Object tracking based on two-dimensional PCA

Abstract

In this paper, we present a novel object tracking method based on two-dimensional PCA. The low quality of images and the changes of the object appearance are very challenging for the object tracking. The representation of the training features is usually used to solve these challenges. Two-dimensional PCA (2DPCA) based on the image covariance matrix is constructed directly using the original image matrices. An appearance model is presented and its likelihood estimation has been established based on 2DPCA representation in this paper. Compared with the state-of-the-art methods, our method has higher reliability and real-time property. The performances of the proposed tracking method are quantitatively and qualitatively shown in experiments.

Appendix

According to Eq. (12), the likelihood of $p\left( {A_{t} } \right)$ can be computed as

$$\begin{aligned} \log \left( {p\left( {A_{t} } \right)} \right) &= - \frac{1}{2}\left\{ {\underbrace {{\log \left( {2\pi } \right){\mathbf{I}} + \log \left| {{\mathbf{X}}_{t} L{\mathbf{X}}_{t}^{T} + \sigma^{2} {\mathbf{I}}} \right|}}_{C}} \right. \\& \quad \left. { + \underbrace {{\left( {A_{t} - u{\mathbf{X}}_{t}^{T} } \right)^{T} \left( {A_{t} - u{\mathbf{X}}_{t}^{T} } \right)\left( {{\mathbf{X}}_{t} L{\mathbf{X}}_{t}^{T} + \sigma^{2} {\mathbf{I}}} \right)^{ - 1} }}_{L}} \right\} \\ \end{aligned}$$

(21)

The polynomial $C$ is a constant, so $\log p\left( {A_{t} } \right)$ is determined by the polynomial $L$. We can redefine the Eq. (21) as:

$$\log \left( {p\left( {A_{t} } \right)} \right) \propto \left( {A_{t} - u{\mathbf{X}}_{t}^{T} } \right)^{T} \left( {A_{t} - u{\mathbf{X}}_{t}^{T} } \right)\left( {{\mathbf{X}}_{t} L{\mathbf{X}}_{t}^{T} + \sigma^{2} {\mathbf{I}}} \right)^{ - 1}$$

(22)

Sherman–Morrison–Woodbury formula [32] is written as:

$$\left( {{\mathbf{A - uv}}^{T} } \right)^{ - 1} = {\mathbf{A}}^{ - 1} + {\mathbf{A}}^{ - 1} {\mathbf{u}}\left( {1 - {\mathbf{v}}^{T} {\mathbf{A}}^{ - 1} {\mathbf{u}}} \right)^{ - 1} {\mathbf{v}}^{T} {\mathbf{A}}^{ - 1}$$

(23)

According to Eq. (23), we can derive

(24)

and since

$$\left( {L^{ - 1} + \frac{1}{{\sigma^{2} }}{\mathbf{I}}} \right)^{ - 1} = \sigma^{2} \left( {{\mathbf{I}} - D^{ - 1} \sigma^{2} } \right)$$

(25)

where $D$ is a diagonal matrix with $D_{ii} = L_{ii} + \sigma^{2}$, we can get

$$\begin{aligned} &\left( {{\mathbf{X}}_{t} L{\mathbf{X}}_{t}^{T} + \sigma^{2} {\mathbf{I}}} \right)^{ - 1} \hfill \\ &= \frac{1}{{\sigma^{2} }}{\mathbf{I}} - \frac{1}{{\sigma^{2} }}{\mathbf{X}}_{t} \sigma^{2} \left( {{\mathbf{I}} - D^{ - 1} \sigma^{2} } \right){\mathbf{X}}_{t}^{T} \frac{1}{{\sigma^{2} }} \hfill \\ &= \frac{1}{{\sigma^{2} }}{\mathbf{I}} - \frac{1}{{\sigma^{2} }}\left( {{\mathbf{X}}_{t} {\mathbf{X}}_{t}^{T} } \right) + \left( {{\mathbf{X}}_{t} D^{ - 1} {\mathbf{X}}_{t}^{T} } \right) \hfill \\ &= \frac{1}{{\sigma^{2} }}\left( {{\mathbf{I}} - {\mathbf{X}}_{t} {\mathbf{X}}_{t}^{T} } \right) + \left( {{\mathbf{X}}_{t} D^{ - 1} {\mathbf{X}}_{t}^{T} } \right) \hfill \\ \end{aligned}$$

(26)

Put Eq. (26) into Eq. (21), we can derive

$$\begin{aligned} \log \left( {p\left( {A_{t} } \right)} \right) &= - \frac{1}{2}\left\{ {\underbrace {{\log \left( {2\pi } \right){\mathbf{I}} + \log \left| {{\mathbf{X}}_{t} L{\mathbf{X}}_{t}^{T} + \sigma^{2} {\mathbf{I}}} \right|}}_{C}} \right. \\& \quad \left. { + \underbrace {{\left( {A_{t} - u{\mathbf{X}}_{t}^{T} } \right)^{T} \left( {A_{t} - u{\mathbf{X}}_{t}^{T} } \right)\left( {\frac{1}{{\sigma^{2} }}\left( {{\mathbf{I}} - {\mathbf{X}}_{t} {\mathbf{X}}^{T} } \right) + \left( {{\mathbf{X}}_{t} D^{ - 1} {\mathbf{X}}_{t}^{T} } \right)} \right)}}_{{L_{1} }}} \right\} \\ \end{aligned}$$

(27)

Similarly to Eq. (21), The polynomial $C$ is a constant and $\log p\left( {A_{t} } \right)$ is determined by the polynomial $L_{1}$. We can redefine the Eq. (22) as:

$$\log \left( {p\left( {A_{t} } \right)} \right) \propto \left( {A_{t} - u{\mathbf{X}}_{t}^{T} } \right)^{T} \left( {A_{t} - u{\mathbf{X}}_{t}^{T} } \right)\left( {\frac{1}{{\sigma^{2} }}\left( {{\mathbf{I}} - {\mathbf{X}}_{t} {\mathbf{X}}_{t}^{T} } \right) + \left( {{\mathbf{X}}_{t} D^{ - 1} {\mathbf{X}}_{t}^{T} } \right)} \right)$$

(28)

According to the Eigen representation, the probability of the observed sub-image for each particle can be expressed as

$$\begin{aligned} p\left( {A_{t} |{\mathbf{Z}}_{t} } \right) &= \exp \left( { - \frac{1}{{\sigma^{2} }}\left\| {\left( {A_{t} - u{\mathbf{X}}_{t(L)}^{T} } \right) - \left( {A_{t} - u{\mathbf{X}}_{t(L)}^{T} } \right){\mathbf{X}}_{t(L)} {\mathbf{X}}_{t(L)}^{T} } \right\|^{2} } \right. \\ & \quad \left. { - \left\| {\left( {A_{t} - u{\mathbf{X}}_{t(L)}^{T} } \right){\mathbf{X}}_{t(L)} \sum_{(L)}^{ - 1/2} } \right\|^{2} } \right) \\ \end{aligned}$$

(29)