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， q^Tis>*u6
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， Dx+ K +(
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？ Bku' H
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ u}jrf Kd E
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function z = zernfun(n,m,r,theta,nflag) H^(L90
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. F>Jg~ FD*
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 1kFjas `g
% and angular frequency M, evaluated at positions (R,THETA) on the YdOUv|tZC
% unit circle. N is a vector of positive integers (including 0), and W"sr$K2m|
% M is a vector with the same number of elements as N. Each element R{3CW^1
% k of M must be a positive integer, with possible values M(k) = -N(k) W cGXp $M
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, n6f3H\/P&
% and THETA is a vector of angles. R and THETA must have the same 4#rAm"H
% length. The output Z is a matrix with one column for every (N,M) :c4kBl%gJ
% pair, and one row for every (R,THETA) pair. 'U )8rR
% K5flit4-
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 9YC&&0 C@
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), rihlae5Kz
% with delta(m,0) the Kronecker delta, is chosen so that the integral 1D1b" o
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, |~ $7X
% and theta=0 to theta=2*pi) is unity. For the non-normalized D Vw Cx^
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. \C/z%Hf7-
% f=ib9WbR#
% The Zernike functions are an orthogonal basis on the unit circle. 'Z[d7P
% They are used in disciplines such as astronomy, optics, and nFX AF!,jj
% optometry to describe functions on a circular domain. 7%CIt?Z%
% zqGYOm$r
% The following table lists the first 15 Zernike functions. oh&Y< d0
% L>nO:`>h
% n m Zernike function Normalization D@ hmO]5c
% -------------------------------------------------- JuJ5qIal
% 0 0 1 1 V\zsDP
% 1 1 r * cos(theta) 2 N(R,8GF5G
% 1 -1 r * sin(theta) 2 c!D> {N
% 2 -2 r^2 * cos(2*theta) sqrt(6) WEC-<fN|Y\
% 2 0 (2*r^2 - 1) sqrt(3) s/ S+ ec3
% 2 2 r^2 * sin(2*theta) sqrt(6) %FS;>;i?
% 3 -3 r^3 * cos(3*theta) sqrt(8) RndOm.TE
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 6Bdyf(t
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) :&$Xe1)i]
% 3 3 r^3 * sin(3*theta) sqrt(8) MVAc8d S
% 4 -4 r^4 * cos(4*theta) sqrt(10) 9p<:LZd~
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) Mf7E72{D
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) >4'21,q
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) n\~yX<;X3
% 4 4 r^4 * sin(4*theta) sqrt(10) I"V3+2e
% -------------------------------------------------- )dg U mN
% w4}(Ab<Y
% Example 1: R6Pz#`n
% {G.{ a d
% % Display the Zernike function Z(n=5,m=1) N7B}O*;
% x = -1:0.01:1; B}5XRgq
% [X,Y] = meshgrid(x,x); *2:Yf7rvI+
% [theta,r] = cart2pol(X,Y); ddMM74
% idx = r<=1; ^V,@=QL3U
% z = nan(size(X)); /O"0L/hc^
% z(idx) = zernfun(5,1,r(idx),theta(idx)); %0(>! SY
% figure MZi8F o '
% pcolor(x,x,z), shading interp ]Hj`2\KD.d
% axis square, colorbar 0C7 " 3l
% title('Zernike function Z_5^1(r,\theta)') -AeHY'T
% A?V<l<EAm
% Example 2: Z{16S=0
% %>]#vQ|
% % Display the first 10 Zernike functions % NwoU%q
% x = -1:0.01:1; sp,(&Y]US
% [X,Y] = meshgrid(x,x); %w%zv2d
% [theta,r] = cart2pol(X,Y); Es,0'\m&
% idx = r<=1; rN'k4V"K
% z = nan(size(X)); gU*I ;s>
% n = [0 1 1 2 2 2 3 3 3 3]; .=aMjrME
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; 6!o/~I#
% Nplot = [4 10 12 16 18 20 22 24 26 28]; :if5z2PE/
% y = zernfun(n,m,r(idx),theta(idx)); ^)'| |Ly
% figure('Units','normalized') _4S7wOq5
% for k = 1:10 -*5yY#fw}
% z(idx) = y(:,k); '4Y*-!9
% subplot(4,7,Nplot(k)) th;]Vo
% pcolor(x,x,z), shading interp )%1&/uN)
% set(gca,'XTick',[],'YTick',[]) /iTH0@Kw;
% axis square q .)^B@}_
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) j[ BgP\&,
% end D`5:
JR-{
% C(ZcR_+r$,
% See also ZERNPOL, ZERNFUN2. yl|R:/2V
,7/\&X<`B
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% Paul Fricker 11/13/2006 |oB]6VS`
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% Check and prepare the inputs: a:wJ/ p
% ----------------------------- I\)N\mov e
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 9 ?[4i'
error('zernfun:NMvectors','N and M must be vectors.') P/HHWiD`D
end r{c5dQ
+ 4++Z
I%C]>ZZh
if length(n)~=length(m) LjX&' ,
error('zernfun:NMlength','N and M must be the same length.') J#_\+G i
end !G@V<'F
%Gnd"SGs
Ni/|C19Z
n = n(:); oTZo[T@zRx
m = m(:); \Gv- sA
if any(mod(n-m,2)) 4h[2C6
\+`
error('zernfun:NMmultiplesof2', ... k{!iDZr&f,
'All N and M must differ by multiples of 2 (including 0).') iFXUKGiV
end =2Pz$q*ub
75' Ua$
BNF++<s
if any(m>n) ||bA
error('zernfun:MlessthanN', ... ](idf ( j
'Each M must be less than or equal to its corresponding N.') _
+u sn.
end z0FR33-
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* <Nk%`
if any( r>1 | r<0 ) OD>u$tI9
error('zernfun:Rlessthan1','All R must be between 0 and 1.') g#pIMA #/
end sf=%l10Fk#
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:C}KI)
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) U<#$w{d:
error('zernfun:RTHvector','R and THETA must be vectors.') -sruxF
end > & 4 I.nA
_%C_uBLi
pb $ An<P
r = r(:); #w[ q.+A
theta = theta(:); w0F:%:/
length_r = length(r); KR+ a Y.
if length_r~=length(theta) hvwn G>m\
error('zernfun:RTHlength', ... 23.y 3t_?
'The number of R- and THETA-values must be equal.') 10a=Y G
end 5G
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!ds"88:5^
% Check normalization: S0X.8Bq
% -------------------- Al;%u0]5
if nargin==5 && ischar(nflag) &eLQ;<qO*|
isnorm = strcmpi(nflag,'norm'); U[H+87zg
if ~isnorm wjw<@A9
error('zernfun:normalization','Unrecognized normalization flag.') ]-+.lR%vd9
end o>QFd x
else N23 +1 h
isnorm = false; ^+Y-=2 u:
end rA>A=,
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% R LeSA\di
% Compute the Zernike Polynomials )SlUQ 7f>
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% v\r7.l:hf
UH.}B3H
~ L i %
% Determine the required powers of r: 6O[wVaC1u
% ----------------------------------- Y ~\`0?ST
m_abs = abs(m); vb80J< 4
rpowers = []; 2rE~V.)%
for j = 1:length(n) dcc%G7w
rpowers = [rpowers m_abs(j):2:n(j)]; v;NZ"1=_
end F"HI>t)>
rpowers = unique(rpowers); 0w a!pE"
(tz_D7 c$F
WP#_qq O
% Pre-compute the values of r raised to the required powers, 0g a1Yr]
% and compile them in a matrix: 6=`m
% ----------------------------- p7ns(g@9
if rpowers(1)==0 3 R$CxRc:
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); odn97,A
rpowern = cat(2,rpowern{:}); ~_^o?NE,
rpowern = [ones(length_r,1) rpowern]; h(N 9RJ}
else <^X'f
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); qyG636i
rpowern = cat(2,rpowern{:}); H--*[3".
end ;Kd {h
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#B.w7y5*
% Compute the values of the polynomials: ,oi`BOh
% -------------------------------------- Xxsnpb>
y = zeros(length_r,length(n)); E[htB><
for j = 1:length(n) DJ2]NA$Q*
s = 0:(n(j)-m_abs(j))/2; ^Hhw(@`qf
pows = n(j):-2:m_abs(j); %(7w Z0Z
for k = length(s):-1:1 Hr8$1I$=
p = (1-2*mod(s(k),2))* ... .8uwg@yD
prod(2:(n(j)-s(k)))/ ... 5}l#zj
prod(2:s(k))/ ... BC0c c[x
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... E+m"yQp{
prod(2:((n(j)+m_abs(j))/2-s(k))); 0)] C&;}_M
idx = (pows(k)==rpowers); re 1k]
y(:,j) = y(:,j) + p*rpowern(:,idx); hhgz =7Y
end GO
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cT Iw A:)D
if isnorm A(@gv8e[H^
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); oJ; O>J@c
end kI[O {<kQ
end _=^hnv
% END: Compute the Zernike Polynomials 5`{;hFl
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% i,b7Ft:F&
';CuJ XAj
)D-.7m.v]
% Compute the Zernike functions: 6Cv2>'{S
% ------------------------------ ZT6X4 Z
idx_pos = m>0; -O> mY)
idx_neg = m<0; @7Rt[2"e
9xS`@ "`
U.j\ u>a
z = y; pZJQKTCG
if any(idx_pos) m ?"%&|
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); }ok
nB
end iYQy#kO
if any(idx_neg) mW(_FS2%,
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ]Q_G /e
end [W|7r
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{ $TB#=G
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% EOF zernfun A%8`zR
