视频前背景分离论文之（2） GOSUS: Grassmannian Online Subspace Updates with Structured-sparsity

1、Abstract:

(1) view online estimation procedure as an approximate optimization on a Grassmannian(格拉斯曼流形);
(2) leverage structured-sparsity to model both homogeneous perturbations of the subspace and structured contiguities of outliers.

2、Introduction

2.1 Model

$$V=B+X$$
where $V\in \mathbb R^{n\times m}$ is input data, $B$ is main effect of $V$, $X$ is outlier.

2.2 Grassmannian

$$B=UW, U\in\mathbf R^{n\times d}$$

$B$ is a linear combination of $d$ sources (subspace basis) in $n$ dimensions, denoted by $U = [u_d]$.

The orthogonal structure of $U$ implies that it lies on a Grassmannian manifold $\mathcal G_{n,d}$ embedded in a n-dimensional Euclidean space.

2.3 Structured-sparsity

Motivation: For the background estimation example, the texture (or color) of the foreground(outliers) is homogeneous

and the outliers should be contiguous in an image.

Group sparsity:

$$\begin{aligned} D_{jj}^i = \begin{cases} 1, & \text{if pixel $j$ is in group } i; \\ 0, & \text{others}. \end{cases} \end{aligned}$$

Penalty function:

$$h(\mathbf x)=\sum_{i=1}^l\mu_i\Vert D^i\mathbf x\Vert$$
where diagonal matrix $D^i$ denotes a “group” $i$, each element corresponds to the presence/absence of the pixel.
$D^i$ is sparse and allows overlap to form groups from overlapping homogeneous regions.
inner norm is $l_2$ or $l_{\infty}$, outer norm is $l_1$ to encourage sparsity.

3、GOSUS

3.1 Model:

$$\min_{U^TU=I_d,\mathbf{w,x}}\sum_{i=1}^l \mu_i\Vert D^i\mathbf x\Vert_2+\frac{\lambda}{2}\Vert U\mathbf w+\mathbf x-\mathbf v\Vert_2^2$$
where $v\in \mathbb R^n$ denotes the input, $x\in \mathbb R^n$ is outlier, $U\in \mathbb R^{n\times d}$.

3.2 Optimization

Objectives: the non-smoothness of the mixed norm and non-convexity of the orthogonal structure of $U$.

（1）Update $\mathbf{w,x}$ at fixed $U^*$.

引入松弛变量$\{\mathbf z^i\}$:

$$\begin{aligned} &\min_{\mathbf{w,x}}\quad\sum_{i=1}^l \mu_i\Vert\mathbf z^i\Vert_2+\frac{\lambda}{2}\Vert U^*\mathbf w+\mathbf x-\mathbf v\Vert_2^2\\ &s.t.\quad \mathbf z^i=D^i\mathbf x, i=1,\ldots,l. \end{aligned}$$
关于 $\{\mathbf z^i\}$和 $\mathbf{w,x}$凸，仿射形约束，通过ADMM算法求解.

（2）Update $U$ with estimated $\mathbf{w^*,x^*}$.