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, 8m%+O#
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, g '2'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? /5cFa
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? q@K8,=/.#
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function z = zernfun(n,m,r,theta,nflag) 7{<:g!
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 1Rrp#E}
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N /.PjHTM<
% and angular frequency M, evaluated at positions (R,THETA) on the )P&>Tc?;z
% unit circle. N is a vector of positive integers (including 0), and A4;EtW+F
% M is a vector with the same number of elements as N. Each element `PML4P[
% k of M must be a positive integer, with possible values M(k) = -N(k) },r30` )Q
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, :[icd2JCw]
% and THETA is a vector of angles. R and THETA must have the same +/!kL0[v
% length. The output Z is a matrix with one column for every (N,M) IQn|0$':Z
% pair, and one row for every (R,THETA) pair. h SGI
% Bw5zh1ALC;
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike qg521o$*
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), vo48\w7[
% with delta(m,0) the Kronecker delta, is chosen so that the integral &f12Q&jY7
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, B[XVTok
% and theta=0 to theta=2*pi) is unity. For the non-normalized &T7|f!y
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. !~X[qT
% Djv0]Sm^!
% The Zernike functions are an orthogonal basis on the unit circle. P-B3<~*i!
% They are used in disciplines such as astronomy, optics, and 21(8/F ~{
% optometry to describe functions on a circular domain. Po+I!TL'
% LOm*=MVex
% The following table lists the first 15 Zernike functions. y"N7r1Pf
% )=,%iL-
% n m Zernike function Normalization TVaD',5_V%
% -------------------------------------------------- Ql~9a
[8T~
% 0 0 1 1 =E{e|(1+u
% 1 1 r * cos(theta) 2 #jY\l&E
% 1 -1 r * sin(theta) 2 +{b!,D3sa*
% 2 -2 r^2 * cos(2*theta) sqrt(6) *g0} pD;r
% 2 0 (2*r^2 - 1) sqrt(3) AH*{Bi[vX
% 2 2 r^2 * sin(2*theta) sqrt(6) ~!PAs_O
% 3 -3 r^3 * cos(3*theta) sqrt(8) vTrjhTa\
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) M5$YFGGR
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) Gk"o/]Sf
% 3 3 r^3 * sin(3*theta) sqrt(8) \*>r[6]*&5
% 4 -4 r^4 * cos(4*theta) sqrt(10) R$[nYw
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) +TA'P$j
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) DV={bcQ
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) x3./
% 4 4 r^4 * sin(4*theta) sqrt(10) U)v['5%
% -------------------------------------------------- >Yr-aDV
% ;3\oU$'
% Example 1:
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% I+qg'mo
% % Display the Zernike function Z(n=5,m=1) 608}-J=3#
% x = -1:0.01:1; ,`4chD
% [X,Y] = meshgrid(x,x); .Vohd@s9l
% [theta,r] = cart2pol(X,Y); Vjv~RNGF
% idx = r<=1; 5m.{ayE
% z = nan(size(X)); z]/;?
% z(idx) = zernfun(5,1,r(idx),theta(idx)); zWN/>~}U\
% figure x2q6y
% pcolor(x,x,z), shading interp ;m/h?Y~
% axis square, colorbar 4CUoXs'
% title('Zernike function Z_5^1(r,\theta)') yH\3*#+
% hDO\Q7
% Example 2: &E{CQ#k
% uL\b*rI
% % Display the first 10 Zernike functions Xv1SRP#
% x = -1:0.01:1; [r[IWy(}
% [X,Y] = meshgrid(x,x); & XS2q0-x
% [theta,r] = cart2pol(X,Y); =r w60B
% idx = r<=1; Qs38VlR_m
% z = nan(size(X)); h8nJt>h
% n = [0 1 1 2 2 2 3 3 3 3]; JbV\eE#KrC
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; qh|t}#DrR
% Nplot = [4 10 12 16 18 20 22 24 26 28]; #hp7@ Tu
% y = zernfun(n,m,r(idx),theta(idx)); $)HD`E
% figure('Units','normalized') `"7}'|
% for k = 1:10 \&{a/e2:S
% z(idx) = y(:,k); {;z{U;j
% subplot(4,7,Nplot(k)) C:*=tD1
% pcolor(x,x,z), shading interp Q9i&]V[`
% set(gca,'XTick',[],'YTick',[]) k-:wM`C
% axis square 3MmpB9l#H
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) _H,xnh#nZ
% end S.<aCN<@
% )bd)noZi
% See also ZERNPOL, ZERNFUN2. 3"*tP+H
w5C$39e\G
?S*Cvr+=4
% Paul Fricker 11/13/2006 O c[F
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.S5%Qa [uW
% Check and prepare the inputs: %9q]
% ----------------------------- Io(*_3V)B
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 6UAn#d9
error('zernfun:NMvectors','N and M must be vectors.') gwA+%]
end EZ"n3#/
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TR<M3,RG#%
if length(n)~=length(m) %Tv2op
error('zernfun:NMlength','N and M must be the same length.') J1s~w`,
end >U~{WM$"Y
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n = n(:); {
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m = m(:); 483/ZgzT`
if any(mod(n-m,2)) 3)6TnY/u6{
error('zernfun:NMmultiplesof2', ... =O1py_m
'All N and M must differ by multiples of 2 (including 0).') y6hb-:
#1
end F3?PlH:Y
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if any(m>n) hZ[(Ik]*Zd
error('zernfun:MlessthanN', ... f?qp*
'Each M must be less than or equal to its corresponding N.') [ ,&O
end m8T< x>
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NT0im%
if any( r>1 | r<0 ) ^H(,^cVN
error('zernfun:Rlessthan1','All R must be between 0 and 1.') F"ua`ercI
end = pCO1<wR
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) MQY}}a-oug
error('zernfun:RTHvector','R and THETA must be vectors.') jU9\BYUg
end F1q6
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r = r(:); Yo*.? Mq'
theta = theta(:); ~PtIq.BY
length_r = length(r); W7` fI*lc
if length_r~=length(theta) -z~;f<+I`
error('zernfun:RTHlength', ... k9_c<TSzu
'The number of R- and THETA-values must be equal.') -<{;.~nI.
end _)U.5f<
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% Check normalization: mUdj2vB$+'
% -------------------- >+FaPym
if nargin==5 && ischar(nflag) vve L|j
isnorm = strcmpi(nflag,'norm'); Rn_FYP
if ~isnorm >X_5o^s2s
error('zernfun:normalization','Unrecognized normalization flag.') d=DQS>Nz
end n)uck5
else ;i,3KJ[L
isnorm = false; %F}i2!\<L
end aCF=Og
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% i@L_[d^|j`
% Compute the Zernike Polynomials -d4|EtN
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% })yB2Q0
!T"jvDYH
8)ykXx/f@
% Determine the required powers of r: x(+H1D\W
% ----------------------------------- IBz)3gj J
m_abs = abs(m); X.GK5Phd
rpowers = []; VC X^D)[-
for j = 1:length(n) E}g)q;0v|2
rpowers = [rpowers m_abs(j):2:n(j)]; JFu9_=%+
end A&S n^mw
rpowers = unique(rpowers); `kYcTFk
#SX-Y)> 1@
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% Pre-compute the values of r raised to the required powers, U[{vA6
% and compile them in a matrix: m0p%R>:5
% ----------------------------- e0ULr!p
if rpowers(1)==0 ~7>D>!!
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ugzrG0=lx
rpowern = cat(2,rpowern{:}); hjq@.5
rpowern = [ones(length_r,1) rpowern]; dwqR,|
else l.xKv$uOGR
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 2&zklXuo:
rpowern = cat(2,rpowern{:}); +-:o+S`q~
end UuCRQN H
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J,*+Ak
~
% Compute the values of the polynomials: 8 .t3`FGH
% -------------------------------------- Ip|=NQL>
y = zeros(length_r,length(n)); U&W/Nj
for j = 1:length(n) j@R"AP}
s = 0:(n(j)-m_abs(j))/2; o8X? 1
pows = n(j):-2:m_abs(j); .q<5OE(f
for k = length(s):-1:1 & zv!cf
p = (1-2*mod(s(k),2))* ... p"0Dl9
prod(2:(n(j)-s(k)))/ ... b(\Mi_J
prod(2:s(k))/ ... 8?1MnjhX10
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... /vI"v4
prod(2:((n(j)+m_abs(j))/2-s(k))); |W*5<2Q9
idx = (pows(k)==rpowers); ^<3{0g-"AW
y(:,j) = y(:,j) + p*rpowern(:,idx); Q^8/"aV\
end 0),fY(D2T
$Lz!04
if isnorm mD%IHzbn
H
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); eV"s5X[$
end z.^_;Vql_
end nrR2U`
% END: Compute the Zernike Polynomials V
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 29zMs9oKPP
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vX_;Y#uD
% Compute the Zernike functions: M)~sL1)
% ------------------------------ kN6jX
idx_pos = m>0; 4K
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idx_neg = m<0; ]cM8TT
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h; 6G~D
z = y; ' e %>Ip
if any(idx_pos) y>cLG5v
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); vl~HV8MAv
end -EP(/CS!
if any(idx_neg) -qBdcbi|x)
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); EQQ@nW{;
end Zs8]A0$
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% EOF zernfun R@#xPv4o%