Motion object tracking based on the low-rank matrix representation

Abstract

Motion object tracking is one of the most important research directions in computer vision. Challenges in designing a tracking method are usually caused by occlusions, noise, or illumination changes. In this paper, a robust visual tracking algorithm is proposed in order to cope with the occlusion by introducing the motion object tracking issue as a low-rank matrix representation problem. First, being the main contribution of this paper, the observation matrix composed by image sequences is decomposed into a low-rank matrix and a sparse matrix. The motion object in the image sequence forms the low-rank matrix and the occlusion on the motion object forms the sparse matrix. Then the motion object tracking is carried out using a Bayesian state under the particle filter framework. Finally, an effective alternating algorithm is utilized to solve the proposed optimization formulation. The proposed algorithm has been examined throughout several challenging image sequences, and experiment results show that it works effectively and efficiently in different situations.

Appendix

Formula (13) is nonconvex and it includes both discrete and continuous variable. It is extremely difficult to solve for the solutions of $T$ and $S$. Hence, we can adopt an alternating algorithm that separates formula (13) over $T$ and $S$ into two steps to solve this problem.

1.1 Estimation of the low-rank matrix $T$

Given an estimate of the occlusion support matrix $\hat{S}$, the minimization in formula (13) over $T$ turns out to be the matrix completion problem [13]:

$$\mathop {\hbox{min} }\limits_{T} \frac{1}{2}\left\| {P_{{\hat{s}^{ \bot } }} \left( {{\mathbf{I}}_{1:n} - T} \right)} \right\|_{F}^{2} + \alpha ||T||_{*}$$

(20)

The optimal $T$ in formula (20) can be computed efficiently through the algorithm of SOFT-IMPUTE [30], which makes use of the lemma proposed in Ref. [39]. The solution is given as:

$$\hat{T} \leftarrow \varTheta_{\alpha } (P_{{\hat{s}^{ \bot } }} ({\mathbf{I}}_{1:n} ) + P_{{\hat{s}}} (\hat{T}))$$

(21)

where $\varTheta_{\alpha }$ means the singular value thresholding

$$\varTheta_{\alpha } \left( {{\mathbf{I}}_{1:n} } \right) = U\varSigma_{\alpha } V^{T}$$

(22)

where $\varSigma_{\alpha } = {\text{diag}}[\left( {d_{1} - \alpha } \right),\left( {d_{2} - \alpha } \right), \ldots ,\left( {d_{r} - \alpha } \right)]$, $U\varSigma V^{T}$ is the SVD of ${\mathbf{I}}_{1:n}$, and $\varSigma = {\text{diag}}[d_{1} ,d_{2} , \ldots ,d_{r} ]$.

1.2 Estimation of the occlusion support matrix $S$

The energy in formula (12) can be rewritten as follows:

$$\begin{aligned} \varepsilon_{t} &= \mathop {\hbox{min} }\limits_{{T,S_{it} \in \{ 0,1\} }} \frac{1}{2}\left\| {P_{{s^{ \bot } }} \left( {{\mathbf{I}}_{1:n} - T} \right)} \right\|_{F}^{2} + \beta ||S||_{1} + \gamma ||{\text{Avec}}(S)||_{1} \hfill \\ &= \frac{1}{2}\sum\limits_{it} {\left( {{\mathbf{I}}_{1:n} - \hat{T}} \right)}^{2} (1 - S_{it} ) + \beta \sum\limits_{it} {S_{it} } + \gamma ||{\text{Avec}}(S)||_{1} \hfill \\ &= \sum\limits_{it} {\left[ {\beta - \frac{1}{2}\left( {{\mathbf{I}}_{1:n} - \hat{T}} \right)^{2} } \right]S_{it} } + \gamma ||{\text{Avec}}(S)||_{1} + C \hfill \\ \end{aligned}$$

(23)

where $C = \frac{1}{2}\sum\nolimits_{it} {\left( {{\mathbf{I}}_{1:n} - \hat{T}} \right)^{2} }$ is a constant when $\hat{T}$ is fixed. The above energy is in the standard form of the first-order MRFs with binary labels [40], which can be solved exactly using graph cuts [41].