2854

IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 23, NO. 7, JULY 2014

Improving Level Set Method for Fast Auroral Oval Segmentation Xi Yang, Xinbo Gao, Senior Member, IEEE, Dacheng Tao, Senior Member, IEEE, and Xuelong Li, Fellow, IEEE Abstract— Auroral oval segmentation from ultraviolet imager images is of significance in the field of spatial physics. Compared with various existing image segmentation methods, level set is a promising auroral oval segmentation method with satisfactory precision. However, the traditional level set methods are time consuming, which is not suitable for the processing of large aurora image database. For this purpose, an improving level set method is proposed for fast auroral oval segmentation. The proposed algorithm combines four strategies to solve the four problems leading to the high-time complexity. The first two strategies, including our shape knowledge-based initial evolving curve and neighbor embedded level set formulation, can not only accelerate the segmentation process but also improve the segmentation accuracy. And then, the latter two strategies, including the universal lattice Boltzmann method and sparse field method, can further reduce the time cost with an unlimited time step and narrow band computation. Experimental results illustrate that the proposed algorithm achieves satisfactory performance for auroral oval segmentation within a very short processing time. Index Terms— Auroral oval segmentation, shape knowledge, reinitialization, lattice Boltzmann method, sparse field method.

I. I NTRODUCTION

S

OLAR wind collides with charged particles from the earth’s magnetosphere, which forms the aurora with

Manuscript received July 14, 2013; revised December 10, 2013 and April 17, 2014; accepted April 18, 2014. Date of publication May 2, 2014; date of current version May 20, 2014. This work was supported in part by the National Basic Research Program of China (973 Program) under Grant 2012CB316400, in part by the National Natural Science Foundation of China under Grant 61125204, Grant 61125106, Grant 61172146, and Grant 41031064, in part by the Australian Research Council Project under Grant DP-140102164, in part by the Fundamental Research Funds for the Central Universities under Grant K5051202048, Grant BDZ021403, and Grant JB149901, in part by the Ocean Public Welfare Scientific Research Project, State Oceanic Administration of the People’s Republic of China under Grant 201005017, in part by the Microsoft Research Asia Project based Funding under Grant FY13-RES-OPP-034, in part by the Program for Changjiang Scholars and Innovative Research Team in the University of China under Grant IRT13088, and in part by the Shaanxi Innovative Research Team for Key Science and Technology under Grant 2012KCT-02. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Chun-Shien Lu. X. Yang and X. Gao are with the State Key Laboratory of Integrated Services Networks, School of Electronic Engineering, Xidian University, Xi’an 710071, China (e-mail: [email protected]; [email protected]). D. Tao is with the Centre for Quantum Computation & Intelligent Systems and the Faculty of Engineering and Information Technology, University of Technology, Sydney, 235 Jones Street, Ultimo, NSW 2007, Australia (e-mail: [email protected]). X. Li is with the Center for OPTical IMagery Analysis and Learning (OPTIMAL), State Key Laboratory of Transient Optics and Photonics, Xi’an Institute of Optics and Precision Mechanics, Chinese Academy of Sciences, Xi’an 710119, China (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIP.2014.2321506

Fig. 1. Examples of UVI aurora images. (a) Full oval images. (b) Gap oval images.

significant value in observing space physical activities [5]. The aurora is regarded as a mirror reflecting the invisible coupling between atmospheric layers, and thus it carries important information for the research on climate changes, such as global warming. Since the study of aurora is significant for scientific research and human life, scientists have used various methods to observe it. Not like polar research stations capturing aurora images from the earth’s surface [44], the “Polar” satellite acquires them from outerspace with the use of ultraviolet imager (UVI) [4]. UVI is a 2D spatial imager which produces images of the earth’s auroral regions in the far ultraviolet wavelength range (130 nm to 190 nm). It has the capability to detect and supply images of very dim emissions with a wavelength resolution never achievable before. The resulting UVI aurora images are generated once every 37 seconds with an angular resolution of 0.036 degrees. The auroral oval [24] in UVI images is an annular ring shape region representing the instantaneous distribution of auroras. It reflects the principle of aurora particle settlings, and its morphological characteristics imply important information of space physical activities. For example, the area changes of polar cap (region within the inner boundary of the auroral oval) reflect the energy storage/release processes in polar magnetosphere. Thus, to explore the ionosphere-thermospheremagnetosphere (ITM) behaviors, it is a must to first segment the auroral oval region. According to the wholeness of auroral oval, UVI aurora images can be categorized into full oval images and gap oval images. Fig. 1 shows examples of these two kinds of UVI aurora images.

1057-7149 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

YANG et al.: IMPROVING LEVEL SET METHOD FOR FAST AURORAL OVAL SEGMENTATION

2855

Fig. 2. Auroral oval segmentation results using the existing methods. (a) Original image. (b) HKM. (c) AMET. (d) PCNN. (e) LLS-RHT. First two rows are results of full oval images, and last two rows are results of gap oval images.

However, auroral oval segmentation is a challenging task. The existing auroral oval segmentation methods include the histogram-based k-means (HKM) method [23], the adaptive minimum error thresholding (AMET) method [28], the pulsecoupled neural network (PCNN) method [17] and the linear least squares based randomized Hough transform (LLS-RHT) method [6]. The first three methods distinguish the auroral oval from background purely depending on the pixel intensities and spatial relationships between pixels. Since the UVI aurora images are of low contrast and contain heavy noise interference, they are easy to regard the background points or noise points as auroral oval. The LLS-RHT method explores the shape information of auroral oval and obtains the dualellipses results. Although it overcomes the shortcoming of noise interference in many existing methods, when comes to gap oval images, its strong ellipse shape constraint loses effect and causes large deviation from the actual situation because of the lack of sampling points. Also, the smooth inner boundary LLS-RHT obtained conflicts with the fact that there are some bulges on auroral ovals. Fig. 2 shows some segmentation results using the existing auroral oval segmentation methods, which demonstrates their problems including low in accuracy, vulnerable to noise interference, and inapplicable to gap oval images. Level set, an implicit active contour model [14], [26], is a promising method for auroral oval segmentation comparing with the existing methods, for it combines the boundary curve information and the aurora image’s pixel information into an energy minimization problem. By initializing the evolving curve as zero set of the level set function, the points inside the evolving curve are regarded as auroral oval and their value are positive, while the outside are background and their value are negative. Then, the curve evolves driven by a partial differential equation (PDE) [38] until convergence [35]. However, the implementation of level set methods is complex and time consuming, which limits their application for mass aurora images segmentation. The high computational

complexity is mainly due to the following four reasons. Firstly, an improper initialization of evolving curve may lead to the increase in the number of iterations, while an initial evolving curve close to the object boundary needs much less iterations than a faraway one. Secondly, it is necessary to keep the level set function as a signed distance function periodically to ensure a stable and precise evolution result. Such a cumbersome reinitialization procedure is definitely a waste of time. Thirdly, the numerical scheme to solve the PDE such as upwind finite difference scheme [32], [36] and finite difference scheme [29] must satisfy the Courant-Friedrichs-Lewy (CFL) condition [42] for numerical stability, which limits the length of time step in each iteration. Finally, the computation is carried out on the full image domain in traditional level set methods. Nevertheless, pixels far away from the evolving curve are meaningless for obtaining the object boundary, and thus their computation results in the increase of total processing time. In this paper, an improving level set algorithm for fast auroral oval segmentation is proposed by solving the aforementioned four problems. Firstly, a saliency morphological map (SMM) is constructed as the shape knowledge for auroral oval from the view of human beings. By regarding its contour which is very close to the auroral oval’s boundary as the initial evolving curve, the number of iterations can be decreased a lot. Secondly, a penalty term is added to the energy functional to ensure the level set function as a signed distance function, thus avoiding the cumbersome re-initialization procedure. Thirdly, a universal lattice boltzmann method (LBM) which has an external force term is proposed to solve the PDE. Because there is no strong time step constraint in LBM, the convergence speed of level set evolution can be accelerated. Finally, by using the sparse field method (SFM), the computation is constricted to a narrow band which is near to the evolving curve. By avoiding the calculation of meaningless points, the CPU time can be reduced greatly. Compared with existing auroral oval segmentation methods, the proposed algorithm improves the segmentation performance greatly. Unlike existing methods

2856

IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 23, NO. 7, JULY 2014

which either only take the pixel intensity into consideration or rely on a strong shape constraint, our algorithm treats the shape knowledge as the initial curve for level set evolution. This soft shape constraint can maintain the auroral oval’s shape meanwhile avoid the loss of detailed information. Additionally, we do not exploit elliptic fitting to extract the shape knowledge from gap oval images, and thus avoid the huge fitting error. The main contributions of this paper are summarized below. 1) We Design a Shape Knowledge Based Initial Evolving Curve: By incorporating the shape knowledge of auroral oval, the proposed initial evolving curve can improve not only the segmentation accuracy, but also efficiency for level set evolution. 2) We Present a Neighbor Embedded Level Set Formulation: The proposed level set formulation takes the neighbor information into account, making its edge stop function and additional speed function adaptive to the auroral oval’s intensity. Therefore, the annoying boundary leakage phenomenon traditional level set method suffered is avoided and the segmentation performance is further improved. Moreover, the adopted penalty term makes the time-consuming re-initialization procedure unnecessary, which accelerates the segmentation process. 3) We Develop a Universal LBM to Efficiently Solve the Level Set Equation: By adding an external force term, the proposed universal LBM can be used to efficiently solve various level set equations without the constraint of time step. Furthermore, we also derive the corresponding parametric form of geodesic active contour model and the proposed level set formulation in LBM equation for their detailed implementation. 4) Compared With the Existing Auroral Oval Segmentation Techniques and Traditional Level Set Methods, the Proposed Algorithm Is High in Accuracy and Fast in Speed: By introducing the shape knowledge and neighbor information into the level set framework, the proposed algorithm can conquer the shortcomings of other methods including the boundary leakage, noise inference and unsuitability for gap oval images, thus achieving a high segmentation accuracy. In addition, thanks to the proposed shape knowledge based initial evolving curve, the neighbor embedded level set formulation, the universal LBM and the adopted SFM, the proposed algorithm breaks the bottleneck of low convergence speed in traditional level set methods and achieves a fast segmentation speed. The remainder of this paper is organized as follows. In Section II, the background of level set methods is reviewed. Section III presents the proposed fast level set algorithm for auroral oval segmentation. The experimental results and analysis are presented in Section IV, and the final is conclusion. II. BACKGROUND A. Level Set Formulations Level set methods [2], [10], [16], [19], [40], [41] implicitly represent the planar closed curve C by the zero level set of

the level set function (LSF) φ(x, y, t), i.e., C(t) = {(x, y)|φ(x, y, t) = 0}.

(1)

The level set equation (LSE) [35] which represents the evolution of LSF can be written as ∂φ + F|∇φ| = 0, ∂t

(2)

where F denotes the speed of the evolution. Caselles et al. proposed the famous geodesic active contour model [8] as ∇φ ∂φ = |∇φ|(divg( ) + νg), ∂t |∇φ|

(3)

where ν is a constant coefficient, g is the edge stop function (ESF) defined by g=

1 , 1 + |∇G σ ∗ I |2

(4)

and I represents the image, G σ is the Gaussian kernel with standard deviation σ . The variational level set methods [29], [30], [33] treat the evolution of LSE as a problem of minimizing certain energy functional defined on the LSF, i.e., ∂E ∂φ =− . ∂t ∂φ

(5)

Therefore, by adding different energy term which represents certain information such as smoothness, the evolution of LSE can change flexibly according to different purposes.

B. Numerical Scheme The traditional level set methods often use upwind scheme [32], [36] for numerical implementation. However, this numerical scheme is complex and computational expensive. Recently, a simple finite difference scheme [9], [29] is applied to the level set methods. In finite difference scheme, the spatial partial derivatives ∂φ/∂ x and ∂φ/∂y in LSE are approximated by the central difference, while the temporal partial derivative ∂φ/∂t is approximated by the forward difference. Therefore, the LSE can be discretized as the following form. k k φi,k+1 j = φi, j + t L(φi, j ),

(6)

where (i, j ) is the spatial index, k is the temporal index, t is the time step, and L(φi,k j ) is the approximation of the right hand side in the evolution equations. To keep numerical stability and obtain accurate approximation results, the time step t must satisfy the CourantFriedrichs-Lewy (CFL) condition [42], i.e., μt < h 2 /4,

(7)

where μ is one of the coefficient in LSE, and h is the space step.

YANG et al.: IMPROVING LEVEL SET METHOD FOR FAST AURORAL OVAL SEGMENTATION

Fig. 3.

2857

Construction of saliency morphological map.

C. Reasons for High Computational Complexity The traditional level set methods may cost tens of seconds to segment an image, which is unacceptable for mass image data processing. There are four reasons giving rise to the high computational complexity. 1) Improper Initial Evolving Curve: The initialization of evolving curve is important for level set methods. If an initial evolving curve is close to the object boundary, it needs much less iterations than a faraway one. Therefore, an improper initial evolving curve may cause the increase of processing time to obtain the final segmentation result. What’s more, an improper initial evolving curve may end up with a wrong result which is far away from the object boundary. 2) Cumbersome Reinitialization Process: The LSF φ(x, y, t) may develop very steep or flat shape during the evolution, which makes the result inaccurate. The traditional level set methods often use the reinitialization remedy to avoid this problem, that is, periodically initialize the LSF as a signed distance function, i.e., ∂φ/∂t = sign(φ0 )(1 − |∇φ|),

(8)

where φ0 is the function to be re-initialized, and sign(φ) is the sign function. However, the re-initialization process is quite complicated, expensive and has an undesirable side effect of moving the zero level set away from its original location. Moreover, this process is conducted in an ad-hoc manner because there is no rule of when and how to re-initialize the LSF to a signed distance function. Some inaccurate segmentation results may occur during the re-initialization process too. 3) Restricted Numerical Scheme: As discussed before, the numerical scheme that the traditional level set methods used must satisfy the CFL condition, which limits the length of time step. Such a small time step will inevitably increase the number of iterations and lead to a high cost of time. 4) Full Domain Computation: The computation of classical level set methods is carried out on the full image domain, which means the value of φ for all of the pixels in an image must be updated in every iteration. However, the calculation of pixels far away from the evolving curve has no meaningful benefit for the approximation to the object boundary. The needless computations of these points will lead to the increase of total processing time.

Fig. 4.

Subdivision of aurora image based on MLT.

III. A FAST AURORAL OVAL S EGMENTATION A LGORITHM BASED ON L EVEL S ET In real applications, the auroral oval segmentation is often carried out in a huge database with hundreds of thousands of aurora images, and the speed of the traditional level set methods cannot satisfy the application demands. To address this problem, a fast level set algorithm for auroral oval segmentation is proposed in this section. Aiming at the four reasons for high computational complexity in traditional level set methods, we put forward four corresponding solutions to speed up the segmentation. A. Shape Knowledge Based Initial Evolving Curve To obtain a proper initial evolving curve, a saliency morphological map (SMM) which reflects the shape cognition of auroral oval from the human view is constructed. As the contour of SMM is close enough to the object boundary, it can be regarded as a proper initial evolving curve. Such a shape knowledge based initial evolving curve can surely decrease the number of iterations. Inspired by the visual saliency models [20], [22], [25] which depict the visual attention in human visual system (HVS), we construct the SMM for auroral oval images by performing the following steps as shown in Fig. 3. Firstly, a magnetic local time (MLT)-adaptive thresholding method is used to binarize the aurora image. The division of aurora image based on MLT is shown in Fig. 4, there are 24 zones converge in the geomagnetic pole, each zone spans 2h of MLT and has a 1h overlap with a neighboring zone. We use minimum error thresholding (MET) [27] in 24 different zones to distinguish between target and background. Then, isolated points are eliminated and region breakages are filled with a multi-structure element morphological filter. Morphological opening operation denoted in (9) is often used to break narrow isthmuses and eliminate isolated points, while closing denoted in (10) is used to eliminate small holes and

2858

IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 23, NO. 7, JULY 2014

B. Neighbor Embedded Level Set Formulation

Fig. 5. Shape knowledge based initial evolving curve. (a) For full oval image. (b) For gap oval image.

fill gaps in the contour. A ◦ B = (AB) ⊕ B,

(9)

A • B = (A ⊕ B)B,

(10)

where  and ⊕ are morphological erosion operator and dilation operator, respectively. To avoid the loss of detail information meanwhile eliminate noise points, a multi-structure element morphological filter is constructed as G = (((A ◦ Bi ) • Bi ) ◦ Bi+1 ) • Bi+1 . . . i = 1, . . . , n, (11) where B is the structure element with a disk shape and a gradually increasing scale (with start scale 3 and end scale 9). Therefore, the output morphological image Im can be obtained from the input aurora binary image Ib , i.e., Im = Ib ∗ G + Ir ∗ G,

(12)

where Ir is the residual part of Ib after convolving with the filter G. Finally, an elliptic fitting is applied to the full oval images. The output morphological image Im is good enough to describe the overall shape for gap oval images as the intensity of auroral oval is equally distributed. For full oval images, the intensity of some oval regions is much lower than other regions, thus Im cannot describe the shape cognition of full oval from human vision. To address this problem, elliptic fitting is operated on full oval images as a strong shape restraint, i.e., Ie = f i telli pses{ pi }, pi ∈ Im ,

(13)

where “ f i telli pses” represents the operation that after fitted an ellipse using pi , set two concentric ellipses with a preset distance (set as 30 in our experiments) from the obtained ellipse as the inner and outer boundary. Thus, the saliency morphological map (SMM) is constructed as  Im gap oval i mages (14) SM M = Ie f ull oval i mages. In practice, the contour of SMM which shown in Fig. 5 is regarded as the shape knowledge based initial evolving curve, and the initial level set function can be expressed as ⎧ ⎪ (x, y) ∈ S M M − ∂ S M M ⎪ ⎨d (15) φ0 = 0 (x, y) ∈ ∂ S M M ⎪ ⎪ ⎩−d (x, y) ∈  − S M M , where d is a positive constant,  is the image domain, and ∂ S M M is the contour of SMM.

We propose a neighbor embedded level set formulation to better segment auroral ovals with weak boundaries, and meanwhile solve the problem of cumbersome re-initialization process. Inspired by the work of Li et al. [29], [30], a penalty term is added to the energy functional to penalize the deviation of LSF φ from a signed distance function, thus avoiding the LSF φ being too sharp or flat and the re-initialization procedure is no longer needed. As far as we know, a signed distance function has the property |∇φ| = 1, and a simple penalty term which satisfies this property can be designed as  1 (|∇φ| − 1)2 d x d y. (16) P(φ) =  2 Therefore, the energy functional E(φ) is E(φ) = μP(φ) + E e (φ),

(17)

where μ > 0 is a parameter controlling the penalization extent, and E e (φ) is the external energy driving the motion of the zero level curve of φ, i.e., E e (φ) = λLe (φ) + v Ae (φ)  =λ gδ(φ)|∇φ|d x d y + v g H (−φ)d x d y, (18) 



where L e (φ) is the length of the interface with coefficient λ, Ae (φ) is the area of the region enclosed by zero level curve with coefficient v, δ(·) and H (·) are univariate Dirac function and Heaviside function, respectively. g is the edge stop function (ESF). In Li’s level set method [29], the ESF has the same form as (4) which only considers the gradient of an image, and the value of v which determines the additional evolution speed is set to a fixed constant in advance. As a result, the performance of auroral oval segmentation is poor because of lacking the neighbor information, and the result is sensitive to noise and easy to omit low gray level boundary. To this end, we propose a neighbor embedded ESF g(x, y) and additional speed function v(x, y) g(x, y) =

1 , 1 + Il (x, y)|∇G σ ∗ I |m

v(x, y) = k

1 , 1 + exp(−a(Il (x, y) − 0.5))

(19) (20)

where Il (x, y) is the mean intensity of pixel (x, y) and its eight neighborhoods after normalized to [0, 1], m is a positive constant which is set as 2 in this paper, k and a are constant coefficients (set as empirical values 5 and 12 in our experiments) controlling the strength and nonlinear degree of additional evolution speed, respectively. With the neighbor information in our g(x, y) and v(x, y), the case that some auroral oval regions with low gray level are omitted or over evolved because of a too fast evolution speed or a too big ESF can be avoided. Fig. 6 shows some segmentation results of Li’s level set method and the proposed neighbor embedded method, both of them use the proposed shape knowledge based initial evolving curve.

YANG et al.: IMPROVING LEVEL SET METHOD FOR FAST AURORAL OVAL SEGMENTATION

2859

Fig. 7.

Commonly used LBM structure. (a) D3Q19. (b) D2Q9.

C. Universal Lattice Boltzmann Method (LBM) for Numerical Scheme

Fig. 6. Results of auroral oval segmentation using Li’s level set method and the proposed neighbor embedded method. (a) Original image. (b) Li’s level set method. (c) The proposed neighbor embedded method.

Substituting (16), (18), (19) and (20) into (17), the energy functional can be rewritten as   1 2 (|∇φ| − 1) d x d y + λ gδ(φ)|∇φ|d x d y E(φ) = μ   2 g H (−φ)d x d y +v   1 =μ (|∇φ| − 1)2 d x d y 2   δ(φ)|∇φ| dxdy +λ y)|∇G σ ∗ I |m  1 + Il (x,   k H (−φ) (1 + exp(−a(Il (x, y) − 0.5))) + d x d y. 1 + Il (x, y)|∇G σ ∗ I |m  (21) By calculus of variations [15], the evolution equation can be obtained from (5) and (21), i.e., ∂φ ∇φ ∇φ = μ[φ − div( )] + λδ(φ)divg( ) + vgδ(φ) ∂t |∇φ| |∇φ| ∇φ = μ[φ − div( )] |∇φ| ∇φ ) +λδ(φ)div( 1 + Il (x, y)|∇G σ ∗ I |m |∇φ|  k (1 + exp(−a(Il (x, y) − 0.5))) +δ(φ) . (22) 1 + Il (x, y)|∇G σ ∗ I |m In conclusion, due to the penalty term P(φ), the level set function can be automatically kept as a signed distance function, thus avoiding the cumbersome re-initialization process. Meanwhile, the proposed neighbor embedded ESF g(x, y) and additional speed function ν(x, y) can guarantee a precise auroral oval segmentation result without weak boundary leakage.

The CFL condition restricts the time step in traditional numerical solutions for level set equation which is a partial differential equation (PDE), thus leading to the increase in the number of iterations. Other than the finite difference scheme approximates the continuous PDE into a discrete form, LBM [11] proceeds from a discrete form and deduces a continuous PDE which has the same form of level set equation. Because of no strong restricted condition of time step and the highly parallelizable property, LBM is a fast numerical solution for level set equation. LBM is proposed as a computational fluid dynamics (CFD) method for fluid simulation [37]. Instead of solving the NavierStokes equations, the discrete boltzmann equation is solved to simulate the flow of a Newtonian fluid with collision models such as Bhatnagar-Gross-Krook (BGK) [12], [21]. The LBM usually works on an equidistant grid of cells, and the commonly used structure are D3Q19 (three dimensions with nineteen directions) and D2Q9 (two dimensions with nine directions) as shown in Fig. 7. In this paper, we use the D2Q9 structure to simulate the evolution of level set function. In LBM, each direction has its velocity vector ei and the particle distribution f i (x, t), and thus the macroscopic fluid density and velocity can be computed as 1 f i (x, t), u = f i (x, t)ei . (23) ρ= i i ρ The general LBM evolution equation can be written as f i (x + ei t, t + t) = f i (x, t) + i ( f (x, t)),

(24)

where i ( f (x, t)) is the collision operator, here we choose the classical BGK operator, i.e., 1 eq (25) i ( f (x, t)) = − ( f i (x, t) − f i (x, t)), τ and the lattice boltzmann BGK equation (LBGK) [3] can be expressed as f i (x + ei t, t + t) = f i (x, t) +

1 eq ( f (x, t) − f i (x, t)), τ i (26)

where τ is the relaxation time determining the kinematic eq viscosity in Navier-Stokes equations, and fi (x, t) is the local equilibrium particle distribution which has the following form. eq

fi

= ρ[ Ai + Bi ei · u + Ci (ei · u)2 + Di u 2 ],

(27)

2860

IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 23, NO. 7, JULY 2014

TABLE I C ORRESPONDING F ORMS OF τ AND F IN D2Q9 LBM E QUATION FOR L EVEL S ET M ETHODS

where Ai , Bi , Ci and Di are constant coefficients decided by the structure of LBM, the concrete form of D2Q9 is eq

f i = ρwi [1 + 3ei · u + 9/2(ei · u)2 − 3/2u 2 ], w0 = 4/9; w1,2,3,4 = 1/9; w5,6,7,8 = 1/36.

(28)

Since the Navier-Stokes equation which is a convection diffusion equation can be obtained from the discrete LBM equation, the level set equation which is also a convection diffusion equation can be solved with LBM too, what needs to do is set the fluid density ρ as the level set function φ. To add the edge stop function (ESF) of the level set method into LBM, Chen et al. [13] added a medium between the nodes of the lattice. The particles can pass through the medium with a possibility of gi (x), and will be punched back where they were with a possibility of 1 − gi (x). Their LBM evolution equation can be written as f i (x + ei t, t + t) = gi (x)( f i (x, t) 1 eq + ( f i (x, t) − f i (x, t)) + σ ) τ +(1 − gi (x)) f i (x + ei t, t), (29) where σ is the convection coefficient. However, their method just considers the ESF based energy term E e (φ) in (17), and lacks other energy terms which do not contain the ESF information, like in (17). In other words, their method can deal with the geodesic active contour (GAC) model [8] well, but is not suitable for other level set methods, like Li’s method [29] and our method introduced in subsection III.B. In this paper, a universal LBM with an external force term is presented to solve the level set equation. The new LBM evolution equation has the following form. 1 eq f i (x + ei t, t + t) = f i (x, t) + ( f i (x, t) − f i (x, t)) τ 2τ − 1 Bi (F · ei ), + (30) 2τ where F is the external force which guides the evolution of eq the curve in level set methods. The equilibrium function f i eq can be simplified as f i = ρ Ai . After using the ChapmanEnskog expansion [45] in (30), we can obtain the convection diffusion equation as ∂ρ = γ ∇ · ∇ρ + F, (31) ∂t where γ is the diffusion coefficient depending on the relaxation time τ , in D2Q9 structure, their relationship can be expressed as 2 (32) γ = (2τ − 1). 9

By replacing ρ with φ in (31), we have the following formula since level set function φ has the signed distance property |∇φ| = 1. ∇φ ∂φ = γ ∇ · ∇φ + F = γ di v( ) + F. ∂t |∇φ|

(33)

This formula is a general form of level set equation which considers both the ESF based energy term and other energy terms without the ESF information. Table I gives the corresponding forms of τ and F in D2Q9 LBM equation for the geodesic active contour (GAC) model [8] and our method introduced in subsection III.B. The steps of implementation in our universal LBM algorithm are as follows. 1. Compute τ and F. 2. Collision (for all nodes): fi (x ∗ , t + t) = f i (x, t) + +

1 (Ai φ − f i (x, t)) τ

2τ − 1 Bi (F · ei ). 2τ

3. Streaming (for each inner nodes): f i (x + ei t, t + t) = f i (x ∗ , t + t). 4. Bounce-back scheme (for each boundary nodes): f i (x + ei t, t + t) = f i (x + e−i t, t + t). 5. Update level set function (for all nodes): φ= fi . i

6. Find the contour: contournew = zer oφ. 7. Go to step 1. In conclusion, our universal LBM numerical solution is simpler to implement than the classical finite difference scheme as it does not need the complex approximations to discretize the level set equation. Meanwhile, because of no strong restricted condition in time step, a large time step can be chosen and the number of iterations can surely be reduced, thus resulting in a short processing time. What is more, since the computation for each nodes and directions are independent, the LBM is very suitable for parallel implementation, and some GPU based LBMs [7], [18] have already been proposed recently.

YANG et al.: IMPROVING LEVEL SET METHOD FOR FAST AURORAL OVAL SEGMENTATION

2861

maintained at full floating point precision, the other is a label map array which is used to record the status of each point and the values are{−3, −2, −1, 0, 1, 2, 3}. 2) Evolving the Curve: Update the level set function φ of points in L 0 and determine the changing status, then scan through the points around L 0 from inner to outer, by combining the neighborhood information, the changing status of these points can be inferred and saved in five new lists, i.e., S0 → Points moving to L 0 Fig. 8. Computation domain of updating level set function in SFM. (a) Full image domain. (b) Computation domain of updating level set function in SFM.

S−1 → Points moving to L −1 S1 → Points moving to L 1 S−2 → Points moving to L −2 S2 → Points moving to L 2 .

Fig. 9.

(35)

Finally, points in the above lists change their status and one move of contour is finished. This process is repeated until convergence is reached. To achieve a more efficient implementation for level set methods, some integral SFM methods [31], [39] have been proposed. Other than maintaining floating point representation of φ within the lists, they use integer value for simplicity. Although a faster rate to convergence is achieved to some extent, they cannot reach the sub-pixel accuracy. Because high segmentation accuracy is the first goal we must achieve, floating SFM is chosen for our auroral oval segmentation.

One example of the initialization in SFM.

D. Sparse Field Method (SFM) for Computation In classical level set methods, the value of level set function φ is updated in the full image domain, which increases the calculated amount and leads to a waste of time. Narrow band methods [1], [34] conquer this problem by only updating pixels near the evolving curve. The width of the narrow band varies in different narrow band level set methods, and the sparse field method (SFM) [43] takes this strategy to the extreme since it computes the updates on a band of the grid points that is only one point wide. Therefore, the number of computations increases with the size of the curve length, rather than the resolution of the grid. Fig. 8 shows the computation domain of updating level set function in SFM. The procedure of SFM can be divided into the following two steps. 1) Initialization: As shown in Fig. 9, the SFM first initializes an image by using the following five lists to represent five different levels, i.e., L 0 → [−0.5, 0.5] L −1 → [−1.5, −0.5) L 1 → (0.5, 1.5] L −2 → [−2.5, −1.5) L 2 → (1.5, 2.5],

(34)

where the right side indicates the value of level set function φ which is the distance from the zero level set (evolving curve). Two arrays are used to save the information of the above lists. One is the φ array which is

E. Algorithm of Fast Level Set Method for Auroral Oval Segmentation The aforementioned four strategies can solve problems which cause the high computational complexity in traditional level set methods. Our fast level set algorithm for auroral oval segmentation exploiting these strategies is described below. 1) Initialize: Use the contour of SMM as the initial evolving curve according to (15). Initialize the label map array of aurora image which has been described in subsection III.D. 2) Update the Zero Level Set: 1) Compute the input of our universal LBM such as τ and F according to Table I and our level set equation (22). 2) Solve the universal LBM. 3) Update the value of φ and its label map in the evolving curve. 3) Update the narrow band: By using the new value in evolving curve, update other points in the narrow band which has been described in subsection III.D. Go to step 2 until convergence. IV. E XPERIMENTAL R ESULTS AND A NALYSIS To validate the effectiveness of our fast level set algorithm for auroral oval segmentation, several experiments are conducted on MATLAB R2010b installed in a computer with a 3.30GHz Intel Core i3 CPU and 2GB of RAM. Our experimental database includes 200 UVI images of size 200 × 228:

2862

IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 23, NO. 7, JULY 2014

100 full oval images and 100 gap oval images. As we know, among the four strategies in the proposed algorithm, the initial evolving curve and level set formulation have effect on both the segmentation accuracy and speed, while the numerical scheme and computational domain only have big influence on segmentation speed. Therefore, we first conduct some experiments to compare the segmentation accuracy and speed without and with our shape knowledge based initial evolving curve, of traditional level set formulation and the proposed neighbor embedded level set formulation. Then, the contribution of the proposed fast level set method with all these four strategies for segmentation speed is demonstrated with some experiments. Finally, we compare the segmentation performance between the proposed fast level set algorithm and the existing techniques, and the results illustrate the effectiveness of the proposed algorithm for auroral oval segmentation.

Fig. 10. Auroral oval segmentation results using different initial evolving curves. (a) Segmentation benchmark. (b) External square (1300 iterations, execution time 27.323s, FS 0.065, MS 0.042, B D E RM S D 2.015, B D E H D 10.329). (c) Internal square (700 iterations, execution time 14.335s, FS 0.010, MS 2.014, B D E RM S D 40.652, B D E H D 200.154). (d) Shape knowledge based (200 iterations, execution time 9.966s, FS 0.015, MS 0.009, B D E RM S D 0.856, B D E H D 2.587).

A. Results of the First Two Strategies in Segmentation Accuracy and Speed First, let us investigate the influence of initial evolving curve on both segmentation speed and accuracy. Iterations and execution time in segmentation procedure are used to measure the speed, while the accuracy is obtained by using various metrics to measure the errors between a segmentation result and the benchmark (manually segmented result from some aurora experts). These error metrics include false segmentation (FS) (i.e., errors of the background pixels regarded as the object), miss segmentation (MS) (i.e., errors of the object pixels regarded as the background) and boundary deviation error (BDE) (i.e., boundary distance between the segmentation result and the benchmark) with a root mean square distance form B D E RM S D and a Hausdorff distance form B D E H D . 1 FS = r (i, j ), (36) A (i, j )∈R / Ben

1 MS = A



[1 − r (i, j )],

(37)

(i, j )∈R Ben

 1 N  b (i ) − b (i )2 , (38) Seg Ben i=1 N B D E H D = max(h(B Seg , B Ben ), h(B Ben , B Seg )),   min b Seg − b Ben  , (39) h(B Seg , B Ben ) = max B D E RM S D =

b Seg ∈B Seg b Ben ∈B Ben

where r (i, j ) is the segmented image with an area A, i.e.,  0, (i, j ) ∈ / R Seg r (i, j ) = (40) 1, (i, j ) ∈ R Seg . R Ben and R Seg are the object region in the benchmark and the segmentation result, respectively. N is the number of boundary deviation pixels. b Seg ∈ B Seg and b Ben ∈ B Ben are the boundary pixel of the benchmark and the segmen tation result, respectively. b Seg − b Ben  denotes the distance between the segmented boundary pixel and its benchmark location. Remarkably, FS and MS represent the segmentation accuracy from the region aspect, while B D E RM S D and B D E H D describe it from the boundary aspect, which can

Fig. 11. Auroral oval segmentation results of geodesic active contour model and the proposed neighbor embedded level set formulation. (a) Initial evolving curve. (b) Segmentation benchmark. (c) Results of geodesic active contour model (full oval image in the first row: 500 iterations, 10 iterations of re-initialization, execution time 25.369s, FS 0.089, MS 0.021, B D E RM S D 3.214, B D E H D 20.007; gap oval image in the second row: 3000 iterations, 10 iterations of re-initialization, execution time 168.354s, FS 0.051, MS 0.098, B D E RM S D 9.647, B D E H D 25.314). (d) Results of the proposed neighbor embedded level set formulation (full oval image in the first row: 90 iterations, no re-initialization, execution time 5.532s, FS 0.038, MS 0.018, B D E RM S D 2.149, B D E H D 5.674; gap oval image in the second row: 200 iterations, no re-initialization, execution time 9.966s, FS 0.015, MS 0.009, B D E RM S D 0.856, B D E H D 2.587).

achieve a comprehensive measurement. In the experiment, the lower value of all these metrics, the higher segmentation accuracy of the result. Fig. 10 shows some results of auroral oval segmentation using different initial evolving curves, from which we can see that our shape knowledge based initial evolving curve has the most precise segmentation result with the least amount of time. The level set formulation applied in this experiment is the proposed neighbor embedded one. To demonstrate the contribution of the proposed neighbor embedded level set formulation for segmentation accuracy and speed, an experiment which compares the traditional level set formulation with re-initialization (geodesic active contour model [8]) and the proposed neighbor embedded formulation without re-initialization is conducted. Fig. 11 shows some results and we can conclude that the proposed neighbor embedded level set formulation needs much less time meanwhile keeps a precise segmentation result. The initial

YANG et al.: IMPROVING LEVEL SET METHOD FOR FAST AURORAL OVAL SEGMENTATION

Fig. 12.

2863

Additional results of auroral oval segmentation using the first two strategies. (a) Original image. (b) Segmentation result.

TABLE II C OMPARISON OF T IME C OST W ITH D IFFERENT

TABLE IV C OMPARISON OF AVERAGE T IME C OST W ITH D IFFERENT

M ETHODS IN S INGLE F ULL OVAL I MAGE

M ETHODS IN 100 F ULL OVAL I MAGES

TABLE III C OMPARISON OF T IME C OST W ITH D IFFERENT M ETHODS IN S INGLE G AP OVAL I MAGE

evolving curve used in this experiment is our shape knowledge based one. As we know, both of the first two strategies in our fast level set algorithm have influence on segmentation accuracy, and the use of them can guarantee a reasonably precise segmentation result. Fig. 12 gives some additional results of auroral oval segmentation using the first two strategies. Therefore, these two strategies are always used in the following experiments to compare the segmentation speed. B. Results of the Proposed Fast Level Set Algorithm in Segmentation Speed In this part, to demonstrate the speedup of the proposed strategies, segmentation experiments on single gap oval image and full oval image which have been shown in Fig. 11 are first conducted. Then, experiments on aurora image database are carried out to test the performance of these strategies on large scale database. From subsection IV.A, we know that the results of auroral oval segmentation with reasonably precise are obtained with

TABLE V C OMPARISON OF AVERAGE T IME C OST W ITH D IFFERENT M ETHODS IN 100 G AP OVAL I MAGES

the use of our shape knowledge based initial evolving curve and neighbor embedded level set formulation. Without these two strategies, the segmentation results are inaccurate and may take as long as dozens of seconds even hundreds of seconds of time. Therefore, to make this a fair comparison, we only compare the spending time of methods which have similar accurate segmentation results, they are method only with the first two strategies (FTS), method with the first two strategies and our universal LBM (FTS+LBM), method with the first two strategies and SFM (FTS+SFM), our fast level set algorithm which has all of these four strategies (FTS+LBM+SFM). Table II and Table III are the comparison of time cost with the above methods in single full oval image and single gap oval image, respectively. While Table IV and Table V are the comparison of average time cost with the above methods in 100 full oval images and 100 gap oval images, respectively. From the comparison results, we can draw the following conclusions. Firstly, our universal LBM can decrease the iterations to convergence and reduce the consuming time by

2864

IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 23, NO. 7, JULY 2014

TABLE VI C OMPARISON OF S EGMENTATION A CCURACY W ITH D IFFERENT M ETHODS IN 100 F ULL OVAL I MAGES

TABLE VII C OMPARISON OF S EGMENTATION A CCURACY W ITH D IFFERENT M ETHODS IN 100 G AP OVAL I MAGES

several times. Secondly, the speedup of SFM is great to more than one hundred times. Thirdly, time cost of full oval segmentation is shorter than gap oval one, this is because the shape constraint of full oval image is larger and its initial evolving curve is nearer to the object boundary. Finally, our fast level set algorithm which combines all the four strategies can decrease the iterations from hundreds even thousandths of times to dozens of times compared with level set methods without these four strategies, while the consuming time is reduced from dozens even hundreds of seconds to a few hundredths of a second. C. Comparison Between the Proposed Fast Level Set Algorithm and Existing Techniques To prove the superiority of the proposed fast level set algorithm for auroral oval segmentation, we make a comparison against existing techniques in terms of segmentation accuracy. Table VI and Table VII are the comparison results for 100 full oval images and 100 gap oval images, respectively. It can be seen that the proposed fast level set algorithm holds the lowest values in FS, MS, B D E RM S D and B D E H D for both full oval and gap oval images. The performances of HKM, AMET and PCNN are relatively poor. LLS-RHT obtains a high accuracy for full oval images. However, its performance for gap oval images is not as good as the proposed method. In addition, the average time cost to segment one image using HKM, AMET, PCNN and LLS-RHT are 0.242s, 0.058s, 0.543s and 0.435s, respectively. Compared with the result of the proposed algorithm which is only a few hundredths of a second, the existing techniques are hard to keep pace with. Thus, we can conclude that the proposed fast level set algorithm for auroral oval segmentation achieves a high accuracy within a really short time.

V. C ONCLUSION Segmenting auroral oval from large aurora images is an arduous task which needs high accuracy and quick processing time. The traditional level set methods are cumbersome in implementation and high in time complexity. In this paper, we present a fast level set algorithm for auroral oval segmentation. By analyzing four reasons for high computational complexity, four corresponding strategies are proposed to resolve them. Our shape knowledge based initial evolving curve resolves the problem of improper initial evolving curve while our neighbor embedded level set formulation resolves the problem of cumbersome re-initialization process. These two strategies can not only speed up the segmentation process but also improve the precision of segmentation results. Our universal LBM which resolves the problem of restricted numerical scheme can be used for various level set methods, and it can reduce the time cost by decreasing the iterations to convergence with an unlimited time step. The use of SFM can resolve the problem of full domain computation, and it cuts down the time consuming greatly. Experiments are conducted to illustrate the speedup of our four strategies, and the results show that the proposed fast level set algorithm for auroral oval segmentation can reduce the time cost from dozens even hundreds of seconds to a few hundredths of a second compared with the traditional level set methods. In the future, because the LBM and SFM are very suitable for parallel implementation, some GPU technologies can be used to further accelerate the auroral oval segmentation using our level set algorithm. R EFERENCES [1] D. Adalsteinsson and J. A. Sethian, “A fast level set method for propagating interfaces,” J. Comput. Phys., vol. 118, no. 2, pp. 269–277, May 1995. [2] M. Alessandrini, T. Dietenbeck, O. Basset, D. Friboulet, and O. Bernard, “Using a geometric formulation of annular-like shape priors for constraining variational level-sets,” Pattern Recognit. Lett., vol. 32, no. 9, pp. 1240–1249, Jul. 2011. [3] P. L. Bhatnagar, E. P. Gross, and M. Krook, “A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems,” Phys. Rev., vol. 94, no. 3, pp. 511–524, May 1954. [4] M. Brittnacher, J. Spann, G. Parks, and G. Germany, “Auroral observations by the polar ultraviolet imager (UVI),” Adv. Space Res., vol. 20, nos. 4–5, pp. 1037–1042, Apr. 1997. [5] M. Brittnacher et al, “Global auroral response to a solar wind pressure pulse,” Adv. Space Res., vol. 25, nos. 7–8, pp. 1377–1385, Jul. 2000. [6] C. Cao, T. S. Newman, and G. A. Germany, “A new shape-based auroral oval segmentation driven by LLS-RHT,” Pattern Recognit., vol. 42, no. 5, pp. 607–618, May 2009. [7] W. Cao, Z. Wang, Z. Li, L. Yao, and Y. X. Wang, “An improved LBM approach for heterogeneous GPU-CPU clusters,” in Proc. 4th Int. Conf. BMEI, Oct. 2011, pp. 2095–2098. [8] V. Caselles, R. Kimmel, and G. Sapiro, “Geodesic active contours,” Int. J. Comput. Vis., vol. 22, no. 1, pp. 61–79, Jan. 1997. [9] T. F. Chan, B. Y. Sandberg, and L. A. Vese, “Active contours without edges for vector-valued images,” J. Vis. Commun. Image Represent., vol. 11, no. 2, pp. 130–141, Jun. 2000. [10] T. F. Chan and L. A. Vese, “Active contours without edges,” IEEE Trans. Image Process., vol. 10, no. 2, pp. 266–277, Feb. 2001. [11] Q. Chang and T. Yang, “A lattice Boltzmann method for image denoising,” IEEE Trans. Image Process., vol. 18, no. 12, pp. 2797–2802, Dec. 2009. [12] S. Chen and G. D. Doolen, “Lattice Boltzmann method for fluid flows,” Annu. Rev. Fluid Mech., vol. 30, no. 1, pp. 329–364, Jan. 1998. [13] Y. Chen, Z. Yan, and Y. Chu, “Cellular automata based level set method for image segmentation,” in Proc. IEEE/ICME Int. Conf. CME, May 2007, pp. 171–174.

YANG et al.: IMPROVING LEVEL SET METHOD FOR FAST AURORAL OVAL SEGMENTATION

[14] C. Chesnaud, P. Refregier, and V. Boulet, “Statistical region snake-based segmentation adapted to different physical noise models,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 21, no. 11, pp. 1145–1157, Nov. 1999. [15] L. Evans, Partial Differential Equations. Providence, RI, USA: AMS, 1998. [16] X. Gao, B. Wang, D. Tao, and X. Li, “A relay level set method for automatic image segmentation,” IEEE Trans. Syst. Man, Cybern. B, Cybern., vol. 41, no. 2, pp. 518–525, Apr. 2011. [17] G. A. Germany et al., “Analysis of auroral morphology: Substorm precursor and onset on January 10, 1997,” Geophys. Res. Lett., vol. 25, no. 15, pp. 3043–3046, Aug. 1998. [18] A. Hagan and Y. Zhao, “Parallel 3D image segmentation of large data sets on a GPU cluster,” in Proc. 5th Int. Symp. Vis. Comput., 2009, pp. 960–969. [19] X. Han, C. Xu, and J. L. Prince, “A topology preserving level set method for geometric deformable models,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 25, no. 6, pp. 755–768, Jun. 2003. [20] J. Harel, C. Koch, and P. Perona, “Graph-based visual saliency,” in Advanced Neural Information Processing System. Cambridge, MA, USA: MIT Press, 2007, pp. 545–552. [21] X. He, Q. Zou, L.-S. Luo, and M. Dembo, “Analytic solutions of simple flows and analysis of nonslip boundary conditions for the lattice Boltzmann BGK model,” J. Statist. Phys., vol. 87, nos. 1–2, pp. 115–136, Feb. 1997. [22] X. Hou and L. Zhang, “Saliency detection: A spectral residual approach,” in Proc. IEEE Conf. CVPR, Jun. 2007, pp. 1–8. [23] C. C. Hung and G. Germany, “K-means and iterative selection algorithms in image segmentation,” in Proc. IEEE Southeastcon, Apr. 2003. [24] R. D. Hunsucker, R. B. Rose, R. W. Adler, and G. K. Lott, “AuroralE mode oblique HF propagation and its dependence on auroral oval position,” IEEE Trans. Antennas Propag., vol. 44, no. 3, pp. 383–388, Mar. 1996. [25] L. Itti, C. Koch, and E. Niebur, “A model of saliency-based visual attention for rapid scene analysis,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 20, no. 11, pp. 1254–1259, Nov. 1998. [26] M. Kass, A. Witkin, and D. Terzopoulos, “Snakes: Active contour models,” Int. J. Comput. Vis., vol. 1, no. 4, pp. 321–331, Jan. 1988. [27] J. Kittler and J. Illingworth, “Minimum error thresholding,” Pattern Recognit., vol. 19, no. 1, pp. 41–47, Jan. 1986. [28] X. Li et al., “Comparing different thresholding algorithms for segmenting auroras,” in Proc. Int. Conf. ITCC, Apr. 2004, pp. 594–601. [29] C. Li, C. Xu, C. Gui, and M. D. Fox, “Level set evolution without reinitialization: A new variational formulation,” in Proc. IEEE Comput. Soc. Conf. CVPR, 2005, pp. 430–436. [30] C. Li, C. Xu, C. Gui, and M. D. Fox, “Distance regularized level set evolution and its application to image segmentation,” IEEE Trans. Image Process., vol. 19, no. 12, pp. 3243–3254, Dec. 2010. [31] J. G. Malcolm, Y. Rathi, A. Yezzi, and A. Tannenbaum, “Fast approximate surface evolution in arbitrary dimension,” Proc. SPIE, vol. 6914, pp. 6914–6923, Mar. 2008. [32] R. Malladi, J. A. Sethian, and B. C. Vemuri, “Shape modeling with front propagation: A level set approach,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 17, no. 2, pp. 158–175, Feb. 1995. [33] A. Mitiche and H. Sekkati, “Optical flow 3D segmentation and interpretation: A variational method with active curve evolution and level sets,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 28, no. 11, pp. 1818–1829, Nov. 2006. [34] S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces. New York, NY, USA: Springer-Verlag, 2002. [35] S. Osher and J. A. Sethian, “Fronts propagating with curvaturedependent speed: Algorithms based on Hamilton–Jacobi formulations,” J. Comput. Phys., vol. 79, no. 1, pp. 12–49, Nov. 1988. [36] N. Paragios and R. Deriche, “Geodesic active contours and level sets for the detection and tracking of moving objects,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 22, no. 3, pp. 266–280, Mar. 2000. [37] Y. H. Qian, D. D’Humières, and P. Lallemand, “Lattice BGK models for Navier–Stokes equation,” Europhys. Lett., vol. 17, no. 6, pp. 479–484, Feb. 1992. [38] G. Sapiro, Geometric Partial Differential Equations and Image Analysis. Cambridge, U.K.: Cambridge Univ. Press, 2001, ch. 2. [39] Y. Shi and W. Karl, “A fast level set method without solving PDEs,” in Proc. IEEE ICASSP, Mar. 2005, pp. 97–100. [40] A. Tsai, A. Yezzi, Jr., and A. S. Willsky, “Curve evolution implementation of the Mumford–Shah functional for image segmentation, denoising, interpolation, and magnification,” IEEE Trans. Image Process., vol. 10, no. 8, pp. 1169–1186, Aug. 2001.

2865

[41] B. Wang, X. Gao, D. Tao, and X. Li, “A unified tensor level set for image segmentation,” IEEE Trans. Syst. Man, Cybern. B, Cybern., vol. 40, no. 3, pp. 857–867, Jun. 2010. [42] J. Weickert, B. M. T. H. Romeny, and M. A. Viergever, “Efficient and reliable schemes for nonlinear diffusion filtering,” IEEE Trans. Image Process., vol. 7, no. 3, pp. 398–410, Mar. 1998. [43] R. T. Whitaker, “A level-set approach to 3D reconstruction from range data,” Int. J. Comput. Vis., vol. 29, no. 3, pp. 203–231, Sep. 1998. [44] H. Yang et al., “Synoptic observations of auroras along the postnoon oval: A survey with all-sky TV observations at Zhongshan, Antarctica,” J. Atmos. Solar-Terrestrial Phys., vol. 62, no. 9, pp. 787–797, Jun. 2000. [45] Y. Zhao, “Lattice Boltzmann based PDE solver on the GPU,” Vis. Comput., vol. 24, no. 5, pp. 323–333, May 2008. Xi Yang received the B.Sc. degree in electronic information engineering from Xidian University, Xi’an, China, in 2010, where she is currently pursuing the Ph.D. degree in pattern recognition and intelligent system. Since 2013, she has been a Visiting Ph.D. Student with the University of Texas at San Antonio, San Antonio, TX, USA. Her current research interests include image/video processing, computer vision, and multimedia information retrieval. Xinbo Gao (M’02–SM’07) received the B.Eng., M.Sc., and Ph.D. degrees in signal and information processing from Xidian University, Xi’an, China, in 1994, 1997, and 1999, respectively. From 1997 to 1998, he was a Research Fellow with the Department of Computer Science, Shizuoka University, Shizuoka, Japan. From 2000 to 2001, he was a Post-Doctoral Research Fellow with the Department of Information Engineering, Chinese University of Hong Kong, Hong Kong. Since 2001, he has been with the School of Electronic Engineering, Xidian University. He is currently a Cheung Kong Professor of Ministry of Education, China, a Professor of Pattern Recognition and Intelligent System, and the Director of the VIPS Laboratory at Xidian University. His research interests are computational intelligence, machine learning, computer vision, pattern recognition, and wireless communications. In these areas, he has authored five books and around 150 technical articles in refereed journals and proceedings, including the IEEE T RANSACTIONS ON I MAGE P ROCESSING, the IEEE T RANSACTIONS ON C IRCUITS AND S YSTEMS FOR V IDEO T ECHNOLOGY, the IEEE T RANSACTIONS ON N EURAL N ETWORKS , the IEEE T RANSAC TIONS ON S YSTEMS , M AN , AND C YBERNETICS , and Pattern Recognition. He is on the Editorial Boards of several journals, including Signal Processing (Elsevier) and Neurocomputing (Elsevier). He served as the General Chair/Co-Chair or Program Committee Chair/Co-Chair or PC Member for around 30 major international conferences. He is currently a fellow of IET. Dacheng Tao (M’07–SM’12) is a Professor of Computer Science with the Centre for Quantum Computation and Intelligent Systems and the Faculty of Engineering and Information Technology, University of Technology, Sydney, Ultimo, NSW, Australia. He mainly applies statistics and mathematics for data analysis problems in data mining, computer vision, machine learning, multimedia, and video surveillance. He has authored and co-authored more than 100 scientific articles at top venues, including the IEEE T RANSACTIONS ON PATTERN A NALYSIS AND M ACHINE I NTELLIGENCE, the IEEE T RANSACTIONS ON N EURAL N ETWORKS AND L EARNING S YSTEMS , the IEEE T RANSACTIONS ON I MAGE P ROCESSING , the Neural Information Processing Systems Conference, the International Conference on Machine Learning, the International Conference on Artificial Intelligence and Statistics, the IEEE International Conference on Data Mining, the IEEE Conference on Computer Vision and Pattern Recognition, the International Conference on Computer Vision, the European Conference on Computer Vision, the ACM Transactions on Knowledge Discovery from Data, Multimedia and Knowledge Discovery and Data Mining. He was a recipient of the Best Theory/Algorithm Paper Runner Up Award at the IEEE International Conference on Data Mining in 2007. Xuelong Li (M’02–SM’07–F’12) is a Full Professor with the Center for OPTical IMagery Analysis and Learning (OPTIMAL), State Key Laboratory of Transient Optics and Photonics, Xi’an Institute of Optics and Precision Mechanics, Chinese Academy of Sciences, Xi’an, China.