Signed L p -distance Fields

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1 Signed L -distance Fields Alexander Belyaev a, Pierre-Alain Fayolle b Alexander Paso c a School of Engineering & Physical Sciences, Heriot-Watt University, Edinburgh, UK b Comuter Grahics Laboratory, University of Aizu, Jaan c The National Centre for Comuter Animation, Bournemouth University, UK Abstract We introduce and study a family of generalized double-layer otentials which are used to build smooth and accurate aroximants for the signed distance function. Given a surface, the value of an aroximant at a given oint is a ower mean of distances from the oint to the surface oints arameterized by the angle they are viewed from the given oint. We analyze mathematical roerties of the otentials and corresonding aroximants. In articular, aroximation accuracy estimates are derived. Our theoretical results are suorted by numerical exeriments which reveal high ractical otential of our aroach. Key words: aroximate distance functions, singular integrals, generalized double otentials 1. Introduction Singular integrals are central to many mathematical and hysical theories and constructions. In this aer, we use singular integrals to construct smooth aroximations of distance functions. The aer is insired by recent wors [15,8,4] where remarable roerties of the mean value normalization function (the sum of the mean value weights in the case of a olygon/olyhedron) were discovered and studied. We introduce signed L -distance fields, simle but useful generalizations of the mean value normalization function, and exloit their roerties to build smooth aroximations to the signed distance function. For many alications, the exact distance field is not needed and smooth aroximations of the distance field are emloyed. For examle, in the area of solid modeling, Shairo and co-worers [2,27] discussed Ricci oerations [24] and used R-functions [26] for building aroximate distance fields from local aroximations. Fayolle et al. [10] develoed smooth alternatives to the classical min/max oerations for constructing signed aroximate distance functions. Very recently, Freytag et al. [12] introduced samled smooth aroximations of a distance function and used them to enhance the Kantorovich method [17,18] for comutational mechanics uroses. In image rocessing and comuter grahics, Ahuja, Chuang, Corresonding author: htt://home.es.hw.ac.u/ belyaev/ addresses: a.belyaev@hw.ac.u (Alexander Belyaev), fayolle@u-aizu.ac.j (Pierre-Alain Fayolle), aaso@bournemouth.ac.u (Alexander Paso). and co-worers [1,6] and Peng et al. [22,23] introduced generalized otential fields and alied them for shae seletonisation and 3D texture modeling, resectively. It is also worth to mention very recent wors [14,7] where the distance function and geodesics are aroximated by solutions of certain artial differential equations. Our aroach is concetually close to that of Ahuja and Chuang (and, therefore, to that of Peng et al. who re-introduced Ahuja-Chuang ideas for comuter grahics alications) but instead of using generalized Newtonian otentials we introduce and study a family of generalized double-layer otentials and corresonding signed L -distance fields. There are several advantages of using generalized doublelayer otentials to comare with their single-layer counterarts. In articular, (i) our signed L -distance fields deliver smooth and accurate aroximations of the signed distance function; (ii) exact formulas for the generalized double-layer otentials generated by triangle meshes (olylines in 2D) can be obtained; (iii) mathematical roerties of the generalized doublelayer otentials and their corresonding signed L - distance fields are easy to analyze; (iv) our generalized double-layer otentials can be considered as a generalization of the mean value normalization function [8,4] and can be used for high-order transfinite interolation uroses. It is interesting that singular integrals similar to the double-layer otentials we deal with serve as a main build-

2 ing bloc for the so-called surface generalized Born model in biomolecular modeling [13,25]. The urose of this aer is to rovide the reader with insights into mathematical roerties of L -distance fields and reveal their high otential for various alications. 2. Generalized double-layer otentials Consider a smooth oriented hyer-surface S R n and a singular integral n y (y x) φ (x) = x y n+ σ(y) ds y, 0, (1) S where x is a oint outside S, y S, n y is the orientation normal at y, ds y is the surface element at y, and σ(y) is a density function defined on S. The classical double-layer otential corresonds to (1) with = 0. If the electric dioles are distributed over S with density σ, then φ 0 (x) is roortional to the electric field generated by the dioles at x. Let us assume that σ 1 (i.e., the dioles are uniformly distributed over S) and introduce a generalized doublelayer otential by ϕ (x) = S n y (y x) x y n+ ds y = Ω dω y x y, (2) where > 0, Ω is the unit shere centered at x, and dω y is the solid angle at which surface element ds y is seen from x. We have used a simle relation ds y = ρ n dω y / h with ρ = x y and h = n y (y x). Here h is the distance from x to the lane tangent to S at y. The two-dimensional analog of (2) can be written in olar coordinates as ϕ (x) = 2π 0 ρ(θ), (3) where θ is the angle between vector y x and a fixed direction. Note that the surface integral in (2) is correctly defined for an arbitrary smooth oriented surface S, while the integral over unit shere Ω is roerly defined if domain D is star-shaed w.r.t. x. In order to dro this limitation for the integral over Ω we follow a standard aroach of alternating signs (see, for examle, [8,4]). Consider a ray originated at x and intersecting S in m oints y 1,..., y m. We set ε i = 1 if the ray [x, y i ) arrives at y i from ositive side of S, ε i = 1 if the ray aroaches y i from negative side of S, and ε i = 0 if the ray is tangent to S at y i. Now let us assume that x y in the denominator of the integral over Ω in (2) means i ε i x y i. See the left and middle images of Fig. 1 below for a visual exlanation. If S is a closed curve and = 1 then (3) defines the normalization function corresonding to the transfinite mean value interolation for the domain bounded by S. It is easy to see that ψ (x) = [1 /ϕ (x)] 1/, 1, (4) Fig. 1. Left and middle: illustrations for the definition of ϕ (x). Right: otential ϕ (x) generated by a segment ab and notations used. vanishes as x aroaches S. 3. Mathematical roerties In this section we formulate our main mathematical results which describe aroximation roerties of (4). See Aendix A for roofs. Denote by v n and a n the volume of the unit ball in R n and the area of the unit shere, resectively. We have a n = nv n, v 0 = 1, v 1 = 2, v 2 = π, v 3 = 4π/3, and a n = 2π n/2/ Γ(n/2), where Γ( ) is the Gamma function. Let D be a domain in R n bounded by a iecewise-smooth surface S. Consider a family of functions d x (y) = x y, y S, defined on S and arameterized by x D. Proosition 1 We always have ψ (x) dist(x), as. In other words, ψ (x) = dist(x). If d x (y) has continuous second-order artial derivatives in a small vicinity of its global minimum, then [ ψ (x)=dist(x) 1+ n 1 2 ln ] +O ( 1 ), as. (5) If D is star-shaed w.r.t. x and 1 < < q <, then a n ψ 1 (x) > (a n ) 1/ ψ (x) > (a n ) 1/q ψ q (x) > dist(x). Note that tyically the distance-function d x (y) is sufficiently smooth in a small neighborhood of its global minimum and, therefore, (5) is valid. A loss of differentiability of d x (y) haens when x is sufficiently close to a concave singularity (concave corner, concave edge, etc.) of S. Although (5) is not fast, below we will see that, after a roer normalization, ψ (x) delivers a very accurate aroximation of dist(x) near S even for small. Fig. 2 illustrates that in the simlest case of n = 1. As seen in the left image, (a n ) 1/ ψ (x) converges to dist(x) from above as. However a different normalization of ψ (x) (no normalization is needed if n = 1) allows us to obtain a much better aroximation dist(x) near the boundary. Fig. 2. Aroximate distance functions (a n) 1/ ψ (x) (left) and ψ (x) (right) defined for [0, 1] (n = 1). See the main text for details. 2

3 Let us introduce the following sequence: c 0 = a n /2, c 1 = v n 1, c +2 = n c (6) and define Ψ (x) = [c ] 1/ ψ (x) = [ϕ (x)/c ] 1/. Proosition 2 Let n be the outer normal for S = D. If x D then Ψ (x) = 0, Ψ (x)/ n = 1. If D contains a lanar art and x is an internal oint of that lanar art, then, in addition, s Ψ (x)/ n s = 0, s = 2, 3,...,. A simle asymtotic analysis of (6) yields [c ] 1/ = 1 n 1 ( ) ln O, as, which, in view of the last statement of Proosition 1, gives Ψ (x) = dist(x) + O (1/), as. (7) Note that, according to Proosition 2, if x is near a flat art of D, Ψ (x) converges to dist(x) much faster than (7). Our last theoretical result is resented below. In articular, it imlies that the shaes of the grahs of ψ (x) and Ψ (x) are similar to that of dist(x). Proosition 3 We have ϕ (x) = ( + n)ϕ +2 (x) and, therefore, ψ (x) has no local minima inside D. Note that the gradient and the other derivatives of ϕ (x) can be easily obtained by differentiating (2) by x as many times as needed. 4. Singular otentials for olylines and meshes Let us start from the 2D case and consider the generalized double-layer otentials generated by oriented segment ab. Given oint x, γ = γ(x) is the angle between xa and xb, where a and b are the distances from x to a and b, resectively, and ρ(θ) is the distance from x to oint y ab. See the right image of Fig. 1 for a visual illustration of the notations used. The area of xab is equal to the sum of areas of xay and xyb. We have ab sin γ = aρ(θ) sin θ + bρ(θ) sin(γ θ), which gives an exlicit exression of ρ(θ) ρ(θ) = ab sin γ a sin θ + b sin(γ θ). Then the generalized double-layer otential ϕ (x) generated by the segment is given by ϕ (x) = γ 0 ρ(θ) = 1 (ab sin γ) γ 0 (a sin θ+b sin(γ θ)). For an arbitrary ositive integer, the value of this integral can be evaluated symbolically (for examle, one can use Male or Mathematica for that). If is odd, the integral can be exressed as a olynomial of degree of the variable t, where t = tan(γ/2). In articular, ϕ 1 (x)= a + 1 b ] t, ϕ 3 (x)= t 3 a b 3 ]+ t + t3 6 a + 1 ] 3. b Note that ϕ 1 (x) is the weight corresonding to ab and induced by the mean value coordinates [11,15]. If is even, then more comlex exressions are obtained for ϕ (x), excet the case when = 0: ϕ 0 (x) = γ(x) is the standard double-layer otential induced by ab. As exected, in the 3D case, the situation is more comlex. Let S be a triangulated surface and abc S be a mesh triangle. While ϕ 0 (x) is just a solid angle at which abc is seen from x and the formula for ϕ 1 (x) generated by the triangle abc is resented in [16], deriving a closed-form exression for ϕ (x), > 1 is a difficult and tedious tas. Fortunately, a clear way to obtain analytical exressions for a family of singular integrals including ϕ (x) was roosed very recently in [5]. Given a olyline in 2D (triangle mesh in 3D), the otential ϕ (x) at oint x is obtained by summing the contributions of the olyline segments (mesh triangles). 5. Numerical exeriments In Fig. 3, we rovide the reader with a visual comarison of the level sets of dist(x) with those of Ψ 1, Ψ 3 and Ψ 5 for sufficiently comlex lanar olylines. We observe that the aroximation quality imroves quicly as increases. Fig. 4 visualizes the relative error between dist(x) and ψ (x) and between dist(x) and Ψ (x) (normalized L - distance), = 1, 3, 5, for a squared domain. As before, the error goes down quicly as increases. One can observe that the aroximation error is relatively high near the corners. Fig. 4. Estimating the distance function inside and outside a unit square. The first row corresonds to the relative error between the exact signed distance and ψ 1, ψ 3 and ψ 5 resectively. The second row uses the normalized functions: Ψ 1, Ψ 3 and Ψ 5 instead. In Fig. 5, we investigate the aroximation roerties of L -distance fields near a corner (as before, we calculate the 3

4 Fig. 3. Polygon aroximations of two shaes used in our exeriments (left-most images). The olylines consist of 201 and 29 segments resectively. Contour lots for: the signed distance, Ψ1, Ψ3 and Ψ5 for a 2D olyline (columns 2 to 5). Ψ5 (right-most images) rovides already a good aroximation of the distance function (second column). Note that the hole for the letter A is roerly handled. liver smooth and accurate aroximations of the distance function. A ractical comutation of L -distance fields can be accelerated by using a hierarchical aroach similar to that develoed in [9]. Another ossibility is to aroximate otentials via fast multiole exansions [19]. The ability of L -distance fields to deliver accurate aroximations of the distance function not only near the boundary of the object but also dee inside the object (a tas which R-functions do not handle well, as demonstreated in [10]) maes them otentially useful for heterogeneous object modeling [3,10]. We also foresee otential alications of the L -distance fields in comutational mechanics where some new romising comutational techniques rely on constructing accurate and efficient aroximate smoothed distance functions and their derivatives [12, Section 3.2]. Fig. 5. Relative errors between the exact signed distance and Ψ5 for various oening angles between two segments. The angles vary between 0 (to row, left column) and 45 (bottom row, right column). relative error). As exected, the aroximation accuracy in a small vicinity of a corner oint is lower than it is near an internal oint of the segment. Further, the relative error increases on the reflex side of the angle, as the angle becomes sharer. However, even for a shar corner, the aroximation accuracy quicly imroves, as increases. Finally, Fig. 6 illustrates our aroach in 3D with the relative error between dist(x) and Ψ2 (x) lotted on cross-sections of 3D objects. Acnowledgements. We would lie to than the anonymous reviewers of this aer for their valuable and constructive comments. Aendix A Proof setch for Proosition 1. Let x D is fixed, Ω is the unit shere centered at x, and. We have according to Lalace s method [28, Chater IX, 5] Z Z dω 1 C1 + o(1) 1 ϕ (x) = = ex ln dω = (n 1)/2, ρ ρmin Ω ρ Ω where ρmin = dist(x). Thus 1/ (n 1)/2 1/ 1 ψ (x) = = dist(x) ϕ (x) C1 + o(1) (n 1)/2 1/ = dist(x) [C2 + o(1)] 1/, as, Fig. 6. Relative errors between dist(x) and Ψ2 (x) for a cube (left) and the rocer-arm model (right). 6. Discussion and conclusion In this aer, we have introduced a family of generalized double-layer otentials and their corresonding L -distance fields. They ossess interesting mathematical roerties, and, in articular, the L -distance fields de- where C1 and C2 are ositive constants. It remains to note that, as, 4

5 ( 1/) n 1 { } 2 (n 1) ln = ex 2 = 1+ n 1 ln 2 +O ( ) 1. Above, we assumed that d x (y) = ρ = x y has continuous second-order artial derivatives in a small vicinity of its global minimum. This may be not true if x D is located near a reflex singularity (e.g., a reflex angle or edge) of S = D and the minimum of d x (y) is achieved for y situated on that singularity. In that case, weaer asymtotics can be derived [21] (see also [20, Theorem 2.1]). However convergence ψ (x) dist(x), as, remains true. Now let us assume that D is star-shaed w.r.t. x D. An integral version of Jensen s inequality states that F µ(r) R ] f(y) dµ(y) 1 µ(r) R F [f(y)] dµ(y), where R is a measurable set with finite ositive measure µ and F (t) is a convex function. Let 1 < < q <, F (t) = t q/, R = Ω be the unit shere in R n, and f(y) = (1/ρ), where ρ = x y. Since F (t) is strictly convex, we have [ 1 a n Ω ] 1 [ ] 1 dω y 1 q dω y < ρ a n Ω ρ q, [a n ] 1 ψ (x) > [a n ] 1 q ψ q (x). Proof setch for Proosition 2. First let us consider the 2D case and study an asymtotic behavior of otential ϕ 1 (x) generated by a single segment ab. See the left image of Fig. 7 for notations used. Assume that h, the distance from x to ab, is small: h 1. We have γ = π α β, a(h) = a + O(h 2 ), α = arctan (h/a) = h/a + O(h 2 ), b(h) = b + O(h 2 ), β = arctan (h/b) = h/b + O(h 2 ), tan γ 2 = 2ab 1 a + b h + O(h), ϕ 1(x) = 2 h + O(h). Fig. 7. Notations used for asymtotic analysis of otentials ϕ (x) generated by a segment (left) and convex and concave circular arcs (middle and right, resectively). In the 2D case, for a single straight segment, we have ϕ (x) = ρ(θ) = h ds y ρ(θ) +2, ρ = x y. Note that ρ = h 2 + z 2, ( h h ρ +2 ρ h = h ρ, ) ( ) 1 ( + 2)h h ρ +2 = ρ +4, = 1 ( + 2)h2 ρ+2 ρ +4 (see the left image of Fig. 7 for the notations used). Thus ( + 2)ϕ +2 = 1 h 2 ϕ 1 ϕ h h. (8) Note that ϕ (x) is odd w.r.t. h. Thus the exansion of ϕ (x) w.r.t. h 1 contains only odd degrees of h. For examle, ϕ 1 (x) = 2h 1 + a 1 h + a 3 h 3 + a 5 h 5 + O ( h 7), h 0. Then (8) imlies that 3ϕ 3 (x) = 4h 3 2a 3 h 4a 5 h 3 + O ( h 5). One can observe that the right-hand side of (8) ills the linear w.r.t. h term in the exansion of ϕ (x) and, therefore, the exansion of ϕ +2 (x) w.r.t. h contains only one growing term. Mathematical induction yields ϕ (x) = c / h + O(h) with c defined by (6), where singular otential ϕ (x) is generated by segment ab and x aroaches an internal oint of ab. For the otential generated by a olygon containing segment ab, we have a slightly bigger remainder: ϕ (x) = c / h + O(1), as h 0, as the contribution of the other segments of the olygon is O(1). Thus Ψ (x) = [c /ϕ (x)] 1/ = h + O ( h +1), as h 0. In articular, if x is an internal oint of ab, we have s Ψ (x)/ n s = 0, s = 2, 3,...,. Now let us consider a domain D with a smooth boundary S oriented counterclocwise (see the middle image of Fig. 7 for the notations used). Consider a oint x D on a distance h 1 from S and assume that the curvature of S is ositive in a vicinity of x. Let ab be a circular arc osculating S at the oint closest to x such that the segment ab is erendicular to the direction from x to its closest neighbor on S, as seen in the middle image of Fig. 7. Let us also consider another circular arc cd such that it is situated inside D, ab and cd are collinear, and the direction from x to its closest neighbor on S is the bisector for cd. Now we aroximate ϕ (x) given by (3) by the sum of integrals of 1/ρ(θ) over the circular arcs ab and cd. Note that, as x aroaches S, the osculating arc ab delivers a better and better local aroximation of S and the leading term in asymtotics ϕ (x), as h 0, is the same as that in the integral of 1/ρ(θ) over the circular arc ab. Let R be the radius of the osculating circle. The law of cosines and simle algebraic maniulations yield π/2 π/2 I (h) = ρ(θ) = π/2 = [ R 2 (R h) 2 sin 2 θ (R h) cos θ [ R 2 (R h) 2 sin 2 θ + (R h) cos θ] / [2Rh h 2 ] = 1 π/2 π/2 ( ) h cos 1 θ + 2R h 1 cos 2 θ sin 2 θ + O h 2 ] 5

6 which imlies that I (h) = c h + d h 1 / R + O ( h 2 ), as h 0. (9) with c = π/2 cos θ and d = c 2 / 2. The case of negative curvature (see the right image of Fig. 7) is considered similarly: / [ ] 1 ρ(θ) = (R + h) cos θ R 2 (R + h) 2 sin 2 θ = [ (R h) cos θ + R 2 (R + h) 2 sin 2 θ] / [2Rh + h 2 ]. The integration limits are α and α, where 2α is the angle made by the two rays originated from x and tangent to the osculating circle, sin α = R/(R + h). This leads to I (h) = c h d h 1 / R + O ( h 2 ), as h 0. (10) Asymtotics (9) and (10) suggest that the second term is roortional to the curvature of S at the oint closest to x. In the nd case, for a single (n 1)-dimensional mesh tetrahedron (a mesh triangle in 3D), we have ϕ (x) = dωy ρ = h dsy ρ +n, ρ = x y. So we arrive at a generalization of (8) for the multidimensional case ( + n)ϕ +2 (x) = 1 h 2 ϕ (x) 1 ϕ (x) h h. It is shown in [4] that ϕ 1 (x) = v n 1 / h + O(1), as h 0, which justifies that c 1 = v n 1 in (6). The multidimensional case of smooth S is much more comutationally demanding and can be analyzed by using osculating araboloids for local aroximations of S. Proof of Proosition 3. Differentiating of (1) w.r.t x yields φ (x) = ( + n)φ +2 (x). For = 1 and σ(y) 1 this formula coincides with that obtained in [4]. Obviously φ (x) is ositive if surface generalized-diole density σ(y) is ositive. Assume that φ (x) has a local maximum at some oint inside the domain bounded by S. Then φ (x) 0 at that oint. We arrive at a contradiction which comletes the roof. References [1] N. Ahuja and J.-H. Chuang. Shae reresentation using a generalized otential field model. IEEE Transactions on Pattern Analysis and Machine Intelligence, 19(2): , [2] A. Biswas and V. Shairo. Aroximate distance fields with non-vanishing gradients. Grahical Models, 66(3): , [3] A. Biswas, V. Shairo, and I. Tsuanov. Heterogeneous material modeling with distance fields. Comut. Aided Geom. Des., 21: , [4] S. Bruvoll and M. S. 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