This paper studies invertible maps, i.e., maps for which an inverse mapping exists. These maps are also sometimes called 1-to-1 maps, since they define a 1-to-1 correspondence between domains. This correspondence is crucial in many fields, for instance in enabling the study of variations of properties across a collection of objects: in the context of computational anatomy, considering different specimens of an organ and studying the variation of corresponding regions aids the classification of the specimen as healthy or unhealthy. Similarly, in the context of evolutionary biology, variation across a collection is instrumental in the construction of phylogenetic trees describing how species evolve.
Mathematicians have long tried to find families of maps that are guaranteed to be invertible, i.e., prove that if some criteria hold for a map, it is invertible. This is also the focus of our work. Namely, this paper concerns harmonic maps between two-dimensional domains (little is known regarding invertibility of harmonic maps into high-dimensional domains, and existing counterexamples indicate that the higher-dimensional case requires much more intricate assumptions). We consider two analogous families of maps for two separate cases - the continuous case, in which one continuous domain is mapped into another continuous domain; and the discrete case, in which a triangulated surface is mapped into a polygonal domain.
One of the largest known families of invertible harmonic maps is characterized by the celebrated Radó-Kneser-Choquet (RKC) theorem [duren2004harmonic, rado1926aufgave, kneser1926losung, choquet1945type]. RKC considers harmonic maps of planar disks into other planar domains (i.e., maps whose coordinate functions are solution of Laplace’s equation). RKC states that given a homeomorphism (i.e., a continuous map with a continuous inverse) of the unit circle onto a simple closed curve bounding a convex region, extending it harmonically to the interior produces a homeomorphism of the unit disk onto the interior of the curve. Though the constraint that the source domain is a disk may appear overly restrictive, taken in conjunction with the Riemann Mapping Theorem, RKC implies that a harmonic map of any compact Riemann surface with disk topology into a convex domain, such that the map between the one-dimensional boundaries is a homeomorphism, is itself a homeomorphism of the two-dimensional domains.
In the discrete case, a closely-related characterization of invertible maps is derived from Tutte’s work on planar embeddings of graphs [tutte1963draw]. Tutte considered planar 3-connected graphs and their barycentric embeddings, in which the position of each vertex is set to the average of its neighbors’ positions. When the outer face is a convex polygon, Tutte proved that the edges of such a barycentric embedding, realized as straight lines, do not cross.
Maps of triangulations have been further studied in [floater1997parametrization, floater2003one, gortler2006discrete]
, which consider discrete-harmonic embeddings of triangulated surfaces with disk topology. As in Tutte’s embedding, maps are specified by prescribing the position of vertices within the target domain, and are extended to the interior of triangles through linear interpolation. These works show that if the image of the boundary of a triangulation is a convex polygon, then the interpolated map is a homeomorphism. This property of discrete-harmonic maps, along with their computational tractability (via solving a single sparse system of linear equations), accounts for their broad applicability in computer graphics, including meshing, shape analysis, parameterization, and texture mapping[floater2005surface, liu2008local, weber2014locally, aigerman2014lifted, wong2015computing, bright2017harmonic, jiang2017simplicial, li2019optcuts, shen2019progressive].
In this paper, we extend the characterization of invertible (planar) harmonic maps. To this end, we consider harmonic and discrete-harmonic maps into non-convex
regions, and study conditions under which such maps are homeomorphisms. This enables the definition of correspondences while providing more degrees of freedom for optimizing the quality of the map (e.g. the extent to which the map distorts the notion of lengths, angles, areas).
One cannot simply omit the convexity requirement: in both the continuous (RKC) and discrete (Tutte) theorems, convexity plays a crucial role. For example, in the continuous case, it is known (Choquet [choquet1945type]) that for any non-convex region bounded by a simple curve, there exists some homeomorphism from the unit circle to the boundary curve for which the harmonic extension to the unit disk fails to be a homeomorphism [choquet1945type, alessandrini2009invertible]. Figure 1 illustrates harmonic and discrete-harmonic maps into non-convex domains.
Hence, our work is driven by the following question: is there a property sufficient to guarantee injectivity, which is trivially satisfied in the case of convex boundaries, but can also be satisfied in non-convex cases? As we will show, the answer is affirmative and the conditions are tractable.
To give an informal intuition for our condition, imagine a system of springs with a fixed boundary; once the system reaches equilibrium, interior forces cancel out (this is the physical interpretation of the harmonicity condition), except along the boundary where the forces acting on the boundary are balanced by external forces holding the boundary in place (the Dirichlet boundary conditions). At any point along the boundary, if the interior forces point inwards, the membrane is pulled inwards and locally should be in the interior of the target domain. If, on the other hand, the internal forces point outward, the membrane is pulled out of the target domain, and folds over the boundary to spill out. See Figure 1 for an illustration.
We formalize this intuition in terms of normal derivatives and prove that the condition requiring forces to point inwards (which is trivially satisfied for convex target domains) is sufficient to prove invertibility, independently of the target boundary’s shape.
In the continuous case, we build on results by Alessandrini and Nesi [alessandrini2009invertible, alessandrini2017errata]. They show that a sufficient and necessary condition for a harmonic map to be a homeomorphism is that it is a local homeomorphism along the boundary. If the boundary map is differentiable, they prove that a harmonic map is a homeomorphism if and only if it is orientation-preserving along the boundary. We extend their result to the continuous case of a boundary curve admitting a finite number of isolated singularities. To that end, we replace their orientation condition and define a simple geometric condition on the “forces” at the non-smooth points of the boundary, which we call the cone condition. We then prove that the cone condition characterizes homeomorphic harmonic maps.
In the discrete case, discrete-harmonic maps into non-convex domains have been characterized in Gortler et al. [gortler2006discrete]. We provide an alternative characterization in terms of a discrete analog of the cone condition, which, as in the continuous case, is an intuitive geometric condition along the boundary. We prove that it suffices to consider the discrete cone condition at the “non-convex boundary” (i.e., reflex) vertices to characterize discrete-harmonic intersection-free embeddings of triangulations. These, in turn, induce piecewise-linear homeomorphisms. Finally, we derive a discrete analog of the result by Alessandrini and Nesi, characterizing discrete-harmonic maps in terms of the orientation of triangles adjacent to the boundary.
The rest of the paper is organized as follows: Section 2 is concerned with harmonic maps in the continuous case and Section 3 with discrete-harmonic maps in the case of triangulations. The analogy between the conditions and results for the continuous and discrete statements is illustrated in Figure 2.
2. Continuous Harmonic Mappings
Let denote the open unit disk and let be a homeomorphism from the unit circle onto a simple closed curve enclosing a bounded set . We consider the harmonic extension of , that is the map given by the solution of the 2-dimensional Dirichlet problem
Note that has two coordinates, as does ; hence the notation of (1) should be understood as being applied to each coordinate separately.
When needed, we shall use the polar coordinates defined by , with and . Note that, as opposed to the classic definition of polar coordinates, in our definition corresponds to the boundary of the disc, and corresponds to its center, . This slight variation will simplify the subsequent equations. Figure 3 depicts the problem setup and notations.
One of the most well-known results on injectivity of harmonic mappings is due to Radó, Kneser and Choquet (RKC), who prove that if is convex then the harmonic extension of is a homeomorphism [duren2004harmonic, rado1926aufgave, kneser1926losung, choquet1945type]:
Theorem 1 (Rado-Kneser-Choquet).
If is convex, then is a homeomorphism of onto .
The RKC theorem is known to be sharp, in the following sense:
Theorem 2 (Choquet [choquet1945type]).
If is not convex, there exists a homeomorphism such that the harmonic extension is not a homeomorphism.
On the other hand, even when encloses a non-convex domain, there always exist boundary maps whose harmonic extension, i.e., the solution to (1) , is a homeomorphism. In fact, for any simple closed curve , there always exists a specific choice of the boundary map that yields a homeomorphic : one such boundary map can be constructed by considering the Riemann mapping [palka1991introduction, p. 420] from the interior of the unit disk to the interior of the domain enclosed by , which is a harmonic homeomorphism. Then, by Caratheodory’s theorem [palka1991introduction, p. 445], there is a continuous extension to the boundary, yielding the desired .
The following theorem, by Alessandrini and Nesi, states specific conditions that ensure that the harmonic extension is injective even when the target boundary is non-convex, assuming the boundary map is .
Theorem 3 (Alessandrini–Nesi [alessandrini2009invertible, alessandrini2017errata]).
Let be an orientation preserving diffeomorphism of class onto a simple closed curve and let be a solution of (1).
The mapping is a diffeomorphism of onto if and only if
2.2. Harmonic extensions to non-smooth boundary maps
As a step towards the discrete case, we consider also mappings that are only piecewise differentiable, and aim to formulate a criterion for guaranteeing the harmonic extension is a homeomorphism.
Towards that end, we draw inspiration from Alessandrini and Nesi (Theorem 3), but replace their determinant condition with a weaker condition, expressed in terms of one-sided derivatives, thus extending their result to the case of a piecewise-smooth boundary. We will assume that the derivative of the boundary map exists and is strictly bounded away from zero in all but finitely many points. We denote the set of boundary points in which the derivative does not exist as . We will also assume that the normal derivative of the harmonic extension is defined at all points in , does not vanish, and is continuous on . For points where the derivative does not exist, we assume that the one-sided derivatives and exist and do not vanish.
For every boundary point we define a cone
as the set resulting from the intersection of the two “inward-pointing” half-planes supporting the vectors of the one-sided derivativesand ; see figure 4 for an illustration in which the cone is colored in dark-grey.
Formally, suppose is an orientation-preserving homeomorphism of the boundary, i.e., it traverses in the counter-clockwise direction. The open cone at point is defined as
where denotes the clockwise rotation of .
Note that for a differentiable boundary point, i.e., , the two one-sided derivatives are equal, . Thus, in this case, is simply a half-space. On the other hand, if the one-sided derivative point in opposite directions (in which case the opening angle is or ) then the cone is empty. Lastly, note that since is an intersection of two half-spaces, it has an opening angle of at most , regardless of the opening angle of at that point (see inset).
Given a map we say it satisfies the cone condition at if the following equation holds
Intuitively, the cone condition requires that the derivative of the harmonic map in the normal direction is contained within the cone.
It turns out that this simple condition can be used to formulate a theorem similar to that of Alessandrini-Nesi’s, in which the cone condition replaces the determinant condition, ensuring that the resulting map is a homeomorphism into . Furthermore, we will show that, as the cone condition permits the boundary map to have derivative discontinuities, it extends Alessandrini-Nesi’s result to harmonic maps into domains with a non-smooth boundary. More precisely,
Assume that is an orientation preserving homeomorphism onto a simple closed curve that is at all but a finite set of points . Also assume that does not vanish on , and that for each point in the one-sided derivatives and exist, do not vanish and equal the limit of on the corresponding side.
A main ingredient of our proof is another theorem by Alessandrini and Nesi:
Theorem 5 ([alessandrini2009invertible], Theorem 1.7).
Let be a homeomorphism onto a simple closed curve and let be the solution of (1).
The mapping is a homeomorphism of onto if and only if, for every point , the mapping is a local homeomorphism at (i.e., there exists a neighborhood of where is injective).
One direction of the proof (“if”) can thus be obtained by showing that for every boundary point , the cone condition implies that is a local homeomorphism at .
The cone condition implies that
at the boundary point .
For differentiable boundary points , (4) along with continuity imply that the normal and tangential derivatives are linearly independent in a neighborhood of . Hence, the differential of the map is invertible and, by the inverse function theorem, there exists a neighbourhood of in which is a local homeomorphism.
For a point , where the derivative is discontinuous, the argument is more subtle, but follows a similar idea. In this case, the cone condition (4) implies that the boundary map is locally monotone in the direction perpendicular to the normal derivative at ; see Figure 5 for an illustration.
Namely, there exists a -neighborhood of and such that for all in we have
(With this notation we assume, without loss of generality, that is sufficiently far away from .) One can leverage this monotonicity to establish a uniform bound on the angle between the normal and tangential derivatives of the harmonic extension, and , in a neighborhood of . Thus, the cone condition implies that, in a sufficiently small neighborhood of , the normal and tangential derivatives are well-behaved and define a consistent local coordinate system. More precisely, we prove (following this line of reasoning) in Appendix A.1 the following Lemma:
Under the conditions of Theorem 4, the harmonic extension is a local homeomorphism around all boundary points , .
Thus we see that if the cone condition holds for all boundary points then (since is a local-homeomorphism at all points on ) Theorem 5 implies is a homeomorphism of onto .
For the other direction (“only if”), if does not satisfy the cone condition then the inner product in (4) is either zero or negative, for at least one of the one-sided derivatives; without loss generality we shall assume the cone condition is violated for . If (4) vanishes then and are colinear, in which case Theorem 5 implies that is not a homeomorphism. Otherwise, if (4) is negative, then a neighborhood of in is mapped outside of , as well implying that is not a homeomorphism. ∎
3. Discrete-Harmonic Mappings
In this section we discuss discrete analogs of the results presented in Section 2.
We consider a triangulation over a finite set of vertices defined by a set of faces (triangles)
and a corresponding set of edges
We think of the set of faces as (non-ordered) subsets of vertex triplets. The edges are ordered pairs of vertices, corresponding to the directed edges connecting the vertices of each face.
A boundary edge of the triangulation is an edge associated with only a single triangle; that is, is a boundary edge if there exists a unique with . The boundary of the triangulation is the union of all boundary edges. We say that a vertex is a boundary vertex if it belongs to a boundary edge and an interior vertex otherwise. We say that a triangulation is 3-connected if it remains connected after the removal of any two vertices and their incident edges.
A drawing of a given triangulation is a mapping of each vertex to a distinct point of the plane and of each edge to a simple curve with endpoints and . A drawing is intersection-free if its edges do not intersect except, possibly, at common endpoints. We say that a drawing is proper if its faces coincide with the faces of and the edges of its external (unbounded) face coincide with the boundary edges of . A triangulation has disk topology if its boundary is a cycle and it has a proper intersection-free drawing. See Figure 6 for illustrations.
Lastly, a straight-line drawing is a drawing in which each edge is realized by a straight line segment connecting its endpoints; we use a pair to denote the straight-line drawing determined by an embedding . If is a proper intersection-free straight-line drawing we denote by the simple polygonal domain enclosed by the boundary of the the straight-line drawing . See Figure 7 for illustrations.
Moving forward, we assume that we are given a proper intersection-free straight-line drawing of a 3-connected triangulation with disk topology and vertex coordinates .
Let be some assignment of vertex coordinates and let be weights associated with the directed edges of ; note that we do not assume that . We define the discrete Laplace operator with weights with respect to the straight-line drawing by
where is the set of indices of the neighbors of in . Namely, is the weighted sum of the edges incident to the ’th vertex in the straight-line drawing of the triangulation .
Let a simple polygonal domain with number of vertices corresponding to the boundary of . We say that an assignment of vertex coordinates is discrete-harmonic into with weights if it satisfies
for all interior vertices and maps the boundary vertices of onto the boundary of .
For an assignment of vertex coordinates, we denote by the continuous piecewise-linear map defined by linearly interpolating the vertex assignments over each triangle.
3.2. The Convex Case
Discrete-harmonic embeddings are often discussed in the case that is convex, following a classic result originating in the work of Tutte [tutte1963draw]:
Let be a discrete-harmonic embedding (7) into a simple polygonal domain . If is convex then for any positive weights the straight-line drawing is intersection-free. Moreover, the piecewise-linear map induced by is a homeomorphism of onto .
The result proved in [tutte1963draw] addresses the intersection-free realization of a more general class of planar 3-connected graphs via the solution of equations similar to (7) with uniform weights. The case of arbitrary positive weights is similar, and is discussed in the context of triangular meshes in [gortler2006discrete]. Floater [floater2003one] shows that the discrete-harmonic piecewise-linear map is a homeomorphism. This result is often seen as a discrete analog of the Rado-Kneser-Choquet Theorem.
3.3. The Non-Convex Case
When is convex, Theorem 7 guarantees an intersection-free straight-line drawing for any choice of positive weights . This guarantee, however, fails to hold whenever is non-convex. While certain choices of weights could lead to an intersection-free drawing, other choices might result in intersections; in fact, a choice of weights that produces an intersection-free drawing might not even exist. See Figure 9 for an illustration. Next, we focus on the non-convex case and propose a simple geometric condition that, when satisfied, provides a similar guarantee.
We begin by associating a cone with the vertices of a simple polygon . For a vertex of we define the cone at as the intersection of the two inward-pointing open half-planes supporting the edges incident to ; see Figure 10 for an illustration.
Formally, let be the vertices of adjacent to and suppose that the triplet traverses the boundary of the simple polygon in counter-clockwise order. We define the two half-planes
where denotes the clockwise rotation of . Then the cone at the vertex is given by
Figure 10 illustrates the cone for a convex vertex whose internal angle is at most , and a reflex vertex whose internal angle is at least . Note that this definition coincides with that of the boundary cone given in Section 2 for the continuous case; see Figure 4 for an illustration.
Non-convex discrete-harmonic embeddings.
Suppose is a discrete-harmonic embedding into a simple non-convex polygonal domain satisfying (7) with positive weights .
In this embedding, the boundary vertices of are mapped to vertices of the polygon . We shall use to denote the boundary cone associated with the embedding of the -th vertex .
We also note that for boundary vertices does not necessarily vanish and can be interpreted as the force applied on the constrained boundary vertices by their neighbors. See Figure 1 for example harmonic maps in which is illustrated by red arrows along the boundary. As such, it plays a role similar to the normal derivative in the continuous case; in fact, is proportional to the gradient of the energy corresponding to the variational form of (7) when it exists.
Our main result is a discrete analog of Theorem 4 and establishes a connection between the boundary cones and at the respective boundary vertices. We say that a boundary vertex satisfies the cone condition if its embedding satisfies . See Figure 11. In turn, we show that satisfying the cone condition for the reflex boundary vertices, i.e., vertices mapped to polygon vertices whose internal angle is greater than , is sufficient to guarantee an intersection-free embedding.
Let be positive weights and let be a discrete-harmonic embedding into a simple polygonal domain satisfying (7). If the cone condition holds for all reflex boundary vertices of then the straight-line drawing is intersection-free.
The proof provided in Appendix B.2 is based on reduction to the convex case: we extend the triangulation by adding new edges (but not new vertices) into a triangulation compatible with , the convex hull of ; then we show how the cone condition guarantees the existence of positive weights on the edges of the extended triangulation, with which the discrete-harmonic embedding into the convex polygon reproduces ; finally, Tutte’s Theorem 7 implies that the straight-line drawing of the extended triangulation is intersection-free, and thus so is .
We may further consider the piecewise-linear map induced by such an assignment . As a corollary of Theorem 8, we prove in Appendix B.3 the following characterization of piecewise-linear homeomorphisms between two straight-line drawings of the same triangulation,
A piecewise-linear mapping is a homeomorphism of onto if and only if there exist positive weights such that is discrete-harmonic into satisfying (7) and the cone condition holds for all reflex boundary vertices of .
Gortler et al. [gortler2006discrete] also formulate sufficient conditions for invertibility in terms of reflex vertices. However, their conditions differ from the cone conditions, and the statement and proof presented there follow an index counting argument which does not readily take the form of a discrete analog of the geometric condition of Theorem 4.
The characterization of Theorem 9 is provided in terms of the cone condition at the reflex boundary vertices of . Theorem 8 can be also used to derive an alternative characterization of homeomorphisms in terms of the differential of the map near the boundary. Note that is piecewise-linear and thus is a constant matrix on the interior of every triangle. As a corollary, we can now prove the discrete analog of Theorem 3 (Alessandrini–Nesi), due to [gortler2006discrete]:
Let be a discrete-harmonic embedding into a simple polygonal domain satisfying (7) with positive weights . Further assume that the boundary map is orientation preserving; namely, that the boundary polygons of and have the same orientation. Then the piecewise-linear map is a homeomorphism of onto if and only if
on (the interior of) all boundary triangles; i.e., triangles that have a vertex on the boundary.
A proof is provided in Appendix B.4.
Appendix A Continuous Harmonic Mappings: Proofs
a.1. Theorem 4: supplemental details
The proof of Theorem 4 provided in Section 2 is complete with the exception of Lemma 6, which argues that under the conditions of Theorem 4, for every point , the mapping is a local homeomorphism at . In what follows we shall prove this Lemma in a few steps:
A Geometric Lemma.
To simplify notations and without loss of generality we consider the following setup: We will consider the boundary point . We further assume it is mapped to , the boundary of the target domain, in such a way that
for some . In what follows we shall focus on (the more challenging) case where the derivative at is discontinuous; the same proof readily applies to smooth boundary points. With this setup, illustrated in Figure 13, the cone condition of (3) implies that the second coordinate of the boundary map is monotone in a neighborhood of ; namely, we have the following Lemma:
Denoting , the cone condition
at implies that for some and all with we have
where depends on the opening angle of boundary and the magnitudes of the one-sided tangential derivatives.
An Analytic Lemma.
Let with parameterized by the (periodized) segment . Let be the harmonic extension defined by
is the Poisson kernel.
If is bounded, for all , and satisfies
for all , then there exists such that for all and all with , the harmonic extension satisfies
Set . Note that . We have
Next, we will show that each term satisfies .
For we have and in turn . Therefore
Hence, if then for any
For the term we use to get
Also note that on the domain of integration and therefore both and are less than . In turn, for we have
Hence we have
If we take then , and thus .
if with . Applying the same argument for shows that with the same condition on .
Therefore, if we choose
then we have for all and all with that
Let be Lipschitz continuous with Lipschitz constant . Then the harmonic extension satisfies
This follows from the definition of the Poisson kernel acting as a convolution. ∎
Without loss of generality, we consider the normalized setup described above (see Figure 13) and show that is a local homeomorphism in a neighborhood of . To that end, we will establish uniform control on the normal and tangential derivatives
in a neighborhood of . By assumption we have
Let us write the tangential derivative as
The previous Lemmas imply that, in a sufficiently small neighborhood, we have that
This shows that
on a sufficiently small neighborhood of . This establishes a uniform bound on the angle between the normal and tangential derivatives, and . Lastly, continuity implies that and are bounded away from zero in a small enough neighborhood of . This ensures that is a local homeomorphism around . ∎
a.2. Proof of Theorem 2
We begin with a simple observation: the solution to the Dirichlet problem
Let be a simple curve enclosing a non-convex bounded domain and let and denote two points whose connecting line segment is not contained in the interior of . We define a homeomorphism as follows: we pick two antipodal points