A new sparse representation-based object segmentation framework

Abstract

In this paper, a novel sparse representation-based object segmentation model is proposed. The model follows from a new energy function that combines the level-set-based sparse representation and the independent component-based shape representation within a unified framework. Before the optimization of the proposed energy, a set of training shapes is firstly projected into the shape space spanned by the independent components. For an arbitrary input shape similar to some of the elements in the training set, the minimization of the energy will automatically recover a sparse shape combination according to the neighbors in the projected shape space to guide the variational image segmentation. We test our model on both public datasets and real applications, and the experimental results show the superior segmentation capabilities of the proposed model.

Appendices

Appendix 1

Proof of Proposition 1

Given a$\gamma \in \left[ {0, 1} \right]$, for the arbitrary variational shapes $\eta ^{1}=\bar{\psi } +\sum _{i=1}^n {b_{i}^{1} m_{i}} \in \xi$ and $\eta ^{2}=\bar{\psi } +\sum _{i=1}^n {b_{i}^{2} m_{i}} \in \xi$, we have

$$\begin{aligned} \hat{\eta }&=\gamma \psi ^{1}+(1-\gamma )\psi ^{2} \\&=\bar{\psi } +\sum \limits _{i=1}^n {\left( {\gamma (b_{i}^{1} -b_{i}^{2})+b_{i}^{2}} \right) m_{i}} \end{aligned}$$

Let $\hat{b}_{i} =\gamma (b_{i}^{1} -b_{i}^{2})+b_{i}^{2}$, we have $\hat{\eta } =\bar{\psi } +\sum _{i=1}^n {\hat{b}_{i} m_{i}} \in \xi$. Thus the shape set $\xi$ is a convex set. $\square$

Appendix 2

Proof of Proposition 2

Since $A=[\mathrm{\mathbf{a}}_{1}, \mathrm{\mathbf{a}}_{2}, \ldots , \mathrm{\mathbf{a}}_{N}]\in {\mathbb {R}}^{C\times N}$ is the projected training set, and $s=[s_{1}, s_{2}, \ldots ,$ $s_{N}]^{\text {T}}\in {\mathbb {R}}^{N}$ be a sparse coefficient, for any input s we have ${As}\in {\mathbb {R}}^{C}$, considering that $b=[b_{1}, b_{2}, \ldots , b_{C}]^{\text {T}}\in {\mathbb {R}}^{C}$ in Proposition 1 is a arbitrary vector, thus all the sparse combination As can be seen as a subset of b and the As-based set $\zeta$ also can be seen as a subset of vector b-based $\xi$.

The proof of convexity of $\zeta$ is similar to Proposition 1, given a $\gamma \in \left[ {0, 1} \right]$, for the arbitrary variational shapes $\vartheta ^{1}=\bar{\psi } +\sum _{i=1}^C {(As^{1})_{i} m_{i}} \in \xi$ and $\vartheta ^{2}=\bar{\psi } +\sum _{i=1}^n {(As^{2})_{i} m_{i}} \in \xi$, we have

$$\begin{aligned} \hat{\vartheta }&=\gamma \vartheta ^{1}+(1-\gamma )\vartheta ^{2} \\&=\bar{\psi } +\sum \limits _{i=1}^n {\left( {A\left( {\gamma \left( {s^{1}-s^{2}} \right) +s^{2}} \right) } \right) m_{i}} \end{aligned}$$

Let $\hat{s}=\gamma \left( {s^{1}-s^{2}} \right) +s^{2}$, we have $\hat{\vartheta } =\bar{\psi } +\sum _{i=1}^n \left( {A\hat{s}} \right) _{i}$ $m_{i} \in \zeta$. Thus the shape set $\zeta$ is also a convex set. $\square$