Implicit Active Contour Model with Local and Global Intensity Fitting Energies

Abstract

We propose a new active contour model which integrates a local intensity fitting (LIF) energy with an auxiliary global intensity fitting (GIF) energy. The LIF energy is responsible for attracting the contour toward object boundaries and is dominant near object boundaries, while the GIF energy incorporates global image information to improve the robustness to initialization of the contours. The proposed model not only can provide desirable segmentation results in the presence of intensity inhomogeneity but also allows for more flexible initialization of the contour compared to the RSF and LIF models, and we give a theoretical proof to compute a unique steady state regardless of the initialization; that is, the convergence of the zero-level line is irrespective of the initial function. This means that we can obtain the same zero-level line in the steady state, if we choose the initial function as a bounded function. In particular, our proposed model has the capability of detecting multiple objects or objects with interior holes or blurred edges.

1. Introduction

Implicit active contour models have been extensively studied and successfully used in image segmentation [1–3]. The basic idea is to evolve a contour under some constraints to extract the desired object. According to the nature of constraints, the existing active contour models can be categorised into two classes: edge-based models [4–7], and region-based models [8–17]. Each of them has its own pros and cons; the choice of them in applications depends on different characteristics of images. In this study, we focus on region-based models and consider images with intensity inhomogeneity.

Unlike edge-based models that utilize typically an edge indicator depending on image gradient to perform contour extraction, region-based models usually use global and/or local statistics inside regions rather than gradient on edges to find a partition of image domain. They generally have better performances in the presence of weak or discontinuous boundaries than edge-based models. Early popular region-based models tend to rely on intensity homogeneous (roughly constant or smooth) statistics in each of the regions to be segmented. Therefore, they either lack the ability to deal with intensity inhomogeneity like the PC (piecewise constant) model [8] or are excessively expensive in computation like the PS (piecewise smooth) model [9].

To handle intensity inhomogeneity efficiently, some localized region-based models [11–16] have been proposed recently. For example, Li et al. [12] recently proposed a region-scalable fitting (RSF) active contour model (originally termed as local binary fitting (LBF) model [11]). The RSF model draws upon intensity information in spatially varying local regions depending on a scale parameter, so it is able to deal with intensity inhomogeneity efficiently. Very recently, Zhang et al. [15] proposed a novel active contour model driven by local image fitting energy, which also can handle intensity inhomogeneity efficiently. The experimental results in [15] show that this model is more efficient than the RSF model, while yielding similar results. However, these two models easily get stuck in local minimums for the most of contour initializations. This makes it need user intervention to define the initial contours professionally.

In this study, based on the PC model [8] and RSF model [12], we propose a new active contour model, which integrates a local intensity fitting (LIF) energy with an auxiliary global intensity fitting (GIF) energy. The LIF energy is responsible for attracting the contour toward object boundaries and is dominant near object boundaries, while the GIF energy incorporates global image information to improve the robustness to initialization of the contours. The proposed model can efficiently handle intensity inhomogeneity, while allowing for more flexible initialization and maintaining the subpixel accuracy.

The remainder of this paper is organized as follows. Section 2 briefly reviews the PC model [8] and RSF model [12]. Section 3 introduces the proposed model. Section 4 presents experimental results using a set of synthetic and real images. This paper is summarized in Section 5.

2. Related Works

2.1. Piecewise Constant Model by Chan and Vese

In [8], Chan and Vese (CV) proposed a region-based active contour model that relies on intensity homogeneous (roughly constant) statistics in each of the regions to be segmented. In the CV model, a contour 𝐶 is represented implicitly by the zero-level set of a Lipschitz function 𝜙∶Ω→𝐑, which is called a level set function. In what follows, we let the level set function 𝜙 take positive and negative values inside and outside the contour 𝐶, respectively.

Let 𝐼∶Ω⊂𝐑^𝟐→𝐑 be an input image, and let 𝐻_𝜀 be the regularized Heaviside function; the energy functional of the CV model is defined as 𝐸^CV𝑐₁,𝑐₂,𝜙=𝜆_1Ω𝐼−𝑐₁²𝐻_𝜀𝜙𝑑𝐱+𝜆₁∫_Ω𝐼−𝑐₂²1−𝐻_𝜀𝜙𝑑𝐱+𝜈_Ω∇𝐻_𝜀𝜙𝑑𝐱, (1) where 𝜆₁,𝜆₂>0, 𝜈>0 are constants. The regularized version of 𝐻(𝑧) is chosen as 𝐻_𝜀(𝑧)=1221+𝜋𝑧arctan𝜀. (2)𝑐₁ and 𝑐₂ are the global averages of the image intensities in the region {𝐱∶𝜙(𝐱)>0} and {𝐱∶𝜙(𝐱)<0}, respectively; that is, 𝑐₁𝜙=∫_Ω𝐼(𝐱)𝐻_𝜀𝜙(𝐱)𝑑𝐱∫_Ω𝐻_𝜀𝜙(𝐱),𝑐𝑑𝐱₂𝜙=∫_Ω𝐼(𝐱)1−𝐻_𝜀𝜙(𝐱)𝑑𝐱∫_Ω1−𝐻_𝜀𝜙(𝐱).𝑑𝐱 (3)

The solution of the CV model in fact leads to a piecewise constant segmentation of the original image 𝐼(𝐱): 𝐼^𝐺(𝐱)=𝑐₁𝐻_𝜀𝜙(𝐱)+𝑐₂1−𝐻_𝜀𝜙(𝐱), (4) where 𝑐₁ and 𝑐₂ are the averages of the image intensities in the region {𝐱∶𝜙(𝐱)>0} and {𝐱∶𝜙(𝐱)<0}, respectively. Such constants may be far away from the original image data, if the intensities outside or inside the contour 𝐶={𝐱∶𝜙(𝐱)=0} are not homogeneous. As a result, the CV model generally fails to segment images with intensity inhomogeneity.

2.2. Region-Scalable Fitting Model

In order to improve the performance of the global CV model [8] and the PS model [9] on images with inhomogeneity, Li et al. [11, 12] recently proposed a novel region-based active contour model. They introduced a kernel function and defined the following energy functional: 𝐸^RSF𝑓₁,𝑓₂,𝜙=𝜆₁𝐾_𝜎(𝐱−𝐲)𝐼(𝐲)−𝑓₁(𝐱)²×𝐻_𝜀𝜙(𝐲)𝑑𝐲𝑑𝐱+𝜆₂𝐾_𝜎(𝐱−𝐲)𝐼(𝐲)−𝑓₁(𝐱)²×1−𝐻_𝜀𝜙(𝐲)𝑑𝐲𝑑𝐱,+𝜈_Ω∇𝐻_𝜀𝜙(𝐱)𝑑𝐱+𝜇_Ω12∇𝜙(𝐱)−1²𝑑𝐱, (5) where 𝐾_𝜎 is a Gaussian kernel with standard deviation 𝜎, and 𝑓₁(𝐱) and 𝑓₂(𝐱) are two smooth functions that approximate the local image intensities inside and outside the contour, respectively. They are computed by 𝑓₁(𝐱)=∫_Ω𝐾_𝜎(𝐱−𝐲)𝐼(𝐲)𝐻_𝜀𝜙(𝐲)𝑑𝐲∫_Ω𝐾_𝜎(𝐱−𝐲)𝐻_𝜀𝜙(𝐲),𝑓𝑑𝐲₂(𝐱)=∫_Ω𝐾_𝜎(𝐱−𝐲)𝐼(𝐲)1−𝐻_𝜀𝜙(𝐲)𝑑𝐲∫_Ω𝐾_𝜎(𝐱−𝐲)1−𝐻_𝜀𝜙(𝐲).𝑑𝐲 (6)

The solution of the RSF model leads to a piecewise smooth approximation of the original image 𝐼(𝐱): 𝐼^𝐿(𝐱)=𝑓₁(𝐱)𝐻_𝜀𝜙(𝐱)+𝑓₂(𝐱)1−𝐻_𝜀𝜙(𝐱), (7) where the smooth functions 𝑓₁(𝐱) and 𝑓₂(𝐱) are computed by (6). 𝑓₁(𝐱) and 𝑓₂(𝐱) are the averages of local intensities on the two sides of the contour, which are different from the constants 𝑐₁ and 𝑐₂ in the CV model, the averages of the image intensities on the two sides of the contour. Therefore, the RSF model can deal with intensity inhomogeneity efficiently. However, it is sensitive to contour initialization (initial locations, sizes, and shapes).

3. The Proposed Model

3.1. Description of Our Model

Given a positive constant 𝜔 (0≤𝜔≤1), from (4) and (7), we define the following energy functional: 𝐸𝜙,𝑐₁,𝑐₂,𝑓₁,𝑓₂=𝜔𝐸^𝐺𝜙+(1−𝜔)𝐸^𝐿𝜙1=𝜔2_Ω𝐼(𝐱)−𝐼^𝐺(𝐱)²+(1𝑑𝐱1−𝜔)2_Ω𝐼(𝐱)−𝐼^𝐿(𝑥)²𝑑𝐱, (8) where 𝐼^𝐺(𝐱)=𝑐₁𝐻_𝜀𝜙(𝐱)+𝑐₂1−𝐻_𝜀𝜙(𝐱),𝐼^𝐿(𝐱)=𝑓₁(𝐱)𝐻_𝜀𝜙(𝐱)+𝑓₂(𝐱)1−𝐻_𝜀𝜙(𝐱). (9)

Keeping 𝜙 fixed and minimizing the functional 𝐸(𝜙,𝑐₁,𝑐₂,𝑓₁,𝑓₂) with respect to 𝑐₁, 𝑐₂, 𝑓₁, and 𝑓₂, we have 𝑐₁=∫_Ω𝐼(𝐱)𝐻_𝜀𝜙(𝐱)𝑑𝐱∫_Ω𝐻_𝜀𝜙(𝐱),𝑐𝑑𝐱₂=∫_Ω𝐼(𝐱)1−𝐻_𝜀𝜙(𝐱)𝑑𝐱∫_Ω1−𝐻_𝜀𝜙(𝐱),𝑓𝑑𝐱₁(𝐱)=∫_Ω𝐾_𝜎𝐱−𝑦𝐼𝑦𝐻_𝜀𝜙𝑦𝑑𝑦∫_Ω𝐾_𝜎𝐱−𝑦𝐻_𝜀𝜙𝑦,𝑓𝑑𝑦₂(𝐱)=∫_Ω𝐾_𝜎𝐱−𝑦𝐼𝑦1−𝐻_𝜀𝜙𝑦𝑑𝑦∫_Ω𝐾_𝜎𝐱−𝑦1−𝐻_𝜀𝜙𝑦.𝑑𝑦 (10)

Keeping 𝑐₁, 𝑐₂, 𝑓₁, and 𝑓₂ fixed and minimizing the functional 𝐸(𝜙,𝑐₁,𝑐₂,𝑓₁,𝑓₂) with respect to 𝜙, we obtain the corresponding gradient descent flow equation: 𝜕𝜙𝜕𝑡=𝛿_𝜀𝜙𝜔𝐼−𝐼^𝐺𝑐₁−𝑐₂+(1−𝜔)𝐼−𝐼^𝐿𝑓₁−𝑓₂=𝛿_𝜀𝜙𝜔𝐹^𝐺𝜙+(1−𝜔)𝐹^𝐿𝜙=𝐹𝜙, (11) where 𝛿_𝜀(𝑧)=𝐻^′_𝜀(𝑧)=1𝜋𝜀𝜀²+𝑧². (12)

Like the CV and RSF models, our model is also implemented using an alternative procedure: for each iteration and the corresponding level set function 𝜙^𝑛, we first compute the fitting values 𝑐_𝑖(𝜙^𝑛) and 𝑓_𝑖(𝜙^𝑛) and then obtain 𝜙^𝑛+1 by minimizing 𝐸(𝜙,𝑐₁(𝜙^𝑛),𝑐₂(𝜙^𝑛),𝑓₁(𝜙^𝑛),𝑓₂(𝜙^𝑛)) with respect to 𝜙. This process is repeated until the zero-level set of 𝜙^𝑛+1 is exactly on the object boundary.

In the following, we first discuss the properties of 𝐹^𝐺(𝜙) and 𝐹^𝐿(𝜙) and then analyze the behavior of (11).

3.2. Properties of 𝐹^𝐺(𝜙) and 𝐹^𝐿(𝜙)

For the sake of simplicity, we state and prove the properties of 𝐹^𝐺(𝜙) and 𝐹^𝐿(𝜙) only for an image consisting of only two distinct gray levels: 𝐼(𝐱)=𝑔₁𝑔,𝐱∈𝜔,₂,𝐱∈Ω⧵𝜔, (13) where 𝑔₁,𝑔₂≥0 with 𝑔₁≠𝑔₂, 𝜔 and Ω⧵𝜔 represent the objects of interest and the background, respectively.

Theorem 1. Let 𝐼(𝐱) be an image by (13). Then one has 𝐹^𝐺𝜙=(𝑀𝑛−𝑁𝑚)𝑔₁−𝑔₂²×(𝑀−𝑁)(𝑁−𝑛)𝐻_𝜀+𝑁((𝑀+𝑛)−(𝑚+𝑁))1−𝐻_𝜀×𝑁²(𝑀−𝑁)²⁻¹,−()𝑔𝑖𝑛𝜔𝑀𝑛−𝑁𝑚₁−𝑔₂²×𝑛(𝑀−𝑁)𝐻_𝜀+𝑁(𝑚−𝑛)1−𝐻_𝜀×𝑁²(𝑀−𝑁)²⁻¹,𝑖𝑛Ω⧵𝜔, (14) and so, sign𝐹^𝐺𝜙=()()+sign𝑀𝑛−𝑁𝑚,𝑖𝑛𝜔,−sign𝑀𝑛−𝑁𝑚,𝑖𝑛Ω⧵𝜔, (15) where |Ω||𝜔|,,𝑀=,𝑚=,𝑁=𝜙>0𝑛=𝜔∩𝜙>0 (16) in which |Ω| is the area of the region Ω and similarly for others.

Remarks. (i) Due to the discrete nature of image, |Ω| is in fact the number of pixels in the image 𝐼(𝐱), and similarly to others.

(ii) The cases of 𝑛=𝑚, 𝑛=𝑁, and 𝑛=0 correspond to the zero-level lines of 𝜙(𝐱) which are encircling the object (𝜔), inside the object and within the background (Ω⧵𝜔), respectively. The cases of 𝑁≠𝑛≠𝑚 and 𝑁=𝑛=𝑚 correspond to the zero-level line of 𝜙(𝐱) that are partially inside the object and exactly on the object edge, respectively.

(iii) The significance of (15) is that the function 𝐹^𝐺(𝜙) has the opposite sign in 𝜔 (object) and Ω⧵𝜔 (background), respectively.

The proof of Theorem 1 is provided in Appendix A. The following result will be used in the proof of Theorem 7, which guarantees that the evolution by (10) converges to a unique stable value after finite time.

Corollary 2. Let 𝐼(𝐱) be an image by (13). Then one has the following.(i)If 𝑀𝑛−𝑁𝑚≥0, then 𝐹^𝐺𝜙≥()𝑔𝑀𝑛−𝑁𝑚¹−𝑔₂²𝑀⁴𝐹/4,𝑖𝑛𝜔,_𝐺𝜙()𝑔≤−𝑀𝑛−𝑁𝑚¹−𝑔₂²𝑀⁴/4,𝑖𝑛Ω⧵𝜔. (17)(ii)If 𝑀𝑛−𝑁𝑚≤0, then 𝐹^𝐺𝜙≤()𝑔𝑀𝑛−𝑁𝑚¹−𝑔₂²𝑀⁴𝐹/4,𝑖𝑛𝜔,_𝐺𝜙()𝑔≥−𝑀𝑛−𝑁𝑚¹−𝑔₂²𝑀⁴/4,𝑖𝑛Ω⧵𝜔. (18)

The proof of Corollary 2 is given in Appendix A.

We call the property in Theorem 1 an adaptive sign-changing property of 𝐹^𝐺(𝜙). Such property also holds for 𝐹^𝐿(𝜙), which can be expressed by the following theorem.

Theorem 3. Let 𝐼(𝐱) be an image given by (13). Then one has 𝐹^𝐿𝜙=𝑃𝑞−𝑄𝑝𝑔₁−𝑔₂²×(𝑃−𝑄)𝑄−𝑞𝐻_𝜀𝜙+𝑄𝑃+𝑞−𝑝+𝑄1−𝐻_𝜀𝜙×𝑄²(𝑃−𝑄)²⁻¹,−𝑔𝑖𝑛𝜔𝑃𝑞−𝑄𝑝₁−𝑔₂²×𝑞(𝑃−𝑄)𝐻_𝜀𝜙+𝑄𝑝−𝑞1−𝐻_𝜀𝜙×𝑄²(𝑃−𝑄)²⁻¹,𝑖𝑛Ω⧵𝜔, (19) and so, sign𝐹^𝐿𝜙=+sign𝑃𝑞−𝑄𝑝,𝑖𝑛𝜔,−sign𝑃𝑞−𝑄𝑝,𝑖𝑛Ω⧵𝜔, (20) where 𝑃=_Ω𝐾_𝜎(𝐱−𝜉)𝑑𝜉,𝑝=_𝜔𝐾_𝜎(𝐱−𝜉)𝑑𝜉,𝑄=_{𝜙>0}𝐾_𝜎(𝐱−𝜉)𝑑𝜉,𝑞=_{𝜔∩{𝜙>0}}𝐾_𝜎(𝐱−𝜉)𝑑𝜉. (21)
This theorem shows that the local force 𝐹^𝐿(𝜙) has exactly opposite sign in 𝜔 (object) and in Ω⧵𝜔 (background).

The following result together with Corollary 2 will be used in the proof of Theorem 7 mentioned later.

Corollary 4. Under the assumption of Theorem 3, one has the following.(i)If 𝑃𝑞−𝑄𝑝≥0, then 𝐹^𝐿𝜙𝑔≥𝑆𝑃𝑞−𝑄𝑝¹−𝑔₂²𝑃⁴𝐹/4,𝑖𝑛𝜔,_𝐿𝜙𝑔≤−𝑆𝑃𝑞−𝑄𝑝¹−𝑔₂²𝑃⁴/4,𝑖𝑛Ω⧵𝜔. (22)(ii)If 𝑃𝑞−𝑄𝑝≤0, then 𝐹^𝐿𝜙𝑔≤𝑆𝑃𝑞−𝑄𝑝¹−𝑔₂²𝑃⁴𝐹/4,𝑖𝑛𝜔,_𝐿𝜙𝑔≥−𝑆𝑃𝑞−𝑄𝑝¹−𝑔₂²𝑃⁴/4,𝑖𝑛Ω⧵𝜔, (23) where 𝑆=min𝐾_𝜎(𝐱−𝜉)∶𝜉∈Ω. (24)

The proofs of Theorem 3 and Corollary 4 are similar to Theorem 1 and Corollary 2, respectively; see Appendix B for details.

3.3. Behavior of Our Model

In this section, we analyze the behavior of our model (10) for image segmentation. We will show that the zero-level line of the evolution function starting with a bounded function finally comes to a unique steady state, which separates the object from the background.

Due to the discrete nature of image, the continuous equation (10) can be discretized in space with space step 1 (implied pixel spacing), but also in time with a time step Δ𝑡 as follows: 𝜙^𝑘+1_𝑖,𝑗=𝜙^𝑘_𝑖,𝑗+Δ𝑡⋅𝐹𝜙^𝑘_𝑖,𝑗,0≤𝑖≤𝑝−1,0≤𝑗≤𝑞−1, (25) where 𝑝×𝑞 corresponds to the image size, 𝐼_𝑖,𝑗=𝐼(𝑖,𝑗) and 𝜙^𝑘_𝑖,𝑗=𝜙(𝑘Δ𝑡,𝑖,𝑗) with 𝑘≥0 and 𝜙⁰_𝑖,𝑗=𝜙₀(𝑖,𝑗).

Corollary 5. Let 𝑀𝑛₀−𝑁₀𝑚=|Ω||𝜔∩{𝜙⁰>0}|−|{𝜙⁰>0}||𝜔|≠0.(i)If 𝑀𝑛₀−𝑁₀𝑚>0, then 𝐹^𝐺𝜙^𝑘>𝑀𝑛₀−𝑁₀𝑚×𝑔₁−𝑔₂²𝑀⁴/4=𝐴>0,𝑖𝑛𝜔𝑘≥1,𝑘∈𝑧⁺,𝐹^𝐺𝜙^𝑘<−𝑀𝑛₀−𝑁₀𝑚×𝑔₁−𝑔₂²𝑀⁴/4=−𝐴<0,𝑖𝑛Ω⧵𝜔𝑘≥1,𝑘∈𝑧⁺. (26)(ii)If 𝑀𝑛₀−𝑁₀𝑚<0, then 𝐹^𝐺𝜙^𝑘<𝑀𝑛₀−𝑁₀𝑚×𝑔₁−𝑔₂²𝑀⁴/4=𝐴<0,𝑖𝑛𝜔𝑘≥1,𝑘∈𝑧⁺,𝐹^𝐺𝜙^𝑘>−𝑀𝑛₀−𝑁₀𝑚×𝑔₁−𝑔₂²𝑀⁴/4=−𝐴>0,𝑖𝑛Ω⧵𝜔𝑘≥1,𝑘∈𝑧⁺, (27) where 𝐴=𝑀𝑛₀−𝑁₀𝑚𝑔₁−𝑔₂²𝑀⁴./8 (28)

We provide the proof of Corollary 5 in Appendix C. Similar analysis can prove the following corollary, Corollary 6.

Corollary 6. Let 𝑃𝑞₀−𝑄₀𝑝≠0.(i)If 𝑃𝑞₀−𝑄₀𝑝>0, then 𝐹^𝐿𝜙^𝑘>𝑃𝑞₀−𝑄₀𝑝×𝑔₁−𝑔₂²𝑃⁴/4=𝐵>0,𝑖𝑛𝜔𝑘≥1,𝑘∈𝑧⁺,𝐹^𝐿𝜙^𝑘<−𝑃𝑞₀−𝑄₀𝑝×𝑔₁−𝑔₂²𝑃⁴/4=−𝐵<0,𝑖𝑛Ω⧵𝜔𝑘≥1,𝑘∈𝑧⁺. (29)(ii)If 𝑃𝑞₀−𝑄₀𝑝<0, then 𝐹^𝐿𝜙^𝑘<𝑃𝑞₀−𝑄₀𝑝×𝑔₁−𝑔₂²𝑃⁴/4=𝐵<0,𝑖𝑛𝜔𝑘≥1,𝑘∈𝑧⁺,𝐹^𝐿𝜙^𝑘>−𝑃𝑞₀−𝑄₀𝑝×𝑔₁−𝑔₂²𝑃⁴/4=−𝐵>0,𝑖𝑛Ω⧵𝜔𝑘≥1,𝑘∈𝑧⁺, (30) where 𝐵=𝑃𝑞₀−𝑄₀𝑝𝑔₁−𝑔₂²𝑃⁴./8 (31)

Theorem 7. If 𝜙⁰(𝐱) is a bounded function with (𝑀𝑛₀−𝑁₀𝑚)(𝑃𝑞₀−𝑄₀𝑝)>0, then there exists a positive integer 𝐾 such that, for 𝑘≥𝐾, 𝜙^𝑘+1_𝑖,𝑗𝜙>0,𝑖𝑛𝜔,_{𝑘+1𝑖,𝑗}<0,𝑖𝑛Ω⧵𝜔,𝑖𝑓𝑀𝑛₀−𝑁₀𝑚>0,𝑃𝑞₀−𝑄₀𝑝>0,𝜙^𝑘+1_𝑖,𝑗𝜙<0,𝑖𝑛𝜔,_{𝑘+1𝑖,𝑗}>0,𝑖𝑛Ω⧵𝜔,𝑖𝑓𝑀𝑛₀−𝑁₀𝑚<0,𝑃𝑞₀−𝑄₀𝑝<0. (32)(33)

We provide the proof of Theorem 7 in Appendix D.

The significance of Theorem 7 is that if we choose 𝜙⁰(𝐱) such that (𝑀𝑛₀−𝑁₀𝑚)(𝑃𝑞₀−𝑄₀𝑝)>0, then the zero-level line of 𝜙(𝐱) starting with such initial function 𝜙⁰(𝐱) can finally come to a unique steady state, which separates object from the background.

Remark. Mathematically there does exist 𝜙⁰(𝐱) such that (𝑀𝑛₀−𝑁₀𝑚)(𝑃𝑞₀−𝑄₀𝑝)≤0; in practice, however, we always can guarantee that 𝑀𝑛₀−𝑁₀𝑚𝑃𝑞₀−𝑄₀𝑝>0. (34)

3.4. Discussion of Initial Function

Theorem 7 guarantees that the proposed model computes a unique steady state regardless of the initialization; that is, the convergence of the zero-level line {𝜙=0} is irrespective of the initial function. This means that we can obtain the same zero-level line in the steady state if we choose the initial function as a bounded function 𝜙⁰ with (𝑀𝑛₀−𝑁₀𝑚)(𝑃𝑞₀−𝑄₀𝑝)>0.

In applications, the initial function 𝜙⁰ can be defined via a simple curve (closed curve or line segment) in image domain. For example, we can choose the initial function as a signed distance to a circle, which is widely used in most of image segmentation models with level set methods. For the proposed model, however, we prefer to define the initial function 𝜙⁰(𝐱) as a piecewise constant or constant function as follows.(1)If the curve 𝐶 is a closed curve (e.g., circle or square), then 𝜙⁰(𝐱) is defined by 𝜙⁰(𝐱)=(𝐱)(𝐶),(𝐱)(𝐶),+𝜌,∈inside−𝜌,∈outside (35) where 𝜌≠0 is a constant.(2)If the curve 𝐶 is a line segment that partitions the image domain Ω into two disjoint regions Ω₁ and Ω₂ (e.g., the left and right half regions of image domain Ω, respectively) with Ω=Ω₁∪Ω₂ and Ω₁∩Ω₂=Φ, then 𝜙₀(𝑥,𝑦) is defined by 𝜙⁰(𝐱)=+𝜌,(𝐱)∈Ω₁,−𝜌,(𝐱)∈Ω₂, (36) where 𝜌≠0 is a constant. (3)We can also define a zero function as follows: 𝜙⁰(𝐱)=0. (37)

Next, we prove the fact that, with 𝜙⁰(𝐱)=0, 𝜙(𝐱) becomes a sign-changing function and satisfy the condition of (𝑀𝑛₁−𝑁₁𝑚)(𝑃𝑞₁−𝑄₁𝑝)>0 after the first iteration.

Theorem 8. If the initial function 𝜙⁰(𝐱)=0 in Ω and 𝑐⁰₁(𝐱)=𝑓⁰₁(𝐱)=2⋅mean_𝐱∈Ω𝐼(𝐱),𝑐⁰₂(𝐱)=𝑓⁰₂(𝐱)=0, (38) then after the first iteration, one has sign(𝐹(0))=−1,𝐱∈Ω₁,1,𝐱∈Ω₂, (39) where Ω₁∩Ω₂=Φ, Ω₁≠Φ, Ω₂≠Φ.

We provide the proof of Theorem 8 in Appendix E.

By (25) and Theorem 8, we have 𝜙¹(𝐱)=𝛿_𝜀(0)mean_𝐱∈Ω𝐼(𝐱)×𝐼−mean_𝐱∈Ω𝐼(𝐱)<0,𝐱∈Ω₁,>0,𝐱∈Ω₂. (40) Therefore, 𝜙¹(𝐱) became a sign-changing function. Then, using two distinct gray levels of (13) and the above demonstration, we have 𝑀𝑛₁−𝑁₁𝑚𝑃𝑞₁−𝑄₁𝑝=𝑚𝑀−𝑁₁𝑝𝑃−𝑄₁=𝑚𝑝𝑀−𝑁₁𝑃−𝑄₁>0. (41)

4. Implementation and Experimental Results

4.1. Implementation

In tradition PDE-based methods, a certain diffusion term is usually included in the evolution equation to regularize the evolving function, but which increases the computational cost. Recently, [16] proposed a novel scheme to regularize the evolving function, that is, Gaussian filtering the evolving function after each of iterations. We adopt this scheme to regularize the evolving function 𝑢 at each of iteration; that is, 𝜙^𝑛=𝐺_𝜌∗𝜙^𝑛, where 𝑝 controls the regularization strength. Such regularization procedure has some advantages over the traditional regularization term, such as computational efficiency and better smoothing effect; see [16] for more explanations.

The main procedures of the proposed algorithm can be summarized as follows.(1)Initialize the evolving function 𝜙⁰(𝐱). (2)Compute 𝑐_𝑖(𝜙) and 𝑓_𝑖(𝜙), (𝑖=1,2) using (10).(3)Evolve the function 𝜙 according to (11).(4)Regularize the function 𝜙 with a Gaussian filter, that is; 𝜙^𝑛=𝐺_𝜌∗𝜙^𝑛. (5)Check whether the evolution is finished. If not, return to step 2.

4.2. Experimental Results

In this section, we show experimentally that the proposed model not only can provide desirable segmentation results in the presence of intensity inhomogeneity but also allows for more flexible initialization of the contour compared to the RSF and LIF models.

In our numerical experiments, for our model, we choose the parameters as follows: 𝜀=1.5 for the regularized Dirac function, Δ𝑡=0.025 (time step), ℎ=1 (space step), 𝜔=0.1, and 𝑛=5. The MATLAB source code of the LIF model algorithm is downloaded at http://www4.comp.polyu.edu.hk/~cslzhang/code/LIF.zip, and the RSF model algorithm is downloaded at http://www.engr.uconn.edu/~cmli/code/RSF_v0_v0.1.rar. All experiments were run under MATLAB 2010b on a PC with Dual 2.7 GHz processor.

We will utilize two region overlap metrics to evaluate the performances of the three models quantitatively. The two metrics are the ratio of segmentation error (RSE) [18] and the dice similarity coefficient (DSC) [14, 18, 19]. If 𝑆₁ and 𝑆₂ represent a given baseline foreground region (e.g., true object) and the foreground region found by the model, respectively, then the two metrics are defined as follows: 𝑁𝑆RSE=¹⧵𝑆₂+𝑁𝑆₂⧵𝑆₁𝑁(Ω),𝑆DSC=2𝑁₁∩𝑆₂𝑁𝑆₁+𝑁𝑆₂, (42) where 𝑁(⋅) indicates the number of pixels in the enclosed region, and Ω is image domain. The closer the DSC value to 1, the better the segmentation. Since 𝑁(𝑆₁⧵𝑆₂)+𝑁(𝑆₂⧵𝑆₁) is the number of the pixels mistakenly classified by the model, lower RSE means that there are fewer pixels misclassified; that is, the image can be segmented more accurately. Thus, a perfect segmentation will give DSC=1, RSE=0.

In the first example (Figures 1–3), we mainly verify the computation of a unique steady state of the zero-level line of 𝜙, starting with three types of representative initial functions, that is, signed distance function, piecewise constant functions by (26)-(27), and zero function, respectively. The top row of Figure 1 shows the evolution of an active contour (i.e., zero-level line {𝜙=0}), with the function 𝜙 initialized to a signed distance function, piecewise constant functions by (26)-(27) with 𝜌=1, and zero function, respectively, while the bottom row shows the corresponding evolution of 𝜙. With such initializations, the zero-level line of 𝜙 converges to the same steady state.

Figure 1 

Segmentations of our model for two real images with 𝜙 initialized as different functions. Columns 1 to 4: signed distance function, piecewise constant functions by (26)-(27) with 𝜌=1, and zero function.

Figure 2 

Applications of our model to four images (slug, cell, ventriculus sinister MR, and real plane images). The curve evolution process from the initial contour (in the first column) to the final contour (in the fourth column) is shown in every row for the corresponding image.

Figure 3 

Comparisons of three models. Rows 1–3: RSF model, LIF model, and our model with 𝜙0=0. For the RSF and LIF models, initial and final contours are shown in green and red color, respectively. Columns 1–4: the 4 pictures indicate (a), (b), (c), (d), respectively.

In Figure 2, we show that our model has the capability of detecting multiple objects or objects with interior holes or blurred edges, only starting with a zero function. The contours (zero-level lines) evolution processes are shown in the second column to the forth column.

We choose the segmentation results of the RSF and LIF models as baseline foreground regions and then compute DSC values for the corresponding images. The RSE and DSC values for the four real images are given in Table 1, from which we can see that the proposed algorithm works as well as the RSF and LIF models for images with intensity inhomogeneity.

Table 1 

RSE and DSC for RSF and LIF models and our model.

The experimental results shown in Figure 4 validate that our method can also achieve subpixel segmentation accuracy as in [15]. As can be seen from Figures 4(b) and 4(d), both models achieve subpixel segmentation accuracy of the finger boundaries. The final contour accurately reflects the true hand shape.

Figure 4 

Segmentation results of both models for a hand phantom. Upper row: LIF model. Lower row: our model with 𝜙0=0. Column 2: zoomed view of the narrow parts in hand fingers.

5. Conclusion

In this study, we propose a new active contour model integrating a local intensity fitting (LIF) energy with an auxiliary global intensity fitting (GIF) energy. The LIF energy is responsible for attracting the contour toward object boundaries and is dominant near object boundaries, while the GIF energy incorporates global image information to improve the robustness to initialization of the contours. The proposed model can efficiently handle intensity inhomogeneity, while allowing for more flexible initialization and maintaining the subpixel accuracy. The utility model has the advantages of allowing for more flexible initialization of the contour and the capability of detecting multiple objects or objects with interior holes or blurred edges. But [14] has not been implemented.

Appendices

A. Proofs of Theorem 1 and Corollary 2

Proof of Theorem 1. Clearly, {𝜙>0}≠Φ and {𝜙<0}≠Φ, owing to the sign-changing property of 𝜙(𝑥,𝑦). Thus, by (3), we have 𝑐₁𝜙=𝑔₁𝑛+𝑔₂(𝑁−𝑛)𝑁=𝑔₁−𝑔₂𝑛+𝑔₂𝑁𝑁,𝑐₂𝜙=𝑔₁(𝑚−𝑛)+𝑔₂[(𝑀−𝑁)−(𝑚−𝑛)]=𝑔𝑀−𝑁₁−𝑔₂(𝑚−𝑛)+𝑔₂(𝑀−𝑁),𝑀−𝑁 (A.1) and so, 𝑐₁𝜙−𝑐₂𝜙=(𝑀−𝑁)𝑔₁−𝑔₂𝑛+𝑔₂𝑁−𝑁𝑔₁−𝑔₂(𝑚−𝑛)+𝑔₂(×(𝑁(𝑀−𝑁)𝑀−𝑁))⁻¹=𝑔₁−𝑔₂(𝑀𝑛−𝑁𝑚)𝑁().𝑀−𝑁 (A.2) Therefore, in 𝜔, we get 𝐹^𝐺=𝑐₁𝜙−𝑐₂𝜙𝑔₁−𝑐₁𝜙𝐻_𝜀𝜙+𝑔₁−𝑐₂𝜙1−𝐻_𝜀𝜙=𝑔₁−𝑔₂(𝑀𝑛−𝑁𝑚)𝑁()⋅𝑔𝑀−𝑁¹−𝑔₂𝑁()×)𝐻𝑀−𝑁(𝑀−𝑁)(𝑁−𝑛_𝜀+𝑁((𝑀+𝑛)−(𝑚+𝑁))1−𝐻_𝜀=(𝑀𝑛−𝑁𝑚)×𝑔₁−𝑔₂²𝑁²(𝑀−𝑁)²×(𝑀−𝑁)(𝑁−𝑛)𝐻_𝜀+𝑁((𝑀+𝑛)−(𝑚+𝑁))1−𝐻_𝜀, (A.3) and in Ω⧵𝜔, we have 𝐹^𝐺𝜙=𝑐₁𝜙−𝑐₂𝜙𝑔₂−𝑐₁𝜙𝐻_𝜀𝜙+𝑔₂−𝑐₂𝜙1−𝐻_𝜀𝜙=𝑔₁−𝑔₂(𝑀𝑛−𝑁𝑚)𝑁()×−𝑔𝑀−𝑁₁−𝑔₂×𝑛(𝑀−𝑁)𝐻_𝜀𝜙+𝑁(𝑚−𝑛)1−𝐻_𝜀𝜙×(𝑁(𝑀−𝑁))⁻¹=−(𝑀𝑛−𝑁𝑚)𝑔₁−𝑔₂²𝑁²(𝑀−𝑁)²×𝑛(𝑀−𝑁)𝐻_𝜀𝜙𝑁(𝑚−𝑛)1−𝐻_𝜀𝜙. (A.4) This completes the proof of (14). The last assertion (15) follows clearly from the following fact: (𝑀−𝑁)(𝑁−𝑛)𝐻_𝜀𝜙+𝑁((𝑀+𝑛)−(𝑚+𝑁))1−𝐻_𝜀𝜙≥𝐻_𝜀𝜙+1−𝐻_𝜀𝜙≥1𝑛(𝑀−𝑁)𝐻_𝜀𝜙+𝑁(𝑚−𝑛)1−𝐻_𝜀𝜙≥𝐻_𝜀𝜙+1−𝐻_𝜀𝜙≥1. (A.5)(A.6) The proof of Theorem 1 is completed.

Now, we give the proof of Corollary 2. (i) Since 𝑀²=(𝑁+(𝑀−𝑁))²=𝑁²+(𝑀−𝑁)²()(),+2𝑁𝑀−𝑁≥4𝑁𝑀−𝑁 (A.7) by (14), (A.5), and (A.7) we have 𝐹^𝐺𝜙=(𝑀𝑛−𝑁𝑚)𝑔₁−𝑔₂²×(𝑀−𝑁)(𝑁−𝑛)𝐻_𝜀+𝑁((𝑀+𝑛)−(𝑚+𝑁))1−𝐻_𝜀×𝑁²(𝑀−𝑁)²⁻¹≥()𝑔𝑀𝑛−𝑁𝑚¹−𝑔₂²𝑀⁴/4,in𝜔, (A.8) and by (14) and (A.6), 𝐹^𝐺𝜙=−(𝑀𝑛−𝑁𝑚)𝑔₁−𝑔₂²×𝑛(𝑀−𝑁)𝐻_𝜀+𝑁(𝑚−𝑛)1−𝐻_𝜀×𝑁²(𝑀−𝑁)²⁻¹()𝑔≤−𝑀𝑛−𝑁𝑚¹−𝑔₂²𝑀⁴/4,inΩ⧵𝜔. (A.9) This completes the proof of (17), and similarly for (18). The proof is completed.

B. Proofs of Theorem 3 and Corollary 4

Proof of Theorem 3. By (6), we have 𝑓₁(𝐱)=∫_𝜙>0𝐾_𝜎(𝐱−𝜉)𝐼(𝜉)𝑑𝜉∫_𝜙>0𝐾_𝜎()=𝐱−𝜉𝑑𝜉_{𝜙>0∩𝜔}𝐾_𝜎()𝐼(𝜉)+𝐱−𝜉𝑑𝜉_{⧵𝜙>0𝜙>0∩𝜔}𝐾_𝜎()𝐼(𝜉)×𝐱−𝜉𝑑𝜉_𝑢>𝐼𝐾_𝜎(𝐱−𝜉)𝑑𝜉⁻¹=𝑔_{1𝜙>0∩𝜔}𝐾_𝜎(𝐱−𝜉)𝑑𝜉+𝑔_{2⧵𝜙>0𝜙>0∩𝜔}𝐾_𝜎()×𝐱−𝜉𝑑𝜉_𝑢>𝐼𝐾_𝜎(𝑥−𝜉)𝑑𝜉⁻¹=𝑔₁𝑞+𝑔₂𝑄−𝑞𝑄,𝑓₂(𝐱)=∫_𝜙≤0𝐾_𝜎(𝐱−𝜉)𝐼(𝜉)𝑑𝜉∫_𝜙≤0𝐾_𝜎()=𝐱−𝜉𝑑𝜉_{𝜙≤0∩𝜔}𝐾_𝜎()𝐼(𝜉)+𝐱−𝜉𝑑𝜉_{⧵𝜙≤0𝜙≤0∩𝜔}𝐾_𝜎()𝐼(𝜉)×𝐱−𝜉𝑑𝜉_𝜙≤0𝐾_𝜎(𝐱−𝜉)𝑑𝜉⁻¹=𝑔_{1𝜙≤0∩𝜔}𝐾_𝜎(𝐱−𝜉)𝑑𝜉+𝑔_{2⧵𝜙≤0𝜙≤0∩𝜔}𝐾_𝜎()×𝐱−𝜉𝑑𝜉_𝜙≤0𝐾_𝜎(𝐱−𝜉)𝑑𝜉⁻¹=𝑔₁𝑝−𝑞+𝑔₂(𝑃−𝑄)−𝑝−𝑞,𝑃−𝑄 (B.1)(B.2) and so, 𝑓₁(𝐱)−𝑓₂(𝐱)=𝑔₁−𝑔₂𝑃𝑞−𝑄𝑝𝑄().𝑃−𝑄 (B.3) Therefore, in 𝜔, we get 𝐹^𝐿𝜙=𝑚₁𝜙−𝑚₂𝜙×𝑔₁−𝑚₁𝜙𝐻_𝜀𝜙+𝑔₁−𝑚₂𝜙1−𝐻_𝜀𝜙=𝑃𝑞−𝑄𝑝𝑔₁−𝑔₂²×(𝑃−𝑄)𝑄−𝑞𝐻_𝜀𝜙+𝑄𝑃+𝑞−𝑝+𝑄1−𝐻_𝜀𝜙×𝑄²(𝑃−𝑄)²⁻¹, (B.4) and in Ω⧵𝜔, we have 𝐹^𝐿=𝑚₁𝜙−𝑚₂𝜙×𝑔₂−𝑚₁𝜙𝐻_𝜀𝜙+𝑔₂−𝑚₂𝜙1−𝐻_𝜀𝜙=−𝑃𝑞−𝑄𝑝𝑔₁−𝑔₂²×𝑞(𝑃−𝑄)𝐻_𝜀𝜙+𝑄𝑝−𝑞1−𝐻_𝜀𝜙×𝑄²(𝑃−𝑄)²⁻¹. (B.5) This completes the proof of (19). The last assertion (20) follows clearly from the following fact: (𝑃−𝑄)𝑄−𝑞𝐻_𝜀𝜙+𝑄𝑃+𝑞−𝑝+𝑄1−𝐻_𝜀𝜙=𝐻_𝜀𝜙_Ω−_𝜙>0𝐾_𝜎()⋅𝐱−𝜉𝑑𝜉_𝜙>0−_{𝜙>0∩𝜔}𝐾_𝜎()+𝐱−𝜉𝑑𝜉1−𝐻_𝜀𝜙_𝜙>0𝐾_𝜎()⋅𝐱−𝜉𝑑𝜉_Ω+_{𝜙>0∩𝜔}−_𝜔+_𝜙>0𝐾_𝜎(𝐱−𝜉)𝑑𝜉=𝐻_𝜀𝜙_𝜙≤0𝐾_𝜎()⋅𝐱−𝜉𝑑𝜉_{⧵𝜙>0𝜙>0∩𝜔}𝐾_𝜎()+𝐱−𝜉𝑑𝜉1−𝐻_𝜀𝜙×_𝜙>0𝐾_𝜎(𝐱−𝜉)𝑑𝜉⋅_{⧵𝜙≤0𝜙≤0∩𝜔}𝐾_𝜎(𝐱−𝜉)𝑑𝜉≥𝐻_𝜀𝜙_𝜙≤0𝑆𝑑𝜉⋅_{⧵𝜙>0𝜙>0∩𝜔}+𝑆𝑑𝜉1−𝐻_𝜀𝜙_𝜙>0𝑆𝑑𝜉⋅_{⧵𝜙≤0𝜙≤0∩𝜔})𝐻𝑆𝑑𝜉=𝑆(𝑀−𝑁)(𝑁−𝑛_𝜀𝜙+𝑁((𝑀−𝑁)−(𝑚−𝑛))1−𝐻_𝜀𝜙≥𝑆,𝑞(𝑃−𝑄)𝐻_𝜀𝜙+𝑄𝑝−𝑞1−𝐻_𝜀𝜙=𝐻_𝜀𝜙_{𝜙>0∩𝜔}𝐾_𝜎(𝐱−𝜉)𝑑𝜉⋅_𝜙≤0𝐾_𝜎()+𝐱−𝜉𝑑𝜉1−𝐻_𝜀𝜙×_𝜙>0𝐾_𝜎(𝐱−𝜉)𝑑𝜉⋅_{𝜙≤0∩𝜔}𝐾_𝜎(𝐱−𝜉)𝑑𝜉≥𝐻_𝜀𝜙_{𝑢>𝐼∩𝜔}𝑆𝑑𝜉⋅_𝑢≤𝐼+𝑆𝑑𝜉1−𝐻_𝜀𝜙_𝑢>𝐼𝑆𝑑𝜉⋅_{𝑢≤𝐼∩𝜔}𝑛()𝐻𝑆𝑑𝜉=𝑆𝑀−𝑁_𝜀𝜙+1−𝐻_𝜀𝜙𝑁(𝑚−𝑛)≥𝑆, (B.6)(B.7) where 𝑆=min{𝐾_𝜎(𝐱−𝜉)∶𝜉∈Ω)}.
Now, we give the proof of Corollary 4. (i) Since 𝑃²≥4𝑃(𝑃−𝑄), (B.8) we have by (19) and (B.6) 𝐹^𝐿𝜙=𝑃𝑞−𝑄𝑝𝑔₁−𝑔₂²×(𝑃−𝑄)𝑄−𝑞𝐻_𝜀𝜙+𝑄𝑃+𝑞−𝑝+𝑄1−𝐻_𝜀𝜙×𝑄²(𝑃−𝑄)²⁻¹≥𝑔𝑃𝑞−𝑄𝑝¹−𝑔₂²𝑃⁴/4,in𝜔, (B.9) and by (19) and (B.7), 𝐹^𝐿𝜙=−𝑃𝑞−𝑄𝑝𝑔₁−𝑔₂²×𝑞(𝑃−𝑄)𝐻_𝜀𝜙+𝑄𝑝−𝑞1−𝐻_𝜀𝜙×𝑄²(𝑃−𝑄)²⁻¹𝑔≤−𝑃𝑞−𝑄𝑝¹−𝑔₂²𝑃⁴/4,inΩ⧵𝜔. (B.10) This completes the proof of (19), and similarly to (20). The proof is completed.

C. Proof of Corollary 5

Proof. (i) By Corollary 2 (i), we have 𝐹^𝐺𝜙⁰≥𝑀𝑛₀−𝑁₀𝑚𝑔₁−𝑔₂²𝑀⁴𝐹/4=𝐴>0,in𝜔,_𝐺𝜙⁰≤−𝑀𝑛₀−𝑁₀𝑚𝑔₁−𝑔₂²𝑀⁴/4=−𝐴<0,inΩ⧵𝜔. (C.1) At the first iteration, by (25) and (C.1), we have 𝜙¹_𝑖,𝑗=𝜙⁰_𝑖,𝑗+Δ𝑡⋅𝐹^𝐺𝜙⁰_𝑖,𝑗≥𝜙⁰_𝑖,𝑗+Δ𝑡⋅𝛿_𝜀𝜙⁰𝐴,in𝜔,≤𝜙⁰_𝑖,𝑗−Δ𝑡⋅𝛿_𝜀𝜙⁰𝐴,inΩ⧵𝜔, (C.2) or equivalently, 𝜙¹(𝐱)≥𝜙⁰(𝐱)+Δ𝑡⋅𝛿_𝜀𝜙⁰𝐴>𝜙⁰(𝐱)𝜙,in𝜔,₁(𝐱)≤𝜙⁰(𝐱)−Δ𝑡⋅𝛿_𝜀𝜙⁰𝐴<𝜙⁰(𝐱),inΩ⧵𝜔, (C.3) which implies that 𝑀𝑛₁−𝑁₁𝑚=|Ω|𝜔∩𝜙¹>0−𝜙¹𝜔|=|+|𝜔𝜙>0|(|Ω⧵𝜔|)𝜔∩¹−𝜙>0𝜔∩¹+)∩𝜙>0(Ω⧵𝜔¹|𝜔|=|𝜙>0Ω⧵𝜔|𝜔∩¹−)∩𝜙>0(Ω⧵𝜔¹𝜔|>|𝜙>0|Ω⧵𝜔|𝜔∩⁰−)∩𝜙>0(Ω⧵𝜔⁰𝜔|=|𝜙>0|Ω⧵𝜔|𝜔∩⁰+𝜙>0𝜔∩⁰𝜔|−𝜙>0|𝜔∩⁰𝜔|−)∩𝜙>0|(Ω⧵𝜔⁰𝜔|=|+|𝜔𝜙>0|(|Ω⧵𝜔|)𝜔∩⁰−𝜙>0𝜔∩⁰+)∩𝜙>0(Ω⧵𝜔⁰|𝜔|=|Ω𝜙>0|𝜔∩₀>0−𝜙₀>0|𝜔|=𝑀𝑛₀−𝑁₀𝑚>0. (C.4) In a manner similar to the steps used to obtain (26)-(27) and (C.1)– (C.4), we obtain 𝐹^𝐺𝐼,𝜙¹≥𝑀𝑛₁−𝑁₁𝑚𝑔₁−𝑔₂²𝑀⁴>/4𝑀𝑛₀−𝑁₀𝑚𝑔₁−𝑔₂²𝑀⁴𝐹/4=𝐴>0,in𝜔,_𝐺𝐼,𝜙¹≤−𝑀𝑛₁−𝑁₁𝑚𝑔₁−𝑔₂²𝑀⁴/4<−𝑀𝑛₀−𝑁₀𝑚𝑔₁−𝑔₂²𝑀⁴𝜙/4=−𝐴<0,inΩ⧵𝜔,₂(𝐱)≥𝜙¹(𝐱)+Δ𝑡⋅𝛿_𝜀𝜙¹𝐴>𝜙¹(𝐱)𝜙,in𝜔₂(𝐱)≤𝜙¹(𝐱)−Δ𝑡⋅𝛿_𝜀𝜙¹𝐴<𝜙¹(𝐱),inΩ⧵𝜔,𝑀𝑛₂−𝑁₂𝑚>𝑀𝑛₁−𝑁₁𝑚>𝑀𝑛₀−𝑁₀𝑚>0. (C.5)

D. Proof of Theorem 7

Proof. We prove (32), and similarly to (33). By (25), Corollary 5(i), and Corollary 6(i), we get 𝜙^𝑘+1_𝑖,𝑗=𝜙^𝑘_𝑖,𝑗+Δ𝑡⋅𝐹𝜙^𝑘_𝑖,𝑗=𝜙^𝑘_𝑖,𝑗⋅()𝐹+Δ𝑡1−𝜔^𝐿𝜙^𝑘_𝑖,𝑗+𝜔𝐹^𝐺𝜙^𝑘_𝑖,𝑗≥𝜙^𝑘_𝑖,𝑗⋅()+Δ𝑡1−𝜔𝐵⋅𝛿_𝜀𝜙^𝑘+1+𝜔𝐴⋅𝛿_𝜀𝜙^𝑘+1=𝜙^𝑘_𝑖,𝑗+((1−𝜔)𝐵+𝜔𝐴)⋅𝛿_𝜀𝜙^𝑘+1⋅Δ𝑡,in𝜔𝑘≥1,𝑘∈𝑧⁺,≤𝜙^𝑘_𝑖,𝑗⋅−+Δ𝑡1−𝜔𝐵⋅𝛿_𝜀𝜙^𝑘+1+𝜔𝐴⋅𝛿_𝜀𝜙^𝑘+1=𝜙^𝑘_𝑖,𝑗−((1−𝜔)𝐵+𝜔𝐴)⋅𝛿_𝜀𝜙^𝑘+1⋅Δ𝑡,inΩ⧵𝜔𝑘≥1,𝑘∈𝑧⁺, (D.1) which implies that 𝜙^𝑘+1_𝑖,𝑗≥𝜙⁰_𝑖+((1−𝜔)𝐵+𝜔𝐴⋅)⋅^𝑘+1_𝑖=0𝛿_𝜀𝜙^𝑖𝜙⋅Δ𝑡,in𝜔,_{𝑘+1𝑖,𝑗}≤𝜙⁰_𝑖−(𝑘+1)𝐴((1−𝜔)𝐵+𝜔𝐴⋅)⋅^𝑘+1_𝑖=0𝛿_𝜀𝜙^𝑖⋅Δ𝑡,inΩ⧵𝜔. (D.2) Because 𝜙₀(𝐱) is bounded in domain Ω, there clearly exists a positive integer 𝐾 such that 𝜙^𝑘+1_𝑖,𝑗𝜙>0,in𝜔,_{𝑘+1𝑖,𝑗}<0,inΩ⧵𝜔,for𝑘≥𝐾. (D.3) The proof is completed.

E. Proof of Theorem 8

Proof. From (11), we have 𝐹(0)=𝛿_𝜀(0)𝜔𝐼−𝐼^𝐺⋅2mean_𝐱∈Ω𝐼(𝐱)+(1−𝜔)𝐼−𝐼^𝐿⋅2mean_𝐱∈Ω𝐼(𝐱)=2𝛿_𝜀(0)mean_𝐱∈Ω𝐼(𝐱)𝐼−2𝐻_𝜀(0)mean_𝐱∈Ω𝐼(𝐱)=𝛿_𝜀(0)mean_𝐱∈Ω𝐼(𝐱)𝐼−mean_𝐱∈Ω𝐼(𝐱). (E.1) Clearly, min_𝐱∈Ω𝐼(𝐱)<mean_𝐱∈Ω𝐼(𝐱)<max_𝐱∈Ω𝐼(𝐱) (E.2) which implies that sign(𝐹(0))=−1,𝐱∈Ω₁,1,𝐱∈Ω₂, (E.3) where Ω₁=𝐼(𝐱)<mean_𝐱∈Ω𝐼(𝐱,Ω₂)=𝐼(𝐱)>mean_𝐱∈Ω𝐼(𝐱). (E.4) The proof of Theorem 8 is completed.

Acknowledgments

The authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve this paper. Besides, this work was supported by the Fundamental Research Funds for the Central University, Grant no. (CDJXS10100006).