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， AJ) N?s-=
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， aIklAj)=
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？ drh,=M\F
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ s|-g)
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function z = zernfun(n,m,r,theta,nflag) )C%S`d<%,
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. ANXN.V
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 0{sYD*gK]
% and angular frequency M, evaluated at positions (R,THETA) on the u Av'%/
% unit circle. N is a vector of positive integers (including 0), and !sav~dB)
% M is a vector with the same number of elements as N. Each element >on' y+
% k of M must be a positive integer, with possible values M(k) = -N(k) V; 1i/{
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, MFs W
% and THETA is a vector of angles. R and THETA must have the same a\Dw*h?b~
% length. The output Z is a matrix with one column for every (N,M) {#H'K*j{
% pair, and one row for every (R,THETA) pair. tnFhL&
% !E9A=u{
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike c$~J7e6$
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), f}{Oj-:"CC
% with delta(m,0) the Kronecker delta, is chosen so that the integral -ZBSkyMGy
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, ?CZ*MMV
% and theta=0 to theta=2*pi) is unity. For the non-normalized Pc=:j(
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. "Sd2VSLg
% BnIZ+ fg=
% The Zernike functions are an orthogonal basis on the unit circle. `&>CK`%Xu
% They are used in disciplines such as astronomy, optics, and m'5rzZP
% optometry to describe functions on a circular domain. J3AS"+]
% 2jH&@g$cl;
% The following table lists the first 15 Zernike functions. $jL+15^N0+
% 0A.9<&Lod
% n m Zernike function Normalization VMV~K7%0
% -------------------------------------------------- bb"x^DtT
% 0 0 1 1 a~O](/+p;
% 1 1 r * cos(theta) 2 y jY}o
% 1 -1 r * sin(theta) 2 JU RJN+)z
% 2 -2 r^2 * cos(2*theta) sqrt(6) N="H
06t
% 2 0 (2*r^2 - 1) sqrt(3) Rb_+C
% 2 2 r^2 * sin(2*theta) sqrt(6) I>45xVA
% 3 -3 r^3 * cos(3*theta) sqrt(8) mY/x|)MmM
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) h/\/dp/tt
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) <!I^ xo [
% 3 3 r^3 * sin(3*theta) sqrt(8) vAo|o *
% 4 -4 r^4 * cos(4*theta) sqrt(10) c9axzg
UA
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) >}> cJh6
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) Xsv^GmP+
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) (vr
v -4
% 4 4 r^4 * sin(4*theta) sqrt(10) hPgDK.R'
% -------------------------------------------------- $_b^p=
% ~Is-^k)y
% Example 1: * 2s(TW
% ^%2S,3*0
% % Display the Zernike function Z(n=5,m=1) 6yPh0n
% x = -1:0.01:1; i`HXBq!|w
% [X,Y] = meshgrid(x,x); xgv&M:%D-
% [theta,r] = cart2pol(X,Y); oM)4""|
% idx = r<=1; yB,{:kq7D
% z = nan(size(X)); IL N0/eH
% z(idx) = zernfun(5,1,r(idx),theta(idx)); rdQ'#}I x
% figure Vh;P,no#
% pcolor(x,x,z), shading interp O7GJg;>?
% axis square, colorbar y|[YEY U)
% title('Zernike function Z_5^1(r,\theta)') O5 ?3 nYHa
% %!QY :[
% Example 2: </7_T<He.
% :h60
% % Display the first 10 Zernike functions ` ]\:%+-
% x = -1:0.01:1; #\r5Q>
% [X,Y] = meshgrid(x,x); 0@*EwI
% [theta,r] = cart2pol(X,Y); _ ^cFdP)8|
% idx = r<=1; pG9qD2C f
% z = nan(size(X)); R7-+@
% n = [0 1 1 2 2 2 3 3 3 3]; #ysSfM6
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; g7nqe~`{
% Nplot = [4 10 12 16 18 20 22 24 26 28]; Zi~-m]9U
% y = zernfun(n,m,r(idx),theta(idx)); u[ 2B0a
% figure('Units','normalized') k&8& D
% for k = 1:10 3 tIno!|
% z(idx) = y(:,k); b<?A
% subplot(4,7,Nplot(k)) qLh[B R
% pcolor(x,x,z), shading interp cp g+-Zf%
% set(gca,'XTick',[],'YTick',[]) 4Hcds9y9
% axis square IL2OVL X
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) &[iunJv:eq
% end NamO5(1C
% (& t8.7O
% See also ZERNPOL, ZERNFUN2. WjsE#9D!of
;H:+w\?8f$
O9(6 ?n
% Paul Fricker 11/13/2006 "=ogO/_Q"
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% Check and prepare the inputs: Q( \2(x\
% ----------------------------- Zn 9ecN
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) ~*,e &I
error('zernfun:NMvectors','N and M must be vectors.') ss>p
end <fm0B3i?
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if length(n)~=length(m) H[KTM 'n
error('zernfun:NMlength','N and M must be the same length.') =ijVT_|u0
end (D 5.NB%@
Gv uX"J
/ %:%la%
n = n(:); c (Gl3^
m = m(:); Jg\1(ix
if any(mod(n-m,2)) EM& ;SQ;C9
error('zernfun:NMmultiplesof2', ... KJ&~z? X
'All N and M must differ by multiples of 2 (including 0).') jWL; ElM'
end ?}g#Mc
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tQ >
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if any(m>n) ; YK{[$F
error('zernfun:MlessthanN', ... ehCZhi~
'Each M must be less than or equal to its corresponding N.') Hg}@2n)/
end +GqV9x 8
7 ,! [oY[
(#"iZv,
if any( r>1 | r<0 ) jJfV_#'N'
error('zernfun:Rlessthan1','All R must be between 0 and 1.') M~/R1\'&j
end MH8 Selnv
_x ;f TW0
b=-LQkcZhK
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) qIIl,!&}A
error('zernfun:RTHvector','R and THETA must be vectors.') hz8Z)xjJ V
end lh?TEQ
>
l@ o\
D >~S-]
r = r(:); cA8"Ft{P)
theta = theta(:); qr~= S
length_r = length(r); ~>]/1JFz
if length_r~=length(theta) c[xH: $G?Y
error('zernfun:RTHlength', ... k}o*=s> M
'The number of R- and THETA-values must be equal.') d].(x)|st
end [8J/# !B
T)QT_ST.9
~dO&e=6Hk
% Check normalization: *`HE$k!
% -------------------- F;&a=R!.
if nargin==5 && ischar(nflag) `?PpzDV7Y
isnorm = strcmpi(nflag,'norm'); 1*>lYd8 _
if ~isnorm Pd[&&!+gV
error('zernfun:normalization','Unrecognized normalization flag.') QNz x(IV@
end <&$:$_ah
else D`G ;kp
isnorm = false; cI Byv I-
end l"-F<^
U
IO4 8sV }
ct3^V M&/
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 9W[ ~c"Ku
% Compute the Zernike Polynomials ;1&7 v
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% du:%{4
#el i_Cxe
vV?=r5j
% Determine the required powers of r: !AGjiP$
% ----------------------------------- X~Yj#@
m_abs = abs(m); ,X2CV INb}
rpowers = []; %Z"I=;=nxI
for j = 1:length(n) dt efDsK
rpowers = [rpowers m_abs(j):2:n(j)]; dIUg
e`O9
end MJ )a Y2
rpowers = unique(rpowers); 9mT;> mE
/4 R|QD
:] viLw\&g
% Pre-compute the values of r raised to the required powers, $ 4&
)
% and compile them in a matrix: hu
G]kv3F:
% ----------------------------- BZP~m=kq
if rpowers(1)==0 -PI_ *
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); =nmvG%.h d
rpowern = cat(2,rpowern{:}); i8tH0w/(M
rpowern = [ones(length_r,1) rpowern]; o $=D`B
else ?1f( @
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 7|"gMw/
rpowern = cat(2,rpowern{:}); >c~ Fg s
end HZ# <+~J
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5 GP,J,J
% Compute the values of the polynomials: qOV6Kh)
% -------------------------------------- z8ox#+l
y = zeros(length_r,length(n)); zuR F6?un
for j = 1:length(n) #Zm%U_$<
s = 0:(n(j)-m_abs(j))/2; P7||d@VW,
pows = n(j):-2:m_abs(j); "2}E ARa
for k = length(s):-1:1 L b'HM-d
p = (1-2*mod(s(k),2))* ... ~;f,Ad`Q
prod(2:(n(j)-s(k)))/ ... ! ] W}I
prod(2:s(k))/ ... Ier0F7]I
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... d0`5zd@S
prod(2:((n(j)+m_abs(j))/2-s(k))); RSNukg
idx = (pows(k)==rpowers); bOi`JJ^
y(:,j) = y(:,j) + p*rpowern(:,idx); & s|& cT
end Z"# /,?|3@
GTw3rD^wg
if isnorm "v"w ER?
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); bGl5=`
end y8$TU;
end j&G*$/lTO6
% END: Compute the Zernike Polynomials oM=Ltxv}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% >lo,0oG
kT!Y~c
M-Az2x;6
% Compute the Zernike functions: )V}u}5
% ------------------------------ DL ^}?Ve
idx_pos = m>0; L
y!!+UM\
idx_neg = m<0; KT)A{i
H$
!78/f
;+dB-g[
z = y; f$lf(brQ:
if any(idx_pos) f?iQ0wv)
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ;_yp@.,\T
end 9` /\|t|V
if any(idx_neg) t\hvhcbL
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); PQmgv&!DP
end z;dD
} Fo
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vGK'U*gGD
% EOF zernfun (f^K\7HM
