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， jZ{ S{"j
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， s~]nsqLt9p
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？ ?Y hua9
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ D 1hKjB&
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function z = zernfun(n,m,r,theta,nflag) yZ!Eu#81
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. ,^<+5TYM7
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N N 0qC/da1
% and angular frequency M, evaluated at positions (R,THETA) on the Iiy:<c
% unit circle. N is a vector of positive integers (including 0), and #63/;o:l$
% M is a vector with the same number of elements as N. Each element rL,)Tc|"
% k of M must be a positive integer, with possible values M(k) = -N(k) wl{p,[]
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, Z?X$8o^Z
% and THETA is a vector of angles. R and THETA must have the same @Op8^8$`
% length. The output Z is a matrix with one column for every (N,M) AQiP2`?
% pair, and one row for every (R,THETA) pair. <m6Xh^Ko;
% yav)mO~QU6
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike "=".ne
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), PCLSY8N
% with delta(m,0) the Kronecker delta, is chosen so that the integral hx2C<;s4
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, KOmP-q=6
% and theta=0 to theta=2*pi) is unity. For the non-normalized |v1 K@
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. | 5L1\O8#
% {//F>5~[
% The Zernike functions are an orthogonal basis on the unit circle. 3 =<iGX"z
% They are used in disciplines such as astronomy, optics, and `-/l$A}
U
% optometry to describe functions on a circular domain. Y( :OfC?
% g~y9j8 8?
% The following table lists the first 15 Zernike functions. n47=eKd70
% =3zn
Ta }
% n m Zernike function Normalization a:| 4q
% -------------------------------------------------- aW6+Up+G*
% 0 0 1 1 "aBd0i&
% 1 1 r * cos(theta) 2 >C-_Zv<!T\
% 1 -1 r * sin(theta) 2 >=`c [=:Z_
% 2 -2 r^2 * cos(2*theta) sqrt(6) n% ` r
% 2 0 (2*r^2 - 1) sqrt(3) QlS5B.h,
% 2 2 r^2 * sin(2*theta) sqrt(6) ATzNV=2s
% 3 -3 r^3 * cos(3*theta) sqrt(8) b$
x"&&
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) op|mRJBq;
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) &53#`WgJ
% 3 3 r^3 * sin(3*theta) sqrt(8) wqwJpWIe
% 4 -4 r^4 * cos(4*theta) sqrt(10) kr*c?^b
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) OhW=F2OIV
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) n>E*g |a
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) YT5>pM-%
% 4 4 r^4 * sin(4*theta) sqrt(10) )PG,K 4z
% -------------------------------------------------- B@;)$1-UT
% rq1kj 8%2
% Example 1: &V?q d{39
% 6|KX8\, A@
% % Display the Zernike function Z(n=5,m=1) VBX#
!K1Q
% x = -1:0.01:1; p\{+l;`
% [X,Y] = meshgrid(x,x); Z M+Hb_6f
% [theta,r] = cart2pol(X,Y); 0lRH
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% idx = r<=1; zkp
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% z = nan(size(X)); 2 &B y q
% z(idx) = zernfun(5,1,r(idx),theta(idx)); 0 v@/I<
% figure F3Y>hs):7
% pcolor(x,x,z), shading interp }R1`ThTM
% axis square, colorbar YSV,q@I&1
% title('Zernike function Z_5^1(r,\theta)') 2*citB{
% 99 !{[gOv
% Example 2: q`aY.dD=O
% O8 r"M8
% % Display the first 10 Zernike functions >-w=7,?'?z
% x = -1:0.01:1; Idlu1g
% [X,Y] = meshgrid(x,x); ^-IsK#r. k
% [theta,r] = cart2pol(X,Y); ? nZ <?
% idx = r<=1; d# 3tQ*G/
% z = nan(size(X)); -m1 60k3
% n = [0 1 1 2 2 2 3 3 3 3]; #eC;3Kq#-
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; w"v'dU ^
% Nplot = [4 10 12 16 18 20 22 24 26 28]; p? ?/r
% y = zernfun(n,m,r(idx),theta(idx)); n r>{ uTa
% figure('Units','normalized') Q$)|/Y))
% for k = 1:10 /Tj"Fl\h
% z(idx) = y(:,k); F36ViN\b
% subplot(4,7,Nplot(k)) b| dCEmFt
% pcolor(x,x,z), shading interp Yg)V*%0n
% set(gca,'XTick',[],'YTick',[]) d=Do@)
m|
% axis square (b%y$D
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) &^IcL !t[
% end + V9B
% z@~&Kwf\}
% See also ZERNPOL, ZERNFUN2. OF&h=1De,
z9 w&uZz i
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% Paul Fricker 11/13/2006 ^QAiySR`0
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% Check and prepare the inputs: 3l%,D:
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% ----------------------------- oM<!I0"gC+
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 14D 7U/zer
error('zernfun:NMvectors','N and M must be vectors.') y|.fR>5
end 7"q+"0G
y-#
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if length(n)~=length(m) tKZ&1E
error('zernfun:NMlength','N and M must be the same length.') Px?Ao0)Z,
end 5!AV!A_J p
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n = n(:); T;!ukGoFP
m = m(:); JA)o@[l F
if any(mod(n-m,2)) T^$g N|
error('zernfun:NMmultiplesof2', ... 1s`)yu^`v
'All N and M must differ by multiples of 2 (including 0).') JzMZB"Z?
end @8nLQh^
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if any(m>n) & Vt2be*
error('zernfun:MlessthanN', ... :)p)=c8%
'Each M must be less than or equal to its corresponding N.') O4EIE) c
end d=XpO*v,[
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if any( r>1 | r<0 ) ]4Y/x i-
error('zernfun:Rlessthan1','All R must be between 0 and 1.') d(fP ECv (
end fw' r.
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) '@TI48 J+
error('zernfun:RTHvector','R and THETA must be vectors.') H&X:!xa5
end JI"/N`-?;b
/vjGjb=3U
%bP~wl~
r = r(:); {l2N&
theta = theta(:); ( =1q!c`
length_r = length(r); 53
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if length_r~=length(theta) (kIz
error('zernfun:RTHlength', ... dhHEE|vrz
'The number of R- and THETA-values must be equal.') -Z%F mv8
end 3J%V%}mD
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HU &)
% Check normalization: yl-fbYH
% -------------------- =}JBA>q(
if nargin==5 && ischar(nflag) GQN98Y+h
isnorm = strcmpi(nflag,'norm'); b5j*xZv
if ~isnorm Lt1U+o[ot
error('zernfun:normalization','Unrecognized normalization flag.') -bypuMQ-p
end SLkuT`*
else lv4(4$T
isnorm = false; ~,ynJ]_aJB
end W`$[j0
!@u&{"{`
\a\= gn
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% .nEs:yn
% Compute the Zernike Polynomials E0QPE5 _
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% fk>l{W}e)
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~r<p@k=.#0
% Determine the required powers of r: A 4j<\xL
% ----------------------------------- R "*R99
m_abs = abs(m); HsnG4OE
rpowers = []; `(!NYx
for j = 1:length(n) GR%{T'ZD`
rpowers = [rpowers m_abs(j):2:n(j)]; ic-IN~J-
end )1f+ld%R
rpowers = unique(rpowers); d$K=c1
wcZbmJ:
I}+;ME|< 2
% Pre-compute the values of r raised to the required powers, f&ytK
% and compile them in a matrix: ==N` !+
% ----------------------------- [Ct=F|
if rpowers(1)==0 H`-=?t
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ExCM< $,
rpowern = cat(2,rpowern{:}); tMFsA`ng
rpowern = [ones(length_r,1) rpowern]; ^ av6HFQ
else aG!
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rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); R}r~p?(M
rpowern = cat(2,rpowern{:}); nUc ;/
end KCUU#t|8V\
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% Compute the values of the polynomials: QR"bYQ
% -------------------------------------- B3mS]
y = zeros(length_r,length(n)); ',ZF5T5z@
for j = 1:length(n) FLZS K:3B]
s = 0:(n(j)-m_abs(j))/2; T%(C-Quh
pows = n(j):-2:m_abs(j); F;u_7OM
for k = length(s):-1:1 ;cKH1
p = (1-2*mod(s(k),2))* ... cy|%sf`
prod(2:(n(j)-s(k)))/ ... L-\ =J
prod(2:s(k))/ ... r` 6:Q&&
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... g9KTn4
prod(2:((n(j)+m_abs(j))/2-s(k))); b,@aqu
idx = (pows(k)==rpowers); gn ?YF`
y(:,j) = y(:,j) + p*rpowern(:,idx); eA=WGy@IcN
end /0lC KU!=
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if isnorm {.CMD9F[
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); -(#-I $z
end 51 b y
end l Y'N4x7n
% END: Compute the Zernike Polynomials CPv iR<ms_
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Z\? E3j
>(3\k iYS
OeElMRU"
% Compute the Zernike functions: ;..o7I
% ------------------------------ pQWHG#?7
idx_pos = m>0; por/^=e{ Y
idx_neg = m<0; j~`\XX{>
v`DI<Lt
3fr ^ T
z = y; >w|*ei:@S
if any(idx_pos) gfy19c 9
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); vl:J40Kfn
end >t <pFh
if any(idx_neg) ^Q.,\TL01
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); YF[f Z
end +(?>-3_z
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k< j"~S1
% EOF zernfun cpZc9;@IC
