1 Introduction
Temporal or dynamic textures (DT) are image sequences that exhibit spatially repetitive and certain stationarity properties in time. This kind of sequences are typically videos of processes, such as moving water, smoke, swaying trees, moving clouds, or a flag blowing in the wind. Study and analysis of DT is important in several applications such as video segmentation [1], video recognition [2], and DT synthesizing [3].
One classical approach is to model dynamic scenes via the optical flow [4]. However, such methods require a certain degree of motion smoothness and parametric motion models [1]. Nonsmoothness, discontinuities, and noise inherence to rapidly varying, nonstationary DTs (e.g. fire) pose a challenge to develop optical flow based algorithms. Another technique, called particle filter [5], models the dynamical course of DTs as a Markov process. A reasonable assumption in DT modeling is that each observation is correlated to an underlying latent variable, or “state”, and then derive the parameter transition operator between these states.
Some approaches directly view each observation as a state, and then focus on transitions between the observations in the time domain. For instance, the work in [6]
treats this transition as an associated probability problem, and other methods construct a spatiotemporal autoregressive model (STAR) or position affine operator for this transition
[7, 8].Differently, featurebased models capture the intrinsic law and underlying structures of the data by projecting the original data onto a lowdimensional feature space via feature extracted techniques, such as principle component analysis (PCA). G. Doretto et al.
[2, 3] model the evolution of the dynamic textured scenes as a linear dynamical system (LDS) under a Gaussian noise assumption. As a popular method in dynamic textures, LDS and its derivative algorithms have been successfully used for various dynamic texture applications [3, 2]. However, constraints are imposed on the types of motion and noise that can be modeled in LDS. For instance, it is sensitive to input variations due to various noise. Especially, it is vulnerable to nonGaussian noise, such as missing data or occlusion of the dynamic scenes. Moreover, stability is also a challenging problem for LDS [9].To tackle these challenges, the approach taken here is to explore an alternative method to model the DTs by appealing to the principle of sparsity. Instead of using the Principle Components (PCs) as the transition “states” in LDS, sparse coefficients over a learned dictionary are imposed as the underlying “states”. In this way, the dynamical process of DTs exhibits a transition course of corresponding sparse events. These sparse events can be obtained via a recent technique on linear decomposition of data, called dictionary learning [10, 11]. Formally, these sparse representations to a signal , can be written as
where is a dictionary, and is sparse, i.e. most of its entries are zero or small in magnitude. That is, the signal can be sparsely represented only using a few elements from some dictionary .
In this work, we start with a brief review of the dynamic texture model from the viewpoint of convex optimization, and then deduce a combined regression associated with several regularizations for a joint process—“states extraction” and “states transition”. Then we treat the solution of the above combined regression as an adaptive dictionary learning problem, which can achieve two distinct yet tightly coupled tasks— efficiently reducing the dimensionality via sparse representation and robustly modeling the dynamical process. Finally, we cast this dictionary learning problem as the optimization of a smooth nonconvex objective function, which is efficiently resolved via a gradient descent method.
2 Adaptive Video Dictionary Learning
In this section, we start with a brief introduction to the linear dynamical systems (LDS) model and develop an adaptive dictionary learning framework for sparse coding.
2.1 Linear Dynamical Systems
Let us denote a given sequence of frames by , where the time is indexed by . The evolution of a LDS is often described by the following two equations
(1) 
where , , and denote the observation, its hidden state or feature, state noise, and observation noise, respectively. The system is described by the dynamics matrix , and the modeling matrix
. Here we are interested in estimating the system parameters
and , together with the hidden states, given the sequence of observations .The problem of learning the LDS (1
) can be considered as a coupled linear regression problem
[9]. Let us denote , , and . The system dynamics and modeling matrix are expected to be caught by solving the following minimization problem,(2) 
where is a small positive constant. In our approach, we assume that all observations admit a sparse representation with respect to an unknown dictionary , i.e.
(3) 
where is sparse. Without loss of generality, we further assume that all columns of the dictionary have unit norm. We then define the set
(4) 
where is the diagonal matrix whose entries on the diagonal are those of ,
denotes the identity matrix. The set
is the product of unit spheres, and is hence a dimensional smooth manifold. Finally, by adopting the common sparse coding framework to problem (2), we have the following minimization problem(5) 
where , denotes the Frobenius norm of matrices, and is the norm, which measures the overall sparsity of a matrix. The parameter weighs the sparsity measurement against the residual errors.
2.2 A Dictionary Learning Model for Dynamical Scene
Solving the minimization problem as stated in Eq. (5) is a very challenging task. In this work, we employ an idea similar to subspace identification methods [9], which treat the state as a function of . Here, we confine ourselves to the sparse solution of an elasticnet problem, which is proposed in [12], as
(6) 
where and are regularization parameters, which play an important role in ensuring stability and uniqueness of the solutions. Let us define the set of indices of the nonzero entries of the solution as
(7) 
Then the solution has a closedform expression as
(8) 
where carries the signs of , is the subset of in which the index of atoms (rows) fall into support . Furthermore, it is known that the solution as given in (8) is a locally twice differentiable function at . By an abuse of notation, we define
(9) 
In a similar way, is defined. Thus, the cost function reads as
(10) 
It is known that an LDS with the dynamic matrix
is said to be stable, if the largest eigenvalue of
is bounded by [9]. Let be the largest eigenvalue of , then Thus, we enforce the small via imposing a penalty on (10), and then end up with the cost function as(11) 
2.3 Development of the Algorithm
In this section, we firstly derive a gradient descent algorithm to minimize (11) and then discuss some details of the choice of the parameters in the final implementation.
We start with the computation of the first derivative of the sparse solution of the elasticnet problem as given in (8). Given the tangent space of at as
(12) 
the orthogonal projection of a matrix onto the tangent space with respect to the inner product is given by
(13) 
Let us denote . The first derivative of in the direction is
(14) 
By the product structure of , the Riemannian gradient of the function is
(15) 
Here, the Euclidean gradient of with respect to is computed as
(16) 
with being the
th standard basis vector of
. Using the shorthand notation, , , and , the Euclidean gradient of with respect to is(17) 
For a gradient search iteration on manifolds, we employ the following smooth curve on through in direction
(18) 
with . It essentially normalizes all columns of . For a detailed overview on optimization on matrix manifold, refer to [13].
Until now, we have computed the gradient of as defined in (11) with respect to its two arguments and . An iterative scheme (such as the gradient descent method or conjugate gradient method) can be used to find the optimal and , using the gradient expression above. The procedure displayed in Algorithm (1) is the version of AVDL based on gradient descent procedure. The learning rate can be computed via the wellknown backtracking line search method, similar to [11]. Here, considering the high coherence among the temporal frames, we prefer nonredundant dictionary, that is, for the dictionary . For parameters in the elastic net, we put an emphasis on sparse solutions and choose , as proposed in [12].
Instance  LDS, (PCs)  AVDL, , (loops)  
64  128  256  1  50  100  200  400  
Compression rate (%)  6.25  12.50  25.00  1.02  3.29  3.41  3.50  3.55 
0.9802  0.9833  0.9849  1.78  1.06  0.9992  0.9994  0.9994  
60.29  71.27  
171.99  75.52  61.96  46.18  
3 Numerical Experiments
We carry out a few experiments on natural image sequences data, and demonstrate the practicality of the proposed algorithm. Our test dataset comprises of videos from DynTex++ [14], and data from internet sources (for instance, YouTube). Firstly, we show the performance on reconstruction and synthesizing with a grayscale video of burning candle, which is corrupted by Gaussian noise or occlusion. This video has 1024 frames with size of , see figure 1. The initial dictionary is . After the acquisition of the dictionary and the transition , the synthesized data can be generated easily by , or more precisely, using a convex formulation
Table 1 shows the performance of synthesizing on burning candle with Gaussian noise. The error pairs are defined as , , and the largest eigenvalue of is denoted by . The compression rate for AVDL is sparsity of to , and for LDS is number of PCs to . Table 1 shows AVDL can obtain the stable dynamic matrix , smaller compression rate and smaller error of cost function (5), by increasing the numbers of main loops in Algorithm 1.
Figure 1 is the visual comparison between LDS and AVDL. AVDL performs well on denoising against corruption by Gaussian noise. In the case of occlusion in figure 1 (d), random 50 frames of the 1024 burning candle video are corrupted by a rectangle. The length of both synthesizing data is 1024, based on first frame of the burning candle. of the synthesizing data from LDS are corrupted by this rectangle, but for AVDL.
Occlusion rate (%)  0  5  15  30 

LDSNN (128PCs)  69.72  45.00  25.14  14.17 
AVDLSRC 
70.28  64.72  44.44  22.36 

The second experiment is about scenes classification on DynTex++, which contains DTs from 36 classes. Each class has 100 subsequences of length 50 frames with
pixels. 20 videos are randomly chosen in each class and total 720 videos are used for our experiments. Classification for LDS is performed using the Martin distance with a nearestneighbor classifier on its parameters pair
[2]. Another classifier is AVDL associated with the sparse representationbased classifier (SRC) [15, 16], in which the class of a test sequence is determined by the smallest reconstruction error and transition error . Table 2 provides the recognition results with increasing occlusion rates for test data. Compared to LDS with nearestneighbor classifier (LDSNN), Table 2 shows the proposed AVDL with SRC (AVDLSRC) performs better while the test videos are corrupted by increasing occlusion.4 Conclusions
This paper proposes an alternative method, called AVDL, to model the dynamic process of DTs. In AVDL, the sparse events over a dictionary are imposed as transition states. The proposed method show a robust performance for synthesizing, reconstruction and recognition on DTs corrupted by Gaussian noise. Especially, AVDL exhibits more powerful in the case of test data with nonGaussian noise, such as occlusion. One possible future extension is to learn a dictionary for large scale DT sequences based on AVDL.
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