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, XBh0=E?qiS
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, si.ZTG9m
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? 9l]+rs+
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? Rr{mD#+
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function z = zernfun(n,m,r,theta,nflag) ; $i{>mDT
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. *:{s|18Pj
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N wDVKp['
% and angular frequency M, evaluated at positions (R,THETA) on the {P&{+`sov
% unit circle. N is a vector of positive integers (including 0), and V|13%aE_v
% M is a vector with the same number of elements as N. Each element nm`[\3R
% k of M must be a positive integer, with possible values M(k) = -N(k) ?\"GT] 5D
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, "v@Y[QI
% and THETA is a vector of angles. R and THETA must have the same Ub2t7MU
% length. The output Z is a matrix with one column for every (N,M) >-*rtiE
% pair, and one row for every (R,THETA) pair. U0 nSI
% O3/][\
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 6!*be|<&
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ikX"f?Q;S2
% with delta(m,0) the Kronecker delta, is chosen so that the integral o$;t
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ^~9fQJNs
% and theta=0 to theta=2*pi) is unity. For the non-normalized q^; SZ^yW5
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 't.IYBHx
% v[{g"C
% The Zernike functions are an orthogonal basis on the unit circle. dWqKt0uh!
% They are used in disciplines such as astronomy, optics, and mvgsf(a*'
% optometry to describe functions on a circular domain. d,8L-pT$FM
% ZP~Mgz{f
% The following table lists the first 15 Zernike functions. [
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% X6)%2TwO
% n m Zernike function Normalization JZI)jIh
% --------------------------------------------------
U*(/eEtd-
% 0 0 1 1 9:
N[9;('
% 1 1 r * cos(theta) 2 Q6)Wh6Cm
% 1 -1 r * sin(theta) 2 BbsgZ4
% 2 -2 r^2 * cos(2*theta) sqrt(6) [U]^:sV)
% 2 0 (2*r^2 - 1) sqrt(3) -@L7!,j
% 2 2 r^2 * sin(2*theta) sqrt(6) 5.! OC5tO
% 3 -3 r^3 * cos(3*theta) sqrt(8) gR1vUad7
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) |>|f?^
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) ?1w{lz(P
% 3 3 r^3 * sin(3*theta) sqrt(8) [$M=+YRHMW
% 4 -4 r^4 * cos(4*theta) sqrt(10) (KxI*
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) \]<eLw-v
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 5|O~
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) /5:2g#S4
% 4 4 r^4 * sin(4*theta) sqrt(10) IUf&*'_
% -------------------------------------------------- Voy1
% 7>.d*?eao\
% Example 1: ^/]w}C#:d
% QiH>!Ssw
% % Display the Zernike function Z(n=5,m=1) ,+2!&"zD
% x = -1:0.01:1; &pHSX
% [X,Y] = meshgrid(x,x); )|3BS`
% [theta,r] = cart2pol(X,Y); wnUuoX(
% idx = r<=1; e~oh%l^C72
% z = nan(size(X)); &s6;2G&L$
% z(idx) = zernfun(5,1,r(idx),theta(idx)); HQ /D )D
% figure GdN9bA&,
% pcolor(x,x,z), shading interp ]#k=VKdV
% axis square, colorbar Z9wKjxu+
% title('Zernike function Z_5^1(r,\theta)') 9K!kU6Gh
% !0-KB#
% Example 2: (A(j.[4a
% ;k?Z,M:
% % Display the first 10 Zernike functions \k4tYL5
% x = -1:0.01:1; LV2#w_^I
% [X,Y] = meshgrid(x,x); RN^<bt{_U
% [theta,r] = cart2pol(X,Y); =csh=V@s
% idx = r<=1; ej91)3AO
% z = nan(size(X)); :2t0//@X
% n = [0 1 1 2 2 2 3 3 3 3]; elJ?g
&"
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; i~3\jD=<
% Nplot = [4 10 12 16 18 20 22 24 26 28]; m^!Kthq
% y = zernfun(n,m,r(idx),theta(idx)); 4Jn+Ot.,d
% figure('Units','normalized') ,V^2Oa
% for k = 1:10 ygK@\JHn
% z(idx) = y(:,k); \LG0
% subplot(4,7,Nplot(k)) >\br8=R
% pcolor(x,x,z), shading interp QM('bbN
% set(gca,'XTick',[],'YTick',[]) dNu?O>=
% axis square X9
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% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ^>Vl@cW0uz
% end 7D(Eo{ue
% VLPPEV-u
% See also ZERNPOL, ZERNFUN2. C5Vlqc;
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% Paul Fricker 11/13/2006 0Q7|2{
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% Check and prepare the inputs: "D,}|
% ----------------------------- e0<Wed
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) q2b>Z6!5
error('zernfun:NMvectors','N and M must be vectors.') %i6/=
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end bMq)[8,N
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if length(n)~=length(m) S5BS![-QK
error('zernfun:NMlength','N and M must be the same length.') dQn,0
end `pb=y}
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n = n(:); >*MGF=.QG
m = m(:); ."Kp6s `k
if any(mod(n-m,2)) DHg)]FQ/
error('zernfun:NMmultiplesof2', ... (gRTSd T?
'All N and M must differ by multiples of 2 (including 0).') :}UjX|D
end wP7
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if any(m>n) NR,R.N^[
error('zernfun:MlessthanN', ... tkYPfUvTE
'Each M must be less than or equal to its corresponding N.') D GL=\
end !hFzIp
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if any( r>1 | r<0 ) .6!cHL3ln
error('zernfun:Rlessthan1','All R must be between 0 and 1.') -d9L
end }uwZS=pw
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) F}.R-j#
error('zernfun:RTHvector','R and THETA must be vectors.') 'rNLh3
end "574%\#4z
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r = r(:);
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theta = theta(:); @QMU$]&i]
length_r = length(r); l_s#7 .9$
if length_r~=length(theta) v^J']p
error('zernfun:RTHlength', ... d/3bE*gr
'The number of R- and THETA-values must be equal.') xS(VgP&YGO
end 9mW
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% Check normalization: 1,Es'
% -------------------- vmv6y*qU
if nargin==5 && ischar(nflag) 3&I3ViAH
isnorm = strcmpi(nflag,'norm'); F~0iJnF
if ~isnorm TS`m&N{i")
error('zernfun:normalization','Unrecognized normalization flag.') ._]*Y`5)d
end p1[|5r5Day
else HWIn.ij
isnorm = false; guVuO
end fRxn,HyV
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% :]4s;q:m
% Compute the Zernike Polynomials r:PYAb=g
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Em4'b1mDX%
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% Determine the required powers of r: #F6<N]i
% ----------------------------------- .AQTUd(_
m_abs = abs(m); mG1!~}[
rpowers = []; i1X!G|Awfv
for j = 1:length(n) RD0*]4>]
rpowers = [rpowers m_abs(j):2:n(j)]; M;W&#Fz%
end M1]w0~G
rpowers = unique(rpowers); i03=Af3
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% Pre-compute the values of r raised to the required powers, lKf kRyO_S
% and compile them in a matrix: 7L!}F;yT
% ----------------------------- mhM;`dl
if rpowers(1)==0 wz@[rMf
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); >Hmho'
rpowern = cat(2,rpowern{:}); j+>[~c;0)
rpowern = [ones(length_r,1) rpowern]; t\]kVo)
else W4qnXD1n
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); fLeHn,*,"
rpowern = cat(2,rpowern{:}); I?nU+t;
end EuA352x
iaQfxQP1w%
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% Compute the values of the polynomials: V6+:g=@U-l
% -------------------------------------- \),zDO+
y = zeros(length_r,length(n)); nET<u;
for j = 1:length(n) QpiDBJCL
s = 0:(n(j)-m_abs(j))/2; I.Xbowl
pows = n(j):-2:m_abs(j); A/&u/?*C
for k = length(s):-1:1 O>I%O^
p = (1-2*mod(s(k),2))* ... G^z>2P
prod(2:(n(j)-s(k)))/ ... M04u>|
,
prod(2:s(k))/ ... @\:@_}Z`_}
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... `Ba?4_>k
prod(2:((n(j)+m_abs(j))/2-s(k))); vR pO0qG
idx = (pows(k)==rpowers); O'(D:D?
y(:,j) = y(:,j) + p*rpowern(:,idx); "r8N-
h/P
end asE.!g?
fGW~xul_
if isnorm +6~zMKp
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); RH$l?j6
end !b+!] 2~g}
end [z*1#lj S
% END: Compute the Zernike Polynomials bSQj=|h1
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Z#l6BXK
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% Compute the Zernike functions: @HE?G
% ------------------------------ a[,p1}!_
idx_pos = m>0; VV#'d
idx_neg = m<0; I.>8p]X
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z = y; hQz1zG`z7
if any(idx_pos) Q
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z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ~$rSy|19
end _;/+8=
if any(idx_neg) c>! ^\
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); |VjD. ]I
end ;>fM?ae5
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% EOF zernfun bPkz= ^-