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, hh!4DHv
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, zb3,2D+P
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? O@HL%ha
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? m`BE{%
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function z = zernfun(n,m,r,theta,nflag) p(>D5uN_}5
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. w?V;ItcL
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N lk*wM?Z
% and angular frequency M, evaluated at positions (R,THETA) on the `*WzHDv5p
% unit circle. N is a vector of positive integers (including 0), and ]TVc 'G;
% M is a vector with the same number of elements as N. Each element #+&"m7
s
% k of M must be a positive integer, with possible values M(k) = -N(k) oP~%7Jt
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, m
yy*rt
% and THETA is a vector of angles. R and THETA must have the same v,|jmv+:
% length. The output Z is a matrix with one column for every (N,M) \1sWmN6
% pair, and one row for every (R,THETA) pair. XTJA"y
% _Un*x5u2O
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike Y0yu,
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), VpX*l3
% with delta(m,0) the Kronecker delta, is chosen so that the integral )>tT""yEl
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, Ax6zx
% and theta=0 to theta=2*pi) is unity. For the non-normalized
RK/>5
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. `-MCI)Fq_R
% 5(thDZ !
% The Zernike functions are an orthogonal basis on the unit circle. [>LO'}%
% They are used in disciplines such as astronomy, optics, and JFdMYb
% optometry to describe functions on a circular domain. .P#t"oW}
% ]?T,J+S
% The following table lists the first 15 Zernike functions. tn;Uaw
% 5 qMP u|A
% n m Zernike function Normalization v}\Fbe
% -------------------------------------------------- Ap~6Vu
% 0 0 1 1 XVF!l>nE
% 1 1 r * cos(theta) 2 g_@b- :$Yq
% 1 -1 r * sin(theta) 2 0ybMI+*
% 2 -2 r^2 * cos(2*theta) sqrt(6) +7{8T{
% 2 0 (2*r^2 - 1) sqrt(3) cv;2zq=T
% 2 2 r^2 * sin(2*theta) sqrt(6) _hgGF9
% 3 -3 r^3 * cos(3*theta) sqrt(8) tr58J%Mu
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 7)RRCsn
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) O>>/2V9
% 3 3 r^3 * sin(3*theta) sqrt(8) {Y3:Y+2X3*
% 4 -4 r^4 * cos(4*theta) sqrt(10) /.(~=6o5
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) uqVarRi$
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) Gzp*Vr
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) dXPTW;w
% 4 4 r^4 * sin(4*theta) sqrt(10) ^U);MH8
% -------------------------------------------------- /]?e^akA
% cfPp>EK
% Example 1: y7,t"XV
% 411z-aS
% % Display the Zernike function Z(n=5,m=1) vXZ
)
% x = -1:0.01:1; %jJIR88
% [X,Y] = meshgrid(x,x); /i
% [theta,r] = cart2pol(X,Y); oBs5xH7@-
% idx = r<=1; \~r_S
% z = nan(size(X)); MwX8F YF
D
% z(idx) = zernfun(5,1,r(idx),theta(idx)); e0]#vqdO
% figure xf?"Q#
% pcolor(x,x,z), shading interp .$1S-+(kV
% axis square, colorbar qC-4X"y+
% title('Zernike function Z_5^1(r,\theta)') p q%inSY
% -v:3#9uX)
% Example 2: <?:h(IZe[
% KpIY>k
% % Display the first 10 Zernike functions |"[;0)dw^
% x = -1:0.01:1; (w`_{%T
% [X,Y] = meshgrid(x,x); R2Lq??XA=
% [theta,r] = cart2pol(X,Y); 1d$wP$
% idx = r<=1; P`S'F_IN
% z = nan(size(X)); L3\(<[
% n = [0 1 1 2 2 2 3 3 3 3]; B`w8d[cL7
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; &XW~l>!+
% Nplot = [4 10 12 16 18 20 22 24 26 28]; }rnu:7
% y = zernfun(n,m,r(idx),theta(idx)); iVo-z#
% figure('Units','normalized') nm)/BK
% for k = 1:10 $oJjgA xcZ
% z(idx) = y(:,k); q^uCZnkb=
% subplot(4,7,Nplot(k)) O|+$9#,
% pcolor(x,x,z), shading interp 7#N
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% set(gca,'XTick',[],'YTick',[]) >;#rK@*&
% axis square UR(i_T&w
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) :2+z_+k}<
% end p<&>1}j=
% Jx Kd
% See also ZERNPOL, ZERNFUN2. ~fs}
J
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% Paul Fricker 11/13/2006 9?^0pR p
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% Check and prepare the inputs: 3lKs>HE0
% ----------------------------- oTr,zRL
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) `=Rxnl,<U
error('zernfun:NMvectors','N and M must be vectors.') Fu% n8
end j3S!uA?
@i#=1)Ze
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if length(n)~=length(m) sn
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error('zernfun:NMlength','N and M must be the same length.') HdTB[(
end 1;!dTh
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n = n(:); sQ&<cBs2
m = m(:); y 5?kv-"c
if any(mod(n-m,2)) fo<nk|i
error('zernfun:NMmultiplesof2', ... |oQhtk8.
'All N and M must differ by multiples of 2 (including 0).') 9JeT1\VvHY
end m63>P4h?
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if any(m>n) l\2"u M#7
error('zernfun:MlessthanN', ... <e wcWr
'Each M must be less than or equal to its corresponding N.') _`Y%Y6O1/
end 7#*`7 K'P!
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if any( r>1 | r<0 ) &V%faa1
error('zernfun:Rlessthan1','All R must be between 0 and 1.') #MviO!@
end z~i>GN_
#miG"2ea..
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) vrh2}biCR
error('zernfun:RTHvector','R and THETA must be vectors.') ~wcp&D
end kX*.BZI}C
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r = r(:); ];eJ'#
theta = theta(:); ;Y`8Ee4vH
length_r = length(r); y>cT{ )E$
if length_r~=length(theta) !,sQB_09C
error('zernfun:RTHlength', ... @Y ?p-&
'The number of R- and THETA-values must be equal.') kLXa1^Lq
end g3 !<A*<
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% Check normalization: \_)mWK,h
% -------------------- @lqI,Ce5
if nargin==5 && ischar(nflag) H1
i+j;RN
isnorm = strcmpi(nflag,'norm'); ^e80S^
if ~isnorm *8/cd0
error('zernfun:normalization','Unrecognized normalization flag.') <d[GGkY]=
end K]^Jl0
else II\}84U2
.
isnorm = false; :>jzL8
end [t*-s1cq
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% `2j"Z.=
% Compute the Zernike Polynomials &$h#9
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 7p{2&YhB
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% Determine the required powers of r: g}BS:#$
% ----------------------------------- {axRq'=
m_abs = abs(m); iE]^6i
rpowers = []; N*KM6j
for j = 1:length(n) vJQ_mz
rpowers = [rpowers m_abs(j):2:n(j)]; ir_X65l/2
end Xa$tW%)
rpowers = unique(rpowers); &}0#(Fa`
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+('=RyoT
% Pre-compute the values of r raised to the required powers, g&/r =U
% and compile them in a matrix: .G/RQn]x}
% ----------------------------- ;F/s!bupCM
if rpowers(1)==0 .|y{1?f_
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); &
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rpowern = cat(2,rpowern{:}); I=k`VI d:
rpowern = [ones(length_r,1) rpowern]; cdg&)
else zB6&),[,v
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ^>s{o5H&
rpowern = cat(2,rpowern{:}); :x!'Eer
n
end .0dx@Sbv
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% Compute the values of the polynomials: )2F%^<gZ#
% -------------------------------------- |+1k7S,
y = zeros(length_r,length(n)); :eSwXDy&
for j = 1:length(n) f%%'M.is
s = 0:(n(j)-m_abs(j))/2; %,udZyO3uR
pows = n(j):-2:m_abs(j); py\/m]
for k = length(s):-1:1 ` yM9XjEl>
p = (1-2*mod(s(k),2))* ... djDE0-QxcR
prod(2:(n(j)-s(k)))/ ... W"s)s
prod(2:s(k))/ ... ?Lr:>
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... $o*p#LU
prod(2:((n(j)+m_abs(j))/2-s(k))); UJ&gm_M+kL
idx = (pows(k)==rpowers); fBPJ8VY
y(:,j) = y(:,j) + p*rpowern(:,idx); ?9z1'6
end ho6,&Bp8
'~pZj"uy
if isnorm /$UWTq/C7
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ~0 L:c&V
end ;!<@Fm9W
end C+-sf
% END: Compute the Zernike Polynomials ]iaQD _'\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ;{"uG>#R
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% Compute the Zernike functions: ?WrL<?r)}U
% ------------------------------ [Ib17#74
idx_pos = m>0; sV`XJ9e|
idx_neg = m<0; 1<wolTf
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