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, pc.0;gN
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, 4O`h%`M
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? *
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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? i&HV8&KygN
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function z = zernfun(n,m,r,theta,nflag) 0iL8i#y*
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. 6=g7|}
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N A;|DQR()
% and angular frequency M, evaluated at positions (R,THETA) on the ZbrE m
% unit circle. N is a vector of positive integers (including 0), and )m'_>-`^:
% M is a vector with the same number of elements as N. Each element <+b:
% k of M must be a positive integer, with possible values M(k) = -N(k) AO]lXa
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, }O>Zu[8a
% and THETA is a vector of angles. R and THETA must have the same @s@
% length. The output Z is a matrix with one column for every (N,M) Orb(xLChJ
% pair, and one row for every (R,THETA) pair. @i68%6H`?
% &q<8tTW5
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike *Vc=]Z2G^
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), +|H'Ij$
% with delta(m,0) the Kronecker delta, is chosen so that the integral < 5PeI
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, M/DTD98'N
% and theta=0 to theta=2*pi) is unity. For the non-normalized p)jxqg
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. /iN\)y#u1
% TkBBHg;
% The Zernike functions are an orthogonal basis on the unit circle.
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% They are used in disciplines such as astronomy, optics, and C5M-MZaS
% optometry to describe functions on a circular domain. 1v4kN
-
% mTPj@F>
% The following table lists the first 15 Zernike functions. D1n2Z:9
% 3aqmK.`H
% n m Zernike function Normalization h+W^k+~(
% -------------------------------------------------- ry\']\k
% 0 0 1 1 "qsNySI
% 1 1 r * cos(theta) 2 2o$8CR;
% 1 -1 r * sin(theta) 2 +o3g]0
% 2 -2 r^2 * cos(2*theta) sqrt(6) (FaT{W{
% 2 0 (2*r^2 - 1) sqrt(3) x-pMT3m\D#
% 2 2 r^2 * sin(2*theta) sqrt(6) X]fw9tZ
% 3 -3 r^3 * cos(3*theta) sqrt(8) <e^/hR4O
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) +i^s\c!3;
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) TUaK:*x*
% 3 3 r^3 * sin(3*theta) sqrt(8) 7&3URglsL"
% 4 -4 r^4 * cos(4*theta) sqrt(10) *Vl
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% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) ;Wa{q.)
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) `Ek !;u>
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) X
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% 4 4 r^4 * sin(4*theta) sqrt(10) (<l2 ^H
% -------------------------------------------------- 4BT`|(7
% LU{Z
% Example 1: wuzz%9;@B
% *r`Yz}
% % Display the Zernike function Z(n=5,m=1) 4I-p/&Q
% x = -1:0.01:1; ^kr)U8
% [X,Y] = meshgrid(x,x); p*0Ve21i,
% [theta,r] = cart2pol(X,Y); o
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% idx = r<=1; i .N1Cvp&
% z = nan(size(X)); 'y?|shV{]
% z(idx) = zernfun(5,1,r(idx),theta(idx));
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% figure L:M9|/
% pcolor(x,x,z), shading interp o"FiM5L^.
% axis square, colorbar 9oP{Al
% title('Zernike function Z_5^1(r,\theta)') Gme$FWa
% f~FehN7
% Example 2: =%\6}xPEl<
% y!gM)9vq
% % Display the first 10 Zernike functions #mhD; .Wg
% x = -1:0.01:1; Qu,k
% [X,Y] = meshgrid(x,x); pV6HQ:y1
% [theta,r] = cart2pol(X,Y); dz|*n'd
% idx = r<=1; n^\;*1%$c@
% z = nan(size(X)); ~5NGDT#L*
% n = [0 1 1 2 2 2 3 3 3 3]; ;8iL,^.A
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; ZZU 8B?)
% Nplot = [4 10 12 16 18 20 22 24 26 28]; Wi?%)hur
% y = zernfun(n,m,r(idx),theta(idx)); 8 %j{4$
% figure('Units','normalized') s[q4K
% for k = 1:10 8-a6Q|
% z(idx) = y(:,k); Z9 m;@<%
% subplot(4,7,Nplot(k)) E`fssd~
% pcolor(x,x,z), shading interp ^|GtO.
% set(gca,'XTick',[],'YTick',[]) 'd^gRH<z
% axis square aNC,ccm
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) 7J;~&x
% end ^<\} Y
% _IV@^v
% See also ZERNPOL, ZERNFUN2. `b ")Bx|
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% Paul Fricker 11/13/2006 SMO%sZ]
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% Check and prepare the inputs: d^5SeCs6
% ----------------------------- 2nU
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if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) :{s%=\k {d
error('zernfun:NMvectors','N and M must be vectors.') P5}[*k%DQw
end o,Zng4NY
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if length(n)~=length(m) !mxh]x<e
error('zernfun:NMlength','N and M must be the same length.') C^" Hj
end bsi q9$F
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n = n(:); Q^Z<RA(C
m = m(:); ^q&wITGI
if any(mod(n-m,2)) >3`ctbe
error('zernfun:NMmultiplesof2', ... |5IY`;+9
'All N and M must differ by multiples of 2 (including 0).') gQh Ccv
end sIRrEea
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if any(m>n) g<iwxF
error('zernfun:MlessthanN', ... k<'vP{
'Each M must be less than or equal to its corresponding N.') 0"^oTmQN
end j t`p<gI
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if any( r>1 | r<0 ) (C l`+ V
error('zernfun:Rlessthan1','All R must be between 0 and 1.') (>LHj]}K
end 6I cM:x
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) &Ev]x2YC
error('zernfun:RTHvector','R and THETA must be vectors.') 0"kE^=
end loC5o|Wh
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r = r(:); 1V**QSZ1
theta = theta(:); 1>@]@ST[:
length_r = length(r); D){"fw+b
if length_r~=length(theta) qsft*&
error('zernfun:RTHlength', ... |.8d,!5w}
'The number of R- and THETA-values must be equal.') M8?#%x6;N
end :nKsZ1b X
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% Check normalization: Tw}?(\ya
% -------------------- Pv7f
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if nargin==5 && ischar(nflag) V|3yZ8lE
isnorm = strcmpi(nflag,'norm'); urT/+deR
if ~isnorm [/AdeR
error('zernfun:normalization','Unrecognized normalization flag.') z<oE!1St
end B;z>Dd,Y_x
else <t[Z9s$n
isnorm = false; 1=/doo{^
end =wIdC3Ph
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% >?>u bM`,
% Compute the Zernike Polynomials 4T==A#Z
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% .Y u<%
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% Determine the required powers of r: >J^7}J
% ----------------------------------- :jk)(=^
m_abs = abs(m); {WYu0J@
rpowers = []; yD3bl%uZ
for j = 1:length(n) 1A%N0#_(Md
rpowers = [rpowers m_abs(j):2:n(j)]; &547`*
end B_SZ?o
rpowers = unique(rpowers); 1N!Oslum
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% Pre-compute the values of r raised to the required powers, fqcU5l[v,
% and compile them in a matrix: DA+A >5/
% ----------------------------- c$,c`H(~
if rpowers(1)==0 uQ[vgNe*m
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 'DCKD4@C/
rpowern = cat(2,rpowern{:}); MekT?KPQ{L
rpowern = [ones(length_r,1) rpowern]; aW0u8Dz
else ,]~u:Y}
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); l-<3{!
rpowern = cat(2,rpowern{:}); v%H"_T
end &Pu+(~'Q
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% Compute the values of the polynomials: ruZYehu1W
% -------------------------------------- t{/:( Nu
y = zeros(length_r,length(n)); Zz"I.$$[M
for j = 1:length(n) a4A`cUt
s = 0:(n(j)-m_abs(j))/2; r+t ,J|V
pows = n(j):-2:m_abs(j); cr76cYq"Q
for k = length(s):-1:1 rQ`\JE&`
p = (1-2*mod(s(k),2))* ... A#v|@sul
prod(2:(n(j)-s(k)))/ ... d{QMST2&
prod(2:s(k))/ ... ?!bd!:(N
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... p&i.)/
prod(2:((n(j)+m_abs(j))/2-s(k))); lo cW_/
idx = (pows(k)==rpowers); ! 9d_Gf-
y(:,j) = y(:,j) + p*rpowern(:,idx); <\ y!3;
end u|(Ux~O
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if isnorm b->eg 8|
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); `%$8cZ-kr
end 1i4WWK7k
end *-?Wcz
% END: Compute the Zernike Polynomials Of-C
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 7)B&(2D&
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% Compute the Zernike functions: j_!bT!8
% ------------------------------ 1)$%Jr
idx_pos = m>0; TNh=4xQ}
idx_neg = m<0; x|.v{tQa
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z = y; 6tOCZ'f
if any(idx_pos) A[RHw<
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); ci`zR9Ks
end uCw>}3
if any(idx_neg) z<a$q3!#
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); i*X{^A73"
end #"::
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% EOF zernfun a* D,*C5}