k)==rpowers); xr<.r4
y(:,j) = y(:,j) + p*rpowern(:,idx); fsxZQ=-PW
end Fm3f/]>k#_
U $ bLt
if isnorm g^qbd$ }
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi);
, 7kS#`P
end D]h~\
end YV 5kzq
% END: Compute the Zernike Polynomials R>YDn|cWI
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% k\J 6WT
>[U.P)7;
V L&5TZtz
% Compute the Zernike functions: p7YfOUo
k
% ------------------------------ mAFVjSa2
idx_pos = m>0; h"-}BjL
idx_neg = m<0; ^z^ UFW
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tl uyx
z = y; s=uWBh3J
if any(idx_pos) Zk4(
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ezOZHY>|#
end J3$Ce%<
if any(idx_neg) -5Km9X8
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); +-|D$@8S
end Fk 5;
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%ys-y?r
% EOF zernfun #{t?[JUn