Computational and qualitative aspects of motion of plane curves with a curvature adjusted tangential velocity
In this paper we investigate a time dependent family of plane closed Jordan curves evolving in the normal direction with a velocity which is assumed to be a function of the curvature, tangential angle and position vector of a curve. We follow the direct approach and analyze the system of governing PDEs for relevant geometric quantities. We focus on a class of the so-called curvature adjusted tangential velocities for computation of the curvature driven flow of plane closed curves. Such a curvature adjusted tangential velocity depends on the modulus of the curvature and its curve average. Using the theory of abstract parabolic equations we prove local existence, uniqueness and continuation of classical solutions to the system of governing equations. We furthermore analyze geometric flows for which normal velocity may depend on global curve quantities like the length, enclosed area or total elastic energy of a curve. We also propose a stable numerical approximation scheme based on the flowing finite volume method. Several computational examples of various nonlocal geometric flows are also presented in this paper.
Key words. Curvature driven flow, nonlocal geometric flows, curvature adjusted tangential velocity, local existence of solutions,
2000 Mathematical Subject Classifications. 35K65, 65N40, 53C80.
In this paper we investigate a time dependent family of plane closed Jordan curves evolving in the direction of the inner normal with a speed , which is assumed to be a function of the curvature , tangential angle and position vector ,
Recall that the evolving family of plane curves having the normal velocity speed of the form (1) can be often found in various applied problems, e.g. dynamics of phase boundaries in thermomechanics, material science (motion of bipolar loops [8, 22]), image and movie processing in computer vision theory (see e.g. ). For a comprehensive overview of industrial applications of the geometric equation having the form of (1) we refer to a book by Sethian .
We analyze a system of governing PDEs for geometric quantities and propose a numerical method for computing the mean curvature flow of plane closed curves with nontrivial tangential redistribution of points along evolving curves. We focus on a class of so-called curvature adjusted tangential velocities for which the tangential velocity depends on the function of curvature and its average along the curve. Following , the tangential speed is constructed as a linear combination of the asymptotic uniform tangential redistribution developed in  and the tangential velocity extracted from the crystalline curvature flow equations by the second author in . Using the theory of abstract parabolic equations due to Angenent [4, 5] we show local existence and uniqueness of a classical smooth solution to the system of governing equations. We furthermore propose a numerical approximation scheme based on the flowing finite volume method. We present several computational examples. In contrast to the simplified numerical algorithm proposed in  (see also ), the curvature and tangent angle are not calculated from the position vector but the parabolic equations for the curvature and tangent angle are solved separately. Such an approach yields a stable and robust numerical approximation scheme derived by Mikula and the first author in [18, 19, 20] for the case of the so-called (asymptotically) uniform tangential redistribution. Recently, a higher order discretization scheme involving asymptotically uniform tangential redistribution has been proposed and analyzed by Mikula and Balažovjech in .
An embedded closed plane Jordan curve can be parameterized by a smooth function such that and . We denote , and where is the Euclidean inner product between vectors and . The unit tangent vector is , where is the arc-length parameter , and the unit inward normal vector is uniquely determined through the relation . A signed curvature in the direction is denoted by . i.e. . Let be the angle of , i.e., and . The problem of evolution of plane curves can be formulated as follows: given an initial curve , find a -parameterized family of plane curves , starting from and evolving according to the normal velocity . We follow the so-called direct approach in which we describe evolution of plane curves by an evolution equation for the position vector : , . Here is the tangential component of the velocity vector. Note that has no effect on the shape of evolving closed curves, and the shape is determined by the value of the normal velocity only. Hence the simplest setting can be chosen. Dziuk  studied a numerical scheme for in this case. For general , however, such a choice of may lead to numerical instabilities caused by undesirable concentration of grid points. In order to obtain stable numerical computation, several nontrivial choices of have been proposed. In  Kimura proposed a redistribution scheme in the case by choosing tangential velocity such that , where is the total length of . Hou, Lowengrub and Shelley  derived the tangentail velocity (10) with and (see Section 2). They also mentioned (10) for general with . But they did not make explicit comments on the importance of redistribution of grid points. Another choice of tangential velocity for the case has been proposed by Deckelnick . In  Mikula and the first author derived (10) with and in general frame work of the so-called intrinsic heat equation for . Moreover, in  and , they proposed a method of asymptotically uniform redistribution for . Besides aforementioned uniform distribution methods, under the so-called crystalline curvature flow, grid points are not uniformly distributed but their redistribution takes into account variations in the curvature. The second author extracted the tangential velocity which is implicitly built in the crystalline curvature flow equation . The asymptotically uniform redistribution is quite effective and valid for a wide range of applications. However, from a numerical point of view, there is no reason to take uniform redistribution automatically. Hence a new way of curvature adjusted tangential redistribution can be considered in order to take into account the shape of evolved curves and variations in the modulus of the curvature.
In this paper we furthermore analyze qualitative and quantitative properties of nonlocal geometric flows in which the normal velocity is given by
where is a local part of the normal velocity of an evolving curve at a point locally depending on the curvature and tangent angle at position and is a nonlocal part of the normal velocity at depending on the entire shape of the curve . Typically is a function depending on the total length , the enclosed area and the total elastic energy of a curve , i.e. .
The paper is organized as follows. In section 2 we investigate the system of governing partial differential equations for relevant geometric quantities. We derive a tangential velocity taking into account variations in the modulus of the curvature along evolving curves. In section 3 we prove, locally in time, existence and uniqueness of classical smooth solutions. In section 4 we discuss the relationship between crystalline curvature adjusted tangential velocity and affine scale space evolution of planar curves studied by Angenent, Sapiro and Tannenbaum in . We show that the tangential velocity implicitly incorporated in the affine intrinsic heat equation is the same as the one present in the crystalline curvature motion. Section 5 we discuss several interesting examples of geometric flows in which the normal velocity depends on the total length and the enclosed area. A special attention is put on analysis of the gradient flow for the isoperimetric ratio. A numerical approximation scheme based on the flowing finite volume method is presented in section 6 together with computational examples (section 7).
2 System of governing equations and curvature adjusted tangential redistribution
In what follows, we shall assume . Then we can express the normal velocity in the form:
where and . By , and we denote partial derivatives of with respect to and , and, for .
According to  (see also [18, 20]), one can derive a system of PDEs governing evolution of plane curves satisfying . It is easy to check the following facts. Indeed, it follows from the transformation ( is the so-called local length), Frenet’s formulae , , commutation relation , expression for the curvature , tangent angle that the curvature , tangent angle , local length and position vector satisfy the following system of PDEs:
In what follows, we shall specify the tangential velocity function by taking into account the curvature variation with respect to the averaged function of the curvature. First, for the total length of a curve evolving in the normal direction with the speed we have the following identity:
(see e.g. ). Recall that the so-called relative local length plays an important role in controlling redistribution of grid points (cf. ). Let be the maximal time of existence of a solution. If uniformly w.r. to as , then redistribution of grid points becomes asymptotically uniform. In general, the uniform redistribution can stabilize numerical computation, and it has no effect on the shape of evolved closed plane curves. Neither uniform nor asymptotically uniform redistribution take into account variations in the curvature along evolved curves. Therefore, a natural question arises: how to redistribute grid points densely (sparsely) on sub-arcs where the modulus of the curvature is large (small) with respect to its averaged value. To answer this question, we shall define the so-called -adjusted relative local length quantity
where is the arc-length average of a quantity . Throughout the paper we will assume the function is smooth and strictly positive, i.e. . As for the computational experiments discussed in section 7 we shall consider for . Notice that if , if . Other choices of are also possible. For example . In [18, 19, 20] the constant function has been utilized in order to construct (asymptotically) uniform tangential redistribution. On the other hand, the function is implicitly built-in the crystalline curvature flow as it was pointed out in . The desired -adjusted asymptotic uniform redistribution can be obtained by choosing in such a way that as . To this end, let us suppose that satisfies the equation:
According to [19, 20] the shape function can be constructed in the form: where are given constants. With regard to (7) we have . In the case when the time of existence is finite and evolved curves shrinks to a point, i.e. as , we can take and . On the other hand, if we can take and . In both cases we have and the convergence is guaranteed. Setting , we conclude is constant w.r. to the time yielding thus uniform tangential redistribution (cf. ).
Taking into account governing equations (3)–(6), definition of the -adjusted relative local length (8) we obtain, after straightforward calculations, that satisfies equation (9) iff the tangential velocity satisfies:
where and . To construct a unique solution , we assume the renormalization condition , i.e. .
3 Local existence and uniqueness of classical solutions
In this section, by following ideas adopted from  (see also [18, 19]), we shall prove a local time existence, uniqueness and continuation of a classical smooth solution to the governing system of equations (3)–(6). However, we have to rewrite the system (3)–(6) into its equivalent form which does not explicitly contains the derivative of the tangential velocity appearing in equation (5) for the local length . Its presence in (5) is a technical obstacle in a direct application of the general result [20, Theorem 5.1] on existence and uniqueness of solutions to (3)–(6). To this end, we note that the auxiliary function given by equation (9) is a solution to the ODE:
where . Therefore the function satisfies the ODE: . With this replacement we obtain the following system of PDEs:
we can express differentials and in terms of and as follows:
In what follows, we recall key ideas of the abstract theory of nonlinear analytic semigroups developed by Angenent [5, Theorem 2.7]. By means of this theory we will be able to prove local existence and uniqueness of smooth solutions to the system of governing equations (11)–(14) which can be rewritten as an abstract nonlinear equation of the form
Now, suppose that the Fréchet derivative belongs to the so-called maximal regularity class for any where is an open neighborhood of the initial condition (cf. ). Here are Banach spaces, is densely embedded in . Then, it follows from [5, Theorem 2.7] that the abstract initial value problem (17) has a unique solution on some small enough time interval . A maximal regularity class consists of those generators of analytic semigroups for which the linear equation , , , has a unique solution , for any and . The pair is the so-called maximal regularity pair. It was also pointed out by Angenent in  (see also ), that the pair of the so-called little Hölder spaces is the maximal regularity pair for the second order differential operator subject to periodic boundary conditions at . The “little” Hölder space with and is the closure of in the topology of the Hölder space (see ). It means . In order to apply the abstract existence result [5, Theorem 2.7] we define the following scale of Banach space
By we denoted the Banach manifold . In other words, it is the space of all tangent angles satisfying the periodic boundary condition .
Henceforth we shall assume and are at least smooth functions such that and is a -periodic function in the variable. Furthermore, by we shall denote a bounded open subset such that for any .
Let be the tangential velocity function given as a unique solution to (10) satisfying the renormalization condition where . Then .
P r o o f: The term appearing in the definition (10) of can be decomposed as follows:
Assuming and are smooth functions, we conclude . Since any averaged quantity (e.g. , , ) is constant in the arc-length variable it belongs to the class . From (10) we have
Hence . Since and we conclude , as claimed.
In [18, 19] the authors proved that for the (asymptotically) uniform tangential velocity we have , i.e. for . Indeed, in this case is constant and therefore . Such a higher smoothness was needed because the authors applied the abstract result [5, Theorem 2.7] to the original system of equations (3)–(6). Hence, in order to treat equation (5) for the tangential velocity should belong to the class .
Assume where is the curvature, is the tangential vector, is the -adjusted relative local length of an initial regular curve . Assume and where and are smooth functions of their arguments such that is a -periodic function in the variable and . The nonlocal part of the normal velocity is assumed to be a smooth function from a neighborhood of into , i.e. .
Here we have used the relations and , where . Furthermore, using Frenet formulae we can rewrite the position vector equation as follows:
where . This “cheap trick” enables us to utilize the parabolic smoothing effect of the parabolic equation for when we compared with an argument based only on the analysis of solutions to the ODE: having not enough smoothness such as .
There exists an open neighborhood of such that , and , for any . Then the mapping is a smooth mapping from into . Its linearization at has the form:
Here is a diagonal matrix, . The coefficients of satisfy: , , , , where is given by (15), i.e. . The coefficients of matrices corespond to the first and zero-th order derivative terms in the second order linear differential operator .
Let be an open subset in such that . With regard to Lemma 1 we have . Furthermore, we assumed . As a consequence we obtain for .
For the second order differential linear operator we have . Moreover, , is a generator of an analytic semigroup on with the domain . It belongs to the maximal regularity pair , i.e (see ).
Since contains differentials of the first order only, it can be extended to a bounded operator from into . Due to boundedness of and taking into account the interpolation inequality between and spaces we conclude the following inequality:
where is a generic positive constant. Using Young’s inequality we can conclude that the linear operator (now considered as a linear operator from into ) has the relative zero norm, i.e. for any there exists a constant such that . (cf. [23, Section 2.1] and also ). By virtue of [5, Lemma 2.5] (see also [23, Theorem 2.1]) the class is closed with respect to perturbations by linear operators with zero relative norm. Thus for any . The proof of the short time existence of a solution now follows from the aforementioned abstract result [5, Theorem 2.7].
The nonlocal quantities are mappings from a neighborhood into . Indeed, as
we have . For the nonlocal term we conclude provided that where is a smooth function of its arguments .
Let us introduce the and norms of a quantity defined on a smooth curve as follows
Let be a family of planar curves evolving in the normal direction with the velocity for which short time existence of smooth solutions is guaranteed by Theorem 1. Suppose that is a maximal solution defined on the maximal time interval . If then either as or and, in this case, .
P r o o f: Similarly as in [4, Theorem 3.1] we can argue by a contradiction. Suppose to the contrary that the sum of norms is bounded on for and . Following the continuation argument due to Gage  and Angenent , we shall prove that there exists a limiting curve where the uniformly w.r. to . Since the tangential velocity has no impact on the shape of and the existence of the limiting curve we can assume in equation (20) for the curvature. With regard to the assumption on boundedness of the curvature and normal velocity we conclude that the reaction term in the curvature equation remains uniformly bounded for and . The parabolic equation (20) has therefore a smooth solution up to the limiting time . Now, it follows from the position vector equation and the tangent angle equation (12) that the limiting curve exists and is smooth. Moreover, using equation (9) and expression for the -adjusted local length we also conclude existence of the limit uniformly w.r. to . Starting from the limiting curve and using local in time existence result from Theorem 1, the family of curves can be prolonged to a larger time interval where . This is a contradiction.
In general, an estimate for is not sufficient in order to control behavior of the norm and vice versa. Indeed, let us consider, for example, . The selfsimilar family of circles parameterized by evolves with the normal velocity provided that . If then and holds but as . On the other hand, if , then stays bounded but holds as . Finally, if , , then with as . Moreover, stays bounded for but .
4 Affine scale space evolution and crystalline curvature tangential velocity
In the theory of image processing, the so-called morphological image and shape multiscale analysis is often used because of its contrast and affine invariance properties. Affine scale space evolution of closed planar curves has been introduced and studied by Angenent, Sapiro and Tannenbaum [3, 25] and Alvarez et al. . They derived a geometric equation of the form (1) with the normal velocity given by
In this section we will show that the geometric equation (22) representing affine scale space evolution is closely related to the crystalline curvature adjusted tangential velocity. For a wide class of tangential velocities, local existence, uniqueness and continuation of smooth solutions to the geometric equation (22) has been shown by Mikula and Ševčovič in .
In order to investigate affine scale space evolution, we have to introduce a notion of the so-called affine arc-length parameterization (cf. ). Consider a planar closed convex curve parameterized by with the arc-length parameterization , i.e. . Such a curve can be re-parameterized by a new parameterization such that where and . Here is a nonnegative function defined on the curve . The parameterization is called affine arc-length parameterization iff
(cf. ). In other words, is a parameterization for which the modulus of the affine curvature is constant along the curve . It is well known (see ) that the parameterization is the affine arc-length iff . Indeed, since