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106 Loris Nanni, Sheryl Brahnam and Alessandra Lumini

tested for generalizability across several well-known benchmark datasets that reflect a
diversity of classification problems. Our experiments show that when different
approaches for transforming a vector into a matrix are combined with several texture
descriptors the resulting system works well on many different problems without requiring
any ad-hoc optimization. Moreover, because texture-based and standard vector-based
descriptors preserve different aspects of the information available in patterns, our
experiments demonstrate that the combination of the two improves overall classification
performance. The MATLAB code for our proposed system will be publicly available to
other researchers for future comparisons.

Keywords: two-dimensional representation

INTRODUCTION

Most machine pattern recognition problems require the transformation of raw sensor
data so that relevant features can be extracted for input into one or more classifiers. A
common first step in machine vision, for instance, is to reshape the sensor matrix by
concatenating its elements into a one dimensional vector so that various feature
transforms, such as principal component analysis (PCA) (Beymer & Poggio, 1996), can
be applied that side step the curse of dimensionality by reducing the number of features
without eliminating too much vital information. Reshaping the data matrix into a vector,
however, is not necessarily the only nor the best approach for representing raw input
values [16]. One problem with vectorizing a data matrix is that it destroys some of the
original structural knowledge (D. Li, Zhu, Wang, Chong, & Gao, 2016; H. Wang &
Ahuja, 2005).
In contrast to vectorization, direct manipulation of matrices offers a number of
advantages, including an improvement in the performance of canonical transforms when
applied to matrices, a significant reduction in computational complexity (Loris Nanni,
Brahnam, & Lumini, 2012; Z. Wang, Chen, Liu, & Zhang, 2008), and enhanced
discrimination using classifiers developed specifically to handle two-dimensional data
(see, for example, (Z. Wang & Chen, 2008) and (Z. Wang et al., 2008)). Moreover, some
of the most powerful state-of-the-art two-dimensional feature extraction methods, such as
Gabor filters (Eustice, Pizarro, Singh, & Howland, 2002) and Local binary patterns
(LBP) (L. Nanni & Lumini, 2008; Ojala, Pietikainen, & Maeenpaa, 2002), and their
variants, extract descriptors directly from matrices. Other methods, such as Two-
Dimensional Principal Component Analysis (2DPCA) (Yang, Zhang, Frangi, & Yang,
2004) and Two-Dimensional Linear Discriminant Analysis (2DLDA) (J. Li, Janardan, &
Li, 2002), allow classic transforms, such as PCA and Linear Discriminant Analysis
(LDA) (Zhang, Jing, & Yang, 2006), to work directly on matrix data. By projecting
matrix patterns via matrices, both 2DPCA and 2DLDA avoid the singular scatter matrix
problem. Classifier systems that are designed to handle two-dimensional data include