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, AG|:mQO
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, q!WiX|P
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? S&F;~
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛? KB$Y8[
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function z = zernfun(n,m,r,theta,nflag) s14ot80)
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. QzY5S0
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N @v/
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% and angular frequency M, evaluated at positions (R,THETA) on the 2tS,q_-=
% unit circle. N is a vector of positive integers (including 0), and oGL2uQXX
% M is a vector with the same number of elements as N. Each element 9O\yIL
% k of M must be a positive integer, with possible values M(k) = -N(k) X.AE>fx*h
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, 6%MM)Vj+u
% and THETA is a vector of angles. R and THETA must have the same |eksvO'~
% length. The output Z is a matrix with one column for every (N,M) 0U!_ o2]
% pair, and one row for every (R,THETA) pair. _pkmHj(
% Ue=1NnRDkA
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike =WK's8FB;8
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), Sc:)H2k`$
% with delta(m,0) the Kronecker delta, is chosen so that the integral mcWN.
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, !gi3J @
% and theta=0 to theta=2*pi) is unity. For the non-normalized bQHJ}aCi
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. OG^#e+
% Kc`#~-`,(
% The Zernike functions are an orthogonal basis on the unit circle. a``Q}.ST
% They are used in disciplines such as astronomy, optics, and w}U'>fj
% optometry to describe functions on a circular domain. A`V:r2hnb
% &H%z1Lp
% The following table lists the first 15 Zernike functions. J"fv5{
% o[g]Va*8
% n m Zernike function Normalization Vg7BK%
% -------------------------------------------------- ,D' bIk
% 0 0 1 1 HG3iK
% 1 1 r * cos(theta) 2 # (-?i\i
% 1 -1 r * sin(theta) 2 0QBK(_O`
% 2 -2 r^2 * cos(2*theta) sqrt(6) G#3$sz
% 2 0 (2*r^2 - 1) sqrt(3) vkLyGb7r<
% 2 2 r^2 * sin(2*theta) sqrt(6) gH0Rd
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% 3 -3 r^3 * cos(3*theta) sqrt(8) Q@rlqWgU
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% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) 5KW
n >n
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) ;pG5zRe
% 3 3 r^3 * sin(3*theta) sqrt(8) Ll`nO;h
% 4 -4 r^4 * cos(4*theta) sqrt(10) bLO^5` 6
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) R.rE+gxO1
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) =_[Ich,}
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) |&~);>Cq2
% 4 4 r^4 * sin(4*theta) sqrt(10) +XAM2uN5_.
% -------------------------------------------------- F
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% 7lAn GP.;
% Example 1: v"dl6%D"
% UZo[]$"Q`
% % Display the Zernike function Z(n=5,m=1) "F?p Y@4
% x = -1:0.01:1; >~uKkQ_p
% [X,Y] = meshgrid(x,x); *a`_,Q{x
% [theta,r] = cart2pol(X,Y); *7C l1o
% idx = r<=1; ~(eD 4"
% z = nan(size(X)); )_K:A(V>
% z(idx) = zernfun(5,1,r(idx),theta(idx)); \n-.gG
% figure ES5a`"H
% pcolor(x,x,z), shading interp [k=LX+w@
% axis square, colorbar <H|]^An!H
% title('Zernike function Z_5^1(r,\theta)') >ajcfG.k(
% D;Y2yc[v
% Example 2: Kp[5"N8
% QS<)*
% % Display the first 10 Zernike functions GX N:=
% x = -1:0.01:1; G.qjw]Llf
% [X,Y] = meshgrid(x,x); /?S,u,R
% [theta,r] = cart2pol(X,Y); ,1i l&
% idx = r<=1; lht :%Ts$
% z = nan(size(X)); on f7V
% n = [0 1 1 2 2 2 3 3 3 3]; *-s':('R
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; :(i=> ~O
% Nplot = [4 10 12 16 18 20 22 24 26 28]; Zc=#Y
% y = zernfun(n,m,r(idx),theta(idx)); hho\e
8
% figure('Units','normalized') Pa/2]) w
% for k = 1:10 gO bP
% z(idx) = y(:,k); Xnxb.{C
% subplot(4,7,Nplot(k)) RY~mQ
% pcolor(x,x,z), shading interp Kj+TPqXb
% set(gca,'XTick',[],'YTick',[]) ||#+ ^p7G
% axis square M"8?XD%
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}'])
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% end d4LH`@SUZ-
% B &)wJG
% See also ZERNPOL, ZERNFUN2. tS[@?qP
`%=!_|
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% Paul Fricker 11/13/2006 j`-9.
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% Check and prepare the inputs: ,sn
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% ----------------------------- m"vWu0/#
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) $WbfRyXi7'
error('zernfun:NMvectors','N and M must be vectors.') %&iWc_"
end NJ(H$tB@
@*JS[w$1
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if length(n)~=length(m) 6"V86b0)h}
error('zernfun:NMlength','N and M must be the same length.') zl$z> z )
end } BnPNc[I
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n = n(:); BtKbX)R$J
m = m(:); S{JBV@@tC
if any(mod(n-m,2)) g#[,4o;
error('zernfun:NMmultiplesof2', ... j8ag}%
'All N and M must differ by multiples of 2 (including 0).') ''B}^yKEW
end U_5\FM
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if any(m>n) uG4Q\,R
error('zernfun:MlessthanN', ... ./}W3
'Each M must be less than or equal to its corresponding N.') HeM-
end ?^
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if any( r>1 | r<0 ) {_X&{dZLX
error('zernfun:Rlessthan1','All R must be between 0 and 1.') G4)X~.Fy
end ]PZ\N~T
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) pAc "Wo(Q
error('zernfun:RTHvector','R and THETA must be vectors.') $(;0;!t.
end L_}F.nbS5
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r = r(:); r<-@.$lf
theta = theta(:); %o#|zaK
length_r = length(r); Y> PC>
if length_r~=length(theta) cy#N(S[ 1
error('zernfun:RTHlength', ... Z_[jah
'The number of R- and THETA-values must be equal.') K?acRi
end n }4L q^$
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% Check normalization: 4GqE%n+ta~
% -------------------- IsP!ZcV;
if nargin==5 && ischar(nflag) D2Dk7//82Y
isnorm = strcmpi(nflag,'norm'); >y
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if ~isnorm *\F,?yU
error('zernfun:normalization','Unrecognized normalization flag.') ;.b^A
end $Z4IPs
else -LEpT$v|
isnorm = false; Qb&gKQtt@
end 3(>NS ?lX
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% f7j9'k
% Compute the Zernike Polynomials '1Q [&
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% #BX^"J{~
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% Determine the required powers of r: F9C3i
% ----------------------------------- ]$?\,`
m_abs = abs(m); "\1QJ
rpowers = []; hS+R/7
for j = 1:length(n) \x+ "1
rpowers = [rpowers m_abs(j):2:n(j)]; m6M:l"u
end E>O1dPZcM
rpowers = unique(rpowers); -87]$ ax
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% Pre-compute the values of r raised to the required powers, RT+_e
% and compile them in a matrix: rty&\u@}
% ----------------------------- Z|qUVD5Ic
if rpowers(1)==0 txXt<]N
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 4+15`
rpowern = cat(2,rpowern{:}); f3HleA&&
rpowern = [ones(length_r,1) rpowern]; LjMhPzCp
else L64cCP*
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); 9!|+GIjn
rpowern = cat(2,rpowern{:}); ,7|Wf
%X
end sn?YD'>k
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% Compute the values of the polynomials: ##clReS
% -------------------------------------- 1rQKHC:|
y = zeros(length_r,length(n)); 6b9&V`
for j = 1:length(n) /f)
#CR0$
s = 0:(n(j)-m_abs(j))/2;
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pows = n(j):-2:m_abs(j); 61{IXx_
for k = length(s):-1:1 5 ]v]^Y'?
p = (1-2*mod(s(k),2))* ... [p[C45d=<
prod(2:(n(j)-s(k)))/ ... /yS/*ET8
prod(2:s(k))/ ... a#4 'X*
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 3Y+
bIz!
prod(2:((n(j)+m_abs(j))/2-s(k))); 2OQDG7#Kc
idx = (pows(k)==rpowers); Y]>Qu f.!
y(:,j) = y(:,j) + p*rpowern(:,idx); zaoC
end ?sm@lDZ\
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if isnorm .WT^L2l%
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ,3E9H&@j
end J=C63YB
end [.`%]Z(
% END: Compute the Zernike Polynomials sCE2 F_xjL
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% J,=:
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[Y=X^"PF
% Compute the Zernike functions: F_&bE@k
% ------------------------------ Oe[qfsdW
idx_pos = m>0; ~ GW8|tw
idx_neg = m<0; &9F(uk=X
4%L-3Ij
Om=*b#k
z = y; lYMNx|PF
if any(idx_pos) ,dO$R.h
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); X ?l F,p
end d$(>=gzBQ
if any(idx_neg) XTOZ]H*^
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); oK3aW6
end \<R.F
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% EOF zernfun /K@{(=n