Compressive Sensing for Vision - Sparse Representations and Compressive Sensing for Imaging and Vision - page 53

4.2.3 Compressive Acquisition of Dynamic Textures

One can explore the use of predictive/generative signal models for video CS that
are characterized by static parameters. Predictive modeling provides a prior for
the evolution of the video in both forward and reverse time. By relating video
frames over small durations, predictive modeling helps to reduce the number of
measurements required at a given time instant. Models that are largely characterized
by static parameters help in eliminating problems arising from the ephemeral nature
of dynamic events. Under such a model, measurements taken at all time instants
contribute towards estimation of the static parameters. At each time instant, it is
only required to sense at the rate sufficient to acquire the dynamic component of the
scene, which could be significantly lower than the sparsity of an individual frame
of the video. One dynamic scene model that exhibits predictive modeling as well
as high-dimensional static parameters is the linear dynamical system (LDS). In this
section, we highlight the methods for the CS of dynamic scenes modeled as the
linear dynamical system. We first give a background on dynamic textures and linear
dynamical systems [127].

4.2.3.1 Dynamic Textures and Linear Dynamical Systems

Linear dynamical systems represent a class of parametric models for time-series
data including dynamic textures [53], traffic scenes [34], and human activities [148],
[144]. Let be a sequence of frames indexed by time t .TheLDS

model parameterizes the evolution of y t as follows:

{ y t , t = 0 ,..., T }

N × N

y t = C x t + w t w t ∼ N ( 0 , R ) , R ∈ R (4.23)

d × d

x t + 1 = A x t + v t v t ∼ N ( 0 , Q ) , Q ∈ R (4.24)

d d × d

where x t ∈ R is the hidden state vector, A ∈ R the transition matrix, and C ∈

N × d is the observation matrix.
Given

R

the observations { y t } , the truncated SVD of the matrix [ y ] 1: T =

[ can be used to estimate both C and A . In particular, an estimate

of the observation matrix C is obtained using the truncated SVD of

y 1 , y 2 ,..., y T ]

[ y ] 1: T . Note that

the choice of C is unique only up to a d × d linear transformation. That is, given

] 1: T , we can define C

[ d matrix. This represents

our choice of coordinates in the subspace defined by the columns of C . This lack
of uniqueness leads to structured sparsity patterns which can be exploited in the
inference algorithms.

y = UL ,where L is an invertible d ×