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, i4s_:%+
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, zPn2
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? J*FUJT
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? xN
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function z = zernfun(n,m,r,theta,nflag) Um2RLM%
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. T;`2t;
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N Kd;Iu\4hv
% and angular frequency M, evaluated at positions (R,THETA) on the yhG%@vSq
% unit circle. N is a vector of positive integers (including 0), and DqN<bu2
% M is a vector with the same number of elements as N. Each element 0Q4i<4 XW
% k of M must be a positive integer, with possible values M(k) = -N(k) -~=?g9fGm6
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, u}QcyG^
% and THETA is a vector of angles. R and THETA must have the same Lh;U2pA
% length. The output Z is a matrix with one column for every (N,M) u/ZV35z
% pair, and one row for every (R,THETA) pair. h#JX$9
% zz**HwRt
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike lv!8)GX|
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 7,7-E&d
% with delta(m,0) the Kronecker delta, is chosen so that the integral 2m{d>
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, T:=ST3#m
% and theta=0 to theta=2*pi) is unity. For the non-normalized )kk10AZV-E
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. )1, U~+JFU
% a>8&B
% The Zernike functions are an orthogonal basis on the unit circle. cf+EQY
% They are used in disciplines such as astronomy, optics, and [M/0 Qx[,
% optometry to describe functions on a circular domain. ,+GS.]8<
% 5`\"UC7?%
% The following table lists the first 15 Zernike functions. =lZtI6tZ
% $eiW2@
% n m Zernike function Normalization LTWiCI
% -------------------------------------------------- %n@ ^$&,&;
% 0 0 1 1 E/@
% 1 1 r * cos(theta) 2 VKMgcfbHr/
% 1 -1 r * sin(theta) 2 ?A]/
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% 2 -2 r^2 * cos(2*theta) sqrt(6) 9!?Ywc>0#
% 2 0 (2*r^2 - 1) sqrt(3) 'PWX19
% 2 2 r^2 * sin(2*theta) sqrt(6) Jt(RF*i
% 3 -3 r^3 * cos(3*theta) sqrt(8) u2
t=*<X
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) Dn[u zY6
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8)
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% 3 3 r^3 * sin(3*theta) sqrt(8) 76(/(v.x
% 4 -4 r^4 * cos(4*theta) sqrt(10) Zdy{e|-Zn
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) >J) 9&?
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) ?M BOd9
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) y&L Lx[8^
% 4 4 r^4 * sin(4*theta) sqrt(10) XImX1GH
% -------------------------------------------------- e4LJ3y&z"
% C Ef*:kr
% Example 1: eZ8DW6 l*
% au#/Q
% % Display the Zernike function Z(n=5,m=1) /*e6('9s
% x = -1:0.01:1; PS$g*x
% [X,Y] = meshgrid(x,x); utU;M*
% [theta,r] = cart2pol(X,Y); &fe67#0r)
% idx = r<=1; 4L/nEZ!Nsu
% z = nan(size(X)); +FH@|~^O
% z(idx) = zernfun(5,1,r(idx),theta(idx)); oS^g "hQ`\
% figure
4 z^7T
% pcolor(x,x,z), shading interp }6"l`$=Ev
% axis square, colorbar
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% title('Zernike function Z_5^1(r,\theta)') 1Vx>\A
% _sAcvKH
% Example 2: \ 0/m$V.
% s1bb2R
% % Display the first 10 Zernike functions :"'*1S*
% x = -1:0.01:1; L~("C
% [X,Y] = meshgrid(x,x); 2$b JMx>
% [theta,r] = cart2pol(X,Y); ^VsE2CX
% idx = r<=1; 4}-G<7*
% z = nan(size(X)); t1ers> h
% n = [0 1 1 2 2 2 3 3 3 3]; <9]J/w+
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; zNM*xPgS
% Nplot = [4 10 12 16 18 20 22 24 26 28]; d)kOW!5\
% y = zernfun(n,m,r(idx),theta(idx)); cb }OjM F
% figure('Units','normalized') nCDG PzJ
% for k = 1:10 a
y$CUw
% z(idx) = y(:,k); ?OFfU 4
% subplot(4,7,Nplot(k)) 4mvnFY}
% pcolor(x,x,z), shading interp ,z~"Mst
% set(gca,'XTick',[],'YTick',[]) l
p|`n
% axis square "u)Le6.
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) VV"w{#XKw
% end DD}YbuO7
% afE8Kqa:H
% See also ZERNPOL, ZERNFUN2. M_h8{
7c83g2|%
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% Paul Fricker 11/13/2006 iL|5}x5\
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NInZ~4:
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% Check and prepare the inputs: ee{K5 G
% ----------------------------- Z|xgZG{
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) C=t9P#g*.
error('zernfun:NMvectors','N and M must be vectors.') =1\mLI}@
end 8x-(7[#e<g
%$N,6}n
k\Y*tY#2
if length(n)~=length(m) : . PRM+
error('zernfun:NMlength','N and M must be the same length.') u7
end a,h]DkD
y"k%Wa`*
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n = n(:); 3|%058bF
m = m(:); I~4!8W-Y
if any(mod(n-m,2)) >z73uKA(
error('zernfun:NMmultiplesof2', ... ^ywDa^;-
'All N and M must differ by multiples of 2 (including 0).') LTuT"}dT[
end m#<Jr:-
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if any(m>n) T.W^L'L`
error('zernfun:MlessthanN', ... ~=9S AJr]
'Each M must be less than or equal to its corresponding N.') `6(Zc"/
\m
end VO~%O.>
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if any( r>1 | r<0 ) .)<l69ZD Z
error('zernfun:Rlessthan1','All R must be between 0 and 1.') 7rG+)kHG
end *JAC+<~d
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R ^ZOcONd-
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) Mkr
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error('zernfun:RTHvector','R and THETA must be vectors.') dptfIBYc+
end 5}a.<
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r = r(:); g _0| `Sm
theta = theta(:); p_vldTIW
length_r = length(r); #CcEI
if length_r~=length(theta) "{Hl! Zq/
error('zernfun:RTHlength', ... (}s& 84!
'The number of R- and THETA-values must be equal.') P=7X+}@
end NKTy!zWh
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% Check normalization: N`4XlD
% -------------------- .m.Ga|;
if nargin==5 && ischar(nflag) >v f-,B
isnorm = strcmpi(nflag,'norm'); p+0gE5
if ~isnorm :4)(Qa(
error('zernfun:normalization','Unrecognized normalization flag.') WJ^]mpH9
end 8l'W[6
else *3s-=.U~
isnorm = false; Bd- &~s^
end :yTr:FoF
%gWQ}QF
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \.>7w 1p
% Compute the Zernike Polynomials *IIA"tC
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% QO>';ul5
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R / ND f`
% Determine the required powers of r: {&L^|X
% ----------------------------------- C6Cr+TScH
m_abs = abs(m); `P <#kt
rpowers = []; ].2t7{64
for j = 1:length(n) "zkQu
rpowers = [rpowers m_abs(j):2:n(j)]; `VvQems
end rz&'wCiOO
rpowers = unique(rpowers); Q [C26U
h<bhH=6~
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% Pre-compute the values of r raised to the required powers, KP
6vb@(6
% and compile them in a matrix: ><xmw=
% ----------------------------- qM6hE.J
if rpowers(1)==0 %I{>H%CjE
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); $G{j[iLY
rpowern = cat(2,rpowern{:}); l<+PA$+}}
rpowern = [ones(length_r,1) rpowern]; 'X6Z:dZY
else C+"c^9[
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); FBvh7D.hV
rpowern = cat(2,rpowern{:}); oE6`]^^
end !"&-k:|g
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wm Ie x
% Compute the values of the polynomials: 587;2
% -------------------------------------- pzaU'y#PM
y = zeros(length_r,length(n)); ^q#[oO
for j = 1:length(n) Ul6|LTY
s = 0:(n(j)-m_abs(j))/2; NHe)$%a=H
pows = n(j):-2:m_abs(j); X Q
CE`m
for k = length(s):-1:1 cP\z*\dS
p = (1-2*mod(s(k),2))* ... sjb.Ezoq3
prod(2:(n(j)-s(k)))/ ... "C(yuVK1G
prod(2:s(k))/ ... B}. :7,/0
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... <QC7HR
prod(2:((n(j)+m_abs(j))/2-s(k))); l9OpaOVfJ
idx = (pows(k)==rpowers); 87W!R<G
y(:,j) = y(:,j) + p*rpowern(:,idx); 9Kgyt
end OU}eTc(FeC
4_sJ0 =z-
if isnorm pLCS\AUTsv
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); <m\<yZ2aa
end 0rz1b6F5,
end H1L)9oa
% END: Compute the Zernike Polynomials AzSu_
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% YllZ5<}
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% Compute the Zernike functions: o0>|
% ------------------------------ =wW M\f`=
idx_pos = m>0; S'W,AkT
idx_neg = m<0; ^suQ7#g
= :zPT;K
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z = y; "Da-e\yA
if any(idx_pos) \8m9^Z7IfK
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); Nnr[@^M5
end sD2,!/'
if any(idx_neg) 4nP4F+
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 9nY|S{L
end x?lRObHK
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% EOF zernfun %3"3V1