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， }g# & Q0
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， oI)GKA_Ng7
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？ 'XY `(3q
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ oAWzYu(v
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function z = zernfun(n,m,r,theta,nflag) 1]IQg;q
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. K=!Bh*
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N lEHzyh}2k
% and angular frequency M, evaluated at positions (R,THETA) on the p.+ho~sC,.
% unit circle. N is a vector of positive integers (including 0), and $zB[B;-!$
% M is a vector with the same number of elements as N. Each element &Ysosy*
% k of M must be a positive integer, with possible values M(k) = -N(k) 1]or UF&_
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, A,r*%&4~
% and THETA is a vector of angles. R and THETA must have the same l;y7]DO
% length. The output Z is a matrix with one column for every (N,M) k}
]T;|h]
% pair, and one row for every (R,THETA) pair. /U o
y/}!
% zC _<(4$-"
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike }y9mNT
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), Y"OG@1V;8
% with delta(m,0) the Kronecker delta, is chosen so that the integral "\0v,!@
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, v1a6?-
% and theta=0 to theta=2*pi) is unity. For the non-normalized (JM4R8fR&
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. }%Bl>M
% ?wnzTbJN
% The Zernike functions are an orthogonal basis on the unit circle. U|g:`v7
% They are used in disciplines such as astronomy, optics, and )(y) A[
% optometry to describe functions on a circular domain. uV 7BK+[O
% /-bO!RTwf
% The following table lists the first 15 Zernike functions. r}uz7}z %"
% ,V*%V;
% n m Zernike function Normalization (@iMLuewK
% -------------------------------------------------- Oft4- 4$E
% 0 0 1 1 n_3O-X(
% 1 1 r * cos(theta) 2 1 "pw
% 1 -1 r * sin(theta) 2 tv!_e$ CR
% 2 -2 r^2 * cos(2*theta) sqrt(6) 5 |jw^s7
% 2 0 (2*r^2 - 1) sqrt(3) [HCAmnb
% 2 2 r^2 * sin(2*theta) sqrt(6) Qg6 W5Hc
% 3 -3 r^3 * cos(3*theta) sqrt(8) s}N#n(
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) } :Z# }8
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) wm+/e#'&
% 3 3 r^3 * sin(3*theta) sqrt(8) u]vQ>Uu
% 4 -4 r^4 * cos(4*theta) sqrt(10) 'uq#ai[5I
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) X^WrccNX
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 0_CN/5F
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) #!)n
{h+
% 4 4 r^4 * sin(4*theta) sqrt(10) tU_y 6
% -------------------------------------------------- C+|b1/N-
% ?JL:CBvCp
% Example 1: ,\qs4&
% _x!7}O#k
% % Display the Zernike function Z(n=5,m=1) A45A:hqs
% x = -1:0.01:1; ei
rz Yt
% [X,Y] = meshgrid(x,x); wC5ee:u C%
% [theta,r] = cart2pol(X,Y); Y5F]:gs@
% idx = r<=1; {'U
Rz[g
% z = nan(size(X)); $z+8<?YD
% z(idx) = zernfun(5,1,r(idx),theta(idx)); +|tC'gCnV
% figure @-+Q#
Zz`
% pcolor(x,x,z), shading interp A<W 6=5h
% axis square, colorbar RIIitgV_
% title('Zernike function Z_5^1(r,\theta)') Y +Fljr*
% NMA}Q$o
s
% Example 2: +zy=50,
% PG,_^QGCX
% % Display the first 10 Zernike functions c q$i
% x = -1:0.01:1; e*L.U~ZR
% [X,Y] = meshgrid(x,x); T8 ^5=/
% [theta,r] = cart2pol(X,Y); [ :zO}r:
% idx = r<=1; j\m_o% 4
% z = nan(size(X)); s>^dxF!+
% n = [0 1 1 2 2 2 3 3 3 3]; / z}~zO
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; X,LD
% Nplot = [4 10 12 16 18 20 22 24 26 28]; {#{DH?=^)u
% y = zernfun(n,m,r(idx),theta(idx)); -=(!g&0
% figure('Units','normalized') Kw#i),M
% for k = 1:10 {RF-sqce
% z(idx) = y(:,k); tz s</2
G,
% subplot(4,7,Nplot(k)) mQY_`&Jq
% pcolor(x,x,z), shading interp $jg *pmR-
% set(gca,'XTick',[],'YTick',[]) f"St&q>[s
% axis square n/h,Lr)Z
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) L: z?Zt)|
% end Y*! qG
% ahPoEh
% See also ZERNPOL, ZERNFUN2. % DdJ ^qHI
Op_RzZP`
ezMI \r6
% Paul Fricker 11/13/2006 ?yj6CL(,
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% Check and prepare the inputs: LU 9A#
% ----------------------------- $_x^lr
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) m'f,_ \'
error('zernfun:NMvectors','N and M must be vectors.') 0A( +ZMd
end ;f"0~D2
$ >EYhLBa
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if length(n)~=length(m) A7n\h-b
error('zernfun:NMlength','N and M must be the same length.') rs ~wv('
end ~t~- A,1
%%4t~XC#
z- b*D}&
n = n(:); xQ@^$_
m = m(:); Cm\6tD
if any(mod(n-m,2)) beu\cV3
error('zernfun:NMmultiplesof2', ... qu-/"w<3$
'All N and M must differ by multiples of 2 (including 0).') QPfc(Z
end ~SnSEhE
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if any(m>n) 1[yq0^\]M[
error('zernfun:MlessthanN', ... v_nj$1dY6
'Each M must be less than or equal to its corresponding N.') 6 C+"`(u%V
end 8f3vjK'
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if any( r>1 | r<0 ) Ba/Yl
error('zernfun:Rlessthan1','All R must be between 0 and 1.') ]~E0gsq
end
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) 7__Q1 > o
error('zernfun:RTHvector','R and THETA must be vectors.') 7IjQi=#:
end 9s_,crq5
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r = r(:); I:d[Q
s
theta = theta(:); n8D xB@DI
length_r = length(r); kVy\b E0o
if length_r~=length(theta) :P(K 2q3
error('zernfun:RTHlength', ...
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'The number of R- and THETA-values must be equal.') T.qNCJmB
end hc'-Dh
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% Check normalization: Z" uY}P3
% -------------------- MC {
2 X
if nargin==5 && ischar(nflag) j7)Ao*WN
isnorm = strcmpi(nflag,'norm'); [Ts"OPb% ~
if ~isnorm n2I V2^ "
error('zernfun:normalization','Unrecognized normalization flag.') T
N!= @Gy
end +fnK /%b
else tT79 p.z B
isnorm = false; izx#3u$P
end Y p:KI7
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {je -I9%OK
% Compute the Zernike Polynomials bpxe znz
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% &zuG81F6
Kk{<@v)
V} zEK0n(6
% Determine the required powers of r: D2,z)O%VK
% ----------------------------------- I'@Ydt2
m_abs = abs(m); jr`Es s
rpowers = []; 6 HlePTf8
for j = 1:length(n) e~"fn*"
rpowers = [rpowers m_abs(j):2:n(j)]; d`(@_czdF
end %bdjBa}
rpowers = unique(rpowers); G7CG~:3h+
~jb"5CX
MX ;J5(Ae
% Pre-compute the values of r raised to the required powers, i}~SDY
% and compile them in a matrix: 0p@k({ ] <
% ----------------------------- E. U _W
if rpowers(1)==0 Q[d}J+l4{
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); hnznp1[#@
rpowern = cat(2,rpowern{:}); +/ &_v^sC;
rpowern = [ones(length_r,1) rpowern]; H`geS
else 0pSmj2/,.
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); =ID
2
rpowern = cat(2,rpowern{:}); A?@@*$&
end <2nZ&M4/s{
p^ pOuy8
Hy R!O>
% Compute the values of the polynomials: Hp(D);0+)
% -------------------------------------- }`NU@O#
y = zeros(length_r,length(n)); L =8+_0
for j = 1:length(n) O%ug@& S{
s = 0:(n(j)-m_abs(j))/2; {Ions~ cO)
pows = n(j):-2:m_abs(j); }>[G5[ \
for k = length(s):-1:1 ^7.h%lSg
p = (1-2*mod(s(k),2))* ... 2"-S<zM
prod(2:(n(j)-s(k)))/ ... Kn?lHH*w7
prod(2:s(k))/ ... `w.AQ?p@
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ez9 q7SpA
prod(2:((n(j)+m_abs(j))/2-s(k))); H&yD*@
idx = (pows(k)==rpowers); ys#i@
y(:,j) = y(:,j) + p*rpowern(:,idx); M1%Dg'}G
end nIvJrAm4k
nA ~E
"*
if isnorm g(| 6~}|o+
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); /NFz4h =>
end c1xrn4f@a
end +L=*:e\j
% END: Compute the Zernike Polynomials 0W%@gs5d&
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% u@3 y&b
%r iK+
W k}AmC
% Compute the Zernike functions: c c
% ------------------------------ NOS>8sy
idx_pos = m>0; \-*eL;qP
idx_neg = m<0; aSP4a+\*
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efY8M2
z = y; O,.!2wVrN
if any(idx_pos) Mzd[fR5a8
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); dgo3'ZO
end DE
IB!n
if any(idx_neg) T{}f HfM
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); WX4;l(P L=
end =@)d5^<5F
,]5Ic.};p
%5* @l vy
% EOF zernfun Krs2Gre}
