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   -> Volume 6, Issue 4

Answer: Smoothest Scaling Function (WD 6.3 #21)
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Author Message (Michael Unser)

PostPosted: Thu Mar 06, 1997 4:02 pm    
Subject: Answer: Smoothest Scaling Function (WD 6.3 #21)
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#18 Answer: Smoothest Scaling Function (WD 6.3 #21)

>What is the smoothest scaling function, w(x), solving a
>'wavelet equation' of type:
>(1) w(x) = c(0)w(2x) + c(1)w(2x-1) + c(2)w(2x-2) +...+ c(m)w(2x-m),
>where the coefficients c(i) satisfy the 'usual conditions':
>(2) sum c(i) over odd i = 1,
>(3) sum c(i) over even i = 1,
>and where w is normalized to have, say, integral = 1.

First, the smoothness of a scaling function cannot be greater than p
(the order of the function = its ability to reproduce polynomials of
degree p-1 = corresponds to the factor (1+z)^p in the refinement
filter C(z)=(1+z)^p*Q(z)).

Second, it is easy to see that the B-spline of order p (or degree p-1)
is the shortest scaling function of order p, because Q(z)=1.

Third, B-splines of order p have a Sobolev smoothness index of p-1/2
(cf. Strang's book). However, B-splines are not the smoothest scaling
functions of order p! For instance, one can take Q(z)=(1+EPSILON+z),
which for EPSILON>0 sufficiently small (but non-zero !) can achieve
the maximum smoothness s_max=p. But this also shows that the B-splines
are the smoothest scaling for a refinement filter of a given length:
the example above with EPSILON=0 is a B-spline of order p+1 which has
a smoothness s_max=p+1/2 --- this is 1/2 better than anything else of
the same length !

Michael Unser

BEIP, Building 13/3N17
National Institutes of Health
Bethesda, MD 20892-5766, USA

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