Comparison of two local discontinuous Galerkin formulations for the subjective surfaces problem

Abstract

Based on the local discontinuous Galerkin method, two substantially different mixed formulations for the subjective surfaces problem are compared using a number of numerical tests of various types. The work also performs the energy stability analysis for both schemes.

Introduction

The discontinuous Galerkin (DG) method first introduced by Reed and Hill [35] represents a very attractive extension of the classical finite element method; its popularity has been rapidly growing ever since especially for CFD applications. A big advantage of this method clearly distinguishing it from the finite volume methods also very popular in the CFD area is the fact that DG schemes fully fit into the Galerkin/Petrov–Galerkin framework and thus allow to exploit the highly developed analysis toolbox that relies on the Sobolev space theory. The original DG formulation was only suited for the first order conservation laws; this drawback was addressed in the 1970s and early 1980s with the introduction of the interior penalty (IP) methods for elliptic problems and in late 1990s with the first local discontinuous Galerkin (LDG) papers for parabolic equations (see [5] for an overview).

All proposed generalization of the original DG framework can be roughly divided into two main classes: Those discretizing the second (as well as higher than second) order operators directly (e.g., IP methods) and the schemes relying on a mixed formulation as in the LDG method introduced by Cockburn and Shu [14] and used in the present work. The latter methodology employs auxiliary variables to represent the PDE as a system of first order equations and may incur higher computational costs; however it offers, in addition to some other advantages, a very flexible framework suitable for a large variety of complex mathematical models represented by non-linear PDEs. This flexibility of the mixed formulation often allows to deconstruct the original non-linear equation into a system of much simpler equations and—as in the present case—to put those equations into divergence form natural for a DG discretization. Aside of that, each equation in the mixed system can be discretized independently, thus allowing to choose a discretization space most appropriate for this particular unknown.

An interesting—however somewhat neglected in the DG literature—issue pertaining to producing a mixed formulation for a given non-linear PDE is the fact that such representations are not unique. Although fully equivalent in the continuous sense, the discrete versions may generally have different properties with regard to computational cost, implementation convenience, discrete stability, etc. The present work attempts to address this deficit by comparing two mixed formulations for the subjective surfaces equation discretized using the LDG method. The first one is closely related to our earlier work on the same application (see [8]), the second is a simplified version of the scheme proposed by Xu and Shu [45] for Willmore flow of graphs. In addition to presenting a number of numerical examples that illustrate the performance of both formulations, an energy stability analysis is carried out in this work.

Solving the image segmentation problem used as the target application in the current work boils down to determining different regions in a given image identified by colors, gray levels, textures, etc. or, alternatively, outlined by some more or less clear contours. Problems of this sort also arise in psychology (see Kanizsa’s triangle [25]) and may involve edges subjectively implied by shapes present on the image.

The origin of the model used in our study can be traced back to the ’Snake’ model proposed by Kass et al. [27]. This active contour method places an initial curve into the image and considers its evolution using the gradient of an energy functional constructed in a way that penalizes curve irregularities and deviations from the object of interest. This method engendered a number of variations (see [6, 10] for an overview of recent developments). The most widely used schemes to represent the evolving curve are based on the level set method [22, 23, 34, 37, 41, 42], whereas handling of the image gradient information turned out to be more difficult due to noisy peaks and troughs often present in the image. One approach proposed in [11, 12] uses a Mumford–Shah segmentation [33] instead of image gradients to improve robustness with respect to noise. Another popular methodology introduced in [9] is called ’Geodesic Active Contours’ since it employs geodesic flow of a level set function that is based on a metric constructed from the input image.

The underlying weakness of the active contour methodology relying on a single curve to detect the image contours is related to the fact that missing image parts (and missing gradient information) may cause the curve evolution to miss some object edges. In order to address this deficiency, Sarti et al. [39, 40], and Sarti and Citti [38] proposed based on the work in [9] the subjective surface method. As opposed to the original active contour method, all level sets of a segmentation function are evolved. This segmentation function is designed in such way that its graph approximates a piecewise constant surface whose levels correspond to the objects present in the image. The evolution of the level sets stops at edges with large gradients, moves in low-gradient parts of the image, and, on top of that, can bridge gaps in the gradient information.

While the first papers on the subjective surface method [38,39,40] used the finite differences, Mikula et al. introduced in [15, 31] complementary volume schemes that combine finite element and finite volume methods. While finite difference schemes are straightforward to formulate and implement and also lead to adequate results, they lack adaptivity capabilities that may potentially result in substantial reductions of computational cost. An adaptive scheme based on the classical finite elements introduced by Mikula and Fried [21] addressed the issue of optimal mesh adaptivity; however, the solution produced there contained spurious peaks in the vicinity of high gradients.

The well known stability of DG schemes when postprocessed by a slope limiting procedure combined with the highly adaptive nature of discontinuous discretizations motivated our interest in DG method for this application. Our previous work [8] was the first one to attempt utilizing the discontinuous Galerkin methodology in the context of the subjective surface method. In addition, a head-to-head comparison utilizing a standard benchmark between classical finite element, finite difference, and DG methods was presented there. This comparison was carried out in the simplest possible configuration (no adaptivity, piecewise linear approximations on triangular meshes for DG and finite elements vs. 5-point-stencil for finite differences) that certainly did not favor such a flexible scheme as the LDG method. However, even in such setting the findings presented in [8] indicated that DG discretizations are a viable alternative to the established numerical methods both in terms of computational performance and solution quality and thus deserve a further study.

Model problem and mixed formulations

For \(T>0\) and a domain \(\varOmega \subset \mathbb {R}^n\) we consider the following evolution equation on \((0,T)\times \varOmega \) which is used for instance in image segmentation (cf. [29]):

figurea

Here, \({\mathcal {G}}\) is a non-negative function in \(L^\infty (\varOmega )\). A common choice in image segmentation is

$$\begin{aligned} {\mathcal {G}}({\varvec{x}}):=\frac{1}{1+K|{\varvec{\nabla }}{\mathcal {I}}({\varvec{x}})|^2}\,\in (0,1] \end{aligned}$$
(1)

with a constant \(K>0\) and a function \({\mathcal {I}}\) that represents a smoothed greyscale image. To obtain a mean curvature flow equation (cf. [19]) one chooses \({\mathcal {G}}\equiv 1\). Since Eq. (MP) can become singular in regions of vanishing \({\varvec{\nabla }}u\), \(|{\varvec{\nabla }}u|\) is often replaced by its regularized approximation \(\sqrt{|{\varvec{\nabla }}u|^2+\varepsilon ^2}\) with \(\varepsilon >0\) resulting in the governing equation considered in this work

figureb

which is combined with an initial condition \(u(0)=u_0\). In order to make second-order Eq. (MP\(_\varepsilon \)) accessible for the LDG method we need to transform it into a first-order system. This transformation is not unique, and in the following we present two different mixed formulations. Introducing notation \({Q_{\varepsilon }}:=\sqrt{|{\varvec{z}}|^2+\varepsilon ^2}\), the first system is very similar to the one used in [8]:

figurec

The main advantage of system (M1\(_\varepsilon \)) is that all equations are in divergence form suitable for a DG discretization.

The second mixed formulation is similar to the set-up proposed by Xu and Shu for Willmore flow of graphs [45] and is given by

figured

System (M2\(_\varepsilon \)) is not in divergence form, but it consists of only three equations versus four in (M1\(_\varepsilon \)).

There exist various reasonable choices of boundary conditions for these problems, but for simplicity we assume sufficient regularity of the solution and the domain and impose homogeneous Neumann boundary conditions \({\varvec{z}}\cdot {\varvec{\nu }}=0\) a. e. on \(\partial \varOmega \), where \({\varvec{\nu }}\) denotes the outward unit normal to \(\partial \varOmega \).

Discretization

Notation

Similarly to our first paper dealing with the application of a DG method to the subjective surfaces problem [8], the MATLAB toolbox FESTUNG [17] was used for our implementation; thus, the notation used here is generally similar to [18, 24, 36]. We assume that \(\varOmega \) is polygonally bounded and let \(\mathcal {T}_h=\{{\mathcal {K}}\}\) denote a regular family of non-overlapping partitions of \({\overline{\varOmega }}\) into K closed triangles \({\mathcal {K}}\) of characteristic size h, i.e., \(\displaystyle {\overline{\varOmega }}=\cup {\mathcal {K}}\). For \({\mathcal {K}}\in \mathcal {T}_h\), \({\varvec{\nu }}_{\mathcal {K}}\) is the unit normal on \(\partial {\mathcal {K}}\) exterior to \({\mathcal {K}}\). Letting \(\mathcal {E}_I\) be the set of interior faces, and \(\mathcal {E}_E\) the set of boundary faces, the set of all faces is given by \(\mathcal {E}:=\mathcal {E}_I\,\cup \,\mathcal {E}_E\) (the subscript h is suppressed here). For an interior face \(F\in \mathcal {E}_I\) shared by triangles \({\mathcal {K}}^-\) and \({\mathcal {K}}^+\), we define the one-sided values of a scalar quantity \(w=w({\varvec{x}})\) on F by

$$\begin{aligned} w^-({\varvec{x}}) :=\!\!\lim _{\tau \rightarrow 0+} \!\!w\left( {\varvec{x}} - \tau \,{\varvec{\nu }}_{{\mathcal {K}}^-}\right) ,\, w^+({\varvec{x}}) :=\!\!\lim _{\tau \rightarrow 0+} \!\!w\left( {\varvec{x}} - \tau \,{\varvec{\nu }}_{{\mathcal {K}}^+}\right) . \end{aligned}$$

For a boundary face \(F\in \mathcal {E}_E\), only the first makes sense. The one-sided values of vector-valued quantities are defined analogously. The average and the jump of scalar w and vector \({\varvec{u}}\) on \(F\in \mathcal {E}_I\) are given then by

$$\begin{aligned} \left\{ \!\left| {w}\right| \!\right\}:= & {} \left( w^- + w^+\right) /2,\quad \left\{ \!\left| {{\varvec{u}}}\right| \!\right\} :=\left( {\varvec{u}}^- + {\varvec{u}}^+\right) /2,\\ \llbracket {w}\rrbracket:= & {} w^- {\varvec{\nu }}_{{\mathcal {K}}^-} + w^+ {\varvec{\nu }}_{{\mathcal {K}}^+}, \quad \llbracket {{\varvec{u}}}\rrbracket :={\varvec{u}}^-\cdot {\varvec{\nu }}_{{\mathcal {K}}^-} + {\varvec{u}}^+\cdot {\varvec{\nu }}_{{\mathcal {K}}^+}, \end{aligned}$$

respectively. Note that \(\llbracket {w}\rrbracket \) is vector-valued, and \(\llbracket {{\varvec{u}}}\rrbracket \) is scalar-valued quantity. Furthermore, we define a vector-valued jump of vector \({\varvec{u}} = (u_x, u_y)^T\) for an element \({\mathcal {K}}\) as

$$\begin{aligned} \llbracket {\llbracket {{\varvec{u}}}\rrbracket }\rrbracket :=\left( \llbracket {u_x}\rrbracket \cdot {\varvec{\nu }}_{{\mathcal {K}}}, \llbracket {u_y}\rrbracket \cdot {\varvec{\nu }}_{{\mathcal {K}}}\right) ^T. \end{aligned}$$

Semi-discrete formulation

For \(\mathbb {P}_p({\mathcal {K}})\) the space of all polynomials of degree not greater than p on \({\mathcal {K}}\in \mathcal {T}_h\), we denote by

$$\begin{aligned} \mathbb {P}_p(\mathcal {T}_h) \;:=\; \Big \{ w_h:{\overline{\varOmega }}\rightarrow \mathbb {R}\,; \forall {\mathcal {K}}\in \mathcal {T}_h, {w_h}|_{\mathcal {K}}\in \mathbb {P}_p({\mathcal {K}})\Big \} \end{aligned}$$

the broken polynomial space on triangulation \(\mathcal {T}_h\). In the following, we utilize \(L^2\)-projections of functions \(u_\text {0}\) and \({\mathcal {G}}\) into the DG space \(\mathbb {P}_p(\mathcal {T}_h)\) marked by h-subscript. For a function f, its \(L^2\)-projection \(f_h\in \mathbb {P}_p(\mathcal {T}_h)\) has to satisfy

$$\begin{aligned} \int _{\mathcal {K}}f\,\varphi \,\mathrm {d}{\varvec{x}} = \int _{\mathcal {K}}f_h\,\varphi \,\mathrm {d}{\varvec{x}},\quad \forall \varphi \in \mathbb {P}_p({\mathcal {K}}),\;\forall {\mathcal {K}}\in \mathcal {T}_h. \end{aligned}$$

Using this together with the boundary conditions and the initial value of \(u_h\) computed as an \(L^2\)-projection of \(u_0\) we can now formulate the semi-discrete problems corresponding to (M1\(_\varepsilon \)) and (M2\(_\varepsilon \)):

Formulation 1

Seek \(({\varvec{z}}_h,{\varvec{v}}_h,{\varvec{y}}_h,u_h)\in \left[ \mathbb {P}_p(\mathcal {T}_h)\right] ^2 \times \left[ \mathbb {P}_p(\mathcal {T}_h)\right] ^2\times \left[ \mathbb {P}_{3p}(\mathcal {T}_h)\right] ^2\times \mathbb {P}_p(\mathcal {T}_h)\) such that the following holds \(\forall {\mathcal {K}}\in \mathcal {T}_h\), \(\forall {\varvec{\varphi _z}},{\varvec{\varphi _v}}\in \left[ \mathbb {P}_p(\mathcal {T}_h)\right] ^2,{\varvec{\varphi _y}}\in \left[ \mathbb {P}_{3p}(\mathcal {T}_h)\right] ^2,\) \({\varphi _u}\in \mathbb {P}_p(\mathcal {T}_h)\), and a.e. \(0<t<T\):

$$\begin{aligned}&\int _{{\mathcal {K}}}{\varvec{z}}_h\cdot {\varvec{\varphi _z}}\,\mathrm {d}{\varvec{x}} - \int _{{\mathcal {K}}} u_h\,{\varvec{\nabla }}\cdot {\varvec{\varphi _z}}\,\mathrm {d}{\varvec{x}} +\int _{\partial {\mathcal {K}}\setminus \partial \varOmega } \left\{ \!\left| {u_h}\right| \!\right\} {\varvec{\varphi _z}}\cdot {\varvec{\nu }}_{{\mathcal {K}}}\,\mathrm {d}s\nonumber \\&\quad +\int _{\partial {\mathcal {K}}\cap \partial \varOmega } u_h {\varvec{\varphi _z}}\cdot {\varvec{\nu }}_{{\mathcal {K}}}\,\mathrm {d}s = 0, \end{aligned}$$
(2a)
$$\begin{aligned} \int _{{\mathcal {K}}}&{\varvec{v}}_h\cdot {\varvec{\varphi _v}}\,\mathrm {d}{\varvec{x}} -\int _{{\mathcal {K}}} {\mathcal {G}}_h\,{\varvec{z}}_h\cdot {\varvec{\varphi _v}}\,\mathrm {d}{\varvec{x}}\nonumber \\&\quad -\alpha \int _{\partial {\mathcal {K}}\setminus \partial \varOmega } \left| \left\{ \!\left| {{\mathcal {G}}_h{\varvec{z}}_h u_h}\right| \!\right\} \right| ^2\left| \left\{ \!\left| {{\varvec{z}}_h}\right| \!\right\} \right| ^2\llbracket {\llbracket {{\varvec{z}}_h}\rrbracket }\rrbracket \cdot {\varvec{\varphi _v}}\,\mathrm {d}s =0,\end{aligned}$$
(2b)
$$\begin{aligned}&\int _{{\mathcal {K}}} {\varvec{y}}_h\cdot {\varvec{\varphi _y}}\,\mathrm {d}{\varvec{x}} + \int _{{\mathcal {K}}}\log \left( {Q_{\varepsilon }}_h^2\right) \,{\varvec{\nabla }}\cdot {\varvec{\varphi _y}}\,\mathrm {d}{\varvec{x}}\nonumber \\&\quad -\int _{\partial {\mathcal {K}}\setminus \partial \varOmega }\left\{ \!\left| {\log \left( {Q_{\varepsilon }}_h^2\right) }\right| \!\right\} {\varvec{\varphi _y}}\cdot {\varvec{\nu }}_{{\mathcal {K}}}\,\mathrm {d}s \nonumber \\&\quad -\int _{\partial {\mathcal {K}}\cap \partial \varOmega }\log \left( {Q_{\varepsilon }}_h^2\right) {\varvec{\varphi _y}}\cdot {\varvec{\nu }}_{{\mathcal {K}}}\,\mathrm {d}s = 0.\end{aligned}$$
(2c)
$$\begin{aligned}&\int _{{\mathcal {K}}} \partial _t u_h\,{\varphi _u}\,\mathrm {d}{\varvec{x}} - \int _{{\mathcal {K}}} {\varvec{v}}_h\cdot {\varvec{\nabla }}{\varphi _u}\,\mathrm {d}{\varvec{x}}\, +\int _{\partial {\mathcal {K}}\setminus \partial \varOmega } \left( \left\{ \!\left| {{\varvec{v}}_h}\right| \!\right\} \nonumber \right. \nonumber \\&\quad \left. +\frac{\beta }{h_F}\llbracket {u_h}\rrbracket \right) \cdot {\varvec{\nu }}_{{\mathcal {K}}} \, {\varphi _u}\, \mathrm {d}s -\frac{1}{2}\int _{{\mathcal {K}}} {\mathcal {G}}_h\,{\varvec{z}}_h\cdot {\varvec{y}}_h\,{\varphi _u}\,\mathrm {d}{\varvec{x}} = 0,\nonumber \\ \end{aligned}$$
(2d)

Formulation 2

Seek \(({\varvec{z}}_h,{\varvec{v}}_h,u_h)\in \left[ \mathbb {P}_p(\mathcal {T}_h)\right] ^2\times \left[ \mathbb {P}_p(\mathcal {T}_h)\right] ^2\times \mathbb {P}_p(\mathcal {T}_h)\) such that the following holds \(\forall {\mathcal {K}}\in \mathcal {T}_h\), \(\forall {\varvec{\varphi _z}},{\varvec{\varphi _v}}\in \left[ \mathbb {P}_p(\mathcal {T}_h)\right] ^2, {\varphi _u}\in \mathbb {P}_p(\mathcal {T}_h)\), and a.e. \(0<t<T\):

$$\begin{aligned}&\int _{{\mathcal {K}}}{\varvec{z}}_h\cdot {\varvec{\varphi _z}}\,\mathrm {d}{\varvec{x}} - \int _{{\mathcal {K}}} u_h\,{\varvec{\nabla }}\cdot {\varvec{\varphi _z}}\,\mathrm {d}{\varvec{x}} +\int _{\partial {\mathcal {K}}\setminus \partial \varOmega } \left\{ \!\left| {u_h}\right| \!\right\} {\varvec{\varphi _z}}\cdot {\varvec{\nu }}_{{\mathcal {K}}} \,\mathrm {d}s\nonumber \\&\quad +\int _{\partial {\mathcal {K}}\cap \partial \varOmega } u_h {\varvec{\varphi _z}}\cdot {\varvec{\nu }}_{{\mathcal {K}}} \,\mathrm {d}s = 0, \end{aligned}$$
(3a)
$$\begin{aligned}&\int _{{\mathcal {K}}}{\varvec{v}}_h\cdot {\varvec{\varphi _v}}\,\mathrm {d}{\varvec{x}}-\int _{{\mathcal {K}}}\frac{{\mathcal {G}}_h}{{Q_{\varepsilon }}_h}\,{\varvec{z}}_h\cdot {\varvec{\varphi _v}}\,\mathrm {d}{\varvec{x}} \;=0, \end{aligned}$$
(3b)
$$\begin{aligned}&\int _{{\mathcal {K}}} \frac{\partial _t u_h}{{Q_{\varepsilon }}_h}\,{\varphi _u}\,\mathrm {d}{\varvec{x}} - \int _{{\mathcal {K}}} {\varvec{v}}_h\cdot {\varvec{\nabla }}{\varphi _u}\,\mathrm {d}{\varvec{x}}\, \nonumber \\&\quad +\int _{\partial {\mathcal {K}}\setminus \partial \varOmega } \left( \left\{ \!\left| {{\varvec{v}}_h}\right| \!\right\} +\frac{\gamma }{h_F}\llbracket {u_h}\rrbracket \right) \cdot {\varvec{\nu }}_{{\mathcal {K}}} \, {\varphi _u}\, \mathrm {d}s \,=\, 0, \end{aligned}$$
(3c)

where \(\alpha ,\beta ,\gamma >0\) denote penalty coefficients, and \(h_F\) is the length of the boundary face \(F \subset \partial {\mathcal {K}}\).

Stability analysis

In this section, we consider the discrete stability of Formulations 1 and 2. Since the problem is non-linear we do not carry out these estimates in some standard norm but rather consider statements of so called "energy stability". We were not able to find literature on discrete stability estimates for DG approaches to the equation at hand; however, there are some results for similar highly non-linear equations. Karasözen et al. [26] and Feng and Li [16] presented fully discrete energy stability estimates for discontinuous Galerkin approximations of the Allen–Cahn equation which is also related to the mean curvature flow as indicated in [16]. The DG schemes used in [16, 26] relied on the symmetric interior penalty method (SIPG) where second order operators are discretized as in the traditional finite element methods (cf. Sect. 1). Xia et al. [43, 44] presented semi-discrete energy estimates for fourth-order Cahn–Hilliard and coupled Allen–Cahn/Cahn–Hilliard systems discretized using LDG methods. Similar estimates for surface diffusion and Willmore flow of graphs—also fourth-order equations—were presented by Xu and Shu [45] and served as an inspiration for the current work. The analysis techniques employed in the proofs below are in part based on our own stability estimates for the 2- and 3D shallow-water [2, 32] and Darcy [4] equations as well as on the work by Cockburn and Dawson [13].

Theorem 1

Let \(({\varvec{z}}_h,{\varvec{v}}_h,{\varvec{y}}_h,u_h)\in \left[ \mathbb {P}_p(\mathcal {T}_h)\right] ^2\times \left[ \mathbb {P}_p(\mathcal {T}_h)\right] ^2\times \left[ \mathbb {P}_{3p}(\mathcal {T}_h)\right] ^2\times \mathbb {P}_p(\mathcal {T}_h)\) be a solution of Formulation 1. In addition, we assume \(0 < {\mathcal {G}}_\text {min} \le {\mathcal {G}}_h \le {\mathcal {G}}_\text {max}\). Then there exists a constant C independent of h (but dependent on the domain \(\varOmega \), problem parameters, etc.) such that the following holds:

$$\begin{aligned}&\int _\varOmega \partial _t |u_h|^2\,\mathrm {d}{\varvec{x}}+\int _\varOmega {\mathcal {G}}_h\left| {\varvec{z}}_h\right| ^2\,\mathrm {d}{\varvec{x}} \\&\quad +\beta \sum _{F\in \mathcal {E}_I}\frac{1}{h_F}\int _{F}\left| \llbracket {u_h}\rrbracket \right| ^2\,\mathrm {d}s \le C \left( 1 + h^{-2}\right) . \end{aligned}$$

Proof

First, we integrate by parts the respective second terms of Eqs. (2a) and (2c) obtaining

figuree

Substituting \({\varvec{\varphi _z}}= {\varvec{v}}_h,\, {\varvec{\varphi _v}}= {\varvec{z}}_h, \, {\varvec{\varphi _y}}= {\mathcal {G}}_h {\varvec{z}}_h u_h,\, {\varphi _u}= u_h\), summing over all elements \({\mathcal {K}}\in \mathcal {T}_h\), and evaluating the meta-equation ’(2a\(^\prime \))–(2b) + 1/2  (2c\(^\prime \)) + (2d)’, produces

$$\begin{aligned}&\frac{1}{2}\int _\varOmega \partial _t |u_h|^2\,\mathrm {d}{\varvec{x}}+\int _\varOmega {\mathcal {G}}_h\left| {\varvec{z}}_h\right| ^2\,\mathrm {d}{\varvec{x}}\nonumber \\&\qquad +\alpha \sum _{F\in \mathcal {E}_I}\int _{F}\left| \left\{ \!\left| {{\mathcal {G}}_h{\varvec{z}}_h u_h}\right| \!\right\} \right| ^2\left| \left\{ \!\left| {{\varvec{z}}_h}\right| \!\right\} \right| ^2\left| \llbracket {\llbracket {{\varvec{z}}_h}\rrbracket }\rrbracket \right| ^2\,\mathrm {d}s\nonumber \\&\qquad + \beta \sum _{F\in \mathcal {E}_I}\frac{1}{h_F}\int _{F}\left| \llbracket {u_h}\rrbracket \right| ^2\,\mathrm {d}s \nonumber \\&\quad = \frac{1}{2} \sum _{{\mathcal {K}}\in \mathcal {T}_h}\int _{{\mathcal {K}}}{\varvec{\nabla }}\log \left( {Q_{\varepsilon }}_h^2\right) \cdot {\mathcal {G}}_h {\varvec{z}}_h u_h \,\mathrm {d}{\varvec{x}} \nonumber \\&\qquad - \frac{1}{2} \sum _{F\in \mathcal {E}_I}\int _{F} \llbracket {\log \left( {Q_{\varepsilon }}_h^2\right) }\rrbracket \cdot \left\{ \!\left| {{\mathcal {G}}_h {\varvec{z}}_h u_h}\right| \!\right\} \,\mathrm {d}s. \end{aligned}$$
(5)

The remainder of the proof deals with estimating the right-hand-side terms in (5). For convenience, we remind the reader of Young’s inequality frequently used in our analysis: \(\forall a,b\ge 0,\sigma >0\) it holds \(a\,b\le \frac{a^2}{2\sigma }+\frac{\sigma b^2}{2}\).

$$\begin{aligned}&\left| \int _{{\mathcal {K}}} {\varvec{\nabla }}\log \left( {Q_{\varepsilon }}_h^2\right) \cdot {\mathcal {G}}_h {\varvec{z}}_h u_h \,\mathrm {d}{\varvec{x}}\right| \le \int _{{\mathcal {K}}} \left| \frac{{\mathcal {G}}_h {\varvec{\nabla }}|{\varvec{z}}_h|^2 \cdot {\varvec{z}}_h u_h}{|{\varvec{z}}_h|^2 + \varepsilon ^2} \right| \mathrm {d}{\varvec{x}}\\&\quad \le \int _{{\mathcal {K}}} \frac{2 {\mathcal {G}}_h |{\varvec{z}}_h|^2 |{\varvec{\nabla }}{\varvec{z}}_h| |u_h|}{|{\varvec{z}}_h|^2 + \varepsilon ^2} \mathrm {d}{\varvec{x}} \le \int _{{\mathcal {K}}} 2 {\mathcal {G}}_h |{\varvec{\nabla }}{\varvec{z}}_h| |u_h| \,\mathrm {d}{\varvec{x}}\\&\quad \le \frac{h^2 {\mathcal {G}}_\text {min}}{2 C_\text {inv} {\mathcal {G}}_\text {max}} \int _{{\mathcal {K}}} {\mathcal {G}}_h |{\varvec{\nabla }}{\varvec{z}}_h|^2 \,\mathrm {d}{\varvec{x}} + \frac{2 C_\text {inv} {\mathcal {G}}_\text {max}}{h^2 {\mathcal {G}}_\text {min}} \int _{{\mathcal {K}}} {\mathcal {G}}_h |u_h|^2 \,\mathrm {d}{\varvec{x}}\\&\quad \le \frac{1}{2} \int _{{\mathcal {K}}} {\mathcal {G}}_h |{\varvec{z}}_h|^2 \,\mathrm {d}{\varvec{x}} + C_1 h^{-2} \Vert u_h\Vert ^2_{L^2({\mathcal {K}})} \end{aligned}$$

using the inverse inequality for regular meshes with constant \(C_\text {inv}\) and limits for \({\mathcal {G}}_h\).

The estimate of the boundary integral term on the right hand side of (5) exploits the Lipschitz continuity with the Lipschitz constant less than 1 of the logarithmic function for argument values \(\ge 1\), i.e., \(\left| \log a - \log b \right| \le |a -b|, \; \forall a,b \ge 1\) and the simple fact that \(\llbracket {const}\rrbracket =0\):

$$\begin{aligned}&\left| \int _{F} \llbracket {\log \left( {Q_{\varepsilon }}_h^2\right) }\rrbracket \cdot \left\{ \!\left| {{\mathcal {G}}_h {\varvec{z}}_h u_h}\right| \!\right\} \,\mathrm {d}s\right| \\&\quad = \left| \int _{F} \llbracket {\log \frac{|{\varvec{z}}_h|^2 + \varepsilon ^2}{\varepsilon ^2} + \log \left( \varepsilon ^2\right) }\rrbracket \cdot \left\{ \!\left| {{\mathcal {G}}_h {\varvec{z}}_h u_h}\right| \!\right\} \,\mathrm {d}s\right| \\&\quad \le \int _{F} \left| \llbracket {\frac{|{\varvec{z}}_h|^2 + \varepsilon ^2}{\varepsilon ^2}}\rrbracket \right| \left| \left\{ \!\left| {{\mathcal {G}}_h {\varvec{z}}_h u_h}\right| \!\right\} \right| \,\mathrm {d}s\\&\quad = \frac{1}{\varepsilon ^2} \int _{F} \left| \llbracket {|{\varvec{z}}_h|^2}\rrbracket \right| \left| \left\{ \!\left| {{\mathcal {G}}_h {\varvec{z}}_h u_h}\right| \!\right\} \right| \,\mathrm {d}s\\&\quad = \frac{2}{\varepsilon ^2} \int _{F} \left| \left\{ \!\left| {{\varvec{z}}_h}\right| \!\right\} \llbracket {\llbracket {{\varvec{z}}_h}\rrbracket }\rrbracket \right| \left| \left\{ \!\left| {{\mathcal {G}}_h {\varvec{z}}_h u_h}\right| \!\right\} \right| \,\mathrm {d}s\\&\quad \le \alpha \int _{F} \left| \left\{ \!\left| {{\mathcal {G}}_h {\varvec{z}}_h u_h}\right| \!\right\} \right| ^2 \left| \left\{ \!\left| {{\varvec{z}}_h}\right| \!\right\} \right| ^2 \left| \llbracket {\llbracket {{\varvec{z}}_h}\rrbracket }\rrbracket \right| ^2 \,\mathrm {d}s + \frac{1}{\alpha \varepsilon ^4} \int _{F} \,\mathrm {d}s. \end{aligned}$$

The final result is then obtained by applying Grönwall’s inequality. \(\square \)

Some remarks on this estimate and the required assumptions are in order.

  • The limits on \({\mathcal {G}}_h\) as well as the ’special’ penalty term in (2a) are needed to perform the stability estimate and can be omitted in the implementation with no negative effects. Whereas the upper limit on \({\mathcal {G}}_h\) is realistic and justified by the application, the lower limit is not. The choice of \({\mathcal {G}}\) in Sect. 2 reveals that in regions of a very large image gradients function \({\mathcal {G}}\) can indeed attain values arbitrary close to zero. Since the lower limit is only needed to estimate the gradient \({\varvec{\nabla }}{\varvec{z}}_h\) by the function itself, one could alternatively attempt to use a weighted inverse inequality. Such results do exist for some specific choices of the weight function (see, e.g., [7]), but this would unnecessarily make the analysis more complicated and less general.

  • The choice of higher polynomial order for the approximation space for \({\varvec{y}}_h\) in Theorem 1 is motivated by the analysis technique used in this study. From the practical standpoint, using approximation spaces of equal order for all unknowns (as was the case in [8]) also works fine.

  • Variable \({\varvec{v}}\) in (M1\(_\varepsilon \)) has been introduced for analysis convenience and can be trivially omitted by modifying the last equation in system (M1\(_\varepsilon \)) as was done in [8], thus equating the number of equations to that in system (M2\(_\varepsilon \)).

Theorem 2

Let \(({\varvec{z}}_h,{\varvec{v}}_h,u_h)\in \left[ \mathbb {P}_p(\mathcal {T}_h)\right] ^2\times \left[ \mathbb {P}_p(\mathcal {T}_h)\right] ^2\times \mathbb {P}_p(\mathcal {T}_h)\) be a solution for Formulation 2. It holds:

$$\begin{aligned}&\int _\varOmega \frac{\left( \partial _t u_h\right) ^2}{{Q_{\varepsilon }}_h}\,\mathrm {d}{\varvec{x}} +\frac{1}{2}\int _\varOmega \frac{{\mathcal {G}}_h \partial _t \left| {\varvec{z}}_h\right| ^2}{{Q_{\varepsilon }}_h}\,\mathrm {d}{\varvec{x}}\\&\quad +\frac{\gamma }{2}\sum _{F\in \mathcal {E}_I}\frac{1}{h_F}\int _{F}\partial _t \left| \llbracket {u_h}\rrbracket \right| ^2\,\mathrm {d}s \;=\;0. \end{aligned}$$

Proof

In the first step, we differentiate (3a) with respect to time and integrate the second term by parts resulting in

figuref

Choosing \({\varvec{\varphi _z}}= {\varvec{v}}_h,\, {\varvec{\varphi _v}}= \partial _t{\varvec{z}}_h,\, {\varphi _u}= \partial _t u_h\), summing over all elements \({\mathcal {K}}\in \mathcal {T}_h\), and subtracting (3b) from the sum of (3a’) and (3c) we arrive at the result of the theorem. \(\square \)

Numerical results

In this section, we compare the numerical results obtained with Formulations 1 and 2 using an analytical test problem, a standard benchmark on subjective contours, and a medical image. Both formulations represent non-linear systems of equations that we solve semi-implicitly in time using the backward/forward Euler method by employing implicit Euler for all linear parts of the system and explicit Euler for non-linear ones. The time step in all runs is chosen small enough to preclude any effects of the time discretization error on the final result.

Table 1 \(L^2(\varOmega )\) discretization errors and orders of convergences for Formulation 1 using penalty \(\alpha =0\), \(\beta =0.02\) for \(p=0\) and \(\beta =0.8\) otherwise
Table 2 \(L^2(\varOmega )\) discretization errors and orders of convergences for Formulation 2 using penalty \(\gamma =0.02\) for \(p=0\) and \(\gamma =0.8\) otherwise

Code verification

In the following, we verify the correctness of our code and determine the experimental orders of convergence of our numerical scheme. Let

$$\begin{aligned} (0,\,T)\times \varOmega =(0,\,0.01)\times [0.5,\,1.5]^2\, \end{aligned}$$

and let \(\displaystyle u(t,{\varvec{x}})=10\,(t+0.1)\,\exp (-|{\varvec{x}}|^2)\) be the exact solution with \({\mathcal {G}}({\varvec{x}})=|{\varvec{x}}|\). Inserting this in Eq. (MP) yields the right-hand-side function

$$\begin{aligned} f(t,{\varvec{x}})=\left( 10+40\,(t+0.1)\right) \,\exp \left( -|{\varvec{x}}|^2\right) \,|{\varvec{x}}| \end{aligned}$$

which goes into (2d) and (3c) as an additional term. We apply Dirichlet boundary conditions in this test case, set \(\varepsilon =0\), and forego any slope limiting because of the expected smoothness of the solution. For \(j\in {\mathbb {N}}\), we denote by \(h_j\) the mesh size of a Friedrich-Keller triangulation and define the time step size as \(\tau _j:=C(p)\,h_j^{p+1}\), where p denotes the polynomial degree, and C(p) is chosen as 0.05 if \(p=0\) and 1 otherwise. In case \(p=1\), the definition of \(\tau _j\) is a standard choice for diffusion problems (see [19, 20, 29], where a classical (piecewise linear) finite element discretization was used). Furthermore, we define the error of our approximation \(u_{h_j}\) in the jth refinement level via \(\textit{err}_j:=\Vert u(t)-u_{h_j}(t)\Vert _{L^2(\varOmega )}\vert _{t=0.01}\) and, for \(j>1\), the experimental order of convergence by

$$\begin{aligned} \textit{EOC}_j:=\log \left( \frac{\textit{err}_j}{\textit{err}_{j-1}}\right) \Bigg /\log \left( \frac{h_j}{h_{j-1}}\right) . \end{aligned}$$

Formulation 1

Table 1 shows that our first method converges with the expected order of \(p+1\). We omitted the first penalty term, i.e. set \(\alpha =0\), since the exact solution is regular enough to prevent instabilities.

Formulation 2

Table 2 shows that also our second method converges with an estimated order of \(p+1\). This convergence behavior is consistent with known LDG results for advection-diffusion equations [14]. We also note that the first formulation appears to have a somewhat better absolute error even though the convergence orders are very similar for both systems.

Kanizsa’s triangle

In this section, we apply our algorithms to a standard image segmentation benchmark called Kanizsa’s triangle [Fig. 1 (left)] whose aim is to find the subjective contours. As already mentioned in Sect. 2, Eq. (MP) is used in image segmentation (cf. [8, 15, 30, 38,39,40] for a detailed description). In a nutshell, it enforces a level set/graph flow of a segmentation function towards a steady and approximately piecewise constant surface. The different plateaus of the segmentation function correspond to "objects" in an image, and its level sets correspond to their edges. Such objects can be described in terms of both physical edges—corresponding to high gradients in the image—and subjective edges such as the triangle in Fig. 1 (left) that is discernible although it is not objectively present in the image. However, the subjective surfaces method (MP) is able to detect both of these two types of edges.

As initial segmentation function with level sets in Fig. 1 (right) we have

$$\begin{aligned} u_0({\varvec{x}}) = C\left( R-|{\varvec{x}}-{\varvec{x}}_0|\right) \chi _{\{|{\varvec{x}}-{\varvec{x}}_0|\le R\}}({\varvec{x}}), \qquad C = 2. \end{aligned}$$

\({\varvec{x}}_0\in \varOmega \) and \(R>0\) are the object’s estimated center of mass and half diameter, respectively; \(\chi _{\{\cdot \}}\) denotes the characteristic function of a set.

Fig. 1
figure1

Kanizsa’s triangle (left), initial level set (right)

Fig. 2
figure2

Evolution of ten level sets, calculated with Formulations 1 (top) and 2 (bottom). After 60 time steps (left), after 120 time steps (right)

Fig. 3
figure3

Graphs of the segmentation functions at the final time. Formulation 1 (left), Formulation 2 (right)

Fig. 4
figure4

Original image (top), level sets for Formulation 1 (middle), level sets for Formulation 2 (bottom)

The regularization parameter is chosen as \(\varepsilon =0\) for Formulation 1 and \(\varepsilon =10^{-6}\) for Formulation 2, respectively. We choose \((0,T)\times \varOmega =(0,0.12)\times [0,1]^2\) and discretize the problem using 120 equidistant time steps and 11,858 elements in the spatial triangulation. The edge detector \({\mathcal {G}}\) is given by (1) with \(K=0.005\), and the stabilization parameters are \(\alpha =0,\,\beta =10\), and \(\gamma =0.1\). The polynomial degree of the spatial approximation is \(p=1\). Similarly to the results presented in [8], the solution is postprocessed after each time step using the vertex-based slope limiter introduced in [1, 3, 28].

Figure 2 shows 10 level lines of the segmentation function equidistributed in the interval \([1/3,2/3]\cdot \max _{\varOmega }u(T)\) for both formulations after 60 and 120 time steps. In addition, the values of the edge detector \({\mathcal {G}}\) for Kanizsa’s triangle are depicted in the background, where blue corresponds to 0 and red to 1.

Obviously, both formulations are able to produce the desired result, i.e. the level sets evolve towards the contours of the subjective triangle that one perceives. However, the level sets in Fig. 2 and the graphs in Fig. 3 indicate that the first formulation produces a less stable evolution than the second. In particular, small over- and undershoots are clearly visible in Fig. 3 (left) in the vicinity of the triangle. It remains to be mentioned that the second formulation must have a strictly positive \(\varepsilon \) to guarantee regularity of the system matrix, whereas the first formulation allows for a vanishing \(\varepsilon \). However, in our numerical experiments, the first formulation required larger values of the penalty parameter (\(\beta \gg \gamma \)).

Medical image segmentation

To validate the applicability of our algorithms to real-world problems, we attempt to segment a single heart chamber in an ultrasonic image of the heart. The same space-time domain is used as in Sect. 5.2 with a somewhat coarser mesh consisting of only ca. 5000 elements. Apart from the parameter K in the edge detector, which is changed to \(K=0.05\), all other parameters have the same values as in Sect. 5.2. Figure 4 shows the original image and 10 equidistributed level sets of the solutions to (M1\(_\varepsilon \)) and (M2\(_\varepsilon \)) in the intervals \([4/10,6/10]\cdot \max _{\varOmega }u(T)\) and \([6/10,8/10]\cdot \max _{\varOmega }u(T)\), respectively.

Conclusions

In this study, we compared two mixed formulations for the subjective surface method discretized using the local discontinuous Galerkin method. Both formulations were evaluated using numerical tests, and energy stability estimates were carried out. Even though our numerical tests showed similar convergence behavior in both cases, a number of notable differences became apparent in the course of our study. These can be summarized as follows:

  • In the setting considered in our stability analysis, system (M1\(_\varepsilon \)) contains one more equation than (M2\(_\varepsilon \)) and requires a higher order polynomial space for one of the unknowns thus producing more degrees of freedom with the corresponding higher computational cost. However, both requirements were introduced to simplify analysis and can be safely omitted in practical computations as illustrated in [8].

  • System (M1\(_\varepsilon \)) is in divergence form whereas (M2\(_\varepsilon \)) is not; implementing the former in the context of a DG scheme is much more straightforward. In addition, system (M2\(_\varepsilon \)) requires division by a non-linear expression in each time step that may cause further numerical difficulties.

  • The stability analysis for system (M1\(_\varepsilon \)) is more technical; however the obtained result is stronger and utilizes a ’typical’ DG norm. For system (M2\(_\varepsilon \)), on the other hand, the energy stability analysis results in an upper bound for the expression that may also contain non-positive terms and thus does not qualify as a norm or a semi-norm.

  • Applied to the Kanizsa’s triangle benchmark, Formulation 1 produced a less stable evolution than Formulation 2. In addition, the contours located by the second formulation appear to be closer to the subjective contours of the Kanizsa’s triangle. This phenomenon persists if one reduces the time step size and may require a more elaborate solution strategy than the pseudo-time stepping used in this work (e.g. some form of Newton’s method). We ascribe this difference between formulations to a somewhat more efficient handling of the non-linearity \({Q_{\varepsilon }}:=\sqrt{|{\varvec{z}}|^2+\varepsilon ^2}\) in (M2\(_\varepsilon \)).

  • Comparing both formulations on a medical imaging application once again led to a somewhat less stable solution using the first scheme. Nonetheless, both methods were able to sufficiently accurately locate the boundary of the heart chamber.

The results obtained in [8] and in this study encourage further investigation of DG methods for the subjective surface problem incorporating spacial adaptivity in order to significantly reduce the computational effort. Another interesting avenue of research concerns combining elements of lower and higher polynomial order in the same adaptive scheme (hp-adaptivity).

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Aizinger, V., Bungert, L. & Fried, M. Comparison of two local discontinuous Galerkin formulations for the subjective surfaces problem. Comput. Visual Sci. 18, 193–202 (2018). https://doi.org/10.1007/s00791-018-0291-4

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Keywords

  • Local discontinuous Galerkin method
  • Image segmentation
  • Subjective surfaces
  • Stability analysis
  • Mixed formulation
  • Divergence form
  • Edge detection