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, O(f&0h
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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, "]{"4qV1=
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? o[CjRQY]P
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? mnWbV\ VY
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function z = zernfun(n,m,r,theta,nflag) wPU<jAQyp
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. @](\cT64i3
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N <E&"]
% and angular frequency M, evaluated at positions (R,THETA) on the 7Ke#sW.HN
% unit circle. N is a vector of positive integers (including 0), and T~'9p`IW
% M is a vector with the same number of elements as N. Each element B[Fuy y?
% k of M must be a positive integer, with possible values M(k) = -N(k) K=C).5=U
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, Lg4I6 G
% and THETA is a vector of angles. R and THETA must have the same hV4B?##O
% length. The output Z is a matrix with one column for every (N,M) }8qsE
% pair, and one row for every (R,THETA) pair. 8q&*tpE
% Z<0+<tt
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike 5&*B2ZBzH
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), A?sU[b6_
% with delta(m,0) the Kronecker delta, is chosen so that the integral #ZRplA~C7]
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, y:+s*x6Vg
% and theta=0 to theta=2*pi) is unity. For the non-normalized g$ oe00b
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. nob^
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% `L=$,7`
% The Zernike functions are an orthogonal basis on the unit circle. lhA<wV1-9G
% They are used in disciplines such as astronomy, optics, and Q35/Sp[;x
% optometry to describe functions on a circular domain. \aO.LwYm;:
% nu#_,x<LS
% The following table lists the first 15 Zernike functions. X K5qE"
% s GP}>w-JZ
% n m Zernike function Normalization :{v:sK
% -------------------------------------------------- #TX=%x6
% 0 0 1 1 /8` S}g+
% 1 1 r * cos(theta) 2 W<D(M.61A
% 1 -1 r * sin(theta) 2 NK@G0p~O
% 2 -2 r^2 * cos(2*theta) sqrt(6)
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% 2 0 (2*r^2 - 1) sqrt(3) wB;'+d&
% 2 2 r^2 * sin(2*theta) sqrt(6) Vhs:X~=qL
% 3 -3 r^3 * cos(3*theta) sqrt(8) z<F.0~)jb
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) :K6(`J3Y"^
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) k&1~yW
% 3 3 r^3 * sin(3*theta) sqrt(8) *?+maK{5+
% 4 -4 r^4 * cos(4*theta) sqrt(10) emV@kN.
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) "kjjq~l
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) nJ4CXSdE
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) N|ut^X+|\
% 4 4 r^4 * sin(4*theta) sqrt(10) ]{[VTjC7rY
% -------------------------------------------------- df7z&{R
% _;BN;].
% Example 1: sQS2U6
% w^&TG3m1~
% % Display the Zernike function Z(n=5,m=1) 2Ax HhD.
% x = -1:0.01:1; 7n~BDqT
% [X,Y] = meshgrid(x,x); RkJ\?
% [theta,r] = cart2pol(X,Y); @:. 6'ji,`
% idx = r<=1; uv2!][
% z = nan(size(X)); |j
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% z(idx) = zernfun(5,1,r(idx),theta(idx)); X3:-+]6,d
% figure 1lNg} !)[K
% pcolor(x,x,z), shading interp s.rS06x
% axis square, colorbar R?Q@)POW
% title('Zernike function Z_5^1(r,\theta)') t
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% BV&}(9z
% Example 2: <)]B$~(a
% By@<N [I@
% % Display the first 10 Zernike functions F^=|NlU&%
% x = -1:0.01:1; >29eu^~nh
% [X,Y] = meshgrid(x,x); T!hU37g h?
% [theta,r] = cart2pol(X,Y); h@z(yB
j:0
% idx = r<=1; |Js96>B:
% z = nan(size(X)); 4.3Bz1p
% n = [0 1 1 2 2 2 3 3 3 3]; nFlj`k<]Y
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; t?v0ylN
% Nplot = [4 10 12 16 18 20 22 24 26 28]; VYhZ0;' '
% y = zernfun(n,m,r(idx),theta(idx)); w<awCp
% figure('Units','normalized') ,7pO-:*g
% for k = 1:10 I ,AI$A
% z(idx) = y(:,k); %t\`20-1<
% subplot(4,7,Nplot(k)) mV;Egm{A\
% pcolor(x,x,z), shading interp hSD)|
% set(gca,'XTick',[],'YTick',[]) S&V5zB""n
% axis square z1LATy
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) E<a~
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% end CPGXwM=
% (G"b)"Qum
% See also ZERNPOL, ZERNFUN2. Ckvm3r\i2
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% Paul Fricker 11/13/2006 9h$-:y3
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% Check and prepare the inputs:
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% ----------------------------- ;'pEzz?k"
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) tZ=BK:39\
error('zernfun:NMvectors','N and M must be vectors.') gW6lMyiLb
end d?&?$qf[
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if length(n)~=length(m) /Yx 1S'5
error('zernfun:NMlength','N and M must be the same length.') cCU'~
end C|W_j&S65
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n = n(:); (1(3:)@S6
m = m(:); iAT&C`,(&
if any(mod(n-m,2)) S_6`.@B}
error('zernfun:NMmultiplesof2', ... pp#Kb 2*
'All N and M must differ by multiples of 2 (including 0).') f<WnPoV
end Z[AJat@H
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if any(m>n) ^=D77 jS
error('zernfun:MlessthanN', ... eJ%~6c`@!
'Each M must be less than or equal to its corresponding N.') %o#D"
end rQ*'2Zf'<
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if any( r>1 | r<0 ) ?GD{}f33
error('zernfun:Rlessthan1','All R must be between 0 and 1.') v>)[NAY9
end }.2pR*W
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) $|6Le;
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error('zernfun:RTHvector','R and THETA must be vectors.') HC4ad0Gs+{
end cGsxfwD
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r = r(:); k'%c| kx8U
theta = theta(:); x;Dr40wD@y
length_r = length(r); yKOf]m>#
if length_r~=length(theta) U`:#+8h-}
error('zernfun:RTHlength', ... dm.?-u;C
'The number of R- and THETA-values must be equal.') *-_` xe
end V)Z*X88:Tv
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% Check normalization: >S5D-)VX
% -------------------- SP
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if nargin==5 && ischar(nflag) }%$9nq3
isnorm = strcmpi(nflag,'norm'); s.C-II?e
if ~isnorm !pw%l4]/t
error('zernfun:normalization','Unrecognized normalization flag.') _h@7>+vl~
end A+Y>1-=JO
else yn!LJT[~2
isnorm = false; 3Eiy/
end dWD9YIYf
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% k@4]s_2
% Compute the Zernike Polynomials B{s[SZ
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% NO`a2HR$
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% Determine the required powers of r: ?h
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% ----------------------------------- R\0]\JEc
m_abs = abs(m); wvT!NN
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rpowers = []; ~O@V;y
for j = 1:length(n) UTin0k
rpowers = [rpowers m_abs(j):2:n(j)]; 0~Yg={IKhK
end I7BfA,mZ7
rpowers = unique(rpowers); 4d0PW#97.
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% Pre-compute the values of r raised to the required powers, QE<63|
% and compile them in a matrix: f} }Bb8
% ----------------------------- H -.3r
if rpowers(1)==0 MfeW|
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); Lk#u^|Eq7=
rpowern = cat(2,rpowern{:}); "-v9V7KCM
rpowern = [ones(length_r,1) rpowern]; {l *ps-fi
else #0G9{./C
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); SGNi~o
rpowern = cat(2,rpowern{:}); a5Xr"-
end t4h05 i
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% Compute the values of the polynomials: Qkr'C
n
% -------------------------------------- qZ_^#%zO
y = zeros(length_r,length(n)); 3e I:$1"Q
for j = 1:length(n) Y&'2/zI6~
s = 0:(n(j)-m_abs(j))/2; ]C)PZZI='
pows = n(j):-2:m_abs(j); m]7yc>uDy
for k = length(s):-1:1 xiA9X]FB
p = (1-2*mod(s(k),2))* ... ih ,8'D4
prod(2:(n(j)-s(k)))/ ... wAk oX
prod(2:s(k))/ ... ^U~YG=!ww
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 7F|T5[*l
prod(2:((n(j)+m_abs(j))/2-s(k))); C@]Z&H;
idx = (pows(k)==rpowers); X5>p~;[9
y(:,j) = y(:,j) + p*rpowern(:,idx); OWOj|jM
end 8{Zgvqbb
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if isnorm +` Em&
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); G_42ckLq
end dVO|q9 /
end iCl,7$[*
% END: Compute the Zernike Polynomials 'ky'GzX,
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% V-7!)&q
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% Compute the Zernike functions: L2Gm0 v
% ------------------------------ ~73YOGiGJH
idx_pos = m>0; zpg*hlv
idx_neg = m<0; }p8a'3@Z
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z = y; Qs;bVlp!H
if any(idx_pos) YM1@B`yWE
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); /7P4[~vw
end +sgishqn9
if any(idx_neg) ^P&y9dC.
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); q'K=Ly+
end lv$tp,+
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% EOF zernfun #1V vK