AMCF
Curvature flow
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We present a new implementation of anisotropic mean curvature flow for contour recognition. Our procedure couples the mean curvature flow of planar closed smooth curves, with an external field from a potential of point-wise charges. This coupling constrains the motion when the curve matches a picture placed as background. We include a stability criteria for our numerical approximation.
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Many applied and theoretical problems have been studied through the analysis of manifold deformations [5, 6, 10]. The description of these deformations by an evolution equation imposed via a geometric quantity are referred to as geometric flows. Applications include, for example, the growth of crystals, the modeling of fluids, and digital image recognition. Since their conception there has been continued interest in the development of numerical approximations to these flows [21].
The difficulty in analyzing these flows numerically depends on the geometric quantity in evolution (e.g. curvature, metric tensor, the manifold itself). In particular, the mean curvature flow (MCF) deforms a hypersurface in the normal direction
with a speed proportional to its mean curvature . This flow has an associated quasilinear parabolic equation in terms of an immersion of the hypersurface into the ambient manifold:(1) |
The numerical methods employed to solve this equation are classified as Eulerian or Lagrangian methods, depending on the discrete representation of the surface or curve in evolution. In Eulerian methods the evolution is tracked by values at fixed positions on a gridded ambient. Conversely, in Lagrangian methods, the object in evolution is tracked explicitly through the position of its points. In consequence, Lagrangian methods have two advantages: they require smaller data storage than Eulerian methods, and the solution is computed explicitly; although generally their error estimates are difficult to estimate precisely. Eulerian methods are now a powerful and frequently used technique for mean curvature flow applications since the development of the Level Set Method
[21, 24]. To our knowledge, there are very few examples of Lagrangian methods that approach this problem [25, 13], our work adds to this list.The disadvantage of Lagrangian methods are discussed in many works. Two problems arise commonly when straightforward discretization of equation ( 1) is performed: (i) a numerical instability as shown in Figure 1 and (ii) loop formation which contradicts the comparison principle (see [7, 18]), as shown in Figure 2 for a cycloid (solid line) and its first iteration result (polygonal curve) compared with a circumference (dashed curve).
Here we present a Lagrangian method for contour parametrization. To accomplish this application, we assume that a planar closed and differentiable curve is drawn on a 2D digital image. Then, we evolve the curve by mean curvature flow, but constraining the motion of the curve by the objects in the image. If only one object is initially inside the curve, as the curve shrinks, it will match parts of the boundary of the object. The main problem using MCF to recognize images is to couple the restriction and the flow, because it implies that the curve is not evolving uniformly. That is, not all points in the curve will move, even when the curvature at those points is different from zero. This kind of flow is called anisotropic. To avoid certain numerical difficulties, our scheme considers a curve motion along tangential and normal directions, as in [13]. The results of [13] correspond to an unconstrained flow, so the stability and convergence results found there can not be compared with ours. Since the numerical procedure is based on certain MCF properties, we provide the necessary technical details about existence and uniqueness of solutions for MCF in Theorems 4 and 9.
Kimura developed a Lagrangian method which numerically reproduces the mean curvature flow for curves, based on a redistribution of points by reparametrization by arc length [13]. That method represents an initial simply closed smooth curve by an ordered set of points. The position of each point is updated to represent the curve after a time . This method assumes a tubular neighborhood in which, for a small time interval , the whole evolution is inside, and therefore may be parameterized as
In this tubular neighborhood the map , given by
is invertible. A correction term is then added to Euler’s formula (see [13]). Figures (3- 5) show numerical examples with Kimura’s method, the evolution in time is represented by the z axis.
Our main theoretical result is the postulation of an anisotropic MCF, for which short time existence of solutions is shown in Theorem (10). Furthermore, we detail a numerical scheme to approximate solutions and apply it to the task of contour parametrization.
Many varieties of an anisotropic MCF can be proposed, Figure (7) compares the MCF with two versions of anisotropic MCF. Our evolution scheme is an adaptation for contour recognition of Kimura’s planar curve evolution. We approach the stability of our scheme through von Neumann’s analysis. In our analysis, the parameters in error propagators (see Equation (38)) are time dependent. Then, the stability condition cannot be determined for all time but only for the next time step. We establish our main stability criterion in Proposition 13.
In contrast to previous approaches, our method is a Lagrangian scheme whose main features are: (I) Estimation of curvature bounds are not required. (II) It is formulated as a Poisson problem with a boundary condition given implicitly. This condition links Poisson’s problem with the MCF. Our proposed Poisson problem couples MCF and the field due to a point charge distribution. (III) Finding a solution requires to solve a linear system. (IV) When it is implemented to a contour parametrization problem, only the pixel value at each point constraints the motion. (V) These constraints are handled through the source function and the boundary condition in Poisson’s equation. Our numerical experiments reveal that this Poisson formulation can match some non-convex shapes perfectly. In general, matching non-convex shapes successfully depends on the charge distribution. Our scheme takes the numerical approximation for curvature and an equidistant distribution of points from [13], and through our proposed Poisson formulation we can handle these constraints. In Figure 6, we present a contour matching with an initial circumference as the evolving curve. The color gradient advances with time from red to green. A high contrast between red and green represents a fast matching, and a smooth color gradient indicates regions which require more iterations to reach the shape boundary in black.
This work is presented as follows: §1 the proof of existence and uniqueness of solutions for MCF and some of its properties, included for completeness sake; §2 the presentation of our conditioned or anisotropic MCF and formulas for its solutions; §3 the numerical implementation of our anisotropic flow; §4 the implementation of the anisotropic MCF for contour parametrization; §5 details the stability properties of our numerical scheme, and §6 contains our conclusions and suggestions for future directions.
The main result of this section proves existence and uniqueness of solutions to an isotropic version of MCF. The proof includes new technical details, in particular bounds on the displacement of the immersed curve (equation (4) in Lemma 3), which are essential for the computations in the associated numerical problem in section 3.
This proof requires to rewrite equation (1) in terms of a reparametrization of the evolving manifold. Recall the following two theorems.
(Xi-Ping Zhu [30]) The mean curvature flow equation for a hypersurface with metric tensor is equivalent to
(2) |
For the reader’s convenience we include a proof here.
Let be the second fundamental form of a manifold and the induced connection by the ambient manifold connection . In local coordinates the right side of mean curvature flow equation expands to
We focus on the individual components and develop further:
∎
(Mantegazza [18]) Let be a compact submanifold of with induced metric and be an immersion which satisfies at every point in and every time in
(3) |
here is in . Then, there exist a family of diffeomorphisms such that satisfies equation (1). Conversely, given an immersion and a reparametrization such that is a mean curvature flow, then there is a field in such that satisfies equation (3).
In the following, we restrict the study to plane curves, compare with [10] and [18]. Set a regular smooth closed plane curve
with unitary tangent vector
, normal vector and curvature .The proof of short-time existence and uniqueness of solutions for the MCF is divided in three steps. We include these details here because they will be relevant to our main result in the analysis of solutions to anisotropic MCF in Theorem (10) below.
Reparametrize the deformations and rewrite the evolution in terms of a scalar quasilinear parabolic problem.
Prove existence and uniqueness of solutions for the linearized problem.
Extend Step 2 to the quasilinear case using the inverse function theorem for Banach Spaces [28, Sec. 4.13].
First, we reparametrize the deformation of the curve in the normal direction for a small time interval.
Let be small enough such that for in the evolution by mean curvature of is within a tubular neighborhood. If the deformations are parameterized by
(4) |
Then, the evolution equation for is given by
(5) |
Denote by the set of functions over whose derivatives are also in . Let an be functions in , consider the inner product
and denote by the set of functions over whose spatial derivatives and time derivatives are also in . We now state the linearized problem:
Let be a function over , , and be a linear parabolic operator. The problem
has a solution.
Before proceeding to the proof of theorem 4, we will need the following:
(Garding’s inequality [29]) Let an integer, suppose that the application of a linear differential operator over a -times differentiable function over is given by
Let be in and , be positive constants. Suppose that the coefficients are bounded and satisfy
(9) |
Then, if is in , there are constants and such that
(Modified Lax-Milgram theorem [27]) Let be a Hilbert space, a dense subspace and a bilinear form in such that
Let , and some
For every and some
Then for every bounded linear operator in and in , there exists in such that .
Let in , we recall the following definitions from [10]:
Here and are the Hilbert spaces resulting from the completion of the space of functions with compact support using the norms above, in that order.
Let be the space of functions over such that for small and large values of , and let be the completion of with the norm associated to the following inner product
Let
endowed with
From the theory of parabolic linear partial differential equations theory we invoke
[8, p. 351-352].Let and be a function over which vanishes at . We say that in is a weak solution of the linearized problem (LP) if
(10) |
The following lemma will be useful to construct a bounded bilinear operator, and then we will be able to apply Lemma 6.
We can now begin the proof of theorem 4
First we will prove the existence of a weak solution.
Let be a bilinear form given by
(12) |
here is a constant, , , and are the coefficients of the linearized problem (LP). Let be a linear operator over given by
(13) |
We will prove that satisfies the hypotheses of lemma 6:
To verify that condition (2) also holds observe that:
Integrate by parts to obtain:
Let be an upper bound for . Using lemma 5 there exists constants and such that:
The last inequality follows from having properly chosen .
From the above arguments we obtain that for all there exists such that . Since Equation (11) can be obtained from (12) and (13), is therefore a weak solution. ∎
We proved the existence of weak solutions. However, the regularity of weak solutions for linear parabolic problems depends on the regularity of the data and . Since the existence of solutions for linear parabolic problems is an auxiliary result in our work, we only refer to [8, Sec. 7.1.3] for the regularity extension.
The uniqueness of weak solutions for linear parabolic operators is also stated in [8] by considering the difference of two weak solutions . Since satisfies the problem with , from lemma 5 and substituting into (10, we obtain
(14) |
Substituting the last inequality into the derivative of , one obtains
Since is non negative, . Thus implies . A similar procedure can be used to prove the uniqueness of solutions for quasilinear parabolic problems. The main hypothesis is an estimate similar to (14), in particular for MCF see [4].
We now proceed to extend the linearized solution of normal deformation (Theorem 4) to the non linear problem
Given a quasilinear parabolic problem
in . Then, there exist such that this problem has a solution.
Define the map by
Notice that a function such that is a solution for the quasilinear problem. To this end, let be the linearization of around , by Theorem 4 the problem
(15) |
has a solution. Let and be a sequence converging to as tends to from above. Suppose that is a solution of (15), and let and be the coefficients of . Additionally, consider the linearization of around , let and be its linearized coefficients. From (15) we have
here . Notice that as tends to from above, implies as .
We can take small enough such that be in a neighborhood of for all in . Solving each linear problem, we end with a sequence such that . Consider by continuity. Invoking the Inverse Function Theorem for Banach spaces [28, Sec. 4.13], is a local diffeomorphism. Therefore, the map is locally invertible. ∎
This section is focused on Lagrangian methods. These methods track the evolving curve by explicitly computing the update of coordinates for each point in the curve.
A curve evolving by MCF will stop its evolution on some point, if the curvature at that point equals zero. If we need to evolve the curve further even though , we need to modify the Equation 1 using another curvature dependent normal flow. The equation that will describe this new flow is the following:
here the speed is assumed to depend on the curvature.
Consider the system:
Let be a given distribution, and be an initial smooth simple closed curve. We are interested in the evolution of by
Notice that in (P1) we have a normal field which drives the evolution, and in (P2), a Poisson equation which defines a field in terms of its potential with Dirichlet’s boundary condition. To link these problems with the MCF, we impose that if equals zero, then the MCF is recovered i.e. . Next, we need to prove the existence and uniqueness of solutions for (P1) and (P2) before we introduce our numerical solution.
The following theorem is our main theoretical result in this paper:
Let be the initial curve, denote by its interior. Let and be a point in such that for all points in a tubular neighborhood of . Let be a function, be the Dirac’s delta distribution, and be an initial smooth simple closed curve. Then, the system (P1)-(P2) has a unique solution.
The problem (P2) is a Poisson equation with a Dirichlet boundary condition. Therefore, the existence and uniqueness of its solution is well known when is a function with compact support (see, for example, [8]).
The problem (P1) represents a curve being deformed in the normal direction with speed . Therefore, we can consider a tubular neighborhood about , and verify the existence and uniqueness of solutions of (P1) for small time. Similarly to Lemma 3, let be the unitary normal at time , we reparametrize the deformations inside the -tubular neighborhood by:
Taking the time derivative of the last equation, and recalling (P1), we obtain
As is a unitary vector, we can state the partial equation for :
(16) |