An Analysis of the Road Signs Classification Based on the Higher-Order Singular Value Decomposition of the Deformable Pattern Tensors - Advanced Concepts for Intelligent Vision Systems - page 194

becomes two-dimensional, i.e. it is a matrix. Moreover, it is worth noting that due to
orthogonality of the core tensor Z in (7), T h are also orthogonal. Hence, T h in
decomposition (6) constitute a basis. This is a very important result which allows
construction of classifiers based on the HOSVD decomposition. Such a scheme is
used in the proposed system for RS classification, although other tensor constructions
with simultaneous data compression are also possible [19]. Nevertheless, in our case
each set of prototypes for a single sign (i.e. a single class) is independently encoded as
a separate tensor T s . This allows different numbers of prototypes in each of the
classes. As alluded to previously, in each case the series (6) is usually truncated to the
first N

N P most prominent components. In other words, a smaller but dominating N
dimensional subspace is used to approximate T .

≤

T T T

T

Fig. 2. Visualization of the tensor decomposition given by (6)

The series of k-mode products (7) can be equivalently represented in a matrix
notation after tensor flattening

T

⎡ ⎤

TSZSS SSS S = ⊗ ⊗⊗ ⊗⊗ ⊗⊗ , (8)

… …

() () ⎣ ⎦

k k + 1 k + 2 P 1 2 k − 1

k k

where denotes the Kronecker product. This provides us with a convenient link to

the matrix representation of tensor equations which is discussed in the next section of
this paper. By the same token, and taking into an account that S k are orthogonal,
computation of the core tensor Z can be expressed as

⊗

T
k

⎡ ⎤

ZSTSS SSS S = ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ .

… … (9)

⎣ ⎦

() ()

k + 1 k + 2 P 1 2 k − 1

k k

The HOSVD successively applies the matrix SVD decomposition to each of the
flattened T ( k ) versions of the input tensor T . In result the S k matrices are
computed [16]. In the 3D case and considering (9), the HOSVD can be written as

() (

)

T

ZSTSS . = ⊗

(10)

()

1 2 3

1 1

As mentioned, in our framework the original tensor T i of a class i is obtained from the
available exemplars of the prototype patterns for that class i . These, in turn, are
obtained from the patterns cropped from the real traffic images which are additionally
rotated in a given range (in our examples this was ±12° with a step of 2°) with