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∈Rn×m is input data, B is main effect of V, X is outlier.

2.2 Grassmannian

B=UW,U∈Rn×d

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

The orthogonal structure of U implies that it lies on a Grassmannian manifold Gn,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:

Dijj={1,0,if pixel j is in group i;others.

Penalty function:

h(x)=∑i=1lμi∥Dix∥

where diagonal matrix Di denotes a “group” i, each element corresponds to the presence/absence of the pixel.
Di is sparse and allows overlap to form groups from overlapping homogeneous regions.
inner norm is l2 or l∞, outer norm is l1 to encourage sparsity.

3、GOSUS

3.1 Model:

minUTU=Id,w,x∑i=1lμi∥Dix∥2+λ2∥Uw+x−v∥22

where v∈Rn denotes the input, x∈Rn is outlier, U∈Rn×d.

3.2 Optimization

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

（1）Update w,x at fixed U∗.

引入松弛变量{zi}:

minw,x∑i=1lμi∥zi∥2+λ2∥U∗w+x−v∥22s.t.zi=Dix,i=1,…,l.

关于{zi}和w,x凸，仿射形约束，通过ADMM算法求解.

（2）Update U with estimated w∗,x∗.