Robust visual tracking with contiguous occlusion constraint

Abstract

Visual tracking plays a fundamental role in video surveillance, robot vision and many other computer vision applications. In this paper, a robust visual tracking method that is motivated by the regularized $\ell$1 tracker is proposed. We focus on investigating the case that the object target is occluded. Generally, occlusion can be treated as some kind of contiguous outlier with the target object as background. However, the penalty function of the $\ell$1 tracker is not robust for relatively dense error distributed in the contiguous regions. Thus, we exploit a nonconvex penalty function and MRFs for outlier modeling, which is more probable to detect the contiguous occluded regions and recover the target appearance. For long-term tracking, a particle filter framework along with a dynamic model update mechanism is developed. Both qualitative and quantitative evaluations demonstrate a robust and precise performance.

Appendix

The APG algorithm is originally designed for solving the following unconstrained program

$$\mathop {\hbox{min} }\limits_{{\mathbf{x}}} F({\mathbf{x}}) + G({\mathbf{x}})$$

(16)

where F(x) is a differentiable convex function with Lipschitz continuous gradient and G(x) is a non-smooth but convex function. Equation (7) is a nonlinear problem with nonnegativity constraint, which can be reformulated as

$$\mathop {\hbox{min} }\limits_{{\mathbf{a}}} \frac{1}{2}||\mathcal{P}_{{{\hat{\mathbf{S}}}^{ \bot } }} ({\mathbf{y}} - {\mathbf{Ta}})||_{2}^{2} + \frac{\lambda }{{\alpha ({\hat{\mathbf{S}}})}}||{\mathbf{a}}||_{1} \; + {\mathbf{1}}_{{{\mathbb{R}} + }} ({\mathbf{a}})$$

(17)

where the additional term ${\mathbf{1}}_{{{\mathbb{R}} + }} ({\mathbf{a}})$ is defined as

$${\mathbf{1}}_{{{\mathbb{R}} + }} ({\mathbf{a}})\left\{ {\begin{array}{*{20}c} 0 & {{\text{for}}\,\,{\mathbf{a}} \ge 0} \\ { + \infty } & {\text{otherwise}} \\ \end{array} } \right.$$

(18)

Then, Eq. (7) can be decomposed as

$$F({\mathbf{a}}) = \frac{1}{2}||{\mathcal{P}}_{\hat{{\mathbf{S}}}^\bot} ({\mathbf{y}} - {\mathbf{Ta}})||_{2}^{2} ,\quad\,G({\mathbf{a}}) = \frac{\lambda }{{\alpha ({\hat{\mathbf{S}}})}}||{\mathbf{a}}||_{1} + {\mathbf{1}}_{{{\mathbb{R}} + }} ({\mathbf{a}})$$

(19)

The gradient of F(a) is Lipschitz continuous because for any ${\mathbf{x}}_{1} ,{\mathbf{x}}_{2} \in {\mathbb{R}}^{\text{n}}$,

$$\begin{aligned} &||\nabla F({\mathbf{x}}_{1} ) - \nabla F({\mathbf{x}}_{2} )|| \\ & = ||{\mathbf{A}}^{\text{T}} ({\mathbf{Ax}}_{1} - {\mathbf{b}}) - {\mathbf{A}}^{\text{T}} ({\mathbf{Ax}}_{2} - {\mathbf{b}})|| \\ & = ||{\mathbf{A}}^{\text{T}} {\mathbf{A}}({\mathbf{x}}_{1} - {\mathbf{x}}_{2} )|| \le ||{\mathbf{A}}^{\text{T}} {\mathbf{A}}||_{2} ||{\mathbf{x}}_{1} - {\mathbf{x}}_{2} || \\ \end{aligned}$$

(20)

where the matrix ${\mathbf{A}} = \mathcal{P}_{{{\hat{\mathbf{S}}}^{ \bot } }} ({\mathbf{T}})$ and ${\mathbf{b}} = \mathcal{P}_{{{\hat{\mathbf{S}}}^{ \bot } }} ({\mathbf{y}})$. Thus, $L = \left\| \mathcal{P}_{\hat{{\mathbf{S}}}^{\bot}} ({\mathbf{T}})^{\text{T}} \mathcal{P}_{\hat{{\mathbf{S}}}^{\bot}} ({\mathbf{T}}) \right\|_{2}$.

In the generic APG algorithm, we need to solve the following optimization

$$\alpha_{k + 1} = \arg \mathop {\hbox{min} }\limits_{{\mathbf{a}}} \frac{L}{2}\left\| {{\mathbf{a}} - \beta_{k + 1} \frac{{\nabla F(\beta_{k + 1} )}}{L}} \right\|_{2}^{2} + G({\mathbf{a}})$$

(21)

For the G(a) defined in Eq. (19), the function is equivalent to

$$\alpha_{k + 1} = \arg \mathop {\hbox{min} }\limits_{{\mathbf{a}}} \frac{L}{2}\left\| {{\mathbf{a}} - \beta_{k + 1} \frac{{\nabla F(\beta_{k + 1} )}}{L}} \right\|_{2}^{2} + \frac{\lambda }{{\alpha ({\hat{\mathbf{S}}})}}||{\mathbf{a}}||_{1} ,{\text{s}} . {\text{t}} .\,\,{\mathbf{a}} \ge 0$$

(22)

The optimization is a truncated quadratic problem and has a closed-form solution

$$\alpha_{k + 1} = \hbox{max} \left\{ 0,\beta_{k + 1} - \frac{{\nabla F(\beta_{k + 1} )}}{L} - \frac{\lambda } L\alpha (\hat{{\mathbf{S}}})\right\}$$

(23)