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, 4["}U1sG
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, AY! zXJ_$
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? VfZ/SByh7p
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? +mF}j=k
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function z = zernfun(n,m,r,theta,nflag) "BzRLg!J
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. +x+H(of.
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N VQ}=7oe%q
% and angular frequency M, evaluated at positions (R,THETA) on the "`&?<82
% unit circle. N is a vector of positive integers (including 0), and ?G8 D6
% M is a vector with the same number of elements as N. Each element Mq*Sp
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% k of M must be a positive integer, with possible values M(k) = -N(k) 1TbKnmTx
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, ;dB=/U>3U
% and THETA is a vector of angles. R and THETA must have the same BJ&>'rc
% length. The output Z is a matrix with one column for every (N,M) 67n1s
% pair, and one row for every (R,THETA) pair. if`/LJsa
% _ H@pYMNH
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike y:W$~<E`p
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), XPY66VC&_
% with delta(m,0) the Kronecker delta, is chosen so that the integral Z#oo8
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, M8g=t[\
% and theta=0 to theta=2*pi) is unity. For the non-normalized <'gCI Ia2
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 0~FX!1;
% ?rv+ydR/q
% The Zernike functions are an orthogonal basis on the unit circle. G=b`w;oL:
% They are used in disciplines such as astronomy, optics, and <:%Iq13D
% optometry to describe functions on a circular domain. B!8]\D
% &Nec(q<
% The following table lists the first 15 Zernike functions. 8, WQ}cC
% c[j3_fn1]
% n m Zernike function Normalization dXdU4YJX
% -------------------------------------------------- .Q?AzU,2D
% 0 0 1 1 ]cA){^.Jz
% 1 1 r * cos(theta) 2 b"f4}b
% 1 -1 r * sin(theta) 2 Yq2mVo
% 2 -2 r^2 * cos(2*theta) sqrt(6) 9MGA#a
% 2 0 (2*r^2 - 1) sqrt(3) %n-LDn
% 2 2 r^2 * sin(2*theta) sqrt(6) }7&;YAt
% 3 -3 r^3 * cos(3*theta) sqrt(8) "E'OPR
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 5))?,YkrrI
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) vKI,|UD&-
% 3 3 r^3 * sin(3*theta) sqrt(8) "5>p]u>
% 4 -4 r^4 * cos(4*theta) sqrt(10) ^:DlrI$
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) +\}]`uS:
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) %r|fuwwJO
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) -`Z5#8P
% 4 4 r^4 * sin(4*theta) sqrt(10) O'!k$iJNb
% -------------------------------------------------- vK$T$SL
% hL8QA!
% Example 1: @YT=-
% Oz n7C?\*
% % Display the Zernike function Z(n=5,m=1) ||/noUK
% x = -1:0.01:1; Fl|u0SY
% [X,Y] = meshgrid(x,x); !H.&"~w@
% [theta,r] = cart2pol(X,Y); HPU7
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% idx = r<=1; gNxnoOY
% z = nan(size(X)); Nf$Y-v?i
% z(idx) = zernfun(5,1,r(idx),theta(idx)); JQ.ZAhv
% figure P=S)V
% pcolor(x,x,z), shading interp *D|6g|Hb
% axis square, colorbar Oj<2_u
% title('Zernike function Z_5^1(r,\theta)') > m5j.GP;
% GR|Vwxs<@P
% Example 2: ){gO b
% u/k#b2BqL
% % Display the first 10 Zernike functions Q}]Q0'X8
% x = -1:0.01:1; Q&n|tQ*4
% [X,Y] = meshgrid(x,x); +z9;BPw%
% [theta,r] = cart2pol(X,Y); g fO.Ky6
% idx = r<=1; .
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% z = nan(size(X)); M,mj{OY~x
% n = [0 1 1 2 2 2 3 3 3 3]; HeF[H\a<
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; (:@qn+
a
% Nplot = [4 10 12 16 18 20 22 24 26 28]; ;ATk?O4T
% y = zernfun(n,m,r(idx),theta(idx)); @++
X H}
% figure('Units','normalized') v[HxO?x^
% for k = 1:10 '6K WobXm
% z(idx) = y(:,k); N8m^h:b
% subplot(4,7,Nplot(k)) PJb_QL!9
% pcolor(x,x,z), shading interp ~tz[=3!1H
% set(gca,'XTick',[],'YTick',[]) AbfLV942
% axis square {uw'7 d/
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) ~1}NQa(
% end 7p2x}[ .\
% 8xL-j2w
% See also ZERNPOL, ZERNFUN2. qjTz]'^BpM
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% Paul Fricker 11/13/2006 sPbtv[bC
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% Check and prepare the inputs: 4W^0K|fq
% ----------------------------- oYOf<J
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) (|bht 0
error('zernfun:NMvectors','N and M must be vectors.') @NX^__sa
end ym1TGeFAq
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if length(n)~=length(m)
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error('zernfun:NMlength','N and M must be the same length.') )(?s=<H
end LscAsq<H<
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n = n(:); h2,AcM
m = m(:); I,?bZ&@8
if any(mod(n-m,2)) u}#rS%SF*
error('zernfun:NMmultiplesof2', ... m,=$a\UC
'All N and M must differ by multiples of 2 (including 0).') +n)(\k{
end OE:t!66
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if any(m>n) Jbs:}]2
error('zernfun:MlessthanN', ... 9>zN 27
'Each M must be less than or equal to its corresponding N.') =U@*adgw
end +R*4`F:QJQ
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if any( r>1 | r<0 ) Up/1c:<J
error('zernfun:Rlessthan1','All R must be between 0 and 1.') k&^Megcb
end 6bqJM#y@
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) R]ppA=1*_l
error('zernfun:RTHvector','R and THETA must be vectors.') RRq*CLj
end D
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r = r(:); gqe
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theta = theta(:); YQ?|Vb
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length_r = length(r); yy#Xs:/
if length_r~=length(theta) Fs&m'g
error('zernfun:RTHlength', ... JgK?j&!hs:
'The number of R- and THETA-values must be equal.') !!` zz
end fM2[wh@
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% Check normalization: E}a3. 6)p
% -------------------- $_)f|\s
if nargin==5 && ischar(nflag) .h*&$c/l
isnorm = strcmpi(nflag,'norm'); I>P</TE7
if ~isnorm X\$M _b>O
error('zernfun:normalization','Unrecognized normalization flag.') ,lN!XP{M6w
end mexI}
else V-X n&s
isnorm = false; *|` ' L
end G\P*zzSq
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=%RDT9T.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 1pz6e8p:m
% Compute the Zernike Polynomials _abVX#5<
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% fSun{?{
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