sparse image representation via combined transforms - Convex ...

3.4. OTHER TRANSFORMS 71
Folding. Another way to obtain localized time-frequency basis, instead of using the
constructive method in Daubechies [34], is to apply folding to an existing orthonormal
basis for periodic functions on a fixed interval. The idea of folding is described in [141]
and [142]. The folding operation is particularly important in discrete algorithms because if
the basis function is from the folding of a known orthonormal basis function (e.g., Fourier
basis function), since the coefficient of the transform associated with the basis is simply the
inner product of the signal with the basis function, we can apply the adjoint of the folding
operator to the signal, then calculate its inner product with the original basis function. If
there is a fast algorithm to do the original transform (for example, for DFT, there is a FFT),
then there is a fast algorithm to do the transform associated with the new time-frequency
atoms.
Overcomplete dictionary. For a fixed real constant, we can construct an orthonormal
basis whose support of basis functions has length exactly equal to this constant. But we
can choose different lengths for support. In the discrete case, for the simplicity of implementation,
we usually choose the length of support equal to a power of 2. The collection of
all the basis functions make an overcomplete dictionary. A consequential question is how
to find a subset of this dictionary. The subset should construct an orthonormal basis, and,
at the same time, be the best representation for a particular class of signal.
Best orthogonal basis. The best orthogonal basis (BOB) algorithm is proposed to achieve
the objective just mentioned. The key idea is to assign a probability distribution on a
class of signals. Usually, the class of signals is a subspace of L 2 (R). Without loss of
generality, we consider signals that are restricted on the interval [0, 1). Suppose we consider
only functions whose support are dyadic intervals that have forms [2 −l (k − 1), 2 −l k), for
l ∈ N,k =1, 2,... ,2 l . At each level l, for the basis functions whose support are intervals
like [2 −l (k − 1), 2 −l k), the associated coefficients can be calculated. We can compute the
entropy of these coefficients. For an interval [2 −l (k−1), 2 −l k), we can consider its immediate
two subintervals: [2 −l−1 (2k −2), 2 −l−1 (2k −1)) and [2 −l−1 (2k −1), 2 −l−1 2k). (Note [2 −l (k −
1), 2 −l k)=[2 −l−1 (2k−2), 2 −l−1 (2k−1))∪[2 −l−1 (2k−1), 2 −l−1 2k).) Note in the discrete case,
a set of the coefficients associated with an interval [2 −l (k − 1), 2 −l k) and a set of coefficients
associated with intervals [2 −l−1 (2k −2), 2 −l−1 (2k −1)) and [2 −l−1 (2k −1), 2 −l−1 2k) have the
same cardinality. In fact, there is a binary tree structure. Each dyadic interval corresponds
to a node in the tree. Suppose a node in the tree corresponds to an interval I. Two
subsequent nodes in the tree should correspond to the two subintervals of the interval I.