XXXIV SIMP ´ OSIO BRASILEIRO DE TELECOMUNICAC ¸ ˜ OES - SBrT2016, AUGUST 30 TO SEPTEMBER 02, SANTAR ´ EM, PA Random Projections and Their Applications in Computer Vision Guilherme Schu, John Soldera, Rafael Medeiros and Jacob Scharcanski Abstract— Computer vision problems often require extracting and handling large volumes of high dimensional data. Commonly, a dimensionality reduction is initially applied to project the fea- tures representing the input data into lower dimensionality spaces before its analysis and/or classification. Techniques like Principal Component Analysis (PCA) have been used for dimensionality reduction, but data structure distortions may result from these projections, often leading to analysis and/or classification errors. On the other hand, random projections can provide dimen- sionality reduction while preserving the data local properties, and sample relative inter-distances in the lower dimensionality space. We review some of the theoretical foundations of random projection methods, and discuss their application in dimensional reduction and in computer vision problems. Also, some recent trends in the field of random projections are discussed. Keywords— random projections, computer vision, dimension- ality reduction. I. I NTRODUCTION Computer vision problems often involve handling large amounts of high dimensionality data obtained by sensors like cameras or other devices. However, handling high dimensional data is computationally costly and can downgrade the perfor- mance of computer vision algorithms. Specially because as the data dimensionality increases, the data samples become more sparsely distributed in the feature space since the number of samples available remains the same. Therefore, dimensionality reduction techniques play an important role, since they can project the input high dimensionality data features to a lower dimensional space, while preserving important data structural characteristics. Among several dimensionality reduction techniques avail- able, Principal Component Analysis (PCA) is a method that is popular within the computer vision community. It allows to project high dimensionality data features into lower dimen- sional spaces relying on the possibility that most variability of the input data is restricted to a few directions in the original feature space. Nevertheless, if the initial space dimensionality is high (e.g, on the order of c*1000, with c ≥ 1), and the final space dimensionality is much smaller (e.g. on the order of c ′ *1000, with c ′ ≪ 1), PCA can be computationally expensive and may change substantially the local data structure [1]. Guilherme Schu is with Graduate Programme on Electrical Engineer- ing, John Soldera formerly was with Graduate Programme on Computer Science and now is with Informatics Institute and Graduate Programme on Electrical Engineering, Rafael Sachett is with Graduate Programme on Computer Science, and Jacob Scharcanski is with Graduate Programme on Computer Science and Graduate Programme on Electrical Engineer- ing, Federal University of Rio Grande do Sul, Porto Alegre, Brazil, E- mail: ’jacobs@inf.ufrgs.br’. This work was partially supported by CAPES (Coordenac ¸˜ ao de Aperfeic ¸oamento de Pessoal de N´ ıvel Superior, Brazil). The random projection (RP) approach is a different dimen- sionality reduction technique, which is becoming increasingly popular within the computer vision and pattern recognition community. RPs can obtain compact representations for high dimensionality data, preserving well the data sample inter- distances in the lower dimensionality space (i.e. are locality preserving transforms) at an accessible computational cost. In this paper, we outline the theoretical foundations of the RP approach in Section II. Also, some relevant applications of RPs in areas such as texture representation, biometrics and video processing are discussed in Section III. Finally, Section IV discusses some future research directions and presents our conclusions. II. THEORECTICAL BASICS OF RANDOM PROJECTIONS Random projection (RP) is a projection technique where random projection matrices and linear matrix operations are used to project the data from a high dimensional space to a lower dimensional space [2], [3], [4]. Usually, RPs obtain compact representations for high dimensionality data, preserv- ing the data sample inter-distances in the lower dimensionality space. There are different ways to construct RP matrices. Different schemes could be used to construct a RP matrix R ∈ R d×p [4], and the values of its elements R(i, j )= r ij can be obtained as follows: r ij = { +1 with probability 1/2, −1 with probability 1/2. Recently, Medeiros et al. [5] proposed an alternative way to construct RP matrices, where each RP matrix element r ij is obtained randomly as follows: r ij = { +1 with probability 1/2, 0 with probability 1/2. Given the d × p RP matrix R, the original p × 1 data vector x ∈ R p is projected into the lower dimensionality space, obtaining y ∈ R d with d<p, by the following linear operation: y = Rx. (1) Therefore, RPs can be interpreted as a linear weighting of the original data vector components x, where the weights r ij are defined randomly [6]. More specifically, considering the original data vector x ∈ R p in the projection given in Equation 1, the random projection operation in Equation 1 can be expressed equivalently as : Published in XXXIV Simpósio Brasileiro de Telecomunicações e Processamento de Sinais - SBrT 2016 August 30 to September 02, Santarém, Brazil