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, y{=>$C[
我输入10阶的n、m,r,theta为38025*1向量,最后得到的z是29525*10阶的矩阵, *y]+dK&-
这个,跟我们用zygo干涉仪直接拟合出的36项zernike系数,有何关系呢? [po "To
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? 1@qgF
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function z = zernfun(n,m,r,theta,nflag) h"W8N+e\
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. i$uN4tVKT
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N \?lz&<
% and angular frequency M, evaluated at positions (R,THETA) on the rx!=q8=0R
% unit circle. N is a vector of positive integers (including 0), and VR0=SE
% M is a vector with the same number of elements as N. Each element a`c:`v2o
% k of M must be a positive integer, with possible values M(k) = -N(k) ^}$O|t
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, {C3Y7<
% and THETA is a vector of angles. R and THETA must have the same bF-"tm
% length. The output Z is a matrix with one column for every (N,M) "![L#)"s
% pair, and one row for every (R,THETA) pair. l=={pb
% j6YiE~
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike m%r/O&g
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), bGmx7qt#
% with delta(m,0) the Kronecker delta, is chosen so that the integral 9pD
7 f`
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, z5 m>H;P
% and theta=0 to theta=2*pi) is unity. For the non-normalized l#qv 5f
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 7Y( 5]A9=
% 7E7dSq
% The Zernike functions are an orthogonal basis on the unit circle. Lx[
,Z,kD
% They are used in disciplines such as astronomy, optics, and k%81f'H
% optometry to describe functions on a circular domain. (<c7<_-H
% ,kM)7!]N
% The following table lists the first 15 Zernike functions. B80aw>M
% 0C$vS`s&
% n m Zernike function Normalization ~)]} 91p
% -------------------------------------------------- rf
K8q'@
% 0 0 1 1 .*/Fucr
% 1 1 r * cos(theta) 2 n1v5Q2xw
% 1 -1 r * sin(theta) 2 SNpi=K!yn
% 2 -2 r^2 * cos(2*theta) sqrt(6) T)iW`vZg8
% 2 0 (2*r^2 - 1) sqrt(3) }}{Yw
% 2 2 r^2 * sin(2*theta) sqrt(6) h2q/mi5{
% 3 -3 r^3 * cos(3*theta) sqrt(8) g P}+wbk
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) gAbD7SE
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) 5Fw - d
% 3 3 r^3 * sin(3*theta) sqrt(8) 2N [=
% 4 -4 r^4 * cos(4*theta) sqrt(10) #f,y&\Xmf
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) c-4STPNQi
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 4=<*Vd`p
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) HIvZQQW|
% 4 4 r^4 * sin(4*theta) sqrt(10) W;_E 4
% -------------------------------------------------- YwDt.6(+,
% #ToK$8
% Example 1: {dWObh
% L"(4R^]
% % Display the Zernike function Z(n=5,m=1) R^&q-M=O[
% x = -1:0.01:1; \?fI t?
% [X,Y] = meshgrid(x,x); y/_XgPfWU
% [theta,r] = cart2pol(X,Y); C(?blv-vM0
% idx = r<=1; g_.^O$}
% z = nan(size(X)); t*S."
q
% z(idx) = zernfun(5,1,r(idx),theta(idx)); FY3IUG
% figure Sv[ 5NZn0&
% pcolor(x,x,z), shading interp ]+Ixi o
% axis square, colorbar 2f:^S/.A
% title('Zernike function Z_5^1(r,\theta)') $.E6S<(h
% R{hf9R ,
% Example 2: S~OhtHwK
% Vm1-C<V9
% % Display the first 10 Zernike functions cntco@
% x = -1:0.01:1; Li{~=S@N*
% [X,Y] = meshgrid(x,x); 2@j";+
% [theta,r] = cart2pol(X,Y); }FqA ppr
% idx = r<=1; oYM3Rgxf9Q
% z = nan(size(X)); 5jcte<
5I_
% n = [0 1 1 2 2 2 3 3 3 3]; v
$({C
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; 9WG{p[
% Nplot = [4 10 12 16 18 20 22 24 26 28]; [\z/Lbn
,.
% y = zernfun(n,m,r(idx),theta(idx)); B9dt=j3j2
% figure('Units','normalized') ()T[$.(
% for k = 1:10 a:STQk V
% z(idx) = y(:,k); BRRj$)u
% subplot(4,7,Nplot(k)) j Ch=@<9
% pcolor(x,x,z), shading interp Ukz;0q
% set(gca,'XTick',[],'YTick',[]) vw>j J
% axis square YUWn;#
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ~p1EF;4 #
% end aBuoHdg;
% [#^#+ |{\
% See also ZERNPOL, ZERNFUN2. G@ \Pi#1
"f.Z}AbP
kma?v B
% Paul Fricker 11/13/2006 YPDf
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r),PtI0X
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mB'3N;~
% Check and prepare the inputs: %:v`EjRD0
% ----------------------------- *~XA'Vw!
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) o89(
h!
error('zernfun:NMvectors','N and M must be vectors.') tA.`k;LT
end :*514N
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if length(n)~=length(m) [IZM.r`Z
error('zernfun:NMlength','N and M must be the same length.') eU+ {*YJg
end U\@A_
B
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h!@,8y[B
n = n(:); )Q;978:
m = m(:); {2d_"lHBt
if any(mod(n-m,2)) 1Nn@L2b 2
error('zernfun:NMmultiplesof2', ... a
dfR!&J
'All N and M must differ by multiples of 2 (including 0).') l~:v
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end R6;Phdh<>
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if any(m>n) nwI3| &
error('zernfun:MlessthanN', ... $"JpFT
'Each M must be less than or equal to its corresponding N.') q Dd~2"er
end Nil}js27
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if any( r>1 | r<0 ) 66&EBX}
error('zernfun:Rlessthan1','All R must be between 0 and 1.') -[7O7'
end gApoX0nrv
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) e9pOisZ;8
error('zernfun:RTHvector','R and THETA must be vectors.') rt7<Q47QE
end 5E\#%K[
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r = r(:); @ T;L$x
theta = theta(:); BbOu/i|
length_r = length(r); 0*%&>
if length_r~=length(theta) z$lF)r:Bc
error('zernfun:RTHlength', ... >QE{O.Z
'The number of R- and THETA-values must be equal.') n^(A=G
end CJknJn3m&
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% Check normalization: 9<Eg}Ic
% -------------------- qem(s</:
if nargin==5 && ischar(nflag) 4R%*Z~
isnorm = strcmpi(nflag,'norm'); $o?@0
if ~isnorm []W;t\h
error('zernfun:normalization','Unrecognized normalization flag.') <lxD}DH=
end [U
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else 0'Z\O
isnorm = false; imL_lw^?
end 7^TV~E#
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' Ie<Y_
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% /?-7Fg+,
% Compute the Zernike Polynomials \,UZX&ip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% zdun,`6
(P|~>k
K ?$#ntp
% Determine the required powers of r: ,1{Ep`
% ----------------------------------- h&@R| N
m_abs = abs(m); ]uL+&(cr
rpowers = []; uwIc963
for j = 1:length(n) gIEl.
rpowers = [rpowers m_abs(j):2:n(j)]; ~}ml*<z@
end S&jesG-F
rpowers = unique(rpowers); \kamcA
&<'n^n
qk(P>q8[
% Pre-compute the values of r raised to the required powers, ?NNn:t iD
% and compile them in a matrix: ~:Uwg+]j
% ----------------------------- 8[%Ao/m
if rpowers(1)==0 ;!@EixN-YH
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 0o&MB
Dp
rpowern = cat(2,rpowern{:}); 7sNw
rpowern = [ones(length_r,1) rpowern]; >k7q
g$
else N)8HR9[!
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); %WFu<^jm
rpowern = cat(2,rpowern{:}); ,38Eq`5&W
end @R~5-m
Rs& @4_D
F9q8SA#"
% Compute the values of the polynomials: p\vMc\
% -------------------------------------- /nx'Z0&+X
y = zeros(length_r,length(n)); -_VG;$,jE
for j = 1:length(n) 9~IQw#<
s = 0:(n(j)-m_abs(j))/2; =dP{ Gh
pows = n(j):-2:m_abs(j); )wXuwdc[
for k = length(s):-1:1 f2)XP$:
p = (1-2*mod(s(k),2))* ... oSb, :^Wl
prod(2:(n(j)-s(k)))/ ... L?&'xzt B
prod(2:s(k))/ ... Ma-\^S=
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... _#$9 y1bd
prod(2:((n(j)+m_abs(j))/2-s(k))); {[Q0qi =
idx = (pows(k)==rpowers); u<ySd?
y(:,j) = y(:,j) + p*rpowern(:,idx); \6|/RFT
end ^
?hA@{T/1
CENVp"C/`
if isnorm v]:=K-1n
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); {y
kYW%3s
end o@>? *=
end %5Kq^]q;Y
% END: Compute the Zernike Polynomials SJ'
%
^
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Z@D*1\TG=
RWq{Ff}Hk
#:fQ.WWO
% Compute the Zernike functions: Vsq8H}K
% ------------------------------ }w-wSkl1
idx_pos = m>0; G)=HB7u[a
idx_neg = m<0; -7>)i
3.
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1r[@(c0
z = y; (3vHY`9
if any(idx_pos) )YW<" $s
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); "7%:sty
end JeH;v0
if any(idx_neg) vy@rQC %9
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); v"u^M-_
end "HMP$)d
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% EOF zernfun Tq<2`*Qs