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， ) Lv{
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， 54-x 14")
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？ -)/>qFj )
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ }z6H xB]$
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function z = zernfun(n,m,r,theta,nflag) =W?c1EPLCx
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. a ?dM8zAnc
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 08Gr
% and angular frequency M, evaluated at positions (R,THETA) on the O`4X[r1LD
% unit circle. N is a vector of positive integers (including 0), and qW9|&GuZ$
% M is a vector with the same number of elements as N. Each element 2 q>4nN
% k of M must be a positive integer, with possible values M(k) = -N(k) 7e4\BzCC
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, l"64w>,
% and THETA is a vector of angles. R and THETA must have the same sz5@=
% length. The output Z is a matrix with one column for every (N,M) t+
@F"[j
% pair, and one row for every (R,THETA) pair. G(L*8U< UG
% Oc1ZIIkh\
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike qH$p]+Rk 5
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), 5 m:nh<)#
% with delta(m,0) the Kronecker delta, is chosen so that the integral d)@M MF
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, r+n&Pp+9
% and theta=0 to theta=2*pi) is unity. For the non-normalized *Z(qk`e.b
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. *)Y;`Yg$
% BFY~::<b
% The Zernike functions are an orthogonal basis on the unit circle. "D+QT+sD
% They are used in disciplines such as astronomy, optics, and =e63>*M|
% optometry to describe functions on a circular domain. CwAl -o
% a^N/N5-Z
% The following table lists the first 15 Zernike functions. g`6S*&8I
% @<P [z[
% n m Zernike function Normalization GIp?}tM
% -------------------------------------------------- IkupW|}rc
% 0 0 1 1 MFVFr "
% 1 1 r * cos(theta) 2 {.ph )8
% 1 -1 r * sin(theta) 2 /dO& r'!:
% 2 -2 r^2 * cos(2*theta) sqrt(6) `7NgQ*g.d/
% 2 0 (2*r^2 - 1) sqrt(3) HH dc[pJ0D
% 2 2 r^2 * sin(2*theta) sqrt(6) 3 Xy>kG}
% 3 -3 r^3 * cos(3*theta) sqrt(8) >Kx l+F
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 9P]T IV.
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) Z@>>ZS 1Do
% 3 3 r^3 * sin(3*theta) sqrt(8) Sng V<J>zR
% 4 -4 r^4 * cos(4*theta) sqrt(10)
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% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) *op7:o_
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) cWm.']
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) f!87JE=<
% 4 4 r^4 * sin(4*theta) sqrt(10) gpPktp2
% -------------------------------------------------- H_ x35|"
% <1ai0 ]
% Example 1: ^b4o 0me
% YO=;)RA
% % Display the Zernike function Z(n=5,m=1) v<O \ l~S
% x = -1:0.01:1; E;N+B34
% [X,Y] = meshgrid(x,x); 4;_.| !LN
% [theta,r] = cart2pol(X,Y); tZ (Wh
% idx = r<=1; A!NT 2YdHZ
% z = nan(size(X)); +IS B" a
% z(idx) = zernfun(5,1,r(idx),theta(idx)); X;- ,3dy
% figure &c A?|(7-
% pcolor(x,x,z), shading interp
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% axis square, colorbar #GTmC|[
% title('Zernike function Z_5^1(r,\theta)') b*,R9
% &Zov9o:gx
% Example 2: v5&WW?IBQ
% Drg'RR><
% % Display the first 10 Zernike functions aPWFb.JO4
% x = -1:0.01:1; 4*'NpqC(_
% [X,Y] = meshgrid(x,x); z\fk?Tj<ro
% [theta,r] = cart2pol(X,Y); E_$ S T 3
% idx = r<=1; S6cSeRmw
% z = nan(size(X)); #Qkl| h
% n = [0 1 1 2 2 2 3 3 3 3]; p<Zf ,F}
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; z8A`BVqI
% Nplot = [4 10 12 16 18 20 22 24 26 28]; EQg
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% y = zernfun(n,m,r(idx),theta(idx)); 1W5YS +pf
% figure('Units','normalized') `OduBUI] ]
% for k = 1:10 B} & C
h
% z(idx) = y(:,k); +1e*>jE
% subplot(4,7,Nplot(k)) S!rUdxO
% pcolor(x,x,z), shading interp T
`N(=T^*
% set(gca,'XTick',[],'YTick',[]) X~lOFH;}q
% axis square a o
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% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) k/C r ^J"
% end X!r!lW
% Y8Mo . v
% See also ZERNPOL, ZERNFUN2. <{e0 i
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d`^j\b>5(
% Paul Fricker 11/13/2006 7GKeqv
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w
W-GBY3
% Check and prepare the inputs: !5 x"d7
% ----------------------------- eQzTb91
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) InRn!~_N
error('zernfun:NMvectors','N and M must be vectors.') AX[/S8|6
end a]75z)X R
*,mbZE=<
^
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if length(n)~=length(m) @@}A\wA-
error('zernfun:NMlength','N and M must be the same length.') ;b(/PH!O
end :s*&_y
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n = n(:); _)lK. 5
m = m(:); sd
Z=3)
if any(mod(n-m,2)) df}B:?Ew.
error('zernfun:NMmultiplesof2', ... vrh}X[JEw'
'All N and M must differ by multiples of 2 (including 0).') $yRbo '-
end |)1"*` z
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, X+0 71.(
if any(m>n) /ZX8gR5x
error('zernfun:MlessthanN', ... JWM/np6
'Each M must be less than or equal to its corresponding N.') O`H[,+v m[
end :x= ZvAvo
bSBI[S
% kKtPrT
if any( r>1 | r<0 ) B-JgXW.\0
error('zernfun:Rlessthan1','All R must be between 0 and 1.') wHdq :,0-!
end bMf +/n
4{*K%pv\
6$2)m;| XY
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) /9W-;l{=z
error('zernfun:RTHvector','R and THETA must be vectors.') d7P|
x
end 7J##IH+z35
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t4 h5R
r = r(:); eRC@b^~
theta = theta(:); zI(b#eUF
length_r = length(r); #2|sS |0 <
if length_r~=length(theta) X2Y-TE T
error('zernfun:RTHlength', ... N(/DC)DJg
'The number of R- and THETA-values must be equal.') SC"=M^E
end \Ui8Sgeei
ZJ u\
8%I4jL<
% Check normalization: r's4-\
% -------------------- $:F] O$A
if nargin==5 && ischar(nflag) ExV>s* y
isnorm = strcmpi(nflag,'norm'); k2p{<SO;
if ~isnorm RwN*/Li
error('zernfun:normalization','Unrecognized normalization flag.') 6d` 6=D:
end M {_`X
else : !J!l u
isnorm = false; e>y"V; Mj
end 7J7uHl`yq`
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% qzsS"=5
% Compute the Zernike Polynomials KGzBK :
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% a{,EX[~b
-0Y8/6](
tb^3-ZUb
% Determine the required powers of r: L0_R2E A
% ----------------------------------- Ptw E[YDu
m_abs = abs(m); Z3T:R"l;
rpowers = []; 67')nEQ9
for j = 1:length(n) sf@g $
rpowers = [rpowers m_abs(j):2:n(j)]; dy#dug6j
end , B h[jb`y
rpowers = unique(rpowers); }=az6cLE2
D 0\
'72Z Ldi}-
% Pre-compute the values of r raised to the required powers, qGr(MDLc
% and compile them in a matrix:
WwPf z<I
% ----------------------------- ~ g~z"!K
if rpowers(1)==0 aZ@Ke$jD
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); _h M3p
rpowern = cat(2,rpowern{:}); XM>ByfD{
rpowern = [ones(length_r,1) rpowern]; S_ e }>-
else & =xm>;`3
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ; :a7r N"(
rpowern = cat(2,rpowern{:}); x$ ? {)EY
end }I9\=jT
l%rwJLN1
CXb)k.L
% Compute the values of the polynomials: 7P.C~,+D%P
% -------------------------------------- jun>(7
y = zeros(length_r,length(n)); Ks{^R`O au
for j = 1:length(n) X-e)w
s = 0:(n(j)-m_abs(j))/2; Cj31'
pows = n(j):-2:m_abs(j); zl=RK
for k = length(s):-1:1 yv[ s)c}
p = (1-2*mod(s(k),2))* ... vn KKK. E
prod(2:(n(j)-s(k)))/ ... /`Y p]l
prod(2:s(k))/ ... w f,7
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 3z!\Z[
prod(2:((n(j)+m_abs(j))/2-s(k))); + i!/J
idx = (pows(k)==rpowers); =k2In_
y(:,j) = y(:,j) + p*rpowern(:,idx); =ugxPgn
end / ~K-0K#w
k]] (I<2
if isnorm ~ub Gx
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); )?:V5UO\
end XA-DJ
end "'~'xaU!=a
% END: Compute the Zernike Polynomials W52AX.Nm
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % tN{
k "LbB#Q
S=n,unn#t
% Compute the Zernike functions: o=X6PoJ N_
% ------------------------------ +>@<'YI<
idx_pos = m>0; e@Q<hb0<eU
idx_neg = m<0; p%jl-CC1
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.|NF8Fj
z = y; :CNHN2 J
if any(idx_pos) SYOND>E
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ?PO~$dUc]
end Z}5 ;K"T/
if any(idx_neg) cP''
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); b?hdWQSW7
end kQxY"HD
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% EOF zernfun =cs; avtL
