Sparse Representation-based Object Recognition - Sparse Representations and Compressive Sensing for Imaging and Vision - page 61

Chapter 5
Sparse Representation-based Object
Recognition

In this chapter, we show how the sparse representation framework can be used to
develop robust algorithms for object classification [156], [112], [106]. In particular,
we will outline the Sparse Representation-based Classification (SRC) algorithm
[156] and present its applications in robust biometrics recognition [156], [112],
[111]. Through the use of Mercer kernels [128], we will show how SRC can be made
nonlinear [163]. We will then present the algorithms for solving nonlinear sparse
coding problems [41], [63], [161], [163], [130]. Finally, we will show how SRC
can be generalized for multimodal multivariate sparse representations and present
its application in multimodal biometrics recognition problems [130]. We first briefly
outline the idea behind sparse representation 1 [17].

5.1 Sparse Representation

As we saw in Chapter 2 representing a signal involves the choice of a basis, where
the signal is uniquely represented as the linear combination of the basis elements.
In the case when we have the orthogonal basis, the representation coefficients are
simply found by computing inner products of the signal with the basis elements.
In the non-orthogonal basis case, the coefficients are found by taking the inner
products of the signal with the bi-orthogonal basis. Due to the limitations of
orthogonal and bi-orthogonal basis in representing complex signals, overcomplete
dictionaries were developed. An overcomplete dictionary has more elements, also
known as atoms, than the dimension of the signal.
Consider the dictionary B

N × L ,where L

=[ b 1 ,···, b L ] ∈ R ≥ N and the columns

N

of B are the dictionary atoms. Representing x ∈ R using B entails solving the

following optimization problem

1 Also known as sparse coding.