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Data-inspired advances in geometric measure theory: generalized surface and shape metrics

Modern geometric measure theory, developed largely to solve the Plateau problem, has generated a great deal of technical machinery which is unfortunately regarded as inaccessible by outsiders. Some of its tools (e.g., flat norm distance and decomposition in generalized surface space) hold interest from a theoretical perspective but computational infeasibility prevented practical use. Others, like nonasymptotic densities as shape signatures, have been developed independently for data analysis (e.g., the integral area invariant). The flat norm measures distance between currents (generalized surfaces) by decomposing them in a way that is robust to noise. The simplicial deformation theorem shows currents can be approximated on a simplicial complex, generalizing the classical cubical deformation theorem and proving sharper bounds than Sullivan's convex cellular deformation theorem. Computationally, the discretized flat norm can be expressed as a linear programming problem and solved in polynomial time. Furthermore, the solution is guaranteed to be integral for integral input if the complex satisfies a simple topological condition (absence of relative torsion). This discretized integrality result yields a similar statement for the continuous case: the flat norm decomposition of an integral 1-current in the plane can be taken to be integral, something previously unknown for 1-currents which are not boundaries of 2-currents. Nonasymptotic densities (integral area invariants) taken along the boundary of a shape are often enough to reconstruct the shape. This result is easy when the densities are known for arbitrarily small radii but that is not generally possible in practice. When only a single radius is used, variations on reconstruction results (modulo translation and rotation) of polygons and (a dense set of) smooth curves are presented.


Overview
This dissertation applies and extends geometric measure theory tools used for currents and densities. In particular, the flat norm is used to measure currents and provides a useful metric in surface space. This notion is discretized to obtain the multiscale simplicial flat norm and a simplicial deformation theorem (Chapter 2, based on [24]) which approximates currents with chains on a simplicial complex via small deformations (as measured by the flat norm).
The multiscale simplicial flat norm can be computed efficiently and, for integral inputs, has guaranteed integral minimizers in several important cases (in particular, for codimension 1 chains). This statement is stronger than what was known for the continuous case (where the statement was for codimension 1 boundaries). Bridging the gap between these statements and extending the discrete results to the continuous case is the goal of Chapter 3 (based on [25]) where it is shown for 1-currents in R 2 with a framework for establishing the result in general assuming suitable triangulation results.
Lastly, the notion of nonasymptotic densities (also known as the integral area invariant) is developed in the plane in Chapter 4 (based on [27]) where uniqueness questions are addressed in light of a certain useful regularity condition (tangent cone graph-like).
This research was supported in part by the National Science Foundation through grants DMS-0914809 and CCF-1064600.

Measure theory
A few concepts from measure theory prove useful in our development. The Hausdorff measure allows us to sensibly measure m-dimensional sets in R n . Definition 1.2.1 (Hausdorff measure). Given a set A ⊂ R n , the m-dimensional Hausdorff measure of A is an outer measure defined by where α m is the volume of the unit ball in R m and the infimum is taken over all countable coverings S = {S 1 , S 2 , . . . } of A with every S j ∈ S having diameter at most δ.
The Hausdorff measure approximates A locally by covering it with small sets which in turn have their m-dimensional volumes approximated by balls of the same radius in R m . This is the natural way to measure m-dimensional volume in R n and agrees with intuitive notions of what this should mean, for example, for an m-dimensional manifold embedded in R n . It also provides sensible results for any nonnegative real dimension by extending the unit ball volume via the Γ function: α m = π m/2 /Γ(m/2 + 1). For any particular nonempty set A, there is a "correct" dimension to use when measuring it with the Hausdorff measure in the sense that using any other value yields a trivial result.

Currents
The following is a brief introduction to currents, largely following Federer [19], Krantz and Parks [29], and Morgan [32] which are recommended as references for some of the details in descending order of difficulty. Currents are the primary objects of study in Chapters 2 and 3 where the definition of various types of currents (general, normal, and integral) and the flat norm on currents play a central role. There is significant machinery to develop which can obscure the intuition which is that (suitably nice) m-currents can be thought of like oriented submanifolds of dimension m. Definition 1.3.1 (m-covectors). Given n and m, the set of m-covectors of R n is denoted by ∧ m (R n ) and contains all φ such that • φ maps a collection of m vectors in R n to a real number: φ : (R n ) m → R.
The most well-known function with these properties is the determinant applied to n-by-n matrices. It is easy to show that the determinant is (up to multiplication) the only member of ∧ n (R n ).
Given the standard basis vectors e i for R n , we define dual basis vectors dx j linearly by dx j (e i ) = and note that any 1-covector can be written in this basis. Definition 1.3.2 (Exterior product, simple covectors). Given a 1 , . . . , a m ∈ ∧ 1 (R n ), we denote the exterior or wedge product of these 1-covectors to be the m-covector a 1 ∧ a 2 ∧ · · · ∧ a m which is defined by (a 1 ∧ a 2 ∧ · · · ∧ a m )(u 1 , . . . , u m ) = det where the vector a i is the representation of the 1-covector a i in the dual basis [dx 1 , . . . , dx n ].
Any element of ∧ m (R n ) that can be written as a wedge product of 1-covectors a i is called simple and every m-covector can be expressed as the sum of simple m-covectors. The wedge product extends to higher degree covectors by means of this decomposition and a distributive law.
The wedge product is m-linear and is negated when any two covectors are transposed because it relies on the determinant. For the same reason, if a particular 1-covector appears more than once in the wedge product, the result is 0. Working an example, we

Definition 1.3.3 (Forms).
Given open U ⊆ R n , a differential m-form on U is a function φ : U → ∧ m (R n ). The set of all m-forms on U is denoted by D m (U ). We say that Observe that any function f : U → R can be considered as a 0-form. Differential m-forms can be used as integrands over m-dimensional surfaces as they can vary both based on location of a point and its tangent plane; this serves as a useful generalization of integration of 1-forms over a curve.
The exterior derivative of the simple m-form φ = f dx i 1 ∧ dx i 2 ∧ · · · ∧ dx im (where the i k are integers from 1 to n) is given by the (m + 1)-form For all other C 1 m-forms, the definition is extended by linearity.
Theorem 1.3.5 (Properties of exterior differentiation, [29] p. 163). If φ and ψ are C 1 m-forms and θ is a C 1 -form, then we have: • If φ is C 2 , then ddφ = 0. Definition 1.3.6 (Currents). The space of m-currents in R n is denoted by D m (R n ) and is dual to the set of C ∞ differential forms of compact support. The mass of a current T ∈ D m (R n ) is given by , φ has compact support, φ ≤ 1}.
Whereas differential forms correspond to integrands, (suitably nice) currents can be intuitively thought of as the linear integration operator itself, representing and generalizing the oriented submanifolds over which we can integrate differential forms. When T ∈ D m (R n ) represents an oriented submanifold, the mass is simply its m-dimensional volume, counting multiplicities (this is the intuition take take away from this definition). Defining ∂T for 0-currents is not universal, but doing so allows us to simplify various statements slightly by omitting special cases (Equation (1.1a), for example). Currents can be created from any oriented m-dimensional rectifiable set R. Define S as the map from points x ∈ R to unit m-vectors corresponding to the oriented tangent plane to R at x. By this, we mean that S(x) is the wedge product of m orthonormal tangent vectors to R at x. Then for any differential form φ ∈ D m (R n ), define an mcurrent T by We allow T to carry integer multiplicities by introducing a function η : The currents which can be constructed via this procedure are called rectifiable currents and their existence justifies the statement that currents generalize oriented submanifolds. Note that nothing prevents normal currents from being "smeared" out in space.
Morgan [32, p. 48] gives an example of a normal 1-current S 2 which covers the unit square in R 2 but with the concentration of mass of the 2-dimensional Hausdorff measure so it has finite mass and boundary. As an aid to understanding the various classes of currents, note that all integral currents are both normal and rectifiable (in fact, this can be taken as the definition of integral current by the closure theorem [19, 4.2.16]). Furthermore, integral and rectifiable currents have integer multiplicities while normal and general currents need not. Suppose T is the current representing the unit circle in R 2 and T n is an inscribed regular n-gon, both oriented clockwise (see Figure 1.2(a)). As n gets large, it is clear that T n intuitively approximates T arbitrarily well. Thus it would be desirable to have a notion of convergence for which T n → T . In particular, the mass norm is not useful on its own here: the current T n − T has mass M(T n ) + M(T ) → 4π since T n and T do not exactly coincide (so there is no cancellation) except on a measure 0 subset. The flat norm is usually defined as a supremum over forms but this definition is equivalent and more immediately useful for our purposes. Of note is that the minimum exists and is attained whenever F(T ) < ∞. This is proved using the Hahn-Banach

Introduction
Currents are standard objects studied in geometric measure theory, and are named so by analogy with electrical currents that have a kind of magnitude and direction at every point. Intuitively, one could think of currents as generalized surfaces with orientations and multiplicities. The mathematical machinery of currents has been used to tackle many fundamental questions in geometric analysis, such as the ones related to area minimizing surfaces, isoperimetric problems, and soap-bubble conjectures [32].
To formally define d-currents in R n , we first let D d be the set of C ∞ differentiable d-forms with compact support. Then the set of d-currents is given by the dual space of D d (denoted D d ) with the weak topology. We denote by R m the set of rectifiable currents, which contains all currents that represent oriented rectifiable sets (i.e., sets which are almost everywhere the countable union of images of Lipschitz maps from R m to R n ) with integer multiplicities and finite total mass (with multiplicities).
The mass M(T ) of a d-dimensional current T can be thought of intuitively as the weighted d-dimensional volume of the generalized object represented by T . For instance, the mass of a 2-dimensional current can be taken as the area of the surface it represents.
Formally, the mass of T is given by The boundary ∂T of a current T is defined by duality with forms. That is, we 1 Previously published as [24] have ∂T (φ) = T (dφ) for every differential form φ ∈ D d . Note that when T represents a smooth oriented manifold with boundary, this corresponds to the usual definition of boundary. We restrict our attention to integral currents T that are rectifiable currents with a rectifiable boundary (i.e., T ∈ R m and ∂T ∈ R m−1 ). The flat norm of a ddimensional current T is given by where E d is the set of d-dimensional currents with compact support. One also uses flat norm to measure the "distance" between two d-currents. More precisely, the flat norm distance between two d-currents T and P is given by Morgan and Vixie [34] showed that the L 1 total variation functional (L 1 TV) introduced by Chan and Esedoḡlu [9] computes the flat norm for boundaries T with integer multiplicity. Given this correspondence, and the use of scale in L 1 TV, Morgan and Vixie defined [34] the flat norm with scale λ ∈ [0, ∞) of an oriented d-dimensional set T as where S varies over oriented (d+1)-dimensional sets, and V d is the d-dimensional volume, used in place of mass.  the setting where T is a boundary, i.e., a loop, embedded in R 2 and the minimizing surface S ∈ R 2 as well, the flat norm could be calculated efficiently, for instance, using graph cut methods [28] -see the work of Goldfarb and Yin [22] and Vixie et al. [45], and references therein. Motivated by applications in image analysis, these approaches usually worked with a grid representation of the underlying space (R 2 ). Pixels in the image readily provide such a representation.
While it is computationally convenient that L 1 TV minimizers give us the scaled flat norm for the input images, this approach restricts us to currents that are boundaries of codimension 1. Correspondingly, the calculation of flat norm for 1-boundaries embedded in higher dimensional spaces, e.g., R 3 , or for input curves that are not necessarily boundaries has not received much attention so far. Similarly, flat norm calculations for higher dimensional input sets have also not been well-studied. Such situations often appear in practice -for instance, consider the case of an input set T that is a curve sitting on a manifold embedded in R 3 , with choices for S restricted to this manifold as well.
Further, computational complexity of calculating flat norm in arbitrary dimensions has not been studied. But this is not a surprising observation, given the continuous, rather than combinatorial, setting in which flat norm computation has been posed so far.
Simplicial complexes that triangulate the input space are often used as representations of manifolds. Such representations use triangular or tetrahedral meshes [17] as opposed to the uniform square or cubical grid meshes in R 2 and R 3 . Various simplicial complexes are often used to represent data (in any dimension) that captures interactions in a broad sense, e.g., the Vietoris-Rips complex to capture coverage of coordinate-free sensor networks [12,13]. It is natural to consider flat norm calculations in such settings of simplicial complexes for denoising or regularizing sets, or for other similar tasks. At the same time, requiring that the simplicial complex be embedded in high dimensional space modeled by regular square grids may be cumbersome, and computationally prohibitive in many cases.

Contributions
We define a simplicial flat norm (SFN) for an input set T given as a subcomplex of the finite oriented simplicial complex K triangulating the set, or underlying space Ω.
More generally, T is the simplicial representation of a rectifiable current with integer multiplicities. The choices of the higher dimensional sets S are restricted to K as well.
We extend this definition to the multiscale simplicial flat norm (MSFN) by including a scale parameter λ. The simplicial flat norm is thus a special case of the multiscale simplicial flat norm with the default value of λ = 1.
This discrete setting lets us address the worst case complexity of computing flat norm.
Given its combinatorial nature, one would expect the problem to be difficult in arbitrary dimensions. Indeed, we show the problem of computing the multiscale simplicial flat norm is NP-complete by reducing the optimal bounding chain problem (OBCP), which was recently shown to be NP-complete [16], to a special case of the multiscale simplicial flat norm problem. We cast the problem of finding the optimal S, and thus calculating the multiscale simplicial flat norm, as an integer linear programming (ILP) problem.
Given that the original problem is NP-complete, instances of this ILP could be hard to solve. Utilizing recent work [14] on the related optimal homologous chain problem Our most significant contribution is the simplicial deformation theorem (Theorem 2.5.1), which states that given an arbitrary d-current in |K| (underlying space), we are assured of an approximating current in the d-skeleton of K. This result is a substantial modification and generalization of the classical deformation theorem for currents on to square grids. Our deformation theorem explicitly specifies the dependence of the bounds of approximation on the regularity and size of the simplices in the simplicial complex.
Hence it is immediate from the theorem that as we refine the simplicial complex K while preserving the bounds on simplicial regularity, the flat norm distance between an arbitrary d-current in |K| and its deformation onto the d-skeleton of K vanishes.
More importantly, such refinement of K does not affect the efficient computability of the multiscale simplicial flat norm by solving the associated ILP in many cases, e.g., when K is orientable or when it is full-dimensional.

Work on Related Problems
The problem of computing multiscale simplicial flat norm is closely related to two other problems on chains defined on simplicial complexes -the optimal homologous chain problem (OHCP) and the optimal bounding chain problem (OBCP). Given a d-chain t of the simplicial complex K, the optimal homologous chain problem is to find a dchain x that is homologous to t such that x 1 is minimal. In the optimal bounding chain problem, we are given a d-chain t of K, and the goal is to find a (d + 1)-chain s of K whose boundary is t and s 1 is minimal. The optimal bounding chain problem is closely related to the problem of finding an area-minimizing surface with a given boundary [32]. Computing the multiscale simplicial flat norm could be viewed, in a simple sense, as combining the objectives of the corresponding optimal homologous chain and optimal bounding chain problem instances, with the scale factor determining the relative importance of one objective over the other.
When t is a cycle and the homology is defined over Z 2 , Chen and Freedman showed that the optimal homologous chain problem is NP-hard [10]. Dey, Hirani, and Krishnamoorthy [14] studied the original version of the optimal homologous chain problem with homology defined over Z, and showed that the problem is in fact solvable in polynomial time when K satisfies certain conditions (when it has no relative torsion). Recently, Dunfield and Hirani [16] have shown that the optimal homologous chain problem with homology defined over Z is NP-complete. We will use their results to show that the problem of computing the multiscale simplicial flat norm is NP-complete (see Section 2.2.1). These authors also showed that the optimal bounding chain problem with homology defined over Z is NP-complete as well. Their result builds on the previous work of Agol, Hass, and Thurston [3], who showed that the knot genus problem is NP-complete, and a slightly different version of the least area surface problem is NP-hard.
The standard simplicial approximation theorem from algebraic topology describes how continuous maps are approximated by simplicial maps that satisfy the star condition [35, §14]. Our simplicial deformation theorem applies to currents, which are more general objects than continuous maps. More importantly, we present explicit bounds on the expansion of mass of the current resulting from simplicial approximation. In his PhD thesis, Sullivan [42] considered deforming currents on to the boundary of convex sets in a cell complex, which are more general than the simplices we work with. But simplicial complexes admit efficient algorithms more naturally than cell complexes. We adopt a different approach for deformation from Sullivan and obtain new bounds on the approximations (see Section 2.5.2). Along with the multiscale simplicial flat norm, our deformation theorem also establishes how the optimal homologous chain problem and optimal bounding chain problem could be used on general continuous inputs by taking simplicial approximations, thus expanding widely the applicability of this family of techniques.

Definition of Simplicial Flat Norm
Consider a finite p-dimensional simplicial complex K triangulating the set Ω, where the simplices are oriented, with p ≥ d + 1. The set T is defined as the integer multiple of an oriented d-dimensional subcomplex of K, representing a rectifiable d-current with integer multiplicity. Let m and n be the number of d-and (d + 1)-dimensional simplices in K, respectively. The set T is then represented by the d-chain m i=1 t i σ i , where σ i are all d-simplices in K and t i are the corresponding weights. We will represent this chain by the vector of weights t ∈ Z m . We use bold lower case letters to denote vectors, and the corresponding letter with subscript to denote components of the vector, e.g., For t representing the set T with integer multiplicity of one, t i ∈ {−1, 0, 1} with −1 indicating that the orientations of σ i and T are opposite. But t i can take any integer value in general. Thus, t is the representation of T in the elementary d-chain basis of K.
We consider (d + 1)-chains in K modeling sets S representing rectifiable (d + 1)-currents with integer multiplicities, and denote them similarly by n j=1 s j τ j in the elementary (d + 1)-chain basis of K consisting of the individual simplices τ j . We denote the chain modeling such a set S using the corresponding vector of weights s ∈ Z n .
Relationships between the d-and (d + 1)-chains of K are captured by its (d + 1)- otherwise it is zero. This nonzero value is +1 if the orientations of σ i and τ j agree, and is −1 when their orientations are opposite. The d-chain representing the set T − ∂ d+1 S is then given as Notice that x ∈ Z m . We define the simplicial flat norm (SFN) of T represented by the d-chain t in the (d + 1)-dimensional simplicial complex K as Since x and s are chains in a simplicial complex, the masses of the currents they represent (as given in Equation 2.1) are indeed given by the weighted sums of the volumes of the corresponding simplices. The integer restrictions x ∈ Z m and s ∈ Z n are important in this definition as we are studying currents with integer multiplicities. The simplicial flat norm is intuitively the problem of deforming an input chain to another chain of least cost, where cost is determined both by the mass of the resulting chain and the size of the deformation (constrained to the complex) used to get it. For instance, in a triangulation of a manifold, we constrain ourselves to only use deformations on the manifold. We generalize the definition of SFN to define a multiscale simplicial flat norm (MSFN) of T in the simplicial complex K by including a scale parameter λ ∈ [0, ∞).
This definition is the simplicial version of the multiscale flat norm defined in Equation The (non-simplicial) flat norm with scale λ > 0 of a d-dimensional current T can be rewritten as F λ (T ) = λ d · F 1 (T /λ). Thus the flat norm with scale can be thought of as the traditional flat norm applied to a scaled copy of the input current. An equivalent statement can be made for the simplicial flat norm, but crucially requires that the simplicial complex be similarly scaled. To avoid this complex scaling issue especially when considering all possible scales, and to simplify our notation, we henceforth study the more general multiscale simplicial flat norm (which also allows us to consider the λ = 0 case).
We assume the d-and (d+1)-dimensional volumes of simplices to be any nonnegative values. For example, when σ i is a 1-simplex, i.e., edge, V 1 (σ i ) could be taken as its Euclidean length. Similarly, V 2 (τ j ) for a triangle τ j could be its area. For ease of notation, we denote V d (σ i ) by w i and V d+1 (τ j ) by v j , with the dimensions d and d + 1 evident from the context.
is lower bounded by zero, as it is the sum of nonnegative entries (we have λ ≥ 0). Notice Hence its minimum indeed exists, which defines the multiscale simplicial flat norm of t.
On the other hand, the proof of existence of minimum in the original definition of flat norm for rectifiable currents employs the Hahn-Banach theorem [19, pg. 367].
We illustrate the optimal decompositions to compute the multiscale simplicial flat norm for two different scales (λ = 1 and λ 1) in Figure 2  The related optimal homologous chain problem (OHCP) was recently shown to be NP-complete [  We now consider attacking the multiscale simplicial flat norm problem using techniques from the area of discrete optimization. Even though the problem is NP-hard, this approach helps us to identify special cases in which we can compute the multiscale simplicial flat norm in polynomial time.

Multiscale Simplicial Flat Norm and Integer Linear Programming
The problem of finding the multiscale simplicial flat norm of the d-chain t (Equation 2.5) can be cast formally as the following optimization problem. (2.7) The objective function is piecewise linear in the integer variables x and s. Using standard modeling techniques from linear optimization [7, pg. 18], we can reformulate the problem as the following integer linear program (ILP). (2.8) The objective function coefficients need to be nonnegative for this formulation to workindeed, we have w i , v j , and λ nonnegative. Integer linear programming is NP-complete [37]. The linear programming relaxation of the ILP above is obtained by ignoring the integer restrictions on the variables.
We are interested in instances of this linear program (LP) that have integer optimal solutions, which hence are optimal solutions for the original ILP (Equation 2.8) as well.
Totally unimodular matrices yield a prime class of linear programming problems with integral solutions. Recall that a matrix is totally unimodular if all its subdeterminants equal −1, 0, or 1; in particular, each entry is −1, 0, or 1. The connection between total unimodularity and linear programming is specified by the following theorem. is cast as the optimization problem given in Equation ( An optimal solution (x + * , x − * , s + * , s − * ) of the multiscale simplicial flat norm LP can be found in polynomial time using an interior point method [7,Chap. 9]. If it happens to be a unique optimal solution, then it will be a vertex, and hence will be integral by If the optimal solution is not unique, then (x + * , x − * , s + * , s − * ) may be nonintegral. But since the optimal cost is finite, there must exist a vertex in its polyhedron that has this minimum cost. Given a nonintegral optimal solution obtained by an interior point method, one can find such an integral optimal solution at a vertex in polynomial time [23]. Hence the multiscale simplicial flat norm ILP can be solved in polynomial time in this case as well.  where I is the 2m + 2n identity matrix, and O is the m × (2m + 2n) zero matrix. Hence A is obtained from A by first adding 2m + 2n rows with a single nonzero entry of +1, and then adding to the resulting matrix 2n + 2m more columns with a single nonzero entry of +1. These operations preserve total unimodularity [37, pg. 280], and hence the new constraint matrix A is totally unimodular when [∂ d+1 ] is so. The new right-hand side vector b ∈ Z 3m+2n consists of the input chain t and the vector of ones from the new upper bound constraints.
Since the efficient computability of the multiscale simplicial flat norm depends on the total unimodularity of the boundary matrix, we study the conditions under which total unimodularity of boundary matrices can be guaranteed.

Simplicial Complexes and Relative Torsion
Dey, Hirani, and Krishnamoorthy [14] have given a simple characterization of the simplicial complex whose boundary matrix is totally unimodular. In short, if the simplicial complex does not have relative torsion then its boundary matrix is totally unimodular.
We state this and other related results here for the sake of completeness, and refer the reader to the original paper [14] for details and proofs. The simplicial complex K in these results has dimension d + 1 or higher. Recall that a d-dimensional simplicial complex is pure if it consists of d-simplices and their faces, i.e., there are no lower dimensional simplices that are not part of some d-simplex in the complex. These authors further describe situations in which the absence of relative torsion is guaranteed. The following two special cases describe simplicial complexes for which the boundary matrix is always totally unimodular. It is appropriate to mention here that the connection between total unimodularity of boundary matrices and torsion in the complex has been observed as early as in 1895 by Poincaré [36]. However, the result in [14] connecting the total unimodularity with relative torsion is different and has led to a polynomial time algorithm for the OHCP problem. Notice that a complex can be torsion-free, but have non-trivial relative torsion.
The Möbius strip is such an example.
We illustrate the implications of the results above for the efficient computation of the multiscale simplicial flat norm by considering certain sets. When the input set T is of dimension 1, and is described on an orientable 2-manifold to which the choices of 2-dimensional set S are also restricted, we can always compute its multiscale simplicial flat norm by solving the multiscale simplicial flat norm LP (Equation 2.9) in polynomial time. A similar result holds when T is a set of dimension 2 described as a subcomplex of a 3-complex sitting in R 3 . For a 1-dimensional set T with choices of S restricted to a 2-complex K, we can always compute the multiscale simplicial flat norm of T efficiently as long as K does not have a 2-dimensional Möbius subcomplex. Notice that K itself need not be embedded in R 3 for this result to work -it could be sitting in some higher dimensional space.

Simplicial Deformation Theorem
When can we use the multiscale simplicial flat norm as a discrete surrogate for the traditional flat norm? That is, if we wish to solve a flat norm problem (for which there are no practical algorithms in general), can we discretize the problem and find a problem close enough to the original one which we can solve?
The deformation theorem [19, is one of the fundamental results of geometric measure theory, and more particularly of the theory of currents. It approximates an integral current by deforming it onto a cubical grid of appropriate mesh size.
On the other hand, we have been studying currents or sets in the setting of simplicial complexes, rather than on square grids. Our proof is a substantial modification of the classical proof of the deformation theorem. We found the presentation of the latter proof by Krantz and Parks [29, Section 7.7] especially helpful. Our proof mimics their proof when possible. The gist of this theorem is the assertion that we may approximate a current with a simplicial current.
Recall that V d (σ) denotes the d-dimensional volume of a d-simplex σ. The perimeter of σ is the set of all its (d − 1)-dimensional faces, denoted as perimeter(σ) = We will also refer to the (d − 1)-dimensional volume of perimeter(σ) as the perimeter of σ, but denote it as P(σ) = τ j ∈perimeter(σ) V d−1 (τ j ). We let diameter(σ) be the diameter of σ, which is the largest Euclidean distance between any two points in σ.
hold, whereB σ is the largest ball inscribed in σ, B σ is the ball with half the radius and same center asB σ , and r σ is the radius of Then there exists a simplicial d-current P supported in the d-skeleton of K whose bound- and the following controls on mass M hold: Remark 2.5.2. It is immediate that the flat norm distance between T and P can be made arbitrarily small by subdividing the simplicial complex to reduce ∆ while preserving the regularity of the refinement as measured by κ 1 and κ 2 . We get this large bound because we generate P through retractions alone, and not using the usual Sobolev-type estimates [29,. And of course, the ∆ in the coefficient of the extra term means that it becomes unimportant as the simplicial complex is appropriately subdivided.

Proof of the Simplicial Deformation Theorem
At the heart of the modification of the deformation theorem (from cubical grid to simplicial complex settings) is the recalculation of an integral over the current and its boundary. This integral appears in a bound on the Jacobian of the retraction, which measures the expansion in mass of the current resulting from the process of retracting it on to the simplices of the simplicial complex. To do this recalculation, we consider the retraction φ one step at a time, building it through independent choices of centers to project from in every simplex and its every face.
We first describe the general set up of retraction within a simplex. We then present certain bounds on the mass expansion resulting from the retraction in Lemmas 2.5.6, 2.5.7, and 2.5.8. In particular, we obtain bounds on the expansion that are independent of the choice of points from which we project. These bounds are independent of the particular current that we retract on to the simplicial complex. But we employ these bounds to subsequently bound the overall expansion of mass of the current resulting from the retraction.

Retracting from a center inside a simplex
We describe the details of retraction for an -simplex σ in the p-dimensional simplicial complex K. This set up is valid for any , but in particular, we will use the bounds thus obtained for d ≤ ≤ p when retracting a d-current onto K. We pick a center a ∈ Int(σ), the interior of σ, and project every x ∈ Int(σ) \ {a} along the ray (x − a)/ x − a to perimeter(σ). Denoting this map as φ(x, a), we get Choose an orthogonal basis {w 1 , ..., w −2 } for span(v 1 , v 2 ) ⊥ . Note that span(w 1 , ...,  Then {v 1 , w 1 , ..., w −2 , v 2 } is an orthogonal basis for R , and φ π is given by where α =r/ĥ. Notice that the above set up works everywhere except when φ(x) = b, in which case we obtain an orthogonal projection for φ π (x) along b − a. Choosing coordinates for the tangent spaces of σ and τ x to be {v 1 , w 1 , ..., w −2 , v 2 } and {w , w 1 , ...,

Bounding the Integral of the Jacobian
We now present a series of bounds on integrals of the dilation of d-volumes induced by the retraction. Since = d implies we are already in the d-skeleton and no retraction is needed, we can assume that > d. We start with a bound on the maximum dilation of d-volumes under the retraction φ. Dφ will denote the tangent map or Jacobian map of φ.

the maximum dilation of d-volumes induced by
Dφ(x, a) at x.
We will use the definitions and results on Dφ π (x, a) in -dimension given above. In Lemma 2.5.6. For any center a and any point x = a in the -simplex σ with d < ≤ Proof. Following Equation (2.14), we seek bounds on Dφ δ (x, a) and Dφ π (x, a). Since Dφ δ (x, a) simply scales byr/r =ĥ/h, the expansion of d-volume of any d-hyperplane by Dφ δ (x, a) is by a factor of (ĥ/h) d . On the other hand, bounding the dilation that Dφ π (x, a) can cause in d-hyperplanes is a little more involved. We seek a bound on for all × d matrices U . Using the generalized Pythagorean theorem [29, Section 1.5], we get Observe that multiplying by Dφ π (x, a) (Equation 2.16) just scales the first row of U by α and removes the last row. Thus α det(U λ ) ≥ det((Dφ π (x, a)U ) λ ), which implies that α is a bound on the ratio in Equation (2.17). Thus we have that h holds for all x and a in σ, where diameter(σ) is the diameter of the -simplex σ.
Next we describe a bound on the integral of J d φ(x, a) over the entire -simplex, for a fixed center a. We will find that this bound is independent of the position of a. Recall that perimeter(σ) and P(σ) denote the perimeter of -simplex σ and the ( − 1)-dimensional volume of the perimeter, respectively, and Int(σ) its interior.
Lemma 2.5.7. For any fixed center a in the -simplex σ with d < ≤ p = d + k, Let σ j denote the -simplex generated by a and τ j . Then Let τ j (h) denote the ( − 1)-simplex formed by the intersection of σ j and the ( − 1)- Using the bound on J d φ(x, a) from Lemma 2.5.6, and noting that diameter(σ j ) ≤ diameter(σ) ∀ j, we get Summing this quantity over all τ j ∈ perimeter(σ) and replacing − d ≥ 1 with 1 gives the overall bound.
We now bound the integral of J d φ(x, a) over centers a with a fixed x that we are retracting onto perimeter(σ). Examination of the corresponding proof for the original deformation theorem [29,Section 7.7] shows that symmetry of the cubical mesh plays a very special role, which cannot be duplicated in the case of simplicial complex. In particular, we must avoid integrating over a close to the perimeter of σ. Hence we integrate over as big a region as we can while still avoiding a neighborhood of the perimeter. As in the statement of the main Theorem 2.5.1, letB σ be the largest ball inscribed in σ, B σ be the ball with half the radius and same center asB σ , and r σ be the radius of B σ .
Proof. Similar to the subsimplices of σ considered in the Proof of Lemma 2.5.7, let σ j now denote the -simplex formed by x and τ j ∈ perimeter(σ). In order to derive an upper bound, we integrate instead over regions that are by construction bigger than these subsimplices of σ. Denoting the simplex σ j as Region 1, we define Regions 2 and 3 as follows. We refer the reader to Figure 2.5 for an illustration of this construction.
Let σ j be the reflection of σ j through x, and similarly, let τ j be the reflection through x of τ j . We define σ j as Region 2. Notice that unlike Region 1, Region 2 need not be contained fully in σ. As defined in Section 2.5.1, let z be the unit vector normal to the ( −1)-hyperplane containing τ j pointing into σ. We define Region 3 as the -dimensional set τ j + [0, diameter(σ)]z, as illustrated in Figure 2.5.
Note that the union of all Region 2's and Region 3's cover σ. By an argument almost identical to that above, we have the following upper bound on the integrand in question.
Integrating the second of these two terms over Region 2 and summing the integral over all such Regions 2 for all faces τ j , we get the upper bound of diameter(σ) P(σ). Here we use the same arguments as the ones employed in Lemma 2.5.7. Region 2 alone is not guaranteed to cover B σ as some of B σ may occupy parts of Region 3. Since a ∈ B σ , we haveĥ > r σ , and h ≤ h when a ∈Region 3, so that Combining the above estimates while integrating over all such Regions 2 and 3 gives us the bound specified in the Lemma.

Bounding the pushforwards of the current
We consider the d-current T , and employ the bounds on the Jacobian of retraction described above to the pushforwards of T and its boundary ∂T on to the simplicial complex K. Our treatment of the pushforwards essentially follows the corresponding results of Krantz and Parks for the case of square grid [29, Pages 218-219]. We denote by T the total variation measure of the current T , which is determined by the identity of the retraction on to the perimeter of an -simplex σ for d < ≤ p (respectively, d ≤ ≤ p) increases the mass of T or ∂T by at most a factor of Proof. Using Fubini's theorem [29,Page 26] and applying the bound in Lemma 2.5.8, .
Similarly we define H ∂T for the pushforward of ∂T and positive measure, with the centers a in this subset satisfying . Hence we can choose centers to retract from in each simplex σ such that the expansion of mass of the current restricted to that simplex is bounded by 4ϑ σ / V (B σ ). The bound specified in the Lemma follows when we consider retracting the entire current over multiple simplices in K, and set ϑ K = max σ∈K ϑ σ as the generic upper bound that holds for all simplices in K.
Bound on complete sequence of retractions. We can apply the bound specified in Lemma 2.5.9 over multiple levels . Pushing T onto the d-skeleton of p-complex K multiplies the mass of T by a factor of at most (4ϑ K ) k . Likewise, pushing ∂T on to the (d − 1)-skeleton multiplies the mass of ∂T by a factor of at most (4ϑ K ) k+1 .

Bounding the distance between the current and its simplicial approximation
In the final step, we construct the simplicial current P approximating the original current T , and bound the flat norm distance between the two. Since we are now considering retraction maps over many simplices simultaneously, we let φ i denote the global projection from the (p − i + 1)−skeleton to the (p − i)−skeleton, suppressing the particular x and a. We denote the composition of all these steps as ψ 1 ≡ φ k • · · · • φ 1 and hence we map T forward by ψ 1 , picking centers (see Lemma 2.5.8) to project from in each step and in each simplex. We pick each of these centers such that the retractions map ∂T with bounded amplification of mass as well (see Lemma 2.5.9).
The homotopy formula [29,Section 7.4.3] states that given a smooth homotopy g we have that the difference in pushforwards of T under f 1 and f 0 is given by Define the homotopy g(γ, Then the homotopy formula gives We define P is a d-current whose boundary ∂P is contained in the (d − 1)-skeleton of K. Define ). Using the homotopy formula, we get Finally, we apply the bounds on the retraction described in Lemma 2.5.9 and the paragraph following this Lemma to the masses of the pushforwards. Noticing that diameter(σ) ≤ ∆ for all σ ∈ K, we get the following bounds, which finish the proof of our simplicial deformation theorem (Theorem 2.5.1).
Remark 2.5.10. The influence of Simplicial regularity as measured by κ 1 and κ 2 is clearly revealed by the statement of our deformation theorem (Theorem 2.5.1). Explicit constants are a simple yet useful part of the result; as observed above in Remark 2.5.2, the statement of this theorem leads to an easy observation that the flat norm distance between T and P can be made a small as desired by subdividing the simplicial complex in a manner that keeps the regularity constants bounded. This can be done, for example, by using the subdivision algorithm of Edelsbrunner and Grayson [18].
Remark 2.5.11. We did not explicitly discuss the case of 0-dimensional currents. In this case, the bounds on mass expansion are all equal to one.

Comparison of Bounds of Approximation
Sullivan studied the deformation of integral currents on to the skeleton of a cell complex, which is composed of compact convex sets. He presented a deformation theorem for deforming integral currents on to the boundary of a cell complex [42,Theorem 4.5]. For ease of comparison, we use our notation to restate the bounds given by Sullivan for deforming a d-current T to a polyhedral current P in the boundary of a cell complex in R q . Recall that in our simplicial deformation theorem (Theorem 2.5.1), the simplicial complex considered has dimension p and is embedded in R q for q ≥ p. Furthermore, κ 1 , κ 2 , ∆, and ϑ K are simplicial regularity constants. We also note that even though Sullivan stated his results for full-dimensional complexes and the standard flat norm, it is straightforward to extend them to lower dimensional complexes and the flat norm with scale: To gain a better understanding of how the two sets of bounds compare, we compute these bounds explicitly for the case of a 2-current in a regular tetrahedral complex (thus, p = 3 and d = 2). Notice that this instance is close to a best case for Sullivan's bounds, as less regular complexes affect them more severely. With this point in mind, we present in Table 2.1 our bounds and Sullivan's bounds on both a regular tetrahedral complex and one on which we stretch the regular tetrahedra by a factor of 10 in a direction normal to one of their faces (i.e., turn them into skinny, spike-like simplices).

Quantity Sullivan's bound Our bound
Regular tetrahedra  (1) 3-complex of congruent regular tetrahedra and (2) a 3-complex of congruent stretched tetrahedra which are created by taking regular tetrahedra and multiplying their height by a factor of 10.
For the regular tetrahedral complex and the M(P ) bound, our coefficient of M(T ) is more than 74 times better, but we do have a second term that can be quite large, but diminishes in importance if the complex is subdivided appropriately (see Remark 2.5.10).
In the stretched complex, our coefficient on M(T ) is 1.5 × 10 5 times better, indicating that our bound is better behaved for irregular complexes.  As compared to Sullivan, we are able to take advantage of our simplicial setting to get better bounds on the mass expansion of T . While our mass expansion bounds involving ∂T are currently inferior to Sullivan's, we suspect our arguments can be tightened and modified to obtain bounds that are better in all cases. More importantly, our bounds are less sensitive to simplicial irregularity. Given the challenges inherent in creating meshes without slivers even in three dimensions [11], bounds that behave well in their presence are highly desirable.

Computational Results
We illustrate computations of the multiscale simplicial flat norm by describing the flat norm decompositions of a 2-manifold with boundary embedded in R 3 (see Figure 2.6).
The input set has the underlying shape of a pyramid, to which several peaks and troughs of varying scale, as well as random noise, have been added. We model this set as a piecewise linear 2-manifold with boundary, and find a triangulation of the same as a subcomplex of a tetrahedralization of the 2 × 2 × 2 cube centered at the origin, within which the set is located. We use the method of constrained Delaunay tetrahedralization [41] implemented in the package TetGen [39] for this purpose. We then compute the multiscale simplicial flat norm decomposition of the input set at various scale (λ) values.
At high values, e.g., when λ = 6, the optimal decomposition resembles the input set with the small kinks due to random noise smoothed out. At the other end, for λ = 0.01, the optimal decomposition resembles a flat "sheet".  Flat norm decomposition of integral currents 1

Introduction
In geometric measure theory, currents represent a generalization of oriented surfaces with multiplicities. Currents were developed in the context of Plateau's problem and have also found application in isoperimetric problems and soap bubble conjectures [32].
Given a d-dimensional current T , we can consider decompositions T = X + ∂S where X is a d-dimensional current and S is a (d + 1)-dimensional current. Over all such decompositions, the minimum total mass (volume) of the two pieces (i.e., M(X)+M(S)) is the flat norm F(T ). More recently, the L 1 TV functional (introduced in the form most relevant to us by Chan and Esedoḡlu [9]) was shown to be related to the flat norm [34]. This connection suggested the flat norm with scale (yielding the objective M(X) + λ M(S) for any fixed scale λ) and a geometric interpretation for the optimal decompositions: varying λ controls the scale of features isolated in the decomposition.
One natural question: must currents in a particular regularity class (in this paper, integral currents) have an optimal flat norm decomposition in the same class? The L 1 TV connection shows this is true for boundaries of codimension 1 (i.e., boundaries of (d + 1)-currents in R d+1 ) since the L 1 TV functional applied to binary (or step function) input is known to have binary (step function) minimizers [9]. This may be taken one step further in the discretized problem where the boundary requirement can be dropped [24]. 1 Based on [25] In the present work, we present a framework to bridge the gap between the continuous and discrete cases, assuming a suitable triangulation result. This allows us to drop the requirement that integral d-currents in R d+1 be boundaries to have a guaranteed integral optimal decomposition. The necessary triangulation result is proved in R 2 by means of Shewchuk's Terminator algorithm [38] for subdividing planar straight line graphs.
This algorithm simultaneously bounds the smallest angles in the complex and tells us where they can occur, allowing us to tailor a simplicial complex to a given set of input currents. We then obtain a simplicial deformation theorem with constant bounds for these currents and simplicial complex, ensuring the sequence of aprroximating discretized problems are well-behaved and solve the continuous problem in the limit. Assuming a suitable triangulation result for higher dimensions (see Conjecture 3.3.4), we show that codimension 1 integral currents have an integral optimal flat norm decomposition.
For the related problem of least area with a given boundary (which can be considered as the flat norm problem with X constrained to be empty), counterexamples of Young [47], White [46], and Morgan [31] provide instances in which the minimizer is not integral.
These negative results are of codimension 3 (i.e., 1-dimensional curves in R 4 ) which may translate into a limit on the flat norm question.   The flat norm of a current T is given by

Definitions
where E d is the set of d-dimensional currents with compact support (see Figure 3.1). For two currents, the flat distance between them is given by F(T, P ) = F(T − P ).
This definition is useful because it is robust to small additions and perturbances (e.g., noise) and reflects when currents are intuitively close. For example, given a current T representing a unit circle in R 2 and an inscribed n-gon T n (both oriented clockwise, see The flat norm can be usefully discretized as well. Given a simplicial (d + 1)-complex K and a d-chain T on K, the simplicial flat norm [24] of T on K is denoted by F K (T ) and defined analogously except that X and S are restricted to be chains on K.

Overview
Our general technique is a standard notion: express the continuous problem as a limit of discrete problems for which the result holds. Theorem 3.2.3 tells us that the simplicial . The current T is the segment from A to B, the complex K n is the arrangement of 2n equilateral triangles of appropriate size stretching from A to B and P n is the top chain from A to B on K n . Clearly, flat norm of an integral chain in codimension 1 has an optimal integral current decomposition; by the compactness theorem from geometric measure theory, the limit of these decompositions is also integral.
In order to show that an integral current T has integral flat norm decomposition, we therefore find suitable simplicial approximations to T and take the limit of their simplicial flat norm decompositions to obtain an integral decomposition for T .
We must also show that this decomposition achieves the flat norm value for T (that is, express T using integral currents in such a way that it remains an optimal flat norm decomposition). This is immediate if our simplicial approximations to T have simplicial flat norm values that converge to the flat norm of T but this is not necessary (see a P δ X δ S δ where P δ is a simplicial approximation to T on some complex K δ with F(P δ − T ) < δ.
This goal prevents us from simply using the simplicial deformation theorem to obtain P δ since we may end up with the situation in Figure 3 The next step is to take an optimal (possibly nonintegral) decomposition of T and approximate it with polyhedral chains (see Figure 3.4). That is, approximate the decomposition T = X + ∂S with polyhedral X δ and S δ . If these approximations naturally form a decomposition (not necessarily optimal) of P δ (i.e., P δ = X δ + ∂S δ ), then we X δ , and S δ . This of course implies Equation (3.1).
However (as in Figure 3.5), we need not have P δ = X δ + ∂S δ . Since we obtained these quantities by polyhedral approximation, it turns out that the extent to which this equation is violated is small (in the continuous flat norm). That is, we have While Equation (3.2) can be viewed as a decomposition of P δ , the added error terms mean it may not be a chain on a simplicial complex. This means it cannot be used directly to bound the simplicial flat norm of P δ .
If we use the simplicial deformation theorem to push the error terms to some complex K δ while preserving a pushed version of Equation (3.2), we can obtain a candidate simplicial decomposition of P δ . In order to use this to bound F K δ , we must know that the deformation theorem didn't make the small error terms large enough to matter. Unfortunately, the simplicial deformation theorem mass bounds rely on simplicial regularity so a sufficiently skinny simplex could mean the error terms become large. If the simplicial irregularity in K δ gets worse as δ ↓ 0, we will not be able to show Equation (3.1).
Since we know exactly which currents we wish to push, the solution is to pick K δ with these in mind: make sure the complex is as regular as possible overall (independently of δ) with any irregularities (which may be required to embed P δ , X δ , and S δ ) isolated in subcomplexes of small measure. By making the irregular portions small enough (so they contain a negligible portion of the error terms, even considering the possible magnification from pushing), we establish a deformation theorem variant (Theorem 3.3.6) with constant mass expansion bounds, assuming a triangulation result that lets us isolate the irregularities as described (Shewchuk's Terminator algorithm [38] provides this in R 2 ).
The pushed version of Equation (3.2) allows us to prove F K δ (P δ ) ≤ F(T ) + O(δ) from which Equation (3.1) and Theorem 3.3.7 follow.

Preliminaries
Our goal is to investigate conditions under which the flat norm decomposition of an integral current can be taken to be integral as well. The corresponding statement for normal currents is true and useful in our development.  properties hold: (a) αT n + βU n → αT + βU for any constants α, β ∈ R, (b) ∂T n → ∂T , Proof. We apply properties of norms to obtain Letting n → ∞ yields the linearity result. Now let X n and S n be m-and (m+1)-currents such that X n + ∂S n is a flat norm decomposition of T n − T for n = 1, 2, . . . , observing The boundary result follows in the limit.
In the case of the simplicial flat norm, an input integral chain is guaranteed an integral chain decomposition whenever the simplicial complex is totally unimodular [24].
This occurs when the complex is free of relative torsion which is the case for any (d + 1)-complex in R d+1 or when triangulating a compact, orientable (d + 1)-dimensional manifold.
Theorem 3.2.3 (Simplicial flat norm integral decomposition [24]). If K is a simplicial (d + 1)-complex embedded in R d+1 , then for any integral d-chain P on K, the optimal simplicial flat norm value for P is attained by an integral decomposition.
We state the simplicial deformation theorem and sketch a portion of its proof. We will later modify it to obtain a multiple current deformation theorem that preserves linearity (Theorem 3.3.1).
Theorem 3.2.4 (Simplicial deformation theorem [24]). Suppose K is a p-dimensional simplicial complex in R q and T is a normal d-current supported on the underlying space of K. There exists a simplicial d-current P supported on the d-skeleton of K with boundary supported on the (d − 1)-skeleton (i.e., a simplicial d-chain) such that T − P = Q + ∂R and there exists a constant ϑ K (depending only on simplicial regularity in K) such that the following controls on mass hold: where ∆ is the diameter of the largest simplex in K. The regularity constant ϑ K is given by where for each l-simplex σ, perimeter(σ) is the (l − 1)-volume of ∂σ and B σ is the l-volume of a ball with radius inradius(σ)/2 in R l .
Proof highlights. The simplicial current P is obtained by pushing T and its boundary to the d− and (d − 1)-dimension skeletons of K respectively. This pushing is done one dimension at a time; that is, T is pushed from the p-skeleton (i.e., the full complex K) to the (p − 1)-skeleton, then to the (p − 2) and so on until the d-skeleton. Pushing the current from the -skeleton to the ( − 1)-skeleton is done by picking a projection center in each -simplex σ and projecting the current in σ outwards to ∂σ via straight-line projection.
A crucial step in the proof is to find a projection center that bounds the expansion of T and ∂T . In particular, this is done by proving that over all possible centers, the average expansion is bounded and then showing that individual centers exist with bounded expansion. We call out this particular step because we modify it to obtain the next theorem.
When projecting onto the skeleton of each simplex σ, we have [24, Lemma 5.9] where B σ is the set of possible centers in σ and ϑ σ is a regularity constant for σ related to ϑ K by ϑ K = sup σ∈K ϑ σ . This shows that in each projection step the mass of T expands by a factor of at most ϑ K averaged over all possible choices of centers. As the average expansion over all centers is ϑ K , we observe that at most 1 4 of the possible centers can expand the mass of T by a factor of 4ϑ K or more. Similarly, at most 1 4 of the centers can expand ∂T by a factor of 4ϑ K or more. Therefore, at least 1 2 of the possible centers bound the expansion of both T and ∂T by at most a factor of 4ϑ K . Choosing a center from this set for each simplex yields the bounds required in the theorem.
The following theorem allows normal (or integral) currents to be approximated by polyhedral chains which are simplicial chains not necessarily contained in an a priori complex. Note in particular that the mass bounds can be made arbitrarily tight by choice of in contrast with the larger bounds of the deformation theorems. If T is integral, then P can be taken to be integral as well.
Proof. This is a slight modification of Federer's theorems which do not state Equa- In the normal current case [19, 4.2.24], these bounds follow from Federer's proof.
In particular, we have currents P 1 , P 2 and Y such that P = P 1 + Y and the following bounds hold: That is, for any fixed > 0, there exist P and f such that From Equations (3.12a) to (3.12c), we obtain mass bounds on P and ∂P : The bounds in Equations (3.10b) and (3.10c) follow by choosing small enough.

Results
The simplicial deformation theorem can be modified to allow multiple currents to be deformed simultaneously by projecting from the same centers. As opposed to using  the T i and S j ) which can be used with every current T i and S j to obtain the bounds: Moreover, if we let π K denote the projection map that uses these centers to push (d−1)−, d−, and (d + 1)-currents to chains on the complex, then we have that: • π K commutes with the boundary operator (i.e., π K (∂A) = ∂π K (A) where A is any dor (d + 1)-current) • π K is linear on the currents T i , ∂T i , S j and ∂S j . That is, for any scalars a i and b j , Then, using the same average-based argument as in Theorem 3.2.4, we have that H (H T i ,k )/H (B σ ) < 1 2m+2n (i.e., the size of the set of poorly behaved centers with respect to each T i is a small fraction of the set B σ of possible centers). We similarly define H ∂T i ,k , H S j ,k , and H ∂S j ,k and obtain the same bound of 1 2m+2m on the bad centers. For each k ∈ Z + , we are interested in the set of centers which are simultaneously good centers for all currents involved (i.e., points in B σ but not any of the H ·,k sets). Call this set G k and observe that it has positive measure: Thus for any k > 1 we have that G k is a nonempty set of possible projection centers which simultaneously attain an expansion bound of at most (2m + 2n + )ϑ σ for all the pertinent currents.
Our result relies on the ability to localize irregularities via subdivision, focusing on localization rather than removal because the latter is not possible. For example, any subdivision of a 2-complex with a very small input angle will have an angle that is at least as small. With that in mind, we require that subdivisions be possible which push the irregularities into the corners. That is, the irregularity should be bounded by a constant (independent of the complex) away from the original complex skeleton and a complexdependent constant (reflecting the necessity of some bad simplices) near the skeleton.   • q ≥ p = 1 Proof. The p = 1 case is trivial as all 1-simplices have the same regularity so we have ϑ K = 8 and can take M = M = K.
For the p = q = 2 case, we proceed in two steps. First we will superimpose a square grid on K (orienting it to bound the minimum angle created between its edges and those of K), creating a cell complex which is a refinement of K. Next we use Shewchuk's Terminator algorithm [38] to further refine the cell complex back into a simplicial complex with bounds on the minimum angle and, crucially, restrictions on where these small angles can be so that we can obtain regularity bounds.
By superimposing a fine enough square grid, we can force the small angles (whether already present in the complex or newly created) to occur only in a small measure subset of the complex. Pick δ > 0 small enough that the set {x ∈ R 2 | y lies on the 1-skeleton of K, x − y < 3δ} has measure less than . Let G be a finite square grid in R 2 whose cells each have diameter δ such that G covers the underlying space of K in any rotation. Note that there are only two directions present in G so if we bound all possible angles created between these directions and the edges of K, we can bound the minimum new angle created by superimposing G.
Let w ∈ R 2 be a fixed unit vector and define E = φ, φ + π 2 | φ is the angle between u − v and w for some edge (u, v) ∈ K .
Further let E θ = {ψ ∈ [0, 2π) | |φ − ψ| < θ for some φ ∈ E}. This is the set of angles to avoid when rotating G in order to guarantee all created angles will be θ or larger.
Denote by N < ∞ the cardinality of E and note that [0, 2π)\E π 2N has positive measure so there exist rotations of the square grid that create no new angles smaller than π 2N . After superimposing a suitably rotated version of G, we obtain a new cellular complex which is a refinement of K. This is a planar straight line graph which can be used as input to Shewchuk's Terminator algorithm [38] We may take β to be this quantity, noting that it is independent of and K. However, this is a bit stronger than required as the projection is a local operation and the bound at each step depends only on the simplicial regularity of the simplex in question. In addition, there is no reason in principle that we cannot subdivide the complex in between steps. That is, after pushing to the -skeleton, we can further subdivide the complex and then push to the newly refined ( − 1)-skeleton. In this case, the subdivision need not preserve the simplicial regularity of the ( + 1)-or higher simplices as all subsequent pushing steps will take place in lower dimensional simplices.
Moreover, for a given portion of current we can use the maximum of the simplicial regularity constants of the simplices it encounters while being pushed (rather than the maximum over all simplices in the complex).
For all > 0 and nonnegative integers k < p, let N k denote the set of all points in the (k + 1)-skeleton of K with positive distance less than from the k-skeleton of K (i.e., all points in the interior of the (k + 1)-simplices of K which are close to the k-skeleton).
Let T ¬ N p−1 denote the restriction of the current T to the set N p−1 and note that   Proof. We let X + ∂S be an optimal flat norm decomposition of T . That is, X is a d-current and S is a (d + 1)-current such that We note by Lemma 3.2.1 that X and S are normal currents.
As a general outline of the proof, for each δ > 0, we will choose a particular simplicial complex K δ on which we have d-chains P δ and X δ and (d + 1)-chain S δ respectively approximating T , X, and S with error at most δ. We convert the (possibly nonintegral) optimal flat norm decomposition of T into a candidate simplicial decomposition of P δ in order to show (Claim 3.3.7.2) the simplicial flat norm of P δ converges to the flat norm of T (this step does not yet show that the flat norm decompositions converge). We can take the optimal simplicial decomposition to be integral for each P δ by Theorem 3.2.3.
The compactness theorem from geometric measure theory along with the above convergence result allows us to take the limit of (a subsequence of) these integral simplicial decompositions and obtain an integral flat norm decomposition of T (Claim 3.3.7.5).
Suppose δ > 0 and apply Theorem 3.2.5 to obtain polyhedral currents P δ , X δ and We also require optimal flat norm decompositions of P δ − T , X − X δ and S − S δ so let U δ i , W δ j and V δ 2 be d-, (d + 1)-and (d + 2)-dimensional currents such that: (3.18c) To clarify the notation, we adopt the convention that variables with a δ subscript are chains on the simplicial complex K δ whereas a δ superscript merely indicates dependence on δ.
Let K δ be any simplicial complex that triangulates P δ , X δ and S δ separately as well as the convex hull of their union. We may assume (applying the subdivision algorithm of Edelsbrunner and Grayson [18] and Theorem 3.3.6 if necessary) that the currents U 0 , Proof of claim. By the triangle inequality and since any simplicial flat norm decomposition is a candidate decomposition for the flat norm, we have The claim follows from letting δ ↓ 0 and noting that F(T − P δ ) → 0.
Recall that X + ∂S = T is an optimal flat norm decomposition of T and X δ and S δ are polyhedral approximations to X and S on our simplicial complex K δ . Using the decompositions in Equations (3.16) and (3.18), we can write: (3.20) Now apply Theorem 3.3.6 with = 1 to the currents U δ i and W δ j for all i ∈ {0, 1} and j ∈ {0, 1, 2} to obtain U i,δ and W j,δ on the simplicial complex K δ with Applying Equations (3.17) to (3.19), we obtain the following from Equation (3.21): We apply the linearity result of Theorem 3.3.6 to Equation (3.20) along with the fact that P δ , X δ , and ∂S δ are fixed by projection to the d-skeleton of K δ to yield which, as all quantities are chains on K δ , is a candidate simplicial flat norm decomposition of P δ . Using this observation, the triangle inequality, and Equations (3.17) and (3.22), we have The claim follows from taking the limit as δ ↓ 0.

Introduction
This work discusses the integral area invariant introduced by Manay et al. [30], particularly with regard to reconstructability of shapes. This topic has been considered previously by Fidler et al. [20] [21] for the case of star-shaped regions. Recent results have shown local injectivity in the neighborhood of a circle [6] and for graphs in a neighborhood of constant functions [8].
The present work does not assume a star-shaped condition but does make use of a tangent-cone graph-like condition which is local to the integral area circle. We also present an interpretation of the integral area invariant as a nonasymptotic density. This is based on a poster presented by the authors [26].
Our tangentially graph-like and tangent-cone graph-like conditions (definitions 4.2.3 and 4.2.5 in section 4.2) restrict our attention to shapes with boundaries that can locally (i.e., within radius r) be viewed as graphs of functions in a Cartesian plane in one particular orientation (in the case of tangentially graph-like) or a particular set of orientations (for tangent-cone graph-like). Intuitively, these conditions guarantee that the boundary does not turn too sharply within the given radius and that working locally in Euclidean space is the same as working locally on the boundary of our shapes (i.e., the shape boundary does not pass through any given invariant circle multiple times, section 1 Previously published as [27] 4.2.2). These simplifying assumptions allow us to explicitly analyze what happens when we move along the boundary and to work locally without worrying about global effects.
We show that the tangent-cone graph-like property can be preserved when approx-  Suppose that for all s we know g(s, r) and its first derivatives with respect to r (disk radius) and s (position along the boundary). This information is sufficient to completely determine Ω up to translation and rotation; that is, we can recover the side lengths and angles of Ω.
The proof of this theorem uses the discontinuities in the s derivative to determine the locations of vertices (and thus the side lengths between them). We combine the r derivative and the one-sided s derivative information when centered on a vertex to recover the angles at which the polygon enters and exits the circle (which might not be the polygon vertex angle if the circle contains another vertex). Doing this with the other one-sided s derivative gives the same thing but using the orientation determined by the other polygon side incident to the vertex. The combination of these yields the polygon's angle at each vertex.
Theorem 4.7.1. Define G ≡ {γ|γ is a C 2 simple closed curve and tangentially graphlike for r =r}. Suppose that, for r =r, for all s ∈ [0, L], and for each γ ∈ G, we know the first-, second-, and third-order partial derivatives of g γ (s, r). Then the set of reconstructible γ ∈ G is C 1 dense in G where reconstructability is modulo reparametrization, translation, and rotation.
The first part of the proof shows that the derivative information can be used to obtain the curvature. However, it is not the curvature at the boundary point where the circle is centered but rather the curvature at each of the points where the boundary enters and exits the circle. Although the Euclidean distance to these points is known, the arc length distances are not and can vary from point to point. Thus the sequences of curvatures we obtain also lose the arc length parameterization of our area invariant.
The rest of the proof is concerned with finding the arc length distance from the center to the entry and exit points which effectively recovers the curvature for all points. This relies on matching up the unique features of exit angle sequences with each other which in turn relies on the existence of unique maxima and minima in these sequences. While this is not true in general, it can be arranged to be so by a suitable small perturbation of the boundary (which is why our result is one of density rather than for all shapes). This is a theoretical paper about a measure that is useful in applications: we do not pretend that the reconstruction techniques in our proofs are practically useful. In fact, the reconstructions we use to show uniqueness would be seriously disturbed by the noise that any practical application would encounter. We do, however, comment on some possible approaches to reconstruction (section 4.8) using the OrthoMads direct search algorithm [2] to successfully reconstruct shapes which are not predicted by our theory.

Notation and Preliminaries
Unless otherwise specified, we will be assuming throughout this paper that Ω ⊂ R 2 is a compact set with simple closed, piecewise continuously differentiable boundary ∂Ω of length L. Let γ : [0, L] → ∂Ω be a continuous arclength parameterization of ∂Ω (see Figure 4.1). We will adopt the convention that γ traverses ∂Ω in a counterclockwise direction so it always keeps the interior of Ω on the left (there is no compelling reason for this particular choice, but adopting a consistent convention allows us to avoid some ambiguities later). Note that γ(0) = γ(L) and that γ restricted to [0, L) is a bijection.
Denote by D(p, r) the closed disk and C(p, r) the circle of radius r centered at the point In geometric measure theory, the m-dimensional density of a set A ⊆ R n at a point p ∈ R n is given by where H m is the m-dimensional Hausdorff measure and α m is the volume of the unit ball in R m [33]. In the current context, the 2-dimensional density of Ω at γ(s) is simply Area(Ω ∩ D(γ(s), r)) πr 2 .
While we can evaluate this for all s ∈ [0, L), just knowing the density at every point along the boundary is generally insufficient to reconstruct the original shape. If γ (s) exists, then Area(Ω ∩ D(γ(s), r)) is approximated arbitrarily well for sufficiently small r by replacing ∂Ω with its tangent line (which gives us an area of exactly πr 2 2 ). Hence, we have Θ 2 (Ω, γ(s)) = 1 2 at any point where γ is differentiable. That is, just knowing Θ 2 (i.e., the limit) is insufficient to distinguish any two shapes with C 1 boundary.
Contrast this with the situation where we know Area(Ω ∩ D(γ(s), r)) for every s ∈ [0, L) and r > 0 (i.e., we have all of the values needed to compute the limit as well).
This added information is sufficient to uniquely identify C 2 curves by recovering their curvature at every point (see Appendix 4.11).
One natural question to ask (and the focus of the present work) is whether failing to pass to the limit (i.e., using some fixed radius r instead of the limit or all r > 0) and collecting the values for all points along the boundary preserves enough information to reconstruct the original shape. That is, can a nonasymptotic density (perhaps along with information about its derivatives) be used as a signature for shapes?  Since we presume that r is fixed and known for the situations we study, it's trivial to convert data between the forms g(s, r) and g(s,r) πr 2 ; we choose to leave out the normalizing factor in the definition of g(s, r) as it is the integral area invariant of Manay et al. [30] and this form proves useful when computing derivatives in section 4.4.

Definitions
We introduce the tangentially graph-like condition as a simplifying assumption for the shapes we consider.
Definition 4.2.3. For a fixed radius r, we say that ∂Ω is graph-like (GL) at a point p ∈ ∂Ω (or graph-like on D(p, r)) if it is possible to impose a Cartesian coordinate system such that the set of points ∂Ω ∩ D(p, r) is the graph of some function f in this coordinate system. Without loss of generality, we adopt the convention that p is the origin so that f (0) = 0. We define tangentially graph-like (TGL) in the same way but further require that ∂Ω be continuously differentiable and f (0) = 0 (noting that f is C 1 because ∂Ω is). This is illustrated in figure 4.2(a). Without loss of generality (and in keeping with our convention that γ traverses ∂Ω counterclockwise), we assume that the interior of Ω is "up" in the circle (i.e., that (0, ) ∈ Ω for sufficiently small > 0). If ∂Ω is (tangentially) graph-like on D(p, r) for all p ∈ ∂Ω, we say that ∂Ω is (tangentially) graph-like for radius r.
It is instructive to consider what is not graph-like or tangentially graph-like. Violations of the graph-like condition are generally due to a radius that is too large (certainly, choosing a radius so large that all of Ω is in the disk will do it). For example, a unit side length square is not graph-like with radius 1 2 + for any > 0 (position the circle at the center of a side; see figure 4.3(a)). Notice that the same square is graph-like with any radius 1 2 or below. A shape can fail to be tangentially graph-like while still being   graph-like if it fails to be a graph in the required orientation but works in some other (see figure 4.3(b)).
We would like to consider shapes with corners but our tangentially graph-like condition requires that the boundary be differentiable everywhere. The following definitions allow us to generalize the tangentially graph-like condition to this situation by using one-sided derivatives. In particular, we let T γ (s) and Γ + = lim t↓s γ (t).  Remark 4.2.7. Note that when ∂Ω is C 1 , there is only one direction in T γ (s) for each s (i.e., the tangent to ∂Ω at γ(s)). Thus, the definitions of tangentially graph-like and tangent-cone graph-like coincide when ∂Ω is C 1 and every tangentially graph-like boundary is tangent-cone graph-like.

Two-Arc Property
The graph-like condition implies (in proof of the following lemma) that Ω will never be entirely contained in the disk, no matter where on the boundary we center it. That is, some part of Ω lies outside of D(p, r) for every p ∈ ∂Ω.
Proof. Suppose by way of contradiction that |∂Ω ∩ C(p, r)| < 2. Since ∂Ω is a simple closed curve, we have ∂Ω ⊆ D(p, r). As ∂Ω is graph-like at p with radius r, there exists some orientation for which ∂Ω ∩ D(p, r) = ∂Ω is the graph of a well-defined function. However, ∂Ω is a simple closed curve so it is not the graph of a function in any orientation, yielding a contradiction.
The next result is the reason we find the tangent-cone graph-like condition useful. It says that if ∂Ω is tangent-cone graph-like with radius r, then, for every p ∈ ∂Ω, the disk D(p, r) has only two points of intersection with ∂Ω and these are transverse. In other words, this means that when working locally in the disk D(p, r) we need only consider a single piece of ∂Ω.
Proof. By Lemma 4.2.8, we have that |∂Ω ∩ C(p, r)| ≥ 2. Note that ∂Ω contains an interior point (p) and at least two boundary points of the disk D(p, r) (since |∂Ω ∩ C(p, r)| ≥ 2). As ∂Ω is connected and simply closed, there must exist an arc of ∂Ω within the disk going from some point on C(p, r) through p to another point on C(p, r). Suppose |∂Ω ∩ C(p, r)| > 2; that is, there are other points of intersection. Letting q denote one of these, there are two cases to consider (illustrated in Figure 4.4).
(a) ∂Ω does not cross C(p, r) at q.
As ∂Ω is tangent-cone graph-like at q, then ∂Ω ∩ C(q, r) is a graph in every orientation in the tangent cone of ∂Ω at q. In particular, note that the tangent line to C(p, r) at q is in this cone. However, the line from p to q is normal to this line and thus ∂Ω ∩ C(q, r) is not graph-like in this orientation, a contradiction. Therefore, this case cannot occur. This argument applies to all points in ∂Ω ∩ C(p, r) so we immediately have the result that ∂Ω always crosses C(p, r) transversely.
There exists q ∈ ∂Ω ∩ C(p, r) such that there is a path along ∂Ω in D(p, r) from q to q . That is, there exist s 1 , s 2 ∈ [0, L) (without loss of generality, s 1 < s 2 ) such that γ(s 1 ) = q, γ(s 2 ) = q and the image of [s 1 , s 2 ] under γ is contained in D(p, r) (but does not include p, since it is on another arc and ∂Ω is simple). Thus γ enters C(p, r) at s 1 and exits at s 2 .
If we can find s ∈ [s 1 , s 2 ] and w in the tangent cone of ∂Ω at γ(s) satisfying w, p − γ(s) = 0, we will contradict that ∂Ω is tangent-cone graph-like.
Note that v(s) is in the tangent cone of ∂Ω at γ(s) so that ∂Ω ∩ D(γ(s), r) is graph-like using the orientation given by v(s). Observe that v (and therefore φ) is piecewise continuous since γ is piecewise C 1 .
Definition 4.2.9. We say that Ω has the two-arc property for a given radius r if for every point p ∈ ∂Ω, we have that D(p, r) divides ∂Ω into two connected arcs: ∂Ω ∩ D(p, r) and ∂Ω\D(p, r). Instead of considering how D(p, r) divides ∂Ω, we can equivalently frame the definition in terms of how ∂Ω divides C(p, r). That is, Ω has the two-arc property if the circle C(p, r) is divided into two connected arcs by ∂Ω for every p ∈ ∂Ω.

Corollary 4.2.10.
If Ω is tangent-cone graph-like for some radius r, then it has the two-arc property.
Proof. This is a trivial consequence of Theorem 4.2.1.

Corollary 4.2.11.
If Ω is tangentially graph-like for some radius r, then it has the two-arc property for radius r.
Remark 4.2.12. While the assumption of the two-arc property for disks of radius r =r does not imply the two-arc property for all r <r (see Figure 4.5), it is the case that TGL for r =r does imply that γ is TGL for all 0 < r <r. The fact that γ is TGL for all 0 < r <r follows easily from the definition of TGL and the fact that D(p, r) D(p,r).

Notation
Suppose that ∂Ω is tangent-cone graph-like with radius r and we have some s ∈ [0, L) such that ∂Ω is tangentially graph-like at γ(s) with radius r. Since ∂Ω is TGL at γ(s), it has two points of intersection with C(γ(s), r) by theorem 4.2.1. In the orientation The notation is motivated by the fact that 0 < s − (s) < s < s + (s) < L in general due to our convention that γ traverses ∂Ω counterclockwise. The only case where this is not true is when γ(L) = γ(0) is in the disk but even then it will hold for a suitably shifted γ that starts at some point outside the current disk.
The quantities θ 1 (s) and θ 2 (s) are the angles that the rays from the origin to the right and left points of intersection, respectively, make with the positive x axis. We can assume θ 1 (s) ∈ (− π 2 , π 2 ) and θ 2 (s) ∈ ( π 2 , 3π 2 ). We define ν 1 (s) as the angle between the vector γ(s + (s)) − γ(s) and the vector lim t↓s + (s) γ (t), the one-sided tangent to ∂Ω at the point of intersection on the right.
That is, we are measuring the angle between the outward normal to the disk at the point of intersection and the actual direction γ is going as it exits the disk. We define ν 2 (s) similarly. We have ν 1 , ν 2 ∈ (− π 2 , π 2 ) due to the fact that all circle crossings are transverse by theorem 4.2.1.
When the proper s to use is implied by context, we will often simply write s + , s − , θ 1 , θ 2 , ν 1 and ν 2 in place of s + (s), s − (s), and so forth.

Calculus on Tangent Cones
The following result is a version of the intermediate value theorem for elements of the tangent cones.
In addition to the intermediate value theorem, we have an analogous mean value theorem for tangent cone elements.
By the convexity of T γ (c), we have w ∈ T γ (c) with w , n = 0 which follows as in the previous case.
The following lemma tells us that the tangent-cone graph-like condition is sufficient to apply lemma 4.2.14.
Lemma 4.2.15. If ∂Ω is tangent-cone graph-like for some radius r, then ∂Ω has no cusps.

TCGL Boundary Properties
The following technical lemmas allow us to bound various distances and areas encountered in tangent-cone graph-like boundaries.
Lemma 4.2.16. Suppose that ∂Ω is tangent-cone graph-like with radius r and points p 1 , p 2 ∈ ∂Ω with d(p 1 , p 2 ) < r. Then one of the arcs (call it P ) along ∂Ω between p 1 and p 2 is such that, for any two points q 1 , q 2 ∈ P , we have d(q 1 , q 2 ) < r.
Proof. Note that p 2 ∈ D(p 1 , r) so that there is an arc along ∂Ω from p 1 to p 2 which is fully contained in the interior of D(p 1 , r) by theorem 4.2.1. We will call this arc P .
For all x on P , let P x denote the subpath of P from p 1 to x (so P = P p 2 ). We claim that P x is contained in D(x, r) for all x on P (thus, P is contained in D(p 2 , r)). Indeed, if this were not the case, then there must be somex on P such that Px is contained in D(x, r) but C(x, r) ∩ Px is nonempty (i.e., we can move the disk along P until some part of the subpath hits the boundary). That is, the subpath Px has a tangency with the disk D(x, r) which is impossible because of theorem 4.2.1.
Let q 1 ∈ P and note that since P x is contained in D(x, r) for all x on P , we have that P is contained in D(q 1 , r). Therefore, d(q 1 , q 2 ) < r for all q 1 , q 2 ∈ P as desired.
Lemma 4.2.17. If q 1 = γ(s 1 ), q 2 = γ(s 2 ) ∈ P where P is as in the previous lemma, then the arc length between q 1 and q 2 along P is at most √ 2d(q 1 , q 2 ). Proof.
Since Ω is tangentially graph-like, for any w 1 ∈ T γ (s 1 ), w 2 ∈ T γ (s 2 ), the angle between w 1 and w 2 is at most π 2 . Since this is true for all q ∈ P , there is a point q = γ(s ) ∈ P and w ∈ T γ (s ) such that the angle between w and tangent vectors for any other point q ∈ P is at most π 4 . This means that P is the graph of a Lipschitz function g of rank 1 in the orientation defined by w . This does not necessarily imply that D(q , r) ∩ ∂Ω, D(p 1 , r) ∩ ∂Ω or D(p 2 , r) ∩ ∂Ω is the graph of a Lipschitz function; we explore a Lipschitz condition for the disks in section 4.3. Let x 1 , x 2 ∈ [−r, r] with p 1 = (x 1 , g(x 1 )), p 2 = (x 2 , g(x 2 )). Then the arclength from p 1 to p 2 is given by

TCGL polygonal approximations
If Ω is tangent-cone graph-like with radius r, it can sometimes be nice to know that there is an approximating polygon to Ω which is also tangent-cone graph-like. The following lemmas explore this idea. If ∂Ω is TCGL with radius r then for each ∈ (0, r), then there exists a polygonal approximation to ∂Ω that is TCGL with radius r − and such that every point on ∂Ω is within distance 6 of the polygon.
Proof. First, choose a finite number of points along the boundary such that the arc length along γ between any two neighboring points is no more than 3 . These will be the vertices of our polygon. Similarly to γ, we let φ be an arclength parameterization of this polygon so that they both encounter their common points in the same order.
The fine spacing between vertices guarantees that we obtain the 6 bound. Indeed, given any point p ∈ ∂Ω and its neighboring vertices v 1 and v 2 , the arc length along ∂Ω from v 1 to p plus that from p to v 2 is at most 3 by assumption. Since Euclidean distance is bounded above by arc length, we have d(p, v 1 ) + d(p, v 2 ) ≤ 3 . This bound in turn implies that at least one of d(p, v 1 ) and d(p, v 2 ) is bounded above by 6 .
Combining the arcs along γ and φ between v 1 and v 2 , we obtain a closed curve with total length at most 2 3 , so that the distance between any two points on the curve is at most 3 . That is, for any s ∈ [s 1 , s 2 ] and t ∈ [t 1 , t 2 ], we have d(γ(s ), φ(t )) ≤ 3 .
. Let a, b be distinct points on the polygon in D(φ(t), r − ) and consider the line connecting them. This line also intersects a , b on γ such that we have a = b , d(a, a ) ≤ since γ is TCGL at γ(s) with radius r and v 2 − v 1 ∈ T γ (s). Thus φ is TCGL at p with radius r − . The case where p = φ(t) is a vertex is similar but we must consider an arbitrary vector w ∈ T φ (t) in the inner product. We wish to show that, for every w ∈ T φ (t), there is a s such that either w or −w ∈ T γ (s ) and d(p, γ(s )) ≤ 3 , after which the proof follows as in the first case with w (or −w) in place of v 2 − v 1 . We let γ(s) = φ(t) = p and let v 1 = φ(t 1 ) = γ(s 1 ) and v 2 = φ(t 2 ) = γ(s 2 ) be the neighboring vertices (so t 1 < t < t 2 and s 1 < s < s 2 ).
As above, there exist s 1 , s 2 such that s 1 ≤ s 1 ≤ s ≤ s 2 ≤ s 2 , γ(s)−γ(s 1 ) ∈ T γ (s 1 ) and γ(s 2 ) − γ(s) ∈ T γ (s 2 ). Note that T φ (t) is exactly the set of positive linear combinations of these vectors. By lemma 4.2.13, for every w ∈ T φ (t), there is a s ∈ [s 1 , s 2 ] such that w ∈ T γ (s ). As d(p, γ(s )) < 3 , the proof is complete. Definition 4.3.2. We say that Ω is tangentially graph-like and Lipschitz (TGLL) with radius r if Ω is tangentially graph-like with radius r and there is some constant 0 < K < ∞ such that for every p ∈ ∂Ω, the arc D(p, r) ∩ ∂Ω is the graph of a Lipschitz function (in the same orientation used by the tangentially graph-like definition) and that the Lipschitz constant is at most K. Remark 4.3.3. Note that tangentially graph-like does not imply tangentially graph-like and Lipschitz: taking γ to be a square with side length 5 whose corners are replaced by quarter circles of radius 1 and then considering disks of radius √ 2 centered on γ yields one example.
We will use the fact that γ always crosses ∂D transversely to prove that γ is in fact TGLL on slightly bigger disks of radius r + δ as long as one takes a somewhat bigger Lipschitz constantK. It is then an immediate result of lemma 4.3.1 that we can find an approximating polygon that is TCGL with radius r. If γ is TGLL with radius r, then it is TGLL with radius r + δ for some δ > 0 and there is an approximating polygon P γ which is TCGL with radius r. Proof.
Step 1: Show that the quantities ν 1 and ν 2 are continuous as a function of s ∈ [0, L].(see Fig. 4

.6)
Define R 2 (s, t) ≡ ||γ(s) − γ(t)|| 2 . Taking the derivative, we get Because ν 1 and ν 2 are both less than π/2 and γ is graph-like in the disk, we have that both elements of this derivative are nowhere zero. By the implicit function theorem, we get that s − (s) and s + (s) are continuous functions of s. From this it follows that ν 1 and ν 2 are continuous on [0, L].
Step 3: TGLL implies that there is a constant K < ∞ such that γ ∩ D(γ(t), r) is the graph of a function whose x-axis direction is parallel to γ (t) and this function is Lipschitz with Lipschitz constant K.
That is, for all t, γ is TGLL with Lipschitz constant 2K for disks of radius r + δ K,r . The result follows by lemma 4.3.1.     The situation is illustrated in figure 4.9 where we can see that the area being added as we go from s to s + ∆s is the shaded region on the right with height r − h 1 and, considering first-order terms only, uniform width ∆s so has area (r − h 1 )∆s. Similarly, we are subtracting the area (r − h 2 )∆s on the left. Therefore, we have In this section, we consider the case where nonasymptotic densities and first derivatives are known along a T-shaped set (i.e., for all s with a fixed radiusr and for all r ≤r with a fixedŝ). We show that this information is sufficient to guarantee reconstructability modulo reparametrizations, translations, and rotations. 4.6 TCGL Polygon Is Reconstructible from g r and g s without tail Theorem 4.6.1. For a tangent-cone graph-like polygon Ω, knowing g(s, r), g r (s, r) and g s (s, r) for all s ∈ [0, L) and a particular r for which ∂Ω is tangent-cone graph-like is sufficient to completely determine Ω up to translation and rotation; that is, we can recover the side lengths and angles of Ω.

Reconstructing shapes from T-like data
Proof. For a given s and r where g r and g s exist, we can use them to obtain r(θ 2 − θ 1 ) as the length of the circular arc between the entry and exit points by Lemma 4.4.1 and r(sin θ 2 − sin θ 1 ) as the difference in heights of the entry and exit points by Lemma 4.4.2.
We can imagine placing a circular arc with angle gr r on our circle and sliding it around until the endpoints have the appropriate height difference, yielding our θ 1 and θ 2 . Note that since Ω is tangent-cone graph-like, one endpoint must be on the left side of the circle and the other must be on the right and we cannot slide either endpoint to or beyond the vertical line through the center of the circle. Therefore, as we slide the right endpoint down, the left endpoint slides up so that the height difference as a function of the slide is strictly monotonic. Therefore, the slide that gives us θ 1 and θ 2 is unique for a given starting arc placement. However, there are two starting arc placements: the first calls the angle for the right endpoint θ 1 and the left endpoint θ 2 (so the interior of Ω is "up" in the circle) and the second swaps these (so the interior of Ω is "down"). Since we have adopted the convention that ∂Ω is traversed in a counterclockwise direction (so the interior of Ω is up in the circles) we therefore pick This procedure works whenever g r and g s exist which is certainly true whenever the density disk does not touch a vertex of Ω either at its center or on its boundary because if we avoid these cases, then there is only one graph-like orientation to deal with and ∂Ω is C ∞ for all the points that enter into the computation. In fact, with a moment's thought, we can make a stronger statement than this: g r always exists and g s exists as long as the center of the density disk is not a vertex of the polygon.
We can identify the s values at which g s (s, r) does not exist to obtain the arc length positions of the vertices (and therefore obtain side lengths). For a given s corresponding to a vertex, we can find g r and the one-sided derivatives g s− and g s+ . These correspond to the graph-like orientations required by the polygon sides adjacent to the current vertex.
Referring to Figure 4.11, the one-sided derivatives along with the argument at the beginning of the proof yield the angles θ 1 , θ 2 , φ 1 , and φ 2 . Thus we can calculate ψ = θ 1 − φ 1 which means that the polygon vertex at s has angle π − ψ.
Doing this for all s corresponding to vertices, we can determine all of the angles of the polygon. With the side lengths identified earlier, this completely determines the polygon Ω up to translation and rotation. 4.7 Simple closed curves are generically reconstructible using fixed radius data We will assume that γ is TGL for the radiusr. We will also assume that we know the first, second, and third derivatives of g(s, r) for r =r. Under these assumptions, γ is generically reconstructible. By generic we mean the admittedly weak condition of density -reconstructible curves are C 1 dense in the space of C 2 simple closed curves.
Theorem 4.7.1. Define G ≡ {γ|γ is a C 2 simple closed curve and TGL for r =r}.
Suppose that, for r =r, for all s ∈ [0, L], and for each γ ∈ G we know the first-, second-, and third-order partial derivatives of g γ (s, r). Then the set of reconstructible γ ∈ G is C 1 dense in G where reconstructability is modulo reparametrization, translation, and rotation.
Proof: We outline the proof without some of the explicit constructions that follow without much trouble from the outline. We have that θ 1 (s − (s)) + ν 1 (s − (s)) = π − θ 2 (s) − ν 2 (s) and θ 1 (s) + ν 1 (s) = π − θ 2 (s + (s)) − ν 2 (s + (s)). All four of these quantities (the left-and right-hand sides of each of the 2 equations) are the turning angles between the tangent to the curve at the center of the disk and the tangent to the curve at a point r away from the center of the disk. Now we use this correspondence between the θ+ν curves to solve for s − (s) and s + (s).
But these curves can differ by a homeomorphism of the domain. Thus, we can only find the correspondence if there is a distinguished point on those curves as well as no places where the values attained are constant. The turning angle curves having isolated critical points and a unique maximum or minimum is sufficient for our purposes.
To get isolated extrema, start by approximating the curve γ with another one,γ, that agrees in C 1 at a large but finite number of points {s i } N i=1 (i.e. agrees in tangent direction as well as position) and has isolated critical points in the derivative of the tangent direction. Now perturbγ to one that is C 1 close (but not C 2 close) by using oscillations about the curve so that the 2nd and 3rd derivatives are never simultaneously below the bounds on the 2nd and 3rd derivatives of the curve we started with. We do this in a way that alternates around the curve. See Figure 4.13. In a bit more detail, suppose that max{d 2γ /ds 2 , d 3γ /ds 3 } < L 1 . Choose a starting point on the curve; s = 0 works. Now begin perturbingγ at the point sr in the positive s direction such that |γ(sr)−γ(0)| =r.
We continue perturbing until we have reached s 2r defined by |γ(s 2r ) −γ(sr)| =r. We begin perturbing again when we reach s 3r . Continue in this fashion aroundγ. The last piece, shown in green in the figure, will require a perturbation that is distinct in size due to the fact that it will interact with the perturbation that starts at sr. On this last piece, we enforce L 2 < max{d 2γ /ds 2 , d 3γ /ds 3 } < L 3 . All these perturbations can be chosen with isolated singularities in derivatives, thus giving us θ + ν curves that are monotonic between isolated singularities. (In fact, we might as well choose all perturbations to be piecewise polynomial perturbations. This immediately gives us the isolated singularities and monotonicity that we want.) Finally, if there is not a distinct maximum, we can choose one of the maxima and add a small twist to the curve at that point. See Figure 4.14. The idea is that a small twist, applied to the leading edge of the tangents we are comparing to get the turning angle, will increase the angle most at the center of the twist. If this corresponds to a nonunique global maximum, we end up with a unique global maximum. L 2 < max{d 2 γ/ds 2 , d 3 γ/ds 3 } < L 3 L 1 < max{d 2 γ/ds 2 , d 3 γ/ds 3 } < L 2 Figure 4.13: In this schematic figure, we illustrate the alternating perturbation around the curve, keeping the curve C 1 close to and messing with the second and third derivatives to eliminate any critical points other than isolated maxima and minima. Here the perturbation is of course greatly exaggerated. Figure 4.14: A twist perturbation. Notice that if the twist is applied precisely at a global max of the turning angle (as measured by the tangent here and the one lagging it in s), we will increase the turning angle there and will end up with a unique global maximum. Now the correspondence scheme works. That is, we know that the global maximums must match, and because the turning angle curves are monotonic between isolated critical points, we can find the homeomorphisms in s that move the turning angle curves into correspondence.
Taken together, the last two claims give us the curvature as a function of arclength.
This determines γ up to translations and rotations.

Numerical experiments
In this section, we consider a numerical curve reconstruction for the situation in which g(s, r) is known for a given radius r but no derivative information is available. This reconstruction is more strict than the scenarios of sections 4.5-4.7. Our motivation is to explore whether any γ can be uniquely and practically reconstructed with this limited information.
We consider γ a (s) ∈ P N , the set of simple polygons of N ordered vertices {(x 1 , y 1 ), . . . , for some coefficients a i,j ∈ R. In this way, the polygon γ is a discrete approximation of a C ∞ curve. The sides of γ a (s) are not necessarily of equal length.
We take the vector signature g a (s, r) ∈ R N to be the discrete area densities of γ a (s) computed at each vertex. Given such a signature for fixed radius r and fixed partition s, we seek a * satisfying a * ∈ arg min b∈R 4m g b (s, r) − g a (s, r) 2 2 s.t. γ b ∈ P N (4.2) Equation (4.2) represents a nonlinearly constrained optimization problem with continuous nonsmooth objective. The constraint ensures that polygons are simple though any optimal reconstruction γ a * is not expected to lie on the feasible region boundary except in cases of noisy signatures. This approach to reconstructing curves seeks a polygon that matches a given discrete signature, rather than an analytic sequential point construction procedure.
We use the direct search OrthoMads algorithm [2] to solve this problem. Mads class algorithms do not require objective derivative information [2,4] and converge to second-order stationary points under reasonable conditions on nonsmooth functions [1].
We implement our constraint using the extreme barrier method [5] in which the objective value is set to infinity whenever constraints are not satisfied. We utilize the standard implementation with partial polling and minimal spanning sets of 4m + 1 directions.
We performed a series of numerical tests using the synthetic shamrock curve shown in black in the upper portion of  of radius r is shown in green along with corresponding location in the signature; note that the shamrock is not tangent-cone graph-like with this radius.
When comparing and interpreting the shamrock curves, it is important to note that the scale of the curves is determined entirely by the fit parameters a i,j . On the other hand, as the density signature is independent of curve rotation, the rotation is eyeball adjusted for easy visual comparison. Also note that the two-arc property does not hold for this example so our reconstructability results do not apply. The accuracies of both the curve reconstruction and area density signature fit suggest that somewhat more general reconstructability results hold. In particular, we speculate that general simple polygons may be reconstructible from g(s, r) for fixed r and no derivative information.

Conclusions
We have studied the integral area invariant with particular emphasis on the tangentcone graph-like condition. In particular, we have shown that all TCGL polygons and a C 1 -dense set of C 2 TGL curves are reconstructible using only the integral area invariant for a fixed radius along the boundary and its derivatives.
We also showed that TCGL boundaries can be approximated by TCGL polygons, determined what the derivatives represented, and commented on other sets of data sufficient for reconstruction (namely, both T-like and all radii in a neighborhood of 0).
These reconstructions are all modulo translations, rotations, and reparametrizations.
The arc length parameterization plays a special role here since any two such parameterizations of a boundary will differ only by a shift and can easily be placed into correspondence. The situation becomes more complicated in higher dimensions as boundaries are no longer canonically parameterized by a single variable which is a fundamental assumption of our results and methods. It is not immediately obvious how to resolve the issues created by higher dimensions except that it may be possible to modify some of the machinery to work with star convex regions which restore some semblance of canonical representation.

Acknowledgments
The authors would like to thank David Caraballo for introducing us to this topic as well as Simon Morgan and William Meyerson for initial discussions and work on related topics that are not in this paper. This research was supported in part by National Science Foundation grant DMS-0914809.

Appendix: Easy Reconstructability
For completeness, we include a short proof of the fact that knowing g(s, r) for all s and r very easily gives us reconstructability. This follows from the fact that knowing the asymptotic behavior of g(s, r) as r → 0 for any s gives us κ(s). That in turn implies that knowing g(s, r) in any neighborhood of the set (s, r) ∈ [0, L] × {r = 0} also gives us κ(s) and therefore the curve. Proof. Fix s ∈ [0, L). If the curvature of γ at s is positive, we consider what happens if the curvature at γ(s) is always given by −3π lim r→0 ∂ ∂r g(s,r) πr 2 . The available data (the values g(s, r) for all s ∈ [0, L) and all r ∈ (0, )) are sufficient to compute the relevant derivative and limit so we can use this process to determine the curvature of every point on the C 2 curve ∂Ω.