From point cloud data to the continuum model of geometry

J. Harrison1,∗

1 University of California, Berkeley, 2823 Benvenue Ave, Berkeley, CA 94705, USA.

The quantum calculus of [6] gives discrete approximations to continuum models of geometry.

1 Introduction

By the continuum model of geometry we refer to smooth manifolds and associated objects such as smooth functions anddifferential forms, Riemannian metrics, curvatures, and operations like wedge product, Hodge star, exterior derivative, Diracoperator, etc. It involves the sea of points with calculus connecting it all together. Many concepts of physics are describedwith the language of differential forms, but not all. Sullivan [3] articulated four properties for a discrete model of geometry: Itshould be determined by a finite amount of data. It should have analogues of all the objects mentioned before. It should havenatural finite dimensional versions which still have all of these structures. It should have some fine scale parameter so thatwhen this parameter goes to zero the discrete model should have a limit which is the smooth continuum.

To help organize and complete the picture, and express more concepts of physics, we propose adding to this collection anew class of objects that are geometric in nature and reverse the variance of differential forms so that the above operators havegeometric counterparts such as pushforward, boundary, and Hodge star. Furthermore, we set up a discrete quantum versionof the continuum model with analogous objects to all of the above and which has structures preserving finite dimensionaltruncations. There is a limit of the discrete model which is isomorphic to the continuum model. This discrete model isobtained from a reassembly of information from point cloud data into an infinitesimal algebraic construction that encodesgeometry and topology.

Currents are higher dimensional versions of distributions. Every object in M that we wish to treat as a domain of integrationmust be a current as it must act linearly on differential forms. Since de Rham first proposed the problem (see [4]), the holygrail has been to find a normed subspace of currents with good categorical properties. Normed subspaces include chains withfinite mass, integral and normal currents [2] , and sharp and flat chains [1], but none has the full set of features sought by deRham.

Chainlets form a proper normed subspace Chk(M) of currents defined by using test forms Brk(M) that have a bound on

each directional derivative Chk(M) := lim−→Br

k(M)′. Chainlets represent objects such as submanifolds with cusps, fractals,soap films, graphs of L1 functions, charged particles. The limit of operator norms | · |Br

on Brk(M)′ is a norm on chainlets

called the natural norm | · |� = limr→∞ | · |Br. Chainlet spaces satisfy a universal property making them the smallest normed

subspace of currents for which calculus is valid. They contain a discrete subspace satisfying the requirements of Sullivan.

2 Pointed chains

Let Λk denote the exterior algebra of k-vectors in Rn. A pointed chain is a free section of the space of tangent k-vectors of

M . That is, a pointed chain P is nonzero except at most finitely many points {p1, . . . , ps} called the support of P . We useformal sum notation P =

∑si=1

(pi; αi) where αi ∈ Λk. Let Pk(M) denote the space of pointed k-chains with mass |P |0 asa norm, and Qk(M) the space of cellular k-chains.

Theorem 2.1 Pointed k-chains Pk(M) and cellular k-chainsQk(M) form dense subspaces of chainlets Chk(M).A simple k-element (p; α) is a pointed chain where p ∈M and α is a simple k-vector. Geometric operators such as

pushforward, exterior product, boundary and geometric Hodge star may be applied to simple k-elements and extended bylinearity to bounded operators on pointed chains. These operators therefore extend to elements of Chk(M) without regard towhether or not the chainlet has tangent spaces anywhere defined.

Let X be a vector field on M and |X |0 its sup norm. Let TX denote translation through the time-one map of X

Theorem 2.2 The natural norm | · |� is the largest seminorm ‖ · ‖ on Pk(M) such that ‖P‖ ≤ |P |0 and ‖TXP − P‖ ≤|X |0|P |0 for all P ∈ Pk(M) and smooth vector fields X .

These two conditions are necessary for calculus, but they are also sufficient. Chainlet spaces Chk(M) are thus the small-est normed subspace of currents for which calculus is valid. Since boundary is continuous with ∂2 = 0, then Ch(M) =

∗ Corresponding author E-mail: harrison@math.berkeley.edu, Phone: +01 510 642 9666, Fax: +01 510 642 8204

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

PAMM · Proc. Appl. Math. Mech. 7, 1026301–1026302 (2007) / DOI 10.1002/pamm.200700889

⊕nk=0Chk(M) forms a differential complex. It yields the largest dual complex of differential forms and contains the fewest

pathological domains.

3 Properties

• Regularity in the category of chainlets is assured since basic operators are closed and bounded in Ch(M).

• The quantum calculus of pointed chains and the continuum model of geometry are unified within the category of chainlets.Using Theorem 0.1 we obtain convergence of the discrete theory of pointed chains to the smooth continuum and viceversa.

• We define discrete cochains on pointed chains by duality and use this result to characterize the Br norm on cochainsusing divided differences. Complete to obtain smooth forms using the chainlet isomorphism theorem of [6].

• We obtain normal bundles, flux, Stokes, Gauss, Green, vector fields, Lie derivative, creation, annihilation, multiplicationby a function (density), calculus of variations, ... all with discrete and smooth counterparts. (See [6].)

• Theorems are simple to state and prove and optimal as a result of Theorem 0.3.

• We obtain discrete analogues of topology and geometry.

4 Pointed chain approximators from point cloud data

We may use pointed chains to formulate discrete versions of the integral theorems of calculus, homology classes and geometry.From point cloud data taken from a k-submanifold M of Euclidean n-space, we construct a Delaunay triangulation. Eachk-simplex σi determines a k-direction, volume and orientation, and therefore a unique k-vector αi. Choose a base point pi

near σi. The pointed chain P =∑

(pi; αi) converges to M in the natural norm as the distance between the data points tendsto zero. (We have discarded the original data.) For k = n− 1, the pointed 1-chain ⊥ P =

∑(pi;⊥ αi) is an approximation

to the unit normal bundle.

5 Shape operator and strain

The shape operator is the derivative of the Gauss map. It measures the local curvature at a point on a smooth surface. Anapproximating matrix can be found by translating an element (pi;⊥ αi) via a local basis using nearby base points p′i andcomparing the translated vector with α′

i. Details shall appear elsewhere.Theorem 5.1 Let F : M → M ′ be a diffeomorphism of surfaces and P and P ′ pointed chain approximations of M and

M ′ with P ′ = F∗P . Let S and S′ be the shape operators on P and P ′, respectively. Then

Tr(F ∗S′) = F ∗Tr(S′).

We may now approximate strain at p by Tr(F ∗S′− S) = Tr(F ∗S′)−Tr(S) = F ∗Tr(S′)− Tr(S). Summing over thesquares (Tr(F ∗S′ − S))2 over all p ∈ supp(P ), we obtain discrete flexural energy. (Compare [5])

6 Convergence to the smooth continuum

Our discrete shape operator is a matrix at each point in the point cloud whose determinant and trace approximate Gaussianand mean curvature, respectively. These quantities converge to the these quantities in the smooth continuum by definition ofthe shape operator in the smooth category and since all operators involved are continuous in the chainlet norm.

In particular, discrete flexural energy converges to continuous flexural energy∫

M ′4(H ′ ◦ F − H)2dA where H and H ′

represent mean curvature.

References[1] H. Whitney, Geometric Integration Theory, Princeton, Princeton University Press, (1957).[2] H. Federer, W. Fleming, Normal and integral currents Annals of Math., Series 2, 458-520 (1960).[3] D. Sullivan, Combinatorial topology and the continuum model of geometry, http://www.msri.org/publications/video/index4.html

(1998)[4] L. Schwartz, A Mathematician Grappling with His Century, Birkhauser, (2001)[5] Grinspun, Hirani, Desbrun, Schroder, Discrete Shells SCA ’03: Proceedings of the 2003 ACM SIGGRAPH / Eurographics Sympo-

sium on Computer Animation July 26–27, pp. 62–67, (2003).[6] J. Harrison, New geometrical methods in analysis, Berkeley preprint, (2007).

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

ICIAM07 Minisymposia – 02 Numerical Analysis 1026302