Page 124 - Data Science Algorithms in a Week

108 Loris Nanni, Sheryl Brahnam and Alessandra Lumini

machine learning (Loris Nanni et al., 2012). In (Z. Wang & Chen, 2008; Z. Wang et al.,
2008) classifiers were developed for handling two-dimensional patterns, and in (Loris
Nanni et al., 2012) it was shown that a continuous wavelet can be used to transform a
vector into a matrix; once in matrix form, it can then be described using standard texture
descriptors (the best performance obtained in (Loris Nanni et al., 2012) used a variant of
the local phase quantization based on a ternary coding).
The advantage of extracting features from a vector that has been reshaped into a
matrix is the ability to investigate the correlation among sets of features in a given
neighborhood; this is different from coupling feature selection and classification. To
maximize performance, it was important that we test several different texture descriptors
and different neighborhood sizes. The resulting feature vectors were then fed into an
SVM.
The following five methods for reshaping a linear feature vector into a matrix were

tested in this paper. Letting ∈ be the input vector, ∈ ℜ 1 × 2 the output matrix

(where d1 and d2 depend on the method), and a ∈ ℜ a random permutation of the
indices [1..s], the five methods are:

1. Triplet (Tr): in this approach d1 =d2 =255. First, the original feature vector q is
normalized to [0,255] and stored in n. Second, the output matrix ∈ ℜ 255×255 is
initialized to 0. Third, a randomization procedure is performed to obtain a random
permutation aj for j=1..100000 that updates M according to the following formula:
M(n(aj(1)), n(aj (2))) = M(n(aj(1)), n(aj (2))) + q(aj (3));
2. Continuous wavelet (CW) (Loris Nanni et al., 2012): in this approach d1 =100 d2
=s. This method applies the Meyer continuous wavelet to the s dimensional feature vector
q and builds M by extracting the wavelet power spectrum, considering the 100 different
decomposition scales;
3. Random reshaping (RS): in this approach d1=d2=s0.5 and M is a random
rearrangement of the original vector into a square matrix. Each entry of matrix M is an
element of q(a);
4. DCT: in this approach the resulting matrix M has dimensions d1 = d2 = s and
each entry M(i, j) = dct(q(aij(2..6)), where dct() is the discrete cosine transform, aij is a
random permutation (different for each entry of the matrix), and the indices 2..6 are used
to indicate that the number of considered features varies between two and six. We use
DCT in this method because it is considered the de-facto image transformation in most
visual systems. Like other transforms, the DCT attempts to decorrelate the input data.
The 1-dimensional DCT is obtained by the product of the input vector and the orthogonal
matrix whose rows are the DCT basis vectors (the DCT basis vectors are orthogonal and
normalized). The first transform coefficient (referred to as the DC Coefficient) is the
average value of the input vector, while the others are called the AC Coefficients. After
several tests we obtained the best performance using the first DCT coefficient;