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， 2aM7zP[Z
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， T^1
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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？ 7pyzPc#_
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ ai/|qYf
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function z = zernfun(n,m,r,theta,nflag) jb.H[n,\
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. Oo|PZ_P
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N 5.9<g>C
% and angular frequency M, evaluated at positions (R,THETA) on the Mqr_w !8d
% unit circle. N is a vector of positive integers (including 0), and u
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% M is a vector with the same number of elements as N. Each element "0An'7'm
% k of M must be a positive integer, with possible values M(k) = -N(k) Wb-C0^dTn
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, s E pI)9
% and THETA is a vector of angles. R and THETA must have the same }4A] x`3
% length. The output Z is a matrix with one column for every (N,M) RRIh;HhX
% pair, and one row for every (R,THETA) pair. } a9Ah:.7/
% 0ra'H/>Ly
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike aTuu",f
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), V\;Xa0
% with delta(m,0) the Kronecker delta, is chosen so that the integral +i &<`ov
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, W,<q!<z\t
% and theta=0 to theta=2*pi) is unity. For the non-normalized q !ZM Wg
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. o.{W_k/n
% vk92j?
% The Zernike functions are an orthogonal basis on the unit circle. Ek_ 5% n
% They are used in disciplines such as astronomy, optics, and l-+=Yk!X
% optometry to describe functions on a circular domain. C`[<6>&y
% {o}U"b<+Ra
% The following table lists the first 15 Zernike functions. p 0Jr{hM
% O[ Vet/^)
% n m Zernike function Normalization @NL cO }
% -------------------------------------------------- 8s1nE_3
% 0 0 1 1 rAH!%~
% 1 1 r * cos(theta) 2 C^J<qq &
% 1 -1 r * sin(theta) 2 Jka>Er
% 2 -2 r^2 * cos(2*theta) sqrt(6) VeYT [Us"
% 2 0 (2*r^2 - 1) sqrt(3) g+ c*VmY
% 2 2 r^2 * sin(2*theta) sqrt(6) nkW})LyB\
% 3 -3 r^3 * cos(3*theta) sqrt(8) J}#gTG( '
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) ?QOU9"@+B
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) yLnQ9BXB&
% 3 3 r^3 * sin(3*theta) sqrt(8) -s3`mc}*
% 4 -4 r^4 * cos(4*theta) sqrt(10) }L\; W:0
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) VdlT+'HF
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) kxMvOB$
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10)
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% 4 4 r^4 * sin(4*theta) sqrt(10) z'YWomfZm
% -------------------------------------------------- YM}a>o
% .-d'*$
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% Example 1: jn<?,UABD
% \P< aK$g
% % Display the Zernike function Z(n=5,m=1) XO +BZB`F
% x = -1:0.01:1; *~vB6V |1
% [X,Y] = meshgrid(x,x); =;Gq:mHi
% [theta,r] = cart2pol(X,Y); _~<sb,W
% idx = r<=1; )1 s5vNVa
% z = nan(size(X)); ,md_eGF
% z(idx) = zernfun(5,1,r(idx),theta(idx)); ,
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% figure K./qu^+k
% pcolor(x,x,z), shading interp PkvW6,lS
% axis square, colorbar 7v5]% %E/
% title('Zernike function Z_5^1(r,\theta)') my (@~'
% daE.y_9y
% Example 2: 3s6obw$ki
% lvW
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% % Display the first 10 Zernike functions ~gDYb#p
% x = -1:0.01:1; cOV j @z
% [X,Y] = meshgrid(x,x); g) Lf^
% [theta,r] = cart2pol(X,Y); mY"7/dw<v
% idx = r<=1; &<A,\ M
% z = nan(size(X)); i2=- su
% n = [0 1 1 2 2 2 3 3 3 3]; 6{h\CU}"
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; /<rv aR
% Nplot = [4 10 12 16 18 20 22 24 26 28]; 4V@%Y,:ee
% y = zernfun(n,m,r(idx),theta(idx)); Pb5yz-?
% figure('Units','normalized') 4^F[G p?
% for k = 1:10 eZ' 8JU]
% z(idx) = y(:,k); @j!,8JQEd
% subplot(4,7,Nplot(k)) Y%KowgP\
% pcolor(x,x,z), shading interp `Fd
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% set(gca,'XTick',[],'YTick',[]) roADC?@r
% axis square FM {f{2j
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) .5|[gBK
% end 3=O [Q :8
% (i~UH04r>s
% See also ZERNPOL, ZERNFUN2. tOIqX0dWd
x[ 0T$
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% Paul Fricker 11/13/2006 ]
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% Check and prepare the inputs: I9X \@ lTf
% ----------------------------- <V ? 2;Gy
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) *:%&z?<Fw
error('zernfun:NMvectors','N and M must be vectors.') S\GWMB!oF
end m{IlRf'
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if length(n)~=length(m) S;~eI8gQ"
error('zernfun:NMlength','N and M must be the same length.') m?e/ MQr
end K#R]of~/
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2F4<3k! &
n = n(:); 5CI {&E
m = m(:); 'uu*DgEr
if any(mod(n-m,2)) de:@/-|
error('zernfun:NMmultiplesof2', ... #V k?
'All N and M must differ by multiples of 2 (including 0).') &^`Wtd~g
end l2F#^=tp
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if any(m>n) GfONm6A
error('zernfun:MlessthanN', ... a 0SZw
'Each M must be less than or equal to its corresponding N.') W@R7CQE@
end UC`h o%OBF
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if any( r>1 | r<0 ) ZF7n]LgSc&
error('zernfun:Rlessthan1','All R must be between 0 and 1.') F_@B ` ,
end x6cG'3&T
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) $ }/tlA&e
error('zernfun:RTHvector','R and THETA must be vectors.') c.>f,vtcn
end o/-RG LzAo
O=%Ht-kOc
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r = r(:); =M1}HF,7>l
theta = theta(:); P'KA-4!
length_r = length(r); @b(@`yz.a
if length_r~=length(theta) 1>*oN
error('zernfun:RTHlength', ... tddwnpnSw
'The number of R- and THETA-values must be equal.') "(=g7,I4
end & AK\Pw)
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E%
% Check normalization: S)hDsf.I
% -------------------- d(^8#4
if nargin==5 && ischar(nflag) qc (e3x
isnorm = strcmpi(nflag,'norm'); YP,,vcut
if ~isnorm kqB# 9
error('zernfun:normalization','Unrecognized normalization flag.') kn:hxdZ
end b%l H=u
else DN%}OcpZ
isnorm = false; vA6`};|
end @ lB{!j&q
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% q
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% Compute the Zernike Polynomials "rMfe> ;FJ
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% l~$)>?ZD
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% Determine the required powers of r: 0[UI'2
% ----------------------------------- h[dJNawL
m_abs = abs(m); sqhMnDn[
rpowers = []; "E+;O,N-
for j = 1:length(n) dEYw_qJ2
rpowers = [rpowers m_abs(j):2:n(j)]; tQ@7cjq8bA
end ?=lb@U
rpowers = unique(rpowers); A{ > w5 T
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4L>8RiiQE;
% Pre-compute the values of r raised to the required powers, ;?q(8^A
% and compile them in a matrix: 8s22VL
% ----------------------------- ObM/~{rKx
if rpowers(1)==0 'A|c\sy
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); 0d2RB^"i
rpowern = cat(2,rpowern{:}); OcUj_Zd
rpowern = [ones(length_r,1) rpowern]; E^J &?-
else -aBhN~
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); y )K Iz
rpowern = cat(2,rpowern{:}); o|>=< l
end 0W zoI2Q
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% Compute the values of the polynomials:
IZrcn
% -------------------------------------- 4x
?NCD=k
y = zeros(length_r,length(n)); Kz
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for j = 1:length(n) u$tst_y-
s = 0:(n(j)-m_abs(j))/2; uKzx >\}?1
pows = n(j):-2:m_abs(j); P, ZQ*Ju
for k = length(s):-1:1 uP l7u 1c
p = (1-2*mod(s(k),2))* ... 'T^Ma LK
prod(2:(n(j)-s(k)))/ ... F3V: B.C
prod(2:s(k))/ ... DI)"F OM6
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... l`~$cK!
prod(2:((n(j)+m_abs(j))/2-s(k))); gK~Z Ch
idx = (pows(k)==rpowers); . AA#
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y(:,j) = y(:,j) + p*rpowern(:,idx); P'iX?+*
end Q}Ah{H0C
M#Z^8 (
if isnorm j)G%I y[`
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); ;Mq'+ 4$
end !.%*Tp#k#
end o#"yFP1
% END: Compute the Zernike Polynomials (]sm9PO
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% =P ,mix|
(XR}U6^v]
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% Compute the Zernike functions: >"UXY)
% ------------------------------ X*#\JF4$i
idx_pos = m>0; xN$V( ZX4
idx_neg = m<0; Q65M(x+oy
%{'[S0 @Z
k6DJ(.n'%a
z = y; _!|$ i
if any(idx_pos) {R(/Usg!=
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); )/f#~$ws
end [jN Vk3
if any(idx_neg) b i-A m/9
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); ^xk4HF
end Ib2& L
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p\I3 f I0i
% EOF zernfun !p ~.Y+
