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, D-KQRe2@
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, ZvVrbj&
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? WAzn`xGxR"
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? Ex`!C]sQ
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function z = zernfun(n,m,r,theta,nflag) @!np
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%ZERNFUN Zernike functions of order N and frequency M on the unit circle. @b]?Gg
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N }<7S%?TY
% and angular frequency M, evaluated at positions (R,THETA) on the dd>
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% unit circle. N is a vector of positive integers (including 0), and BXj]]S2
% M is a vector with the same number of elements as N. Each element OA?pBA
% k of M must be a positive integer, with possible values M(k) = -N(k) @o/126(k
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, DnI31!+y
% and THETA is a vector of angles. R and THETA must have the same >3SZD
% length. The output Z is a matrix with one column for every (N,M) r0'6\MS13
% pair, and one row for every (R,THETA) pair. `{v!|.d<
% jMUN|(=Y
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike Tj3xK%K_r3
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), G\4*6iw:
% with delta(m,0) the Kronecker delta, is chosen so that the integral y7Sey;
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, 'jr[
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% and theta=0 to theta=2*pi) is unity. For the non-normalized yd+.hg&J
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. ")xd 'V
%
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% The Zernike functions are an orthogonal basis on the unit circle. s%OPoRE
% They are used in disciplines such as astronomy, optics, and PN"s^]4
% optometry to describe functions on a circular domain. fC<pCdsg
% Smc=-M}
% The following table lists the first 15 Zernike functions. IizPu4|
% Rv=rO|&]
% n m Zernike function Normalization qy\Z2k
% -------------------------------------------------- @SX-=Nr
% 0 0 1 1 XcH_Y
% 1 1 r * cos(theta) 2 [!'fE#"a
% 1 -1 r * sin(theta) 2 ,)beK*Iw
% 2 -2 r^2 * cos(2*theta) sqrt(6) }\Ri:&?
% 2 0 (2*r^2 - 1) sqrt(3) 6-6ha7]s
% 2 2 r^2 * sin(2*theta) sqrt(6) #*|Gp_l+%
% 3 -3 r^3 * cos(3*theta) sqrt(8) G.l
~!;
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) l'm\*=3
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) o-7,P
RmKN
% 3 3 r^3 * sin(3*theta) sqrt(8) 8nKb
mjM
% 4 -4 r^4 * cos(4*theta) sqrt(10) 24b?6^8~k
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) aEvW<jHh
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) M:/)|fk
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ih\=mB
% 4 4 r^4 * sin(4*theta) sqrt(10) M9ACaf@
% -------------------------------------------------- `"RT(` m
% mLb>*xt$b@
% Example 1: U_.9H
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% FA7q
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% % Display the Zernike function Z(n=5,m=1) 6(=>!+xpRr
% x = -1:0.01:1; <Y"h2#M "
% [X,Y] = meshgrid(x,x); `-)Hot)
% [theta,r] = cart2pol(X,Y); Q*K31Ln
% idx = r<=1; qC:QY6g$N
% z = nan(size(X)); {1Hs5bg@
% z(idx) = zernfun(5,1,r(idx),theta(idx)); 7Bs:u
% figure Ax{C ^u
% pcolor(x,x,z), shading interp Uw5AHq).
% axis square, colorbar LZ-&qh
% title('Zernike function Z_5^1(r,\theta)') +rN&@}Jt.
% <4%cKW0
% Example 2: "fN=Y$G
% t;/s^-}
% % Display the first 10 Zernike functions tcD DX'S
% x = -1:0.01:1; 8H@] v@Z2
% [X,Y] = meshgrid(x,x); $ts1XIK%
% [theta,r] = cart2pol(X,Y); W<tw],M-#
% idx = r<=1; h*B7UzCg
% z = nan(size(X)); 5e|yW0o
% n = [0 1 1 2 2 2 3 3 3 3]; -.t/c}a#
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; 8m"(T-wb6{
% Nplot = [4 10 12 16 18 20 22 24 26 28]; D4IP$pAD
% y = zernfun(n,m,r(idx),theta(idx)); -POsbb>
% figure('Units','normalized')
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% for k = 1:10 Zp qb0ro
% z(idx) = y(:,k); /^rJ`M[;
% subplot(4,7,Nplot(k)) X')t6DQ( I
% pcolor(x,x,z), shading interp GJj} |+|
% set(gca,'XTick',[],'YTick',[]) 3;Y9<
% axis square eo&^~OVT
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 5v_vv'~
% end @wPyXl
% F9Co m}
% See also ZERNPOL, ZERNFUN2. d3jzGJrU}
aNDpCpy
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% Paul Fricker 11/13/2006 `NB6Of*/
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PFI^+';
% Check and prepare the inputs: H84Zg/ ^
% ----------------------------- b-?d(-
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) }F4%5go
error('zernfun:NMvectors','N and M must be vectors.') S(#v<C,hd
end vf0
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if length(n)~=length(m) 9r=yfc!cS
error('zernfun:NMlength','N and M must be the same length.') vB Vg/
end Zt
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n = n(:); XE($t2x,M
m = m(:); vn1*D-?
if any(mod(n-m,2)) XDyFe'1I
error('zernfun:NMmultiplesof2', ... {xu~Dx
'All N and M must differ by multiples of 2 (including 0).') ( q}{;
end zT+ "Z(oz,
s)~Wcp'+M:
AB=Wj*fr
if any(m>n) P X>>h}%
error('zernfun:MlessthanN', ... [vn"r^P
'Each M must be less than or equal to its corresponding N.') ~u-_DOA
end #3}!Q0
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if any( r>1 | r<0 ) YJB/*SV^
error('zernfun:Rlessthan1','All R must be between 0 and 1.') `N;O6
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end 6QePrf
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) I/_,24[
error('zernfun:RTHvector','R and THETA must be vectors.') 2Q)pT$
end v(`5exWV
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y
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r = r(:); ?o`fX
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theta = theta(:); sNs Hl
length_r = length(r); Sh(XFUJ
if length_r~=length(theta) 91|~KR)
error('zernfun:RTHlength', ... R_gON*9
'The number of R- and THETA-values must be equal.') n0b{Jg *
end :LLz$[c8
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% Check normalization: ^~W s4[Guo
% -------------------- Y@MFH>*
if nargin==5 && ischar(nflag) UQO?hZ!y/.
isnorm = strcmpi(nflag,'norm'); S4D~`"4$/
if ~isnorm Xp~O?2:3l
error('zernfun:normalization','Unrecognized normalization flag.') V`xE&BI
end !yu-MpeG
else C
A $R
isnorm = false; )TOKHN
end #K\;)z(?
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zv~b-Tp
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (``|5;T\
% Compute the Zernike Polynomials Oee>d<
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ZG)6{WS
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% Determine the required powers of r: vg\fBHzn
% ----------------------------------- W <M\b#
m_abs = abs(m); &?M'(` ~
rpowers = []; Y*YV/E.
for j = 1:length(n) pXf5/u8&
rpowers = [rpowers m_abs(j):2:n(j)]; QA#Jx
end :s#&nY
rpowers = unique(rpowers); jN{+$ @cI
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% Pre-compute the values of r raised to the required powers, =z9FjK
% and compile them in a matrix: 7vEZb.~4z
% ----------------------------- YiC_,8A~
if rpowers(1)==0 ~i=5NUE
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 2fG[q3`
rpowern = cat(2,rpowern{:}); j]
rpowern = [ones(length_r,1) rpowern]; +A<7:`sO
else 4n/CSAT1
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); XT\Q"=FD
rpowern = cat(2,rpowern{:}); ^vc#)tm5p
end GL3olKnL
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% Compute the values of the polynomials: A~#w gLGn
% -------------------------------------- 3/*<i
y = zeros(length_r,length(n)); @^^,VgW[
for j = 1:length(n) zN>tSdNkI-
s = 0:(n(j)-m_abs(j))/2; T@j@IEGH
pows = n(j):-2:m_abs(j); ]t;bCD6*
for k = length(s):-1:1 T4x[
\v5d
p = (1-2*mod(s(k),2))* ... O],]\M{GL
prod(2:(n(j)-s(k)))/ ... 9FmX^t$T
prod(2:s(k))/ ... N>',[4pJ|
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... @mu=7_$U
prod(2:((n(j)+m_abs(j))/2-s(k))); ,{sCI/
idx = (pows(k)==rpowers); ;(;{~1~
y(:,j) = y(:,j) + p*rpowern(:,idx); YHI@Cj
end 8&++S> <
AHdh]pfH
if isnorm nHIW_+<Mf
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi);
ui1h M
end pR7 D3Q:^7
end {WN??eys,
% END: Compute the Zernike Polynomials |v= */e
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% q|kkdK|N/Y
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I2*\J)|f
% Compute the Zernike functions: 9Xeg&Z|!
% ------------------------------ o|c%uw
idx_pos = m>0; Ugv"A;l
idx_neg = m<0; L=<{tzTc
zn/b\X/
@M8vPH
z = y; dS~#Lzm
if any(idx_pos) zM++Z*
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); U$AV"F&!&}
end Z)RV6@(
if any(idx_neg) k+y>xI,
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); d(;Qe}ok>
end o :_'R5
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% EOF zernfun KcF#c_f