Multiresolution analysis (MRA). Be able to define
Mallat's multiresolution analysis.
- Be able to derive the scaling relation
$\phi(x)=\sum_{k=-1}^\infty p_k \phi(2^jx-k)$.
- Given the scaling relation, be able to show that the wavelet
$\psi(x) = \sum_{k=-\infty}^\infty (-1)^k p_{1-k} \phi(2^jx-k)$
- Be able to discuss the details of
the Haar MRA case the Shannon MRA (exercise 8 in §
5.4; be sure to know the Whittaker-Shannon Sampling Theorem,
§2.4.)
- Be able to derive the decomposition formulas. Be able to state
the reconstruction formulas. Know the high-pass and low-pass decomposition and
reconstruction filters, down sampling and up sampling. Know how to
implement both decomposition and reconstruction algorithms in terms of
filters.