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, [dFxW6n
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, =zXpeo&|m
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? rb%P30qc4
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? ghd~p@4
?cr;u~-=
d{&+xl^ll
!q~s-~d^
%j=dKd>
function z = zernfun(n,m,r,theta,nflag) $K^"a
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. g[Ah>
5
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N NylN-X7[#
% and angular frequency M, evaluated at positions (R,THETA) on the Woa5Ov!n0
% unit circle. N is a vector of positive integers (including 0), and aWek<Y~+
% M is a vector with the same number of elements as N. Each element )0`;leli
% k of M must be a positive integer, with possible values M(k) = -N(k) 6NJ"ty9Bp
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, !> b>"\b
% and THETA is a vector of angles. R and THETA must have the same qa#Fa)g*
% length. The output Z is a matrix with one column for every (N,M) 6PT ,m
% pair, and one row for every (R,THETA) pair. K"Vv=
% t3u"2B7oG
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike HZCEr6}(
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), Nkn0G_
% with delta(m,0) the Kronecker delta, is chosen so that the integral s `xp6\$
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, QE}S5#_"
% and theta=0 to theta=2*pi) is unity. For the non-normalized uSbOGhP
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. m8$6FN
% +o(t5O[G
% The Zernike functions are an orthogonal basis on the unit circle. W%b<(T;
% They are used in disciplines such as astronomy, optics, and 0z/tceW'F
% optometry to describe functions on a circular domain. Lx,"jA/
% hXM8`iFW5
% The following table lists the first 15 Zernike functions. jV8mn{<
% CeS8I-,
% n m Zernike function Normalization )u ]J`.OA
% -------------------------------------------------- #)q}Jw4]j
% 0 0 1 1 1;3oGuHj8
% 1 1 r * cos(theta) 2 +l@H[r;$
% 1 -1 r * sin(theta) 2 OGg9e
% 2 -2 r^2 * cos(2*theta) sqrt(6) z*ZEw
% 2 0 (2*r^2 - 1) sqrt(3) sp0&"&5
% 2 2 r^2 * sin(2*theta) sqrt(6) 7!w@u6Q
% 3 -3 r^3 * cos(3*theta) sqrt(8) 1qbd6D|t
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) WGKN>nV
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) fL
ng[&
% 3 3 r^3 * sin(3*theta) sqrt(8) P482D)
% 4 -4 r^4 * cos(4*theta) sqrt(10) &+6XdhX
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) #rMMOu9r2
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) i0 {pm q
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) sY7:Lzs.,
% 4 4 r^4 * sin(4*theta) sqrt(10) >T;"bcb
% -------------------------------------------------- H`]nY`HYg
% mm/U9hbp%
% Example 1: >WE3$Q>bi
% ?|TVz!3
% % Display the Zernike function Z(n=5,m=1) Ks@S5:9sp
% x = -1:0.01:1; LdI)
% [X,Y] = meshgrid(x,x); /:>qhRFJA:
% [theta,r] = cart2pol(X,Y); ^~-i>gTD
% idx = r<=1; 4Cke(G
% z = nan(size(X)); \2-!%i,
% z(idx) = zernfun(5,1,r(idx),theta(idx)); $EW31R5h<s
% figure GBtBmV/`
% pcolor(x,x,z), shading interp 'e02rqip{
% axis square, colorbar mA(K`"Bfh
% title('Zernike function Z_5^1(r,\theta)') 'P32G?1C&p
% l-_voOP
% Example 2: VF!?B>
% \hQ[5>
% % Display the first 10 Zernike functions E}c(4RY
% x = -1:0.01:1; <i^Bq=E<rJ
% [X,Y] = meshgrid(x,x); XD{U5.z>y
% [theta,r] = cart2pol(X,Y); vmAMlgZ8{<
% idx = r<=1; 8wwqV{O7
% z = nan(size(X)); gC;y>YGP
% n = [0 1 1 2 2 2 3 3 3 3]; X/lLM`
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; hEsCOcEG
% Nplot = [4 10 12 16 18 20 22 24 26 28]; \
lP
c,8)
% y = zernfun(n,m,r(idx),theta(idx)); eHF#ME
% figure('Units','normalized') QUb#;L@okn
% for k = 1:10 \v9IbU*js
% z(idx) = y(:,k); )b"H]"
% subplot(4,7,Nplot(k)) Im{50%Y
% pcolor(x,x,z), shading interp \ :8~na+(
% set(gca,'XTick',[],'YTick',[]) uTA
/E9OY
% axis square TU$/3fp*
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) &zlwV"W
% end tq}sXt
% qg:R+`z
% See also ZERNPOL, ZERNFUN2. @}!1Uk3ud
%lbSV}V)
wg^#S
% Paul Fricker 11/13/2006 ;{ XKZ}
T2Z;)e$m_
i]Lt8DiRq
<?&GBCe
9'o!9_j
% Check and prepare the inputs: b9)%,3-
% ----------------------------- M<r'j $g
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 7_.z3Km:
error('zernfun:NMvectors','N and M must be vectors.') Fo3[KW)8I
end }Ga@bY6
Q mOG2
@R9zLL6#7
if length(n)~=length(m) 6b9D db*
error('zernfun:NMlength','N and M must be the same length.') '$ ~.x|
end }C/u>89%q
sDK
lbb
M]!R}<]{
n = n(:); Kw3fpNd
m = m(:); Z_}vjk~s
if any(mod(n-m,2)) p H5IBIf'
error('zernfun:NMmultiplesof2', ... DOaEz?2)
'All N and M must differ by multiples of 2 (including 0).') "V&2g?
end OwwH 45
jx!)N>
=<_xUh.
if any(m>n) W*QD'
error('zernfun:MlessthanN', ... *SzP7]1m
'Each M must be less than or equal to its corresponding N.') @(JcM=
end ]3
YJEP
U;D!m+.HK
u
?7(A%
if any( r>1 | r<0 ) zawU
error('zernfun:Rlessthan1','All R must be between 0 and 1.') HLg/=VF7?
end miCt)Qd
WiH%URFB
-TU7GCb=
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) U/>f" F
error('zernfun:RTHvector','R and THETA must be vectors.') d;Z<")
end %RL\t5TV
8JAA?0L"'
fa=#S
r = r(:); c ^G\w+_
theta = theta(:); /wK5YN.em
length_r = length(r); j2cLb
if length_r~=length(theta) U
u(ysN4`
error('zernfun:RTHlength', ... KwN o/x|
v
'The number of R- and THETA-values must be equal.') &32qv`
V_
end 4;M
mn{8"@Z
nZfTK>)A0
% Check normalization: +uM1#-+h
% -------------------- tE]g*]o
if nargin==5 && ischar(nflag) 9r
fR
isnorm = strcmpi(nflag,'norm'); s{NEP/QQJ
if ~isnorm zid?yuP
error('zernfun:normalization','Unrecognized normalization flag.') #StD]d
end GD}3r:wDs
else " 6~pTHT
isnorm = false; r62x*?/
end Hig=PG5I
79~,KFct
>a%NC'~rc
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ;wbQTp2
% Compute the Zernike Polynomials ~=Z&l
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0Tp?ED_
HPCzh
)?%FU?2jrn
% Determine the required powers of r: "z69jxXo
% ----------------------------------- xp7,0'(;
m_abs = abs(m); aj20, w
rpowers = []; A]Zp1XEG
for j = 1:length(n) /R''R:j
rpowers = [rpowers m_abs(j):2:n(j)];
@\i6m]\X
end "monuErg&
rpowers = unique(rpowers); +%>s\W+?]
si/F\NDT
j$Vv'on
% Pre-compute the values of r raised to the required powers, P~#!-9?
% and compile them in a matrix: {dg3 qg~
% ----------------------------- a{L`C"rJ
if rpowers(1)==0 C:hfI;*7
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); @@*->
rpowern = cat(2,rpowern{:}); %+w>`k3(N
rpowern = [ones(length_r,1) rpowern]; +#6WORH0S
else ci+Pg9sS
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); j^1T3 +
rpowern = cat(2,rpowern{:}); e=%7tK*
end `V w9j,G
'P)xY-15
j+*VP
% Compute the values of the polynomials: V(L~t=k$
% -------------------------------------- 8!TbJVR
y = zeros(length_r,length(n)); BgA\l+
for j = 1:length(n) ba%[!
s = 0:(n(j)-m_abs(j))/2; 29Kuq ;6
pows = n(j):-2:m_abs(j); =oluw|TCe7
for k = length(s):-1:1 A~ '2ki5$g
p = (1-2*mod(s(k),2))* ... 1UJ(._0hR
prod(2:(n(j)-s(k)))/ ... Bo`fy/x#
prod(2:s(k))/ ... E,xCfS)
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... ~r]ZD)
prod(2:((n(j)+m_abs(j))/2-s(k))); J,;;`sf
idx = (pows(k)==rpowers); Fz?ON1\
y(:,j) = y(:,j) + p*rpowern(:,idx); `tVBV:4\
end K^J;iu 4
N ]}Re$5
if isnorm 'an{<82i
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); 7Hf6$2Wh
end |E53
[:p
end K
*{C:Y
% END: Compute the Zernike Polynomials #Jy+:|jJ
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% D?}LKs[
<!y_L5S|
/Or76kE
% Compute the Zernike functions: J%aW^+O
% ------------------------------
3cT
idx_pos = m>0; Yl&eeM
idx_neg = m<0; ZB`!@/3X
kC01s
5`^"<wNI
z = y; ,G"?fQ7z R
if any(idx_pos) x)BG%{h
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); csRba;Z[
end 7vNS@[8
if any(idx_neg) 6:v8J1G(<
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 0w< iz;30
end k,X)PQc
aMm`G}9n
1ikkm7
% EOF zernfun s<E_74q1