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Slides14 - Smoothness of wavelet bases: Convergence of the...

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Unformatted text preview: Smoothness of wavelet bases: Convergence of the cascade algorithm (Condition E); Splines. Bases vs. frames. Use eigenvalue analysis to study convergence of the cascade algorithm and smoothness of resulting scaling function. The cascade algorithm revisited: �(i + 1) (t) = 2� h0[k] �(i)(2t - k) k Consider the behavior of the inner products � a(i)[n] = � �(i)(t) �(i)(t + n) dt -� as i � � to understand convergence. 2 � a(i + 1)[n] = �{2�h0[k]�(i)(2t - k)}{2�h0[�]�(i)(2t + 2n - �)}dt -� k = 2� h0[�] � h0[k] a(i)[k + 2n - �] � � k ��� ��� = 2� h0[�] � h0[m œ 2n + �] a(i)[m] ��� m � m = 2� h0[2n œ r] � h0[- (r œ m)]a(i) [m] r -r m Filter with h0[n] Filter with h0[-n] and then downsample 3 In matrix form: a(i+1) = (�2) 2 H0H0T a(i) ; ��� H0 � Toeplitz matrix T Iteration converges if the eigenvalues of the transition matrix T satisfy ��� � 1 ��� with only a simple eigenvalue at � = 1. 4 Splines Splines are scaling functions whose filters only have zeros at � i.e. H0(�) = ( h0[n] = 1 + e-i� p 2) 1p () 2p n Consider p = 1 �(t) = ; n = 0, 1, …, p binomial coefficients 1 0 ^ �(�) = e-i�/2 1 sin �/2 �/2 5 What happens when p = 2 ? H0(�) = ( 1 + e-i� 1 + e-i� 2) 2 )( H1 (�) H2(�) 0 0 ��� ��� ��� ��� � ^(�) = � H (�/2j) � 0 j=1 � =� j=1 H1(�/2j) 0 � 2 . � H0(�/2j) j=1 = ^1(�) . ^2(�) � � = �(t) = -i�/2 sin �/2 )2 (e � 0 � 0 1 1 = 0 1 2 6 More generally �(�) = (e-i�/2 sin �/2 p ) �/2 �(t) = �box(t) � �box(t) � … � �box(t) 0 1 p=1 0 1 p=2 2 01 23 p=3 (p terms) 0 123 4 p=4 �(t) is piecewise polynomial of degree p œ 1. The derivatives, �(s)(t), exist for s � p œ 1 and they are continuous for s � p œ 2. e.g. Cubic spline (p = 4) is C2 continuous. 7 Alternatively, measure smoothness in L2 sense: ^ ||�(s)(t)||2 = = � ^ ��(i�)s �(�)�2 d� 2� -� 1 1� p 2p 2s 4 �sin �/2� � �2p 2� -� � �� (by Plancherel) d� when 2s œ 2p � -1 � Note: � 1 -� � d� is limiting case So, �(t) has s derivatives in the L2 sense for all s � smax, where smax = p - ² Valid for splines 8 Non-spline Scaling Functions In general, we have H0(�) = ( so that 1 + e-i� p ) 2 Q(�) �(t) = �p(t) � �q(t) ��� ��� ��� ��� pth order ugly spline Notice that the approximation power of �(t) comes entirely from �p(t): Suppose that we write � ck�p(t œ k) = t� for some � (0 � � � p). k 9 Then we have � ck �(t œ k) = �q(t) � t� k � = � �q(�)(t - �)� d� -� � = �( i=0 � � i )- � �q(�)(� �i �)�-i d� . ti ����� ����� = polynomial of degree �. 10 What about smoothness (in L2 sense)? Smoothness is given by smax = p - ² log2��max(TQ)� where TQ = (�2)2QQT Transition matrix for Q(�) Alternatively, look at the transition matrix for H0(�), T = (�2)2H0H0T T has 2p special eigenvalues due to the zeros at �: � = 1, ², ³, …, ( 1 )2p-1 2 Disregard these eigenvalues and look at the largest non-special eigenvalue, �max. Then the smoothness is given by 1 smax = - 2 log2��max(T)� �max(T) = 4-smax 11 ...
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