sparse image representation via combined transforms - Convex ...

2.3. DISCUSSION 21
tail compact. Then
α ∗ =1/p ∗ − 1/2. (2.4)
Moreover, coder-decoder pairs achieving the optimal exponent of code length can be derived
from simple uniform quantization of the coefficients (θ i ), followed by simple run-length
coding.
A proof is given in [47].
2.2.3 Summary
From above analysis, we can draw the following conclusions:
1. Critical index measures the sparsity of a sequence space, which usually is a subspace
of l 2 . In an asymptotic sense, the smaller the critical index is, the faster the sequence
decays; and, hence, the sparser the sequence is.
2. Optimal exponent measures the efficiency of the best possible coder in a sequence
space. The larger the optimal exponent is, the fewer the bits required for coding in
this sequence space.
3. The optimal exponent and the critical index have an equality relationship that is
described in (2.4). When the critical index is small, the optimal exponent is large.
Together with the previous two results, we can draw the main conclusion: the sparser
the sequences are in the sequence space, the fewer the bits required to code in the
same space.
2.3 Discussion
Based on the previous discussion, in an asymptotic sense, instead of considering how many
bits are required in coding, it is equivalent to study the sparsity of the coefficient sequences.
From now on, for simplicity, we only consider the sparsity of coefficients. Note that the
sparsity of coefficients can be empirically measured by the decay of sorted amplitudes. For
infinite sequences, the sparsity can be measured by the weak l p norm.
We required the decoding scheme to be computationally cheap, while the encoding
scheme can be computationally expensive. The reason is that in practice (for example, in