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September 28, 2015 16:10 Sparse Coding and Its Applications in Computer Vision – 9in x 6in b2310 page 5
Introduction 5
[van den Berg and Friedlander (2009); Cotter et al. (2005); Duarte et al.
(2005); Mishali and Eldar (2008, 2009)]. In many situations, the sparse
coefficients tend to cluster and a clustering prior exploiting correlations be-
tween neighboring coefficients is enforced in the optimization algorithms
in order to obtain a better representation. In group sparsity, the data is
inherently represented by a small number of pre-defined groups of data
samples, thus a sparsifying term over the groups is used to promote this
property. In the case of multiple measurements, a specific group sparsity
called joint structured sparsity is explored for joint sparse representation
and heterogeneous feature fusion [Zhang et al. (2012b,a)], where not only
the sparsity property for each measurement is utilized, but the structural
information across the multiple sparse representation vectors for the mul-
tiple measurements is exploited as well.
Discrimination Although sparse coding was originally proposed as a
generative model, it also performs surprisingly well in many classification
problems. The Sparse Representation-based Classification (SRC) proposed
in [Wright et al. (2009)] is a pioneering work in this direction. In SRC, a
signal x from class c is assumed to lie in or near a low-dimensional subspace
spanned by the atoms in the class-specific dictionary D
c
.Ifwetryto
represent x using the composite dictionary D=[D
1
, ..., D
C
] for all the C
classes, the resulting sparse code α=[α
1
; ...; α
C
] is supposed to have non-
zero coefficients concentrating in α
c
, which is associated with its class.
In the original work of [Wright et al. (2009)], D
c
consists of all the train-
ing samples from class c, which is not practical if the total class number
or the training set is large. There are quite a few recent papers trying to
learn dictionaries with a more compact form and a better discriminative
capability by augmenting the reconstruction objective in (1.4) with addi-
tional discrimination terms such as Fisher discriminant criterion [Yang et al.
(2011c)], structural incoherence [Ramirez et al. (2010)], class residual dif-
ference [Mairal et al. (2008b); Yang et al. (2011b)] and mutual information
[Qiu et al. (2011)]. Sparse codes generated by discriminative dictionaries
are also used as the input features of general classification models other
than SRC [Bradley and Bagnell (2008); Yang et al. (2010b); Jiang et al.
(2011); Mairal et al. (2012)]. In addition to natural images, discrimina-
tive sparse representation learning has also been actively applied in other
imageries such as hyperspectral classification [Wang et al. (2014a, 2015a)].