Effective Two-Stage Image Segmentation: A New Non-Lipschitz Decomposition Approach with Convergent Algorithm

Abstract

Image segmentation is an important median level vision topic. Accurate and efficient multiphase segmentation for images with intensity inhomogeneity is still a great challenge. We present a new two-stage multiphase segmentation method trying to tackle this, where the key is to compute an inhomogeneity-free approximate image. For this, we propose to use a new non-Lipschitz variational decomposition model in the first stage. The minimization problem is solved by an iterative support shrinking algorithm. By assuming that the subproblem at each iteration is exactly solved, we show the global convergence of the iterative algorithm and a lower bound theory of the image gradient of the iterative sequence, which indicates that the generated approximate image (inhomogeneity-corrected component) is with very neat edges and suitable for the following thresholding operation. Implementation details based on the alternating direction method of multipliers for the strongly convex subproblems are also given. In the second stage, the segmentation is done by applying a widely used simple thresholding technique to the piecewise constant approximation. Numerical experiments indicate good convergence properties and effectiveness of our method in multiphase segmentation for either clean or noisy homogeneous and inhomogeneous images. Both visual and quantitative comparisons with some state-of-the-art approaches demonstrate the performance advantages of our non-Lipschitz-based method.

Appendix

Definition 7.1

(Subdifferentials [50]) Let $\sigma : \mathbb {R}^d \rightarrow (-\infty ,+\infty ]$ be a proper and lower semicontinuous function. The domain of $\sigma$ is defined as $\mathrm {dom}\sigma =\{u \in \mathbb {R}^d: \sigma (u)<+\infty \}$. For a point $u \in \mathrm {dom}\sigma$,

• 1. the regular subdifferential of $\sigma$ at u is defined as

  $$\begin{aligned} \widehat{\partial }\sigma (u) =\left\{ w\in \mathbb {R}^d: \lim _{v \ne u}\inf _{v \rightarrow u}\frac{\sigma (v)-\sigma (u)-\langle w,v-u\rangle }{\Vert v-u\Vert }\ge 0 \right\} ; \end{aligned}$$

• 2. the subdifferential of $\sigma$ at u is defined as

  $$\begin{aligned} \partial \sigma (u)= & {} \{w\in \mathbb {R}^d: \exists u^k\rightarrow u,\sigma (u^k)\rightarrow \sigma (u) \\&\qquad \qquad \text{ and } w^k \in \widehat{\partial }\sigma (u^k)\rightarrow w \text{ as } k\rightarrow \infty \}. \end{aligned}$$

Remark 7.2

From Definition 7.1, it is clear that if $\sigma$ is differentiable at u, then $\widehat{\partial }\sigma (u) = \partial \sigma (u) = \{\nabla \sigma (u)\}$. If $0 \in \partial \sigma (u)$, then we call $u \in \mathbb {R}^d$ a critical point of $\sigma$.

Definition 7.3

(Kurdyka–Łojasiewicz (KL) property [57])

• 1. The function $\sigma : \mathbb {R}^d \rightarrow (-\infty ,+\infty ]$ is said to have the Kurdyka-Łojasiewicz property at $\overline{u} \in \mathrm {dom}\partial \sigma := \{u \in \mathbb {R}^d: \partial \sigma (u) \ne \emptyset \}$ if there exist $\eta \in (0,+\infty ]$, a neighborhood U of $\overline{u}$, and a continuous concave function $\psi :[0,\eta ) \rightarrow (0,+\infty ]$ such that

  • (i) $\psi (0)=0$;

  • (ii) $\psi$ is continuously differentiable on $(0,\eta )$;

  • (iii) for all $s \in (0,\eta )$, $\psi '(s)>0$;

  • (iv) for all $u \in U \cap \{v \in \mathbb {R}^d: \sigma (\overline{u})< \sigma (v) < \sigma (\overline{u})+ \eta \}$, the Kurdyka-Łojasiewicz (KL) inequality holds:

    $$\begin{aligned} \psi '(\sigma (u)-\sigma (\overline{u})) \mathrm {dist}(0,\partial \sigma (u)) \ge 1, \end{aligned}$$

    where $\mathrm {dist}(0, \partial \sigma (u)):= \inf \{\Vert v\Vert : v \in \partial \sigma (u)\}$.

A function $\sigma$ is called a KL function, if $\sigma$ satisfies the KL property at each point of $\mathrm {dom}\partial \sigma$. A rich class of KL functions of great interests are in a so-called o-minimal structure defined in [66]. The following definition is from [57, Definition 4.1].

Definition 7.4

(o-minimal structure on $\mathbb {R}$) Let $\mathscr {O} = \{\mathscr {O}_n\}_{n \in \mathbb {N}}$ such that each $\mathscr {O}_n$ is a collection of subsets of $\mathbb {R}^n$. The family $\mathscr {O}$ is an o-minimal structure on $\mathbb {R}$, if it satisfies the following axioms:

• (i) Each $\mathscr {O}_n$ is a boolean algebra. Namely $\emptyset \in \mathscr {O}_n$ and for each A, B in $\mathscr {O}_n$, $A \cup B$, $A \cap B$, and $\mathbb {R}^n {\setminus } A$ belong to $\mathscr {O}_n$.

• (ii) For all A in $\mathscr {O}_n$, $A \times \mathbb {R}$ and $\mathbb {R} \times A$ belong to $\mathscr {O}_{n+1}$.

• (iii) For all A in $\mathscr {O}_{n+1}$, $\Pi (A):= \{(x_1,\ldots ,x_n)\in \mathbb {R}^n: (x_1,\ldots ,x_n,x_{n+1}) \in A\}$ belongs to $\mathscr {O}_{n}$.

• (iv) For all $i \ne j$ in $\{1,2,\ldots ,n\}$, $\{(x_1,\ldots ,x_n) \in \mathbb {R}^n: x_i = x_j\}$ belongs to $\mathscr {O}_n$.

• (v) The set $\{(x_1,x_2) \in \mathbb {R}^2: x_1<x_2\}$ belongs to $\mathscr {O}_2$.

• (vi) The elements of $\mathscr {O}_1$ are exactly finite unions of intervals.

Let $\mathscr {O}$ be an o-minimal structure on $\mathbb {R}$. We call a set $A \subseteq \mathbb {R}^n$ definable on $\mathscr {O}$ if $A \in \mathscr {O}_n$, and a map $f: \mathbb {R}^n \rightarrow \mathbb {R}^m$ definable on $\mathscr {O}$ if its graph $\{(x,y) \in \mathbb {R}^n \times \mathbb {R}^m: y \in f(x)\}$ is definable on $\mathscr {O}$. A definable function is a special definable map. Some useful properties of definable functions [38, 57] are listed as follows:

• (i) compositions of definable functions are definable;

• (ii) finite sums of definable functions are definable;

• (iii) indicator functions of definable sets are definable.

We have a very useful class of o-minimal structure, i.e., the log-exp structure [66, Example 2.5]. By this, the following functions are all definable:

• (1) semi-algebraic functions [58, Definition 5], such as real polynomial functions, and $f: \mathbb {R} \rightarrow \mathbb {R}$ defined by $x \mapsto |x|$.

• (2) $x^r : \mathbb {R} \rightarrow \mathbb {R}$ defined by

  $$\begin{aligned} a \mapsto {\left\{ \begin{array}{ll} a^r, &{}\quad a >0 \\ 0, &{}\quad a \le 0, \end{array}\right. } \end{aligned}$$

  where $r \in \mathbb {R}$.

We know that any proper lower semicontinuous function definable on an o-minimal structure is a KL function; see [55] and [57, Theorem 14]. For F(u, v) in this paper, $\Vert f-u-v\Vert ^2$, $\Vert D_i u\Vert$, $\Vert Hv\Vert ^2$ and $\Vert v\Vert ^2$ are all semi-algebraic functions. In addition, from examples (1)(2) and the elementary properties (i)(ii) of definable functions, we know that F(u, v) is definable. Thus, F(u, v) is a KL function.