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, ?OFl9%\ V
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, Gn7P` t*.
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? qTD^Vz
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那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? W]U},g8Z
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function z = zernfun(n,m,r,theta,nflag) {'e%Hx
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. /
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% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N F#^<t$5t
% and angular frequency M, evaluated at positions (R,THETA) on the z6+D=<
% unit circle. N is a vector of positive integers (including 0), and vl67Xtk4
% M is a vector with the same number of elements as N. Each element 1*o=I-nOa
% k of M must be a positive integer, with possible values M(k) = -N(k) xJSK"
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, D8S3YdJ
% and THETA is a vector of angles. R and THETA must have the same @;K-@*k3
% length. The output Z is a matrix with one column for every (N,M) QaYUcma~n
% pair, and one row for every (R,THETA) pair. eG05}
% m}oqs0xx
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike LqA&@
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), U1!#TD)@
% with delta(m,0) the Kronecker delta, is chosen so that the integral ?cRGdLP'D
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, yoc;`hO-
% and theta=0 to theta=2*pi) is unity. For the non-normalized /-v6jiM
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. UBZ37P
% q*E<~!jL
% The Zernike functions are an orthogonal basis on the unit circle. #lld*I"d
% They are used in disciplines such as astronomy, optics, and 5y`n8. (?
% optometry to describe functions on a circular domain. X@j.$0eK
% +thkx$o
% The following table lists the first 15 Zernike functions. ].e4a;pt
% A)j',jE&1
% n m Zernike function Normalization 2/ES.>K!.
% -------------------------------------------------- h]{V/
% 0 0 1 1 7yM "G $
% 1 1 r * cos(theta) 2 l1?$quM^V
% 1 -1 r * sin(theta) 2 tW)KpX
% 2 -2 r^2 * cos(2*theta) sqrt(6) , A@uSfC(
% 2 0 (2*r^2 - 1) sqrt(3) <QcQ.b
% 2 2 r^2 * sin(2*theta) sqrt(6) d[7B,l:RN
% 3 -3 r^3 * cos(3*theta) sqrt(8) 'Jl |-RUd
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) + <4gJoI
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) iS]4F_|vd
% 3 3 r^3 * sin(3*theta) sqrt(8) ah9P
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% 4 -4 r^4 * cos(4*theta) sqrt(10) *?v_AZ
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) b:6NVHb%
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) kQt#^pO)
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) b$W~w*O
% 4 4 r^4 * sin(4*theta) sqrt(10) Xvr7qowL
% -------------------------------------------------- }8e_
% R|u2ga~
% Example 1: \Hs*46@TC
% bMp[:dw`y
% % Display the Zernike function Z(n=5,m=1) XTro;R=#
% x = -1:0.01:1; ]o<&Q52 |
% [X,Y] = meshgrid(x,x); jo}yeGbU
% [theta,r] = cart2pol(X,Y); L%Mj{fJ>Wm
% idx = r<=1; ;b6h/*;'
% z = nan(size(X)); !+(c/ gwBh
% z(idx) = zernfun(5,1,r(idx),theta(idx)); d"0=.sA
% figure 3[Xc:;+/
% pcolor(x,x,z), shading interp &`^PO$
% axis square, colorbar @yj$
% title('Zernike function Z_5^1(r,\theta)') ~cL)0/j}
% fuQk}OW{
% Example 2: #M5pQ&yZy
% ?;xL]~Q~1
% % Display the first 10 Zernike functions kE`Fg(M
% x = -1:0.01:1; ;ZtN9l
% [X,Y] = meshgrid(x,x); 5>!I6[{
% [theta,r] = cart2pol(X,Y); _X]\#^UiO2
% idx = r<=1; /:. p{y
% z = nan(size(X)); 8quH#IhB
% n = [0 1 1 2 2 2 3 3 3 3]; %F2T`?t:
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; &y&pjo6v1
% Nplot = [4 10 12 16 18 20 22 24 26 28]; -SlAt$IJ
% y = zernfun(n,m,r(idx),theta(idx)); zb,YYE1
% figure('Units','normalized') {TVQ]G%'b
% for k = 1:10 !~_6S*~
% z(idx) = y(:,k); 'A{B[
% subplot(4,7,Nplot(k)) wvcj*{7[
% pcolor(x,x,z), shading interp m88(f2Ch
% set(gca,'XTick',[],'YTick',[]) JKY
% axis square [U@;EeS
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ZU68\cL
% end <0btwsv}
% E0*62OI~O
% See also ZERNPOL, ZERNFUN2. k!0vpps
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% Paul Fricker 11/13/2006 8'-E>+L
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fi
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% Check and prepare the inputs: ^~kfo|
% ----------------------------- RHu4cK!5
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) orZwm9#].
error('zernfun:NMvectors','N and M must be vectors.') )CoJ9PO7
end >>T,M@s-:
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if length(n)~=length(m) MZ:Ty,pw:O
error('zernfun:NMlength','N and M must be the same length.') },%,v2}
end Ij?Qs{V
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n = n(:); T!1Np'12zF
m = m(:); nn8uFISb
if any(mod(n-m,2)) H8A=]Gq
error('zernfun:NMmultiplesof2', ... M!Ywjvw*)3
'All N and M must differ by multiples of 2 (including 0).') }+fBJ$
end $xK(bc'{
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if any(m>n) p]Q(Z
error('zernfun:MlessthanN', ... F$HL\y
'Each M must be less than or equal to its corresponding N.') *fp4u_:`
end 3A'9=h,lVK
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if any( r>1 | r<0 ) 97n,^t2F\
error('zernfun:Rlessthan1','All R must be between 0 and 1.') 9=9R"X>L
end @5\/L6SRfL
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) *lyRy/POB
error('zernfun:RTHvector','R and THETA must be vectors.') [(iJj3s!
end U?8X]
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r = r(:); v;:. k,E0
theta = theta(:); 4~e6z(
length_r = length(r); }b/G{92
if length_r~=length(theta) puK /;nns
error('zernfun:RTHlength', ... k-8$43
'The number of R- and THETA-values must be equal.') | (: PX
end [p96H)8YU
=%0r_#F%=
Ombvp;
% Check normalization: p2j=73$
% -------------------- TN.&FDqC9
if nargin==5 && ischar(nflag) ^w~Utx4
isnorm = strcmpi(nflag,'norm'); qdwjg8fo4Z
if ~isnorm $jN,]N~
error('zernfun:normalization','Unrecognized normalization flag.') 5uD'Kd$H
end \q:PU6q
else ;op8r u
isnorm = false; xmwH~UWp
end htHnQ4Q
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% r8vF I6J
% Compute the Zernike Polynomials H:`[$
^
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% vDVE#Nm_
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% Determine the required powers of r: y0cB@pWp
% ----------------------------------- 84YZT+TEN
m_abs = abs(m); >TwL&la
rpowers = []; ^
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for j = 1:length(n) ns9a+QQ
rpowers = [rpowers m_abs(j):2:n(j)]; r?wE ;gH
end nnBl:p>< k
rpowers = unique(rpowers); ewv[nJD$
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% Pre-compute the values of r raised to the required powers, ?o.Q
% and compile them in a matrix: L
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% ----------------------------- F<N{ x^
if rpowers(1)==0 3NC-)S
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); VH5Vg We
rpowern = cat(2,rpowern{:}); yf7$m_$C'
rpowern = [ones(length_r,1) rpowern]; exL<cN
else XV*uu "F
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); b+ J)
rpowern = cat(2,rpowern{:}); mqb6 MnK -
end V-%Am
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% Compute the values of the polynomials: cQldBc
% -------------------------------------- k-a3oLCR,
y = zeros(length_r,length(n)); l*z.20^P
for j = 1:length(n) RE}$(T=
s = 0:(n(j)-m_abs(j))/2; 'hl4cHk14
pows = n(j):-2:m_abs(j); WZJ}HHePr
for k = length(s):-1:1 1b-_![&]1
p = (1-2*mod(s(k),2))* ... mo-
Y %
prod(2:(n(j)-s(k)))/ ... `+O7IyTMA
prod(2:s(k))/ ... yZ]u{LJS
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... o$-!E(p
prod(2:((n(j)+m_abs(j))/2-s(k))); L.) 0!1
idx = (pows(k)==rpowers); ]:vo"{*C
y(:,j) = y(:,j) + p*rpowern(:,idx); &p#$}tm
end ]EZiPW-uy
dy^ zOqc
if isnorm _}(ej&'f
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); o7;#B)jWS
end O$,MdhyXC
end 9k[>(LC
% END: Compute the Zernike Polynomials 'lD"{^
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% :gJ?3LwTf
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- r!sY+Z>
% Compute the Zernike functions: bI"_hvcFp
% ------------------------------ >2w^dI2
idx_pos = m>0; a2'f#[as
idx_neg = m<0; ,aBo
p#
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z = y; pZg}7F{$
if any(idx_pos) aEW sru
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); e=m=IVY#W
end CFU'-
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if any(idx_neg) e7^B3FOx
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); @ = M:RA
end da/Tms`T
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% EOF zernfun $+$S}i=