Sparse Representation-based Object Recognition - Sparse Representations and Compressive Sensing for Imaging and Vision - page 63

Define the mutual coherence of the matrix B as follows

Definition 5.1. [17] The mutual coherence of a given matrix B is the largest
absolute normalized inner product between different columns from B . Denoting the
k th column in B by b k , the mutual coherence is given by

b k b j

| |

μ (

˜ )=

b j 2 .

B max

b k 2 . 1 ≤ k , j ≤ L , k = j

With this definition, one can prove the following theorem

N × L full

Theorem 5.1. [50], [69] For the system of linear equations x = B α ( B ∈ R

rank with L ≥ N), if a solution α exists obeying

1

1
2

1

α 0 < + ,

μ ( ˜ B )

( ) ( )

that solution is both unique solution of .

In the rest of the chapter we show how the variants of ( 5.1 ) can be used to develop
robust algorithms for object classification.

P 1 and the unique solution of P 0

5.2 Sparse Representation-based Classification

In object recognition, given a set of labeled training samples, the task is to
identify the class to which a test sample belongs to. Following [156] and [112],
in this section, we briefly describe the use of sparse representations for biometric
recognition, however, this framework can be applied to a general object recognition
problem.
Suppose that we are given L distinct classes and a set of n training images
per class. One can extract an N -dimensional vector of features from each of these
images. Let B k =[ n matrix of features from the k th

class, where x kj denote the feature from the j th training image of the k th class. Define
a new matrix or dictionary B , as the concatenation of training samples from all the
classes as

x k 1 ,..., x kj ,..., x kn ] be an N ×

N × ( n . L )

B =[ B 1 ,..., B L ] ∈ R

=[ x 11 ,..., x 1 n | x 21 ,..., x 2 n |......| x L 1 ,..., x Ln ] .

N

We consider an observation vector y ∈ R of unknown class as a linear combination

of the training vectors as

L
i = 1

n
j = 1 α ij x ij

y = (5.4)