| jssylttc |
2012-04-23 19:23 |
如何从zernike矩中提取出zernike系数啊
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， ,& \&::R
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， W ? 6
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？ Z IGbwL
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ 8Pnqmjjj
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function z = zernfun(n,m,r,theta,nflag) {U+9,6.`
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. _Oaso >
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N "=v J }
% and angular frequency M, evaluated at positions (R,THETA) on the =_H*fhXS
% unit circle. N is a vector of positive integers (including 0), and T{ v <
% M is a vector with the same number of elements as N. Each element M 25z<Y
% k of M must be a positive integer, with possible values M(k) = -N(k) %3@RZe
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, '6Z/-V 4k
% and THETA is a vector of angles. R and THETA must have the same D;
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% length. The output Z is a matrix with one column for every (N,M) :a^ ,Ei-&
% pair, and one row for every (R,THETA) pair. =":V
WHf
% k*UR# z(I
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike w$<fSe7
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi),
a F4V|?+
% with delta(m,0) the Kronecker delta, is chosen so that the integral sL[(cX?;2
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, Br. $L
% and theta=0 to theta=2*pi) is unity. For the non-normalized R;Ix<y{U
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 2=UTH% 1D
% gC,0+Y~
% The Zernike functions are an orthogonal basis on the unit circle. V }r_
% They are used in disciplines such as astronomy, optics, and {H7$uiq3:B
% optometry to describe functions on a circular domain. X
G@>1 /
% v'2OHb#
% The following table lists the first 15 Zernike functions. CH R?i 1e
% $4ZDT]n
% n m Zernike function Normalization _c7
% -------------------------------------------------- H&>>]DD
% 0 0 1 1 gWU(uBS
% 1 1 r * cos(theta) 2 yrv SbqR
% 1 -1 r * sin(theta) 2 E )Zd{9A5)
% 2 -2 r^2 * cos(2*theta) sqrt(6) e^l+ #^fR
% 2 0 (2*r^2 - 1) sqrt(3) O.40^u~
% 2 2 r^2 * sin(2*theta) sqrt(6) ]6c2[r?g{
% 3 -3 r^3 * cos(3*theta) sqrt(8) >=q!!'$:
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) dQ2i{A"BKz
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) x.4)p6
% 3 3 r^3 * sin(3*theta) sqrt(8) @0@'6J04
% 4 -4 r^4 * cos(4*theta) sqrt(10) GT`<jzAi Q
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) .qU%SmQ^
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) 6;=wuoJi
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) !E. l yz
% 4 4 r^4 * sin(4*theta) sqrt(10) `5 v51TpH
% -------------------------------------------------- ~>-;(YU"t
% PFn[[~5V
% Example 1: }Us$y0W\
% `LHfAXKN
% % Display the Zernike function Z(n=5,m=1) EpS8,[w
% x = -1:0.01:1; =rtA{g$)+
% [X,Y] = meshgrid(x,x); +H^V},dBp!
% [theta,r] = cart2pol(X,Y); $4~}_phi
% idx = r<=1; M&\ ?)yG
% z = nan(size(X)); j!8+|eA kk
% z(idx) = zernfun(5,1,r(idx),theta(idx)); s$y#Ufz
% figure Cot\i\]jv
% pcolor(x,x,z), shading interp P HL@1K{)
% axis square, colorbar J,M5<s[Xqt
% title('Zernike function Z_5^1(r,\theta)') IF?B`TmZ
% aiX; D/t?
% Example 2: O? J:+L(
% qA\kx#v]P
% % Display the first 10 Zernike functions -v+^x`HR
% x = -1:0.01:1; pxnUe1=
% [X,Y] = meshgrid(x,x); Y,Zv0-"
% [theta,r] = cart2pol(X,Y); )PA Tz
#
% idx = r<=1; CH+&
% z = nan(size(X)); 7wEG<,D
% n = [0 1 1 2 2 2 3 3 3 3]; J t,
4@
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; HaN _}UMP
% Nplot = [4 10 12 16 18 20 22 24 26 28]; DczF0Ow
% y = zernfun(n,m,r(idx),theta(idx)); M[N.H9
% figure('Units','normalized') eu|q
{p
% for k = 1:10 iBW6<2@oZF
% z(idx) = y(:,k); J0W).mD_H
% subplot(4,7,Nplot(k)) g~D6.OZU
% pcolor(x,x,z), shading interp w=>m G-
% set(gca,'XTick',[],'YTick',[]) s ^@Cq=
% axis square (eE}W~Z
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) /<pQ!'/G
% end zi[M{bm
% =v=! x
% See also ZERNPOL, ZERNFUN2. ]<z(Rmn`Q
+( (31l
\ OIN zfbr
% Paul Fricker 11/13/2006 (SVr>|Db
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=0yJ2[R7Do
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{Y TF]J $
% Check and prepare the inputs: nv
Gd :]Z
% ----------------------------- O +}EE^*a
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) Y rnqi-P
error('zernfun:NMvectors','N and M must be vectors.') &V{,D))6[
end 9 yAu<a
(,y/nc=GN
7f,W zv V
if length(n)~=length(m) qyfxT Q5
error('zernfun:NMlength','N and M must be the same length.') * %BI*p
end R*C+Yk)Tkt
"
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z5'nS&x
n = n(:); P;/wb /
m = m(:); C>VZf,JE1
if any(mod(n-m,2)) 4x=Y9w0?8
error('zernfun:NMmultiplesof2', ... 6+#cyKj
'All N and M must differ by multiples of 2 (including 0).') *lO+^\HXD
end ?tQv|x
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if any(m>n) =6YO!B>7
error('zernfun:MlessthanN', ... n9-[z2n
'Each M must be less than or equal to its corresponding N.') <ft9B05*
end Y\\nJ uJo
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if any( r>1 | r<0 ) =(\!,S '
error('zernfun:Rlessthan1','All R must be between 0 and 1.') X#< Sv>c^
end 6LQ O> k
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bOKNWI
if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) _3[BS9
error('zernfun:RTHvector','R and THETA must be vectors.') FR"^?z?}p
end Q6> ( Z
s$6# 3%h
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r = r(:); tn1aH
+
theta = theta(:); RfRaWbn
length_r = length(r); {NDP}UATw
if length_r~=length(theta) _" V0vV
error('zernfun:RTHlength', ... rd{( E
'The number of R- and THETA-values must be equal.') wv -8\)oA
end !o!04_
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% Check normalization: N/VIP0Kb
% -------------------- 1[]cMyV
if nargin==5 && ischar(nflag) 4[q * 7m
isnorm = strcmpi(nflag,'norm'); =T]OYk
if ~isnorm &@ -glF5
error('zernfun:normalization','Unrecognized normalization flag.') 'h6RZK G T
end gId+hxFa:r
else V
""
isnorm = false; oMbCljUC
end =
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% Compute the Zernike Polynomials tS7u#YMh
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% rge/jE,^~Z
,}0pK\Y>$
M<Mr (z
% Determine the required powers of r: 9iE66N>z
% ----------------------------------- =Wa\yBj_;m
m_abs = abs(m); L?fv5 S3
rpowers = []; (1^(V)@
for j = 1:length(n) -tQ|&fl
rpowers = [rpowers m_abs(j):2:n(j)]; i}19$x.D`
end -E7\ .K3
rpowers = unique(rpowers); T_WQzEL^
}Urt DXhA
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% Pre-compute the values of r raised to the required powers, Qp ,l> k
% and compile them in a matrix: vkK+
C~"
% ----------------------------- 0bE_iu>f'
if rpowers(1)==0 x3 Uv&
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); ?x @khzk
rpowern = cat(2,rpowern{:}); )[1m$>
rpowern = [ones(length_r,1) rpowern]; OBZj-`fq J
else "^H+A-R[
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); &%lh ov
rpowern = cat(2,rpowern{:}); $H^6I8>
end QVpZA,
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% Compute the values of the polynomials: FyEl @ }W
% -------------------------------------- uOQ5.S+
y = zeros(length_r,length(n)); CS/-:>s%
for j = 1:length(n) TI332,eL
s = 0:(n(j)-m_abs(j))/2; N mQ]qv
pows = n(j):-2:m_abs(j); ($wYaw z
for k = length(s):-1:1 RC 48e._t
p = (1-2*mod(s(k),2))* ... 3jN cL{
prod(2:(n(j)-s(k)))/ ... m"*: XfOL
prod(2:s(k))/ ... ezn> 3?S
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... 7XNfH @
prod(2:((n(j)+m_abs(j))/2-s(k))); X'c5s~9
idx = (pows(k)==rpowers); [3.rG!Na
y(:,j) = y(:,j) + p*rpowern(:,idx); OjN]mp-q
end jnTl%aQ Yc
89 paR[
if isnorm {x8`gP\H
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); +K?h] v]%
end F, sT[C
end 4Qv|Z+$i
% END: Compute the Zernike Polynomials i"'k|TGW^
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% E Vf'1^f
T DoYp
R/#*~tPi8
% Compute the Zernike functions: 7,1idY%cy
% ------------------------------ `G'V9Xs(
idx_pos = m>0; Ur`v*LT}~
idx_neg = m<0; 3
*G=U
Lg^m?~{
mT.F$Y9
z = y; <n>< A+D
if any(idx_pos) ct ZW7
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); % ;< FfS
end 0^m02\Li
if any(idx_neg) /$n${M5!
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); E|>I/!{u7`
end |3i ~?]
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% EOF zernfun C>+UZ
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