sparse image representation via combined transforms - Convex ...

12 CHAPTER 2. SPARSITY IN IMAGE CODING
closest to x. This is written as
q(x) = argmin |x − y i |.
y i ∈ C
Each cell c i (c i = {x : q(x) =y i }, where the quantization output is y i ) is called a
Voronoi cell.
• Centroid condition: When the partitioning R = ∪ i c i is fixed, the MSE distortion is
minimized by choosing a representer of a cell as the statistical mean of a variable
restricted within the cell:
y i = E[x|x ∈ c i ],
i =1, 2,... ,K.
Given a source with a known statistical probability density function (p.d.f.), designing a
quantizer that satisfies both of the optimality conditions is not a trivial task. In general, it
is done via an iterative scheme. A quantizer designed in this manner is called a Lloyd-Max
quantizer [94, 104].
A uniform quantizer simply takes each cell c i as an interval of equal length; for example,
for fixed ∆, c i is (i∆ − ∆ 2 ,i∆+ ∆ 2
]. (Note that the number of cells here is not finite.) The
uniform quantizer is easy to implement and design, but in general not optimal.
A detailed discussion about quantization theory is easy to find and beyond the scope of
this thesis. We skip it.
Entropy Coding
Entropy coding means compressing a sequence of discrete values from a finite set into a
shorter sequence combined from elements of the same set. Entropy coding is a lossless
coding, which means that from the compressed sequence one can perfectly reconstruct the
original sequence.
There are two important approaches in entropy coding. One is a statistical approach.
This category includes Huffman coding and arithmetic coding. The other is a dictionary
based approach. The Lempel-Ziv coding belongs to this category. For detailed descriptions
of these coding schemes we refer to some textbooks, notably Cover and Thomas 1991 [33],
Gersho and Gray 1992 [70] and Sayood 1996 [126].