Reconstruction of n-dimensional convex bodies from surface tensors

Reconstruction of -dimensional convex bodies from surface tensors

Astrid Kousholt111kousholt@math.au.dk Department of Mathematics, Aarhus University, Denmark
Abstract

In this paper, we derive uniqueness and stability results for surface tensors. Further, we develop two algorithms that reconstruct shape of -dimensional convex bodies. One algorithm requires knowledge of a finite number of surface tensors, whereas the other algorithm is based on noisy measurements of a finite number of harmonic intrinsic volumes. The derived stability results ensure consistency of the two algorithms. Examples that illustrate the feasibility of the algorithms are presented.

\keywords

Convex body, surface tensor, harmonic intrinsic volume, uniqueness, stability, reconstruction algorithm

MSC2010: 52A20, 44A60, 60D05

1 Introduction

Recently, Minkowski tensors have succesfully been used as shape descriptors of spatial structures in materials science, see, e.g., [3, 13, 14]. Surface tensors are translation invariant Minkowski tensors derived from surface area measures, and the shape of a convex body with nonempty interior in is uniquely determined by the surface tensors of . In this context, the shape of is defined as the equivalence class of all translations of .

In [9], Kousholt and Kiderlen develop reconstruction algorithms that approximate the shape of convex bodies in from a finite number of surface tensors. Kousholt and Kiderlen describe two algorithms. One algorithm requires knowledge of exact surface tensors and one allows for noisy measurements of surface tensors. For the latter algorithm, it is argued that it is preferable to use harmonic intrinsic volumes instead of surface tensors evaluated at the standard basis. The purpose of this paper is threefold. Firstly, the reconstruction algorithms in [9] are generalized to an -dimensional setting. Secondly, stability and uniqueness results for surface tensors are established, and the stability results are used to ensure consistency of the generalized algorithms. Thirdly, we illustrate the feasibility of the reconstruction algorithms by examples. The generalizations of the reconstruction algorithms are developed along the same lines as the algorithms for convex bodies in . However, there are several non-trivial obstacles on the way. In particular, essentially different stability results are needed to ensure consistency.

The input of the first generalized algorithm is exact surface tensors up to a certain rank of an unknown convex body in . The output is a polytope with surface tensors identical to the given surface tensors of the unknown convex body. The input of the second generalized algorithm is measurements of harmonic intrinsic volumes of an unknown convex body in , and the output is a polytope with harmonic intrinsic volumes that fit the given measurements in a least squares sense. When , a convex body that fits the input measurements of harmonic intrinsic volumes may not exist, and in this case, the algorithm based on harmonic intrinsic volumes does not have an output. However, this situation only occurs when the measurements are too noisy, see Lemma 6.3.

The consistency of the algorithms described in [9] is established using the stability result [9, Thm. 4.8] for harmonic intrinsic volumes derived from the first order area measure. This result can be applied as the first order area measure and the surface area measure coincide for . However, for , the stability result is not applicable. Therefore, we establish stability results for surface tensors and for harmonic intrinsic volumes derived from surface area measures. More precisely, first we derive an upper bound of the Dudley distance between surface area measures of two convex bodies. This bound is small, when is large and the distance between the harmonic intrinsic volumes up to degree of the convex bodies is small (Theorem 4.3). From this result and a known connection between the Dudley distance and the translative Hausdorff distance, we obtain that the translative Hausdorff distance between convex bodies with identical surface tensors up to rank becomes small, when is large (Corollary 4.4). The stability result for surface tensors and the fact that the rank surface tensor of a convex body determines the radii of a ball containing and a ball contained in (Lemma 5.4) ensure consistency of the generalized reconstruction algorithm based on exact surface tensors (Theorem 5.5). The consistency of the reconstruction algorithm based on measurements of harmonic intrinsic volumes are ensured by the stability result for harmonic intrinsic volumes under certain assumptions on the variance of the noise variables (Theorems 6.4 and 6.5).

The described algorithms and stability results show that a finite number of surface tensors can be used to approximate the shape of a convex body, but in general, all surface tensors are required to uniquely determine the shape of a convex body. However, there are convex bodies where a finite number of surface tensors contain full information about the shapes of the convex bodies. More precisely, in [9], it is shown that the shape of a convex body in with nonempty interior is uniquely determined by a finite number of surface tensors only if the convex body is a polytope. We complement this result by showing that the shape of a polytope with facets is uniquely determined by the surface tensors up to rank . This result is optimal in the sense that for each there is a polytope with facets and a convex body that is not a polytope, such that and have identical surface tensors up to rank . This implies that the rank cannot be reduced. An earlier and weaker result in this direction is [9, Thm. 4.3] stating that the shape of a polytope with facets is determined by the surface tensors up to rank .

The paper is organized as follows. General notation, surface tensors and harmonic intrinsic volumes are introduced in Section 2. The uniqueness results are derived in Section 3 and are followed by the stability results in Section 4. The two reconstruction algorithms are described in Sections 5 and 6.

2 Notation and preliminaries

We work in the -dimensional Euclidean vector space , with standard inner product and induced norm . The unit sphere in is denoted , and the surface area and volume of the unit ball in is denoted and , respectively.

In the following, we give a brief introduction to the concepts of convex bodies, surface area measures, surface tensors and harmonic intrinsic volumes. For further details, we refer to [12] and [9]. We let denote the set of convex bodies (convex, compact and nonempty sets) in , and let be the set of convex bodies with nonempty interior. Further, is the set of convex bodies contained in a ball of radius , and likewise, is the set of convex bodies that contain a ball of radius and are contained in a concentric ball of radius . The set of convex bodies is equipped with the Hausdorff metric . The Hausdorff distance between two convex bodies can be expressed as the supremum norm of the difference of the support functions of the convex bodies, i.e.

for .

In the present work, we call the equivalence class of translations of a convex body the shape of . Hence, two convex bodies are of the same shape exactly if they are translates. As a measure of distance in shape, we use the translative Hausdorff distance,

for . The translative Hausdorff distance is a metric on the set of shapes of convex bodies, see [6, p. 165].

For a convex body , the surface area measure of is defined as

for a Borel set , where is the -dimensional Hausdorff measure, and is the set of boundary points of with an outer normal belonging to . For a convex body there is a unit vector and an , such that is contained in the hyperplane . The surface area measure of is defined as

where is the surface area of and is the Dirac measure at . Notice that for , and for .

The surface tensors of are the Minkowski tensors of derived from the surface area measure of . Hence for , the surface tensor of of rank is given as

where is the -fold symmetric tensor product of when is identified with the rank tensor . Due to multilinearity, the surface tensor of rank can be identified with the array of components of , where is the standard basis of . Notice that the the components of are scaled versions of the moments of , where the moments of order of a Borel measure on are given by

for with .

By [9, Remark 3.1], the surface tensors of are uniquely determined by and for . More precisely, if has same parity as , say, then can be calculated from by taking the trace consecutively and multiplying with the constant

(1)

We let

be the number of different components of and , and we use the notation for the -dimensional vector of different components of the surface tensors of of rank and .

To a convex body , we further associate the harmonic intrinsic volumes that are the moments of with respect to an orthonormal sequence of spherical harmonics (for details on spherical harmonics, see [8]). More precisely, for , let be the vector space of spherical harmonics of degree on . The dimension of is denoted , and . We let be an orthonormal basis of . Then, the harmonic intrinsic volumes of of degree are given by

for . For a convex body , we let be the -dimensional vector of harmonic intrinsic volumes of up to degree . The vector only depends on through the surface area measure of , and we can write . Likewise, for an arbitrary Borel measure on , we write for the vector of harmonic intrinsic volumes of up to order , that is the vector of moments of up to order with respect to the given orthonormal basis of spherical harmonics. The harmonic intrinsic volumes and the surface tensors of a convex body are closely related as there is an invertible linear mapping such that .

3 Uniqueness results

The shape of a convex body is uniquely determined by a finite number of surface tensors only if the convex body is a polytope, see [9, Cor. 4.2]. Further, in [9, Thm. 4.3] it is shown that a polytope in with nonempty interior and facets is uniquely determined up to translation in by its surface tensors up to rank . In Theorem 3.2, we replace with , and in addition, we show that the rank cannot be reduced.

We let denote the cone of finite Borel measures on . Further, we let be the set of convex polytopes in with at most facets. The proof of Lemma 3.1 is an improved version of the proof of [9, Thm. 4.3].

Lemma 3.1.

Let and have finite support .

  1. The measure is uniquely determined in by its moments up to order .

  2. If the affine hull of is , then is uniquely determined in by its moments up to order .

Proof.

We first prove . Since , we have and the support of can be pared down to vectors, say , such that . For each , the affine hull

is a hyperplane in , so there is a and such that

Now define the polynomial

for . The degree of is , and for . Let and assume that . Then for , so in particular where . We may assume that . Since , this implies that is an affine combination of , so

This is a contradiction, and we conclude that .

Now let and assume that and have identical moments up to order . Since the polynomial is of degree , we obtain that

(2)

where we have used that is of the form

for some . Equation (2) yields that for -almost all as the polynomial is non-negative. Then, the continuity of implies that

so is of the form

(3)

with for .

For , define the polynomial

for . Then is of degree and for . If , then and we obtain a contradiction as before. Hence . Due to (3) and the assumption on coinciding moments, we obtain that

(4)

Since , Equation (4) implies that for .

For , define the polynomial

for . Then is of degree and for . If , then for , which is a contradiction. Hence, . By arguments as before, we obtain that for . Hence , which yields (ii).

The statement (i) can be proved in a similar manner using the polynomials

and

for and . ∎

Theorem 3.2.

Let . A polytope with nonempty interior is uniquely determined up to translation in by its surface tensors up to rank . If , then the result holds for any .

The rank is optimal as there is a polytope and a convex body having identical surface tensors up to rank .

Proof.

Let have facet normals and nonempty interior. Then, and , so is uniquely determined in by its moments up to order due to Lemma 3.1 (ii). Since the surface tensors of are rescaled versions of the moments of , the first part of the statement follows as a convex body in with nonempty interior is uniquely determined up to translation by its surface area measure. Now assume that is a polytope in with empty interior. Then is contained in an affine hyperplane and for some . By Lemma 3.1 (i), the surface area measure of is uniquely determined by its moments up to second order. The second part of the statement then follows since any convex body in is uniquely determined up to translation by its surface area measure.

To show that the rank cannot be reduced, we first consider the case . For , let be a regular polytope in with outer normals for and facet lengths for . Then, and the unit disc in have identical surface tensors up to rank . This is easily seen by calculating and comparing the harmonic intrinsic volumes of and .

Now, counter examples in can be constructed inductively. Essentially, if and are counter examples in , counter examples and in are obtained as bounded cones with scaled versions of and as bases. More precisely, for a fixed , define by for , and let

and

By Minkowski’s existence theorem, the measures and are surface area measures of convex bodies and , respectively. Direct calculations show that if and have identical surface tensors in up to rank , then and have identical surface tensors in up to the same rank. Thus, we obtain that the rank is optimal in the sense that it cannot be reduced. ∎

Due to the one-to-one correspondence between surface tensors up to rank and harmonic intrinsic volumes up to degree of a convex body, the uniqueness result in Theorem 3.2 also holds if surface tensors are replaced by harmonic intrinsic volumes.

4 Stability results

The shape of a convex body is uniquely determined by the set of surface tensors of , but as described in the previous section, only the shape of polytopes are determined by a finite number of surface tensors. However, for an arbitrary convex body, a finite number of its surface tensors still contain information about its shape. This statement is quantified in this section, where we derive an upper bound of the translative Hausdorff distance between two convex bodies with a finite number of coinciding surface tensors.

The cone of finite Borel measures on is equipped with the Dudley metric

for , where

for any function . It can be shown that the Dudley metric induces the weak topology on (the case of probability measures is treated in [5, Sec. 11.3] and is easily generalized to finite measures on ) The set of real-valued functions on with is denoted . Further, we let the vector space of square integrable functions on with respect to the spherical Lebesgue measure be equipped with the usual inner product and norm .

As in [1, Chap. 2.8.1], for , we define the operator on the space by

(5)

for where the constant

satisfies

(6)

As is a polynomial in of order , it follows from the addition theorem for spherical harmonics (see, e.g., [8, Thm. 3.3.3]) that the function for can be expressed as a linear combination of spherical harmonics of degree or less, see also [1, pp. 61-62]. More precisely, there are real constants such that

(7)

where is the projection of onto the space of spherical harmonics of degree . The constants in the linear combination (7) are given by

see [1, p. 62]. By [1, Thm. 2.30], for any continuous function , the sequence converges uniformly to when . When , Lemma 4.1 provides an upper bound for the convergence rate in terms of and .

Lemma 4.1.

Let and . For , we have

(8)
Proof.

We proceed as in the proof of [1, Thm. 2.30]. Let . Using (5) and (6), we obtain that

for and , where

and

Since and

we obtain that

(9)

To derive the upper bound on , we have used that for .

Now let . From the mean value theorem, we obtain that

for some . Hence,

(10)

Combining (9) and (10) yields the assertion. ∎

Remark 4.2.

Stirling’s formula, for , implies that

for . Hence, the upper bound in (8) converges to zero for . The choice of in the proof of Lemma 4.1 is optimal in the sense that if we use with a constant , then the derived upper bound in does not converge to zero. This follows as

for , when .

For functions satisfying , Lemma 4.1 yields an uniform upper bound, only depending on and the dimension , of . In the following theorem, this is used to derive an upper bound of the Dudley distance between the surface area measures of two convex bodies where the harmonic intrinsic volumes up to a certain degree are close in .

Theorem 4.3.

Let for some and let . Let and . If

(11)

then

(12)

where is a constant depending on and .

Due to the addition theorem for spherical harmonics, the condition (11) is independent of the bases of , that are used to derive the harmonic intrinsic volumes.

Proof of Theorem 4.3.

Let with and define the signed Borel measure . Then, by (7),

where . Since , we obtain from Cauchy-Schwarz’ inequality and a discrete version of Jensen’s inequality that

Hence,

where we used Lemma 4.1 and that . For , the convergence of to zero is faster than the convergence of , see Remark 4.2. This implies the existence of a constant only depending on and satisfying (12). ∎

Corollary 4.4.

Let for some and let . If for , then

where is a constant depending on and .

Proof.

The assumption that and have coinciding surface tensors up to rank implies that . The result then follows from Theorem 4.3 with . ∎

The translative Hausdorff distance between two convex bodies in admits an upper bound expressed by the ’th root of the Prokhorov distance between their surface area measures, see [12, Thm. 8.5.3]. Further, the Prokhorov distance between two Borel measures on can be bounded in terms of the square root of the Dudley distance between the measures. Therefore, Corollary 4.4 in combination with [12, Thm. 8.5.3] and [7, Lemma 9.5] yield the following stability result.

Theorem 4.5.

Let for some and let . If for , then

for a constant depending on and .

5 Reconstruction of shape from surface tensors

In this section, we derive an algorithm that approximates the shape of an unknown convex body from a finite number of surface tensors of for some . The reconstruction algorithm is a generalization to higher dimension of Algorithm Surface Tensor in [9] that reconstructs convex bodies in from surface tensors. The shape of a convex body in is uniquely determined by the surface tensors of , when has nonempty interior, see [9, Sec. 4, p. 10]. For , the surface tensors of determine the shape of even when is lower dimensional. Therefore, the algorithm in [9] can be used to approximate the shape of arbitrary convex bodies in , whereas the algorithm described in this section only allows for convex bodies in with nonempty interior. A non-trivial difference between the algorithm in the two-dimensional setting and the generalized algorithm is that in higher dimension, it is crucial that the first and second order moments of a Borel measure on determine if is the surface area measure of a convex body. Therefore, this is shown in Lemma 5.2 that is based on the following remark.

Remark 5.1.

Let be a Borel measure on the unit sphere . Then,

(13)

for all if and only if the support of is full-dimensional (meaning that the support of is not contained in any great subsphere of ). As the integral in (13) is determined by the second order moments

of , these moments determine if the support of is full-dimensional. More precisely, the support of is full-dimensional if and only if the matrix of second order moments is positive definite as

for .

Lemma 5.2.

Let be a Borel measure on with .

  1. The measure is the surface area measure of a convex body , if and only if the first order moments of vanish and the matrix of second order moments of is positive definite.

  2. The measure is the surface area measure of a convex body if and only if the first order moments of vanish and the matrix of second order moments of has one positive eigenvalue and zero eigenvalues.

In the case, where (ii) is satisfied, the measure is the surface area measure of every convex body with surface area contained in a hyperplane with normal vector , where is a unit eigenvector of corresponding to the positive eigenvalue (u is unique up to sign).

Proof.

Remark 5.1 implies that the interior of a convex body is nonempty if and only if the matrix of second order moments of is positive definite, so the statement (i) follows from Minkowski’s existence theorem, [12, Thm. 8.2.2].

If is the surface area measure of , then is of the form

for some . Then, the first order moments of vanish, and the matrix of second order moments of is . Hence, has one positive eigenvalue with eigenvector and zero eigenvalues.

If the matrix is positive semidefinite with one positive eigenvalue and zero eigenvalues, then , where is a unit eigenvector (unique up to sign) corresponding to the positive eigenvalue. Assume further that the first order moments of vanish, and define the measure . Then and have identical moments up to order , and Lemma 3.1 (i) yields that . Therefore, is the surface area measure of any convex body with surface area contained in a hyperplane with normal vector . ∎

5.1 Reconstruction algorithm based on surface tensors

Let be fixed. We consider as unknown and assume that the surface tensors of are known up to rank for some natural number . The aim is to construct a convex body with surface tensors identical to the known surface tensors of . We proceed as in [9, Sec. 5.1].

Let

(14)

and consider the minimization problem