| 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， fWc|gq
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， TL$E V>Nr
这个，跟我们用zygo干涉仪直接拟合出的36项zernike系数，有何关系呢？ 6VP`evan
那些系数是通过对29525*10阶的矩阵每列的值算出来的嘛？ :<Y}l-x
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function z = zernfun(n,m,r,theta,nflag) 4/ kv3rv
%ZERNFUN Zernike functions of order N and frequency M on the unit circle. ?bZovRx
% Z = ZERNFUN(N,M,R,THETA) returns the Zernike functions of order N s >k Uh
% and angular frequency M, evaluated at positions (R,THETA) on the &6 s) X
% unit circle. N is a vector of positive integers (including 0), and 3f " %G\
% M is a vector with the same number of elements as N. Each element n79QJl /
% k of M must be a positive integer, with possible values M(k) = -N(k) VErv;GyV
% to +N(k) in steps of 2. R is a vector of numbers between 0 and 1, n\Fp[9+Z\
% and THETA is a vector of angles. R and THETA must have the same Jj~EiA
% length. The output Z is a matrix with one column for every (N,M) tWTKgbj(
% pair, and one row for every (R,THETA) pair. EN{]Qb06A
% 8dD2
% Z = ZERNFUN(N,M,R,THETA,'norm') returns the normalized Zernike -K,-h[ o
% functions. The normalization factor sqrt((2-delta(m,0))*(n+1)/pi), ^, l_{
% with delta(m,0) the Kronecker delta, is chosen so that the integral |Fm6#1A@
% of (r * [Znm(r,theta)]^2) over the unit circle (from r=0 to r=1, L Mi:%i%\
% and theta=0 to theta=2*pi) is unity. For the non-normalized Uoya3#4 G
% polynomials, max(Znm(r=1,theta))=1 for all [n,m]. 5 uq3\a
% 6u`F
d#
% The Zernike functions are an orthogonal basis on the unit circle. F' U 50usV
% They are used in disciplines such as astronomy, optics, and y@2epY?{
% optometry to describe functions on a circular domain. UYk>'\%H0
% p4IZ
% The following table lists the first 15 Zernike functions. 7 n]65].t
% C%*k.$# r!
% n m Zernike function Normalization O# wpbrJ
% -------------------------------------------------- O }9KJU
% 0 0 1 1 -jgysBw+Xb
% 1 1 r * cos(theta) 2 L
%ip>
% 1 -1 r * sin(theta) 2 8+]hpa,q
% 2 -2 r^2 * cos(2*theta) sqrt(6) 3lV^B[$
% 2 0 (2*r^2 - 1) sqrt(3) f\'{3I29
% 2 2 r^2 * sin(2*theta) sqrt(6) EbeI{ -'aF
% 3 -3 r^3 * cos(3*theta) sqrt(8) K{n{KB&_&
% 3 -1 (3*r^3 - 2*r) * cos(theta) sqrt(8) q-nSLE+ _;
% 3 1 (3*r^3 - 2*r) * sin(theta) sqrt(8) @
'@:sM_
% 3 3 r^3 * sin(3*theta) sqrt(8) ~~/xR s
% 4 -4 r^4 * cos(4*theta) sqrt(10) KH\b_>wU2
% 4 -2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) 1@u2im-O
% 4 0 6*r^4 - 6*r^2 + 1 sqrt(5) ~GE$myUT\p
% 4 2 (4*r^4 - 3*r^2) * cos(2*theta) sqrt(10) qE'9QQ>:b
% 4 4 r^4 * sin(4*theta) sqrt(10) V8eB$in
% -------------------------------------------------- ]9}HEu;1M
% =rdY
@
% Example 1: tXJU vish
% QwhRNnE=
% % Display the Zernike function Z(n=5,m=1) l5l>d62
% x = -1:0.01:1; ikE<=:pe
% [X,Y] = meshgrid(x,x); Fnk_\d6Ma
% [theta,r] = cart2pol(X,Y); n| GaV
% idx = r<=1; hOhS)
% z = nan(size(X)); .0R v(Y
% z(idx) = zernfun(5,1,r(idx),theta(idx)); _ gKe%J&
% figure 2pd vWWh3l
% pcolor(x,x,z), shading interp u?s VcD[
% axis square, colorbar r`c_e)STO
% title('Zernike function Z_5^1(r,\theta)') R/" x}B1d
% -
0?^#G}3}
% Example 2: 5*[2yKsTi
% (98Nzgxgx}
% % Display the first 10 Zernike functions eY{+~|KZ
% x = -1:0.01:1; qj cp65^
% [X,Y] = meshgrid(x,x); '!f5?O+E
% [theta,r] = cart2pol(X,Y); bc
, p }
% idx = r<=1; 2lL,zFAq
% z = nan(size(X)); ? FfC
% n = [0 1 1 2 2 2 3 3 3 3]; EGl^ ! .'
% m = [0 -1 1 -2 0 2 -3 -1 1 3]; VLBE'3Qg 1
% Nplot = [4 10 12 16 18 20 22 24 26 28]; r>GZ58i
% y = zernfun(n,m,r(idx),theta(idx)); sB69R:U;
% figure('Units','normalized') OFje +S
% for k = 1:10 }w4QP+ x
% z(idx) = y(:,k); V. wqZ {G
% subplot(4,7,Nplot(k)) dMR3)CO
% pcolor(x,x,z), shading interp W2uOR{
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% set(gca,'XTick',[],'YTick',[]) U-n;xX0=
% axis square I(=V}s2
% title(['Z_{' num2str(n(k)) '}^{' num2str(m(k)) '}']) Ts~L:3oaQ
% end l }XU 59
% |lv|!]qAma
% See also ZERNPOL, ZERNFUN2. Zw
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!%dN<%Ah
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% Paul Fricker 11/13/2006
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% Check and prepare the inputs: eW| ^tH
% ----------------------------- %kgkXc~6|x
if ( ~any(size(n)==1) ) || ( ~any(size(m)==1) ) 3]?#he
error('zernfun:NMvectors','N and M must be vectors.') 1
hg}(Hix
end aZbw]0q@o
_#vrb;.+
7t.!lh5G%
if length(n)~=length(m) /PsnD_s]5
error('zernfun:NMlength','N and M must be the same length.') epgPT'^
end 3j3N!T9
|F#1C9]P
~)( \6^&=|
n = n(:); ]vuwkn+)
m = m(:); GKcv<G208
if any(mod(n-m,2)) h,"4SSL
error('zernfun:NMmultiplesof2', ... A/`%/0e
'All N and M must differ by multiples of 2 (including 0).') q{+_
<2U|
end &IlU|4`R%
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if any(m>n) 2h%z ("3/
error('zernfun:MlessthanN', ... CW<N: F.9
'Each M must be less than or equal to its corresponding N.') 2U-3Q]/I}
end ]5%/ 3P,/
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if any( r>1 | r<0 ) jbR0%X2
error('zernfun:Rlessthan1','All R must be between 0 and 1.') yV^s,P1
end n9s iX
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if ( ~any(size(r)==1) ) || ( ~any(size(theta)==1) ) ^%` wJ.c
error('zernfun:RTHvector','R and THETA must be vectors.') hdVdcn M
end Q?X>E3=U
MM j9{ou
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r = r(:); }K3x
theta = theta(:); D1 &A,2wO
length_r = length(r); Bm]8m=p
if length_r~=length(theta) 'R_g">B.
error('zernfun:RTHlength', ... XQS9,Hl
'The number of R- and THETA-values must be equal.') p ]d] QMu
end *E{2J:`
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+
% Check normalization: x.q+uU$^
% -------------------- :^(>YAyHj^
if nargin==5 && ischar(nflag) -{ZWo:,r~q
isnorm = strcmpi(nflag,'norm'); w}=5ElB
if ~isnorm \<g*8?yFs
error('zernfun:normalization','Unrecognized normalization flag.') ~s5SZK*
end 2p " WTd
else :>=\. \
isnorm = false; YY!Rz[/
end ,TFIG^Dvq
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Z4\=*ic@
% Compute the Zernike Polynomials QqU!Najf
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% !50[z:
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Y0X-Zqk'
% Determine the required powers of r: r9dyA5o D
% ----------------------------------- bcYF\@};
m_abs = abs(m); hvaSH69*m
rpowers = []; ukUGvK
for j = 1:length(n) q|),`.eh\
rpowers = [rpowers m_abs(j):2:n(j)]; )+6MK(<"
end F|!){=
rpowers = unique(rpowers); LEtG|3Dx
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?F3h)(}
% Pre-compute the values of r raised to the required powers, r
>nG@ A
% and compile them in a matrix: m |G'K[8
% ----------------------------- $b mLu=9
if rpowers(1)==0 yYfs y?3
rpowern = arrayfun(@(p)r.^p,rpowers(2:end),'UniformOutput',false); g
pO C`=
rpowern = cat(2,rpowern{:}); 1aTB %F
rpowern = [ones(length_r,1) rpowern]; OtNd,U.dE
else _D+J!f^
rpowern = arrayfun(@(p)r.^p,rpowers,'UniformOutput',false); ;&)-;l7M
rpowern = cat(2,rpowern{:}); FIsyiSY<j
end \7'+h5a
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?pd8w#O
% Compute the values of the polynomials: :n-]>Q>5=k
% -------------------------------------- Uw7h=UQh
y = zeros(length_r,length(n)); mV pMh#zw
for j = 1:length(n) y9Us n8
s = 0:(n(j)-m_abs(j))/2; b"{'T]"*j
pows = n(j):-2:m_abs(j); 2_Z ? #Y
for k = length(s):-1:1 <Pi|J-Y
p = (1-2*mod(s(k),2))* ... :w^Ed%>y7
prod(2:(n(j)-s(k)))/ ... K>@+m
prod(2:s(k))/ ... Bn &Ws
prod(2:((n(j)-m_abs(j))/2-s(k)))/ ... &e X!#nQ_.
prod(2:((n(j)+m_abs(j))/2-s(k))); W*I(f]8:y`
idx = (pows(k)==rpowers); Iepsz
y(:,j) = y(:,j) + p*rpowern(:,idx); ZNvEW
end ys kO
OD!& .%
if isnorm |3KLk ?2
y(:,j) = y(:,j)*sqrt((1+(m(j)~=0))*(n(j)+1)/pi); TtTj28 k7
end "[ZB+- |[0
end '?p<lu^^B
% END: Compute the Zernike Polynomials ~XmLX)vO/
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% dx@-/^.
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t, +nQ9
% Compute the Zernike functions: |$
lM#Ua
% ------------------------------ F}/S:(6LF2
idx_pos = m>0; Su/6Q$0 t
idx_neg = m<0; \6 Zr
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A"dR{8&0
z = y; |#cm`v
if any(idx_pos) D rY:9[LP
z(:,idx_pos) = y(:,idx_pos).*sin(theta*m(idx_pos)'); eEv@}1~
end GQUe!G9
if any(idx_neg) U"^kH|
z(:,idx_neg) = y(:,idx_neg).*cos(theta*m(idx_neg)'); 4i(JZN?
end SPY|K
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% EOF zernfun ) 9"_J9G
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