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, _5SA(0D#9
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, 8jm\/?k|
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? ;sfk@ec
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? iVqa0Gl+}
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function z = zernfun(n,m,r,theta,nflag) ?j-;;NNf
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. G,JK$j>*l
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N z9IJ%=R
% and angular frequency M, evaluated at positions (R,THETA) on the q+}Er*r
% unit circle. N is a vector of positive integers (including 0), and QP0[
% M is a vector with the same number of elements as N. Each element G2e0\}q
% k of M must be a positive integer, with possible values M(k) = -N(k) eK'ztqQ
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, v#o<.
Ig
% and THETA is a vector of angles. R and THETA must have the same d@0&
% length. The output Z is a matrix with one column for every (N,M) Q2 @Ugt$
% pair, and one row for every (R,THETA) pair. P1Chmg
% s2M|ni=
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike SM@RELA'Lb
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), kG70j{gf
% with delta(m,0) the Kronecker delta, is chosen so that the integral Q^z&;%q1
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, M~#%
[?iU
% and theta=0 to theta=2*pi) is unity. For the non-normalized CxW-lU3G`
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. O]N
8QH
% ';OZP2
% The Zernike functions are an orthogonal basis on the unit circle. ;^Y]nsd
% They are used in disciplines such as astronomy, optics, and !
fSM6Vo
% optometry to describe functions on a circular domain. a0=5G>G9c
% T{Rhn V1
% The following table lists the first 15 Zernike functions. 2Ed
% 2h^9lrQcQG
% n m Zernike function Normalization -L)b;0%
% -------------------------------------------------- Nq=r404
% 0 0 1 1 A-XWG9nL
% 1 1 r * cos(theta) 2 qH-':|h7
% 1 -1 r * sin(theta) 2 Bk9? =
% 2 -2 r^2 * cos(2*theta) sqrt(6) .<|.nK` 6
% 2 0 (2*r^2 - 1) sqrt(3) S|HnmkV66
% 2 2 r^2 * sin(2*theta) sqrt(6) mFu0$N6]H
% 3 -3 r^3 * cos(3*theta) sqrt(8) u"*Wo'3I|
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) aO]FQ#l2b
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) b3RCsIz
% 3 3 r^3 * sin(3*theta) sqrt(8) _]~= Kjp
% 4 -4 r^4 * cos(4*theta) sqrt(10) 4:S?m(ah/
% 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) (aa}0r5
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) qqR8E&Y{
% 4 4 r^4 * sin(4*theta) sqrt(10) gkN
)`/`*
% -------------------------------------------------- _Bhm\|t
% j/+e5.EX/
% Example 1: aur4Ky> :
% y8QJ=v* B
% % Display the Zernike function Z(n=5,m=1) $pOgFA1'
% x = -1:0.01:1; d:V6.7>,
% [X,Y] = meshgrid(x,x); x!@P|c1nKC
% [theta,r] = cart2pol(X,Y); -g;cg7O#(
% idx = r<=1; "2~%-;c
% z = nan(size(X)); WMw]W&
% z(idx) = zernfun(5,1,r(idx),theta(idx)); !04zWYHo
% figure TUaW'
% pcolor(x,x,z), shading interp L[s8`0
% axis square, colorbar %oY=.Ok ]
% title('Zernike function Z_5^1(r,\theta)') nD8CP[bRo
% _jr'A -M
% Example 2: h72#AN
% '3MCb
% % Display the first 10 Zernike functions D)*OQLHW
% x = -1:0.01:1; >TqMb8e_
% [X,Y] = meshgrid(x,x); #mDeA >b
% [theta,r] = cart2pol(X,Y); k-uwK-B}v+
% idx = r<=1; lIlmXjL0
% z = nan(size(X)); (,5,}
% n = [0 1 1 2 2 2 3 3 3 3]; KNw{\Pz~w
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; dY'mY ~Tv
% Nplot = [4 10 12 16 18 20 22 24 26 28]; RXF%A5FXh
% y = zernfun(n,m,r(idx),theta(idx)); 609_ZW;)
% figure('Units','normalized') UD@u hL
% for k = 1:10 _CDl9pP36#
% z(idx) = y(:,k); v>&sb3I
% subplot(4,7,Nplot(k)) !PIpvx{aX
% pcolor(x,x,z), shading interp =Q!)xEK
% set(gca,'XTick',[],'YTick',[]) ?B!=DC @?H
% axis square #r ;;d(
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ,py:e>+^t
% end k]<E1 c/
% RTgR>qI&)
% See also ZERNPOL, ZERNFUN2. }>|M6.n "
8.'[>VzBL
! 9U
% Paul Fricker 11/13/2006 RrPo89o
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8a;I,DK=j
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% Check and prepare the inputs: W=9Zl(2C
% ----------------------------- 4R~f
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ~baVS-v
error('zernfun:NMvectors','N and M must be vectors.') lOc!KZHUp
end \
M_}V[1+
79?%g=#=
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if length(n)~=length(m) aNLkkkJg<;
error('zernfun:NMlength','N and M must be the same length.') I,:R~^qJ8v
end jv
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n = n(:); EP|OKXRltA
m = m(:); DeAi'"&
if any(mod(n-m,2)) (|F } B
error('zernfun:NMmultiplesof2', ... FHu
-';
'All N and M must differ by multiples of 2 (including 0).') Ev R6^n/
end l|O)B #
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if any(m>n) ^zMME*G
error('zernfun:MlessthanN', ... huu v`$~y
'Each M must be less than or equal to its corresponding N.') i09w(k?
end b~1]}9TJ
G9/5KW}-
q Z,7q
if any( r>1 | r<0 ) U8T"ABvFP
error('zernfun:Rlessthan1','All R must be between 0 and 1.') KVvzVQ1
end _msV3JBr
#DN5S#Ic
%SwN/rna
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ?3{R'Buv]
error('zernfun:RTHvector','R and THETA must be vectors.') 4TBK:Vm5
end wO&edZ]zb^
j#C1+Us
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r = r(:); x}`]9XQ
theta = theta(:); .)7r /1o
length_r = length(r); fVU9?^0/)9
if length_r~=length(theta) yC]xYn)
error('zernfun:RTHlength', ... R3G+tE/Y
'The number of R- and THETA-values must be equal.') uA}w?;
end ]y4(WG;:
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],fu#pi=]
% Check normalization: =?*6lS}gy
% -------------------- PFqc_!Pm
if nargin==5 && ischar(nflag) gbf-3KSp^
isnorm = strcmpi(nflag,'norm'); 6O`s&T,t
if ~isnorm Y4\BHFq
error('zernfun:normalization','Unrecognized normalization flag.') 62R94
end eYER"E
else 8$|<`:~J
isnorm = false; Z$0+jpG_s
end 4>uy+"8PO
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% X}Ey6*D:
% Compute the Zernike Polynomials Y:~A-_
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% o)X(;o
D^?jLfW8
hnQDm$k
% Determine the required powers of r: J3]W2m2Zw
% ----------------------------------- 6I$laHx?
m_abs = abs(m); 9@Iz:!oqb
rpowers = []; >q'xW=Y
j\
for j = 1:length(n) YWV"I|Z
rpowers = [rpowers m_abs(j):2:n(j)]; $`2rtF
end +<G |Ru-
rpowers = unique(rpowers); ;g3z?Uz)
N2?o6)
m3apeIEi[
% Pre-compute the values of r raised to the required powers, KjrUTG0oA
% and compile them in a matrix: pJ` M5pF
% ----------------------------- 'IorjR@40
if rpowers(1)==0 O8;`6r
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); F:PaVr3q
rpowern = cat(2,rpowern{:}); Z~g I )
rpowern = [ones(length_r,1) rpowern]; Ub0hISA
else /Hox]r]'e
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); y:U'3G-
rpowern = cat(2,rpowern{:}); (,5oqU9s@
end r/X4Hy0!lT
Ywj=6 +;
b`NXe7A
% Compute the values of the polynomials: K[wOK
% -------------------------------------- DCJmk6p%0
y = zeros(length_r,length(n)); z (N3oBW
for j = 1:length(n) E8TJ*ZU
s = 0:(n(j)-m_abs(j))/2; hSxlj7Eo^T
pows = n(j):-2:m_abs(j); !]%M
for k = length(s):-1:1 IETdL{`~
p = (1-2*mod(s(k),2))* ... o/EN3J
prod(2:(n(j)-s(k)))/ ... i+/:^tc;
prod(2:s(k))/ ... Kn9O=?Xh;
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... zW`Zmt\T2
prod(2:((n(j)+m_abs(j))/2-s(k))); W\(u1>lj
idx = (pows(k)==rpowers); jlmP1b9
y(:,j) = y(:,j) + p*rpowern(:,idx); ;j#$d@VG"
end BrW1:2w
>\
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if isnorm 5ryzAB O\2
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); '}3m('u
end 'Zq$W]i
end l!n<.tQW
% END: Compute the Zernike Polynomials sU
{'
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% f@ &?K<
'%W'HqVcG1
;z6Gk&?
% Compute the Zernike functions: Wvhg:vup
% ------------------------------ T&?0hSYt
idx_pos = m>0; >28.^\?H4
idx_neg = m<0; G1;.\ i
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z = y; <
q6z$c)K
if any(idx_pos) <Tq&Va_w
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); OD,"8JF
end ?/mk FDN
if any(idx_neg) ryz
[A:^G
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); O"otzla
end %5X}4k!p
~R`Rj*Q2Y
dg%Orvuz
% EOF zernfun &&iZ?JteZ