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， 4 tt=u]:
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， ZI!;~q
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？ TU2MG VYy
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ |L)qH"Eo
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function z = zernfun(n,m,r,theta,nflag) ! 6+V
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. %)r1?H} #%
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N [!#;QQ&M
% and angular frequency M, evaluated at positions (R,THETA) on the ;4vx+> -
% unit circle. N is a vector of positive integers (including 0), and _ =(v? 2:?
% M is a vector with the same number of elements as N. Each element 6A} 45
% k of M must be a positive integer, with possible values M(k) = -N(k) 0te[i*G
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, * ^%ohCU i
% and THETA is a vector of angles. R and THETA must have the same !`dn# j
% length. The output Z is a matrix with one column for every (N,M) Eo{js?1G_
% pair, and one row for every (R,THETA) pair. WZ@$bf}f0
% )5U7 w
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike {' zs4)vw
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), `$Vn B
% with delta(m,0) the Kronecker delta, is chosen so that the integral ]!N|3"Ls
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, lHgmljn5u
% and theta=0 to theta=2*pi) is unity. For the non-normalized wIQt
f|ZI>
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. .ffb*gZ 4
% YR~ )07
% The Zernike functions are an orthogonal basis on the unit circle. R M`iOV,Y
% They are used in disciplines such as astronomy, optics, and OxVe}Fym
% optometry to describe functions on a circular domain. yLvU@V@~
% Qb1hk*$=
% The following table lists the first 15 Zernike functions. !2g*=oY
% DIc -"5~
% n m Zernike function Normalization safI`b w1
% -------------------------------------------------- T C._kAm
% 0 0 1 1 <~.1>CI9D3
% 1 1 r * cos(theta) 2 v1s0kdR,>
% 1 -1 r * sin(theta) 2 &;%LTF@I,
% 2 -2 r^2 * cos(2*theta) sqrt(6) .u 9,w
% 2 0 (2*r^2 - 1) sqrt(3) ncij)7c)u
% 2 2 r^2 * sin(2*theta) sqrt(6) )L7h:%h#
% 3 -3 r^3 * cos(3*theta) sqrt(8) ~@VyJT%
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) $)M 5@KT
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) \w@ "`!%
% 3 3 r^3 * sin(3*theta) sqrt(8) @avG*Mr^
% 4 -4 r^4 * cos(4*theta) sqrt(10) IaR D"oCH
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) #; >v,Jo
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) p+1kU1F 0
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) .|3& lb6
% 4 4 r^4 * sin(4*theta) sqrt(10) HY7#z2L
% -------------------------------------------------- ^/$bd4, z
% |`ZW(} ~
% Example 1: XXPpj< c
% [-JU(:Rh
% % Display the Zernike function Z(n=5,m=1) f5&K=4khn
% x = -1:0.01:1; b*"%E, ?
% [X,Y] = meshgrid(x,x); _{YUWV50}
% [theta,r] = cart2pol(X,Y); : ]~G9]R`
% idx = r<=1; m 3 W
% z = nan(size(X)); Q)\4 .d
% z(idx) = zernfun(5,1,r(idx),theta(idx)); E(Y}*.\]#s
% figure c\(CbC
% pcolor(x,x,z), shading interp Meo.
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% axis square, colorbar /X97dF)zt
% title('Zernike function Z_5^1(r,\theta)') 4oRDvn7f&
% <Is~DjIav
% Example 2: 5Ls
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% _ "H&
% % Display the first 10 Zernike functions ~k'SP(6#C
% x = -1:0.01:1; jZ> x5 W
% [X,Y] = meshgrid(x,x); 1gDsL
% [theta,r] = cart2pol(X,Y); h7F5-~SpD
% idx = r<=1; |#`qP^E
% z = nan(size(X)); FWDAG$K@0
% n = [0 1 1 2 2 2 3 3 3 3]; 9._owKj
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; vAjvW&'g
% Nplot = [4 10 12 16 18 20 22 24 26 28]; 8(""ui 8
% y = zernfun(n,m,r(idx),theta(idx)); [,/~* L;7
% figure('Units','normalized') bGe@yXId5
% for k = 1:10 xv>]e <":
% z(idx) = y(:,k); N)^`
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% subplot(4,7,Nplot(k)) ' yR)z\)
% pcolor(x,x,z), shading interp Ud'/
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% set(gca,'XTick',[],'YTick',[]) )lrmP(C*.a
% axis square &'<e9
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) LF\HmKM,
% end 6$A>%Jtwe
% x /E<@?*:
% See also ZERNPOL, ZERNFUN2. .*Ylj2nM
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% Paul Fricker 11/13/2006 IcP\#zhEv
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% Check and prepare the inputs: 6g &E v'
% ----------------------------- + Un(VTD
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 3
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error('zernfun:NMvectors','N and M must be vectors.') aGq1 YOD[$
end r6gfxW5
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if length(n)~=length(m) 6t F_u D
error('zernfun:NMlength','N and M must be the same length.') X_aC$_b
end U;#9^<^
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n = n(:); Zi= /w
m = m(:); lgQ"K(zY
if any(mod(n-m,2)) dpSNh1
error('zernfun:NMmultiplesof2', ... !\5w<*p8
'All N and M must differ by multiples of 2 (including 0).') ^Fpc8D,
end B"?ivxM:U
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if any(m>n) zv0bE?W9
error('zernfun:MlessthanN', ... E.eUd4XG
'Each M must be less than or equal to its corresponding N.') 1Y'NG<d _
end wl7 (|\-
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if any( r>1 | r<0 ) #ADm^UT ^
error('zernfun:Rlessthan1','All R must be between 0 and 1.') {; vLM*
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end gE: ?C2
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) `3:Q.A_?
error('zernfun:RTHvector','R and THETA must be vectors.') dVe,;?+A
end $Da?)Hz'F
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r = r(:); .F},Z[a&
theta = theta(:); qWM+!f
length_r = length(r); f 0&%
if length_r~=length(theta) @ Fkhida
error('zernfun:RTHlength', ... pZz\o
'The number of R- and THETA-values must be equal.') 4-m6e$p;
end {B-*w%}HU
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% Check normalization: LwYWgT\e
% -------------------- ! k 1 Ge+
if nargin==5 && ischar(nflag) YS:p(jtd
isnorm = strcmpi(nflag,'norm'); y9b%P]i
if ~isnorm nF
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error('zernfun:normalization','Unrecognized normalization flag.') f@[qS 7ok
end wJj :hA}
else |j~l%d*<w
isnorm = false; T@A Qe[U'v
end H *e +
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m!5Edo-;<
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 1mD)G55Ep
% Compute the Zernike Polynomials 'o~gT ;T#
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 1YK(oRSDn
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% Determine the required powers of r: 2VzYP~Jg
% ----------------------------------- 5|5p -B
m_abs = abs(m); 1ktxG1"1
rpowers = []; 2RQ- L
for j = 1:length(n) /,`OF/%
rpowers = [rpowers m_abs(j):2:n(j)]; H@1}_ d
end C;j& Vbf
rpowers = unique(rpowers); &r\8VEZq"
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% Pre-compute the values of r raised to the required powers, 2vWn(6`
% and compile them in a matrix: c]zFZJ 6M
% ----------------------------- 3~VV2O
if rpowers(1)==0 C~R
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rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); J#t-." f6^
rpowern = cat(2,rpowern{:}); w@<II-9L)<
rpowern = [ones(length_r,1) rpowern]; +IO>%
else m7DKC,
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); tj$[szo
rpowern = cat(2,rpowern{:}); @ $~IPg[J
end ?w+ V:D
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% Compute the values of the polynomials: 5'c#pm\Q
% -------------------------------------- 2;u
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y = zeros(length_r,length(n)); $ dF3@(p
for j = 1:length(n) :eSsqt9]9
s = 0:(n(j)-m_abs(j))/2; 2j}DI "|h
pows = n(j):-2:m_abs(j); R3;%ey u
for k = length(s):-1:1 3H`{
A/r
p = (1-2*mod(s(k),2))* ... a6-.|tt#t
prod(2:(n(j)-s(k)))/ ... /0 4US5En
prod(2:s(k))/ ... QW $p{ zo
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Zskj? +1
prod(2:((n(j)+m_abs(j))/2-s(k))); |-G2 pu;
idx = (pows(k)==rpowers); !nCq8~#
y(:,j) = y(:,j) + p*rpowern(:,idx); HC/z3b;
end |/vJ+aKq
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if isnorm naW!b& :
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); y?3. W
end //_H _ue$
end 31@Lr[!
% END: Compute the Zernike Polynomials tKeTHj;jO
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% s<)lC;# e
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9&5<ZC -D
% Compute the Zernike functions: f+Sb> $
% ------------------------------ }&t>j[
idx_pos = m>0; UhpJG O
idx_neg = m<0; ?UZt3 0|1
\1Xk[%
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z = y; BUJ \[/
if any(idx_pos) 8v4 o+w P
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); yB2h/~+
end acR|X@ \3
if any(idx_neg) b1Kt SRLV
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); CMaph
end {PcJuRTHB
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% EOF zernfun 0{|HRiQH9+
