# Video Segmentation

We partition the video volume into $$C^N$$ non-overlapping regions using GBH segmentation. The segmentation is based on appearance and motion similarity between the local regions. Each segment $$c_i \in C^N$$ is comprised of arbitrary shape & sized cloud of points $$x_i=\{x^0_i, x^1_i, ...., x^P_i\}$$ in video volume space $$\mathbb{R}^3$$.

The practical challenge is to represent segment $$c_i$$ efficiently without comprimising on the memory and accuracy. Because it is difficult to fit regular structure such as 3D bounding box or ellipsiod. So we came up with solution to divide the video into regular $$m \times m \times m$$ sized cells and construct the representation based on such structure. It does reduce the memory load by $$m^3$$ times. Also such cell can act like a building block to construct arbitrary shape and sized 3D regions.

# Vertex

A region $$c_i$$ will constitute a vertex $$v_i$$ in the video graph $$G(V,E)$$. Then cardinality of $$|V|$$ is equal to the size of segmented regions, $$|C|$$. The BoVW histogram $$h_i$$ of local features $$f_i \in c_i$$ represent an unary potential for vertex $$v_i$$.

## Local feature histogram

Each node $$v_i$$ will be associated with two components of histograms: foreground histogram $$h_i^{fg}$$ and background histogram $$h_i^{bg}$$:

• $$h_i^{fg} = \sum_{j \in c_i} bow(f_j)$$ - a frequency of quantized local features $$f_j$$ extracted inside the region $$c_i$$.

• $$h_i^{bg} = \sum_{j \notin c_i} bow(f_j)$$ - a frequency of quantized local features $$f_j$$ extracted outside the region $$c_i$$.

Hence, the histogram representation of node $$v_i$$ can defined as:

$h_i = \|h_i^{fg}\| + \alpha_{bg} \|h_i^{bg}\|$

### UCF-Sport Dataset

$$mAP$$ performance with different types of kernel functions vs background histogram weight values, $$\alpha_{bg}$$

 Kernel Type $$\alpha_{bg} = 0$$ $$\alpha_{bg} = 0.5$$ $$\alpha_{bg} = 0.75$$ $$Linear$$ 31.91 % 56.69 % 65.29 % $$Intersection$$ 35.65 % 58.98 % 60.89 % $$Chi-Square$$ 39.03 % 63.08 % 65.96 % $$Jenson-Shannon$$ 39.50 % 63.53 % 66.38 %

The runtime evaluation (in mins) with different types of kernel functions vs background histogram weight values, $$\alpha_{bg}$$

 Kernel Type $$\alpha_{bg} = 0$$ $$\alpha_{bg} = 0.5$$ $$\alpha_{bg} = 0.75$$ $$Linear$$ 2.5 22.3 15.1 $$Intersection$$ 3.3 20.3 16.8 $$Chi-Square$$ 2.4 14.8 11.2 $$Jenson-Shannon$$ 3.0 78.7 58.8

# Edge

Edge $$E$$ will govern the relationship between the segmented regions $$C = \{c_0, c_1,...,c_N\}$$, i.e vertex $$V$$ of video graph $$G(V,E)$$. In essence, $$e_{ij} \in E$$ should reflect the likelihood of vertex $$v_i$$ and $$v_j$$ belong to the same action category, i.e $$\mathrm{P_{ij}}(l_i = l_j)$$ where $$l_i, l_j \in L$$.

## Video Graph Data Structure

A video sequence is partitioned into a set of non-overlapping supervoxel regions $$S = \{s_0,s_1,..,s_N\}$$. The supervoxel is mapped into graph: $$f_{map}: S \mapsto G(V,E)$$.

Each node $$v_i \in V$$ has following attributes:

• $$size_i$$ : the total number of cells making up the region $$s_i$$, $$\mathbb{R}$$.

• $$min_i$$ : the minimum location of 3D bounding box point in cell-coordinate $$\mathbb{R}^3$$.

• $$max_i$$ : the maximum location of 3D bounding box point in cell-coordinate $$\mathbb{R}^3$$.

• $$mean_i$$: the mean cell location, $$\mathbb{R}^3$$.

• $$hist_i$$: the bow histogram vector over the local descriptor $$f_i \in s_i$$, $$\mathbb{R}^{4k} \times 5$$ (Used 5 different descriptor with dictionary size of $$4k$$ ).

• $$sparsity_i$$: the number of non-zeros bins in the bow histogram $$hist_i$$.

• $$hist^c_i$$: the concatenated color histogram vector of s_i region over 3-channels (RGB), $$\mathbb{R}^{10} \times 3$$

• $$hist^g_i$$: the orientation histogram vector of s_i region with 50 directional bins, $$\mathbb{R}^{50}3$$

• $$li$$: the action label of $$s_i$$ region, $$l_i \in L$$.

Each edge $$e_{ij} \in E$$ between node $$u_i$$ and $$u_j$$ has following attribute:

• \(ints