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, 8@;]@c)m
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, se\f be ^0
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? >G:Q/3jh
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? x"{aO6M
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function z = zernfun(n,m,r,theta,nflag) xvV";o
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. )O]6dd
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N ]xQv\u
% and angular frequency M, evaluated at positions (R,THETA) on the k
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% unit circle. N is a vector of positive integers (including 0), and UDHWl_%L
% M is a vector with the same number of elements as N. Each element ;=y"Z^
% k of M must be a positive integer, with possible values M(k) = -N(k)
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% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, `gX|q3K\s
% and THETA is a vector of angles. R and THETA must have the same CIx(SeEF
% length. The output Z is a matrix with one column for every (N,M) hZx&j{
% pair, and one row for every (R,THETA) pair. 8M99cx*K
% WO_Uc_R
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike *4}_2"[
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), B?! L~J@p
% with delta(m,0) the Kronecker delta, is chosen so that the integral U?UU]>Q
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, M]s\F(*ib
% and theta=0 to theta=2*pi) is unity. For the non-normalized Vh^y6U<
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. $fmTa02q>
% e$Ksn_wEq
% The Zernike functions are an orthogonal basis on the unit circle. 4j#y?^s
% They are used in disciplines such as astronomy, optics, and ZwkUd-=0i
% optometry to describe functions on a circular domain. BpZ~6WtBq
% ?{ N,&d
% The following table lists the first 15 Zernike functions. ./#YUIC
% =SJ#6uFS
% n m Zernike function Normalization jE*{^+n
% -------------------------------------------------- *'>_XX
% 0 0 1 1 >Zb!?ntN`t
% 1 1 r * cos(theta) 2 lU{)%4e`
% 1 -1 r * sin(theta) 2 q&25,zWD
% 2 -2 r^2 * cos(2*theta) sqrt(6) Xs~'M/>
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% 2 0 (2*r^2 - 1) sqrt(3) QTy=VLk43
% 2 2 r^2 * sin(2*theta) sqrt(6) <tD,Uu{P
% 3 -3 r^3 * cos(3*theta) sqrt(8) gXxi; g
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) #L*\ ^ c
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) "`>6M&`U
% 3 3 r^3 * sin(3*theta) sqrt(8) 2_q/<8t
% 4 -4 r^4 * cos(4*theta) sqrt(10) 32wtN8kx
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) MgeC-XQM
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) g-eJan&]N
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) (/A.,8Ad
% 4 4 r^4 * sin(4*theta) sqrt(10) ;z'&$#pA
% -------------------------------------------------- fx;rMGa
% hY`<J]-'`
% Example 1: ~/L:$
% S%iK);
% % Display the Zernike function Z(n=5,m=1) =\<NTu
% x = -1:0.01:1; 6u, g
% [X,Y] = meshgrid(x,x); 8,U~ p<Gz
% [theta,r] = cart2pol(X,Y); y\T$) XGV
% idx = r<=1; ZC?~RXL(
% z = nan(size(X)); +F)EGB%LXs
% z(idx) = zernfun(5,1,r(idx),theta(idx)); ~<[+!&<U
% figure }j/\OY _&
% pcolor(x,x,z), shading interp #Zdh<.
% axis square, colorbar 2P"643tz
% title('Zernike function Z_5^1(r,\theta)') UD-+BUV
% r8EJ@pOF2w
% Example 2: Jh-yIk
% C
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% % Display the first 10 Zernike functions ~O}r<PQ
% x = -1:0.01:1; xrf|c
% [X,Y] = meshgrid(x,x); %3`*)cp@
% [theta,r] = cart2pol(X,Y); k8s)PN
% idx = r<=1; <f>77vh0
% z = nan(size(X)); nt2b}u>*
% n = [0 1 1 2 2 2 3 3 3 3]; Qw0k-t0=4
% m = [0 -1 1 -2 0 2 -3 -1 1 3];
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% Nplot = [4 10 12 16 18 20 22 24 26 28]; HZ9 >4G3
% y = zernfun(n,m,r(idx),theta(idx)); u`XRgtI{g?
% figure('Units','normalized') tj;47UtH
% for k = 1:10 5iw\F!op:
% z(idx) = y(:,k); ^(q .f=I!a
% subplot(4,7,Nplot(k)) -HF?1c
% pcolor(x,x,z), shading interp /dCsZA
% set(gca,'XTick',[],'YTick',[]) 7m#EqF$P
% axis square uH89oA/H
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) bc(MN8b ]j
% end PhAfEsD
% 5Ew( 0K[
% See also ZERNPOL, ZERNFUN2. 3eUi9_s+
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% Paul Fricker 11/13/2006 &b (*
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% Check and prepare the inputs: |eRE'Wd0
% ----------------------------- T={!/y+
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) vd%AV(]<LJ
error('zernfun:NMvectors','N and M must be vectors.') ozY$}|sjDT
end X@kgc&`0
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if length(n)~=length(m) :Quep-:fy<
error('zernfun:NMlength','N and M must be the same length.') Ar)EbGId
end 3FvVM0l"
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b}(c'W*z%
n = n(:); k{r<S|PK0
m = m(:); S/ oD`
if any(mod(n-m,2)) +s<6eHpm
error('zernfun:NMmultiplesof2', ... ]EK(k7nH
'All N and M must differ by multiples of 2 (including 0).') Lx_Jw\YO
end k9eyl)
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if any(m>n) s/ABT.ZO
error('zernfun:MlessthanN', ... GJWGT`"
'Each M must be less than or equal to its corresponding N.') w7`pbcY,
end 4M%|N
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if any( r>1 | r<0 ) -~c-mt
error('zernfun:Rlessthan1','All R must be between 0 and 1.') Z'A 3\f
end yf*'=q
&w9*pJR %
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ]Sj;\Iz
error('zernfun:RTHvector','R and THETA must be vectors.') )@9Eq|jMC
end ZklO9Ox(
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r = r(:); )*_G/<N)|
theta = theta(:); z
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length_r = length(r); [f:&aS+
if length_r~=length(theta) 7(D)U)9h
error('zernfun:RTHlength', ... /*;a6S8q
'The number of R- and THETA-values must be equal.') [PN2^
end T}{zh
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z#Qe$`4&
% Check normalization: +@uA
% -------------------- 4RctYMz
if nargin==5 && ischar(nflag) db_Qt' >
isnorm = strcmpi(nflag,'norm'); #)n$Q^9&
if ~isnorm 0Sk~m4fj(
error('zernfun:normalization','Unrecognized normalization flag.') iOfO+3'Z_U
end rMVcoO@3
else Q\zaa9P
isnorm = false; `oe=K{aX
end ^O<'Qp,[:
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% X*MK(aV3
% Compute the Zernike Polynomials J0vQqTaT
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% /pkN=OBR
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% Determine the required powers of r: iz;5:
% ----------------------------------- 4pMp@b
m_abs = abs(m); v Cej( ))
rpowers = []; ysi=}+F.
for j = 1:length(n) s]e`q4ip
rpowers = [rpowers m_abs(j):2:n(j)]; tq,^!RSbZ
end wEq&O|Vj
rpowers = unique(rpowers); k?HdW(HA
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+j
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% Pre-compute the values of r raised to the required powers, Py#EjF12
% and compile them in a matrix: ,<!*@xy7v
% ----------------------------- OLt0Q.{
if rpowers(1)==0 5nBJj
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); t$,G%micj
rpowern = cat(2,rpowern{:}); U/PNEGuQ
rpowern = [ones(length_r,1) rpowern]; A`M-N<T
else &ZMQ]'&
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); MCTJ^ g"D
rpowern = cat(2,rpowern{:}); [z\baL|
end M
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% Compute the values of the polynomials: AvZ5?rN$
% -------------------------------------- q2F`q. j
y = zeros(length_r,length(n)); PA803R74
for j = 1:length(n) 7xB]Z;:
s = 0:(n(j)-m_abs(j))/2; %'g)MK!e
pows = n(j):-2:m_abs(j); ud(0}[
for k = length(s):-1:1 z&n2JpLY7
p = (1-2*mod(s(k),2))* ... )c*xKij
prod(2:(n(j)-s(k)))/ ... Gjq7@F'
prod(2:s(k))/ ... vO$cF*
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... Z'9 |
prod(2:((n(j)+m_abs(j))/2-s(k))); 4 a&8G
idx = (pows(k)==rpowers); :sK4mR F
y(:,j) = y(:,j) + p*rpowern(:,idx); I6;6x
end raOuD3
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if isnorm )N~ p4kp
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi);
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end [*It' J^
end NwOV2E6@OW
% END: Compute the Zernike Polynomials y@$E5sz
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 0+1!-Wo
zJ(DO>,p&
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% Compute the Zernike functions: G"m0[|XH
% ------------------------------ ;{H Dz$
idx_pos = m>0; ?(R#
idx_neg = m<0; p*g)-/mA
p{_*<"cfYn
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]
z = y; OESKLjFt
if any(idx_pos) S?`0,F
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); h*y+qk-!\g
end stfniV
if any(idx_neg) z]hRc8g}d
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); B%u[gNZ
end o~y{9Q
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% EOF zernfun Q\IViM