Signed Lp-distance Fields

Alexander Belyaev a Pierre-Alain Fayolle b Alexander Pasko c

a School of Engineering & Physical Sciences, Heriot-Watt University, Edinburgh, UKb Computer Graphics Laboratory, University of Aizu, Japan

c The National Centre for Computer Animation, Bournemouth University, UK

Abstract

We introduce and study a family of generalized double-layer potentials which are used to build smooth and accurate approximantsfor the signed distance function. Given a surface, the value of an approximant at a given point is a power mean of distances from thepoint to the surface points parameterized by the angle they are viewed from the given point. We analyze mathematical propertiesof the potentials and corresponding approximants. In particular, approximation accuracy estimates are derived. Our theoreticalresults are supported by numerical experiments which reveal high practical potential of our approach.

1. Introduction

Singular integrals are central to many mathematical andphysical theories and constructions. In this paper, we usesingular integrals to construct smooth approximations ofdistance functions. The paper is inspired by recent works[15,7,4] where remarkable properties of the mean valuenormalization function (the sum of the mean value weightsin the case of a polygon/polyhedron) were discovered andstudied. In this paper, we introduce signed Lp-distancefields, simple but useful generalizations of the mean valuenormalization function, and exploit their properties tobuild smooth approximations to the signed distance func-tion.

Distance and distance-based scalar fields are widely usedin modeling, computer graphics, scientific visualization,robotics, medical imaging, and many other disciplines[16,8,28,23]. In particular, fast and reliable estimation ofthe distance function is important for efficient implemen-tations of level-set methods [20, Chapter 7].

For many applications, the exact distance field is notneeded and smooth approximations of the distance fieldare employed. For example, in the area of solid modeling,Shapiro and co-workers [2,27] discussed Ricci operations[24] and used R-functions [26] for building approximatedistance fields from local approximations.

Fayolle et al. [9] developed smooth alternatives to theclassical min/max operations for constructing signed ap-proximate distance functions. Very recently, Freytag et al.[11] introduced sampled smooth approximations of a dis-tance function and used them to enhance the Kantorovich

method [18,19] for computational mechanics purposes. Inimage processing and computer graphics, Ahuja, Chuang,and co-workers [1,6] and Peng et al. [21,22] introduced gen-eralized potential fields and applied them for shape skele-tonisation and 3D texture modeling, respectively. It is alsoworth to mention works [14,13] where a diffusion-based dis-tance function was introduced and used for static and dy-namic shape analysis.

Our approach is conceptually close to that of Ahuja andChuang (and Peng et al. who re-introduced the Ahuja-Chuang ideas for computer graphics applications) but in-stead of using generalized Newtonian potentials we intro-duce and study a family of generalized double-layer poten-tials and corresponding signed Lp-distance fields.

There are several advantages of using generalized double-layer potentials to compare with their single-layer counter-parts. In particular,

(i) our signedLp-distance fields deliver smooth and accu-rate approximations of the signed distance function;

(ii) exact formulas for the generalized double-layer poten-tials generated by triangle meshes (polylines in 2D)can be obtained;

(iii) mathematical properties of the generalized double-layer potentials and their corresponding signed Lp-distance fields are easy to analyze;

(iv) our generalized double-layer potentials can be con-sidered as a generalization of the mean value normal-ization function [7,4] and can be used for high-ordertransfinite interpolation purposes.

It is interesting that singular integrals similar to thedouble-layer potentials we deal with serve as a main build-

ing block for the so-called surface generalized Born modelin biomolecular modeling [12,25].

2. Generalized double-layer potentials

Consider a smooth oriented surface S ⊂ Rn and a singu-lar integral

φk(x) =∫S

ny · (y− x)

|x− y|n+kσ(y) dSy, k ≥ 0, (1)

where x is a point outside S, y ∈ S, ny is the orientationnormal at y, dSy is the surface element at y, and σ(y) isa density function defined on S. The classical double-layerpotential corresponds to (1) with k = 0. If the electricdipoles are distributed over S with density σ, then φ0(x) isproportional to the electric field generated by the dipolesat x.

Let us assume that σ ≡ 1 (i.e., the dipoles are uniformlydistributed over S) and introduce a generalized double-layer potential by

ϕk(x) =∫S

ny · (y− x)

|x− y|n+kdSy =

∫Ω

dΩy|x− y|k

, (2)

where k > 0, Ω is the unit sphere centered at x, and dΩy isthe solid angle at which surface element dSy is seen fromx. We have used a simple relation dSy = ρndΩy/h, whereh = ny · (y− x) is the distance from x to the plane tangentto S at y. The two-dimensional analog of (2) can be writtenin polar coordinates as

ϕk(x) =∫ 2π

0

dθ

ρ(θ)k, ρ(θ) = |x− y| , (3)

where θ is the angle between vector y − x and a fixed di-rection.

Note that the surface integral in (2) is correctly definedfor an arbitrary smooth oriented surface S, while the in-tegral over unit sphere Ω is properly defined if domain Dis star-shaped w.r.t. x. In order to drop this limitation forthe integral over Ω we follow a standard approach of alter-nating signs (see, for example, [7,4]). Consider a ray origi-nated at x and intersecting S in m points y1, . . . ,ym. Weset εi = 1 if the ray [x,yi) arrives at yi from positive sideof S, εi = −1 if the ray approaches yi from negative sideof S, and εi = 0 if the ray is tangent to S at yi. Now letus assume that |x− y|k in the denominator of the integralover Ω in (2) means

∑i εi |x− yi|

k. See the left and middleimages of Fig. 1 below for a visual explanation.

If S is a closed curve and k = 1 then (3) defines the nor-malization function corresponding to the transfinite meanvalue interpolation for the domain bounded by S.

It is easy to see that

ψp(x) = [1 /ϕp(x) ]1/p , p ≥ 1, (4)

vanishes as x approaches S. In the next section, we showthat (4) delivers a smooth approximation for the signed dis-tance function dist(x) from S and, furthermore, the bigger

Fig. 1. Left and middle: illustrations for the definition of ϕk(x). Right:

potential ϕk(x) generated by a segment ab and notations used.

p is, the better signed Lp-distance field ψp(x) approximatesdist(x).

3. Mathematical properties of approximatedistance functions

In this section we formulate our main mathematical re-sults which describe approximation properties of (4). SeeAppendix A for proofs.

Denote by vn and an the volume of the unit ball in Rnand area of the unit sphere, respectively. We have an =nvn, v0 = 1, v1 = 2, v2 = π, v3 = 4π/3, and an =2πn/2

/Γ(n/2), where Γ(·) is the Gamma function. Con-

sider a domain D ⊂ Rn bounded by surface S.

Proposition 1 Assume that D is star-shaped w.r.t.x ∈ D. Let 1 < p < q <∞. Then

anψ1(x) > (an)1/pψp(x) > (an)1/q

ψq(x) > dist(x).

Further ψk(x) → dist(x), as k → ∞. In other words,ψ∞(x) = dist(x). In addition, if x ∈ D does not belongto the medial axis of D, then

ψk(x) = dist(x)[1 +

n− 12

ln kk

]+O

(1k

), as k →∞.

Although this convergence is not fast, below we will seethat, after a proper normalization, ψk(x) delivers a veryaccurate approximation of dist(x) near S even for smallk. Fig. 2 illustrates that in the simplest case of n = 1. Asseen in the left image, (an)1/k

ψk(x) converges to dist(x)from above as k → ∞. However a different normalizationof ψk(x) (no normalization is needed if n = 1) allows usto obtain a much better approximation dist(x) near theboundary.

Fig. 2. Approximate distance functions (an)1/k ψk(x) (left) andψk(x) (right) defined for [0, 1] (n = 1). See the main text for details.

Let us introduce a sequence

2

c0 = an/2, c1 = vn−1, ck+2 =k + 1k + n

ck (5)

and define

Ψk(x) = [ck]1/k ψk(x) = [ϕk(x)/ck]−1/k.

Proposition 2 Let n be the outer normal for S = ∂D.If x ∈ ∂D then

Ψk(x) = 0, ∂Ψk(x)/ ∂n = −1.

If ∂D contains a planar part and x is an internal pointof that planar part, then, in addition,

∂sΨk(x)/ ∂ns = 0, s = 2, 3, . . . , k.

A simple asymptotic analysis of (5) yields

[ck]1/k = 1− n− 12

ln kk

+O

(1k

), as k →∞,

which, in view of the last statement of Proposition 1, gives

Ψk(x) = dist(x) +O (1/k) , as k →∞, (6)

for x outside the medial axis of D. Note that, accordingto Proposition 2, near a flat part of ∂D the convergence ofΨk(x) to dist(x) is much faster than (6).

Our last theoretical result is presented below. In partic-ular, it implies that the shapes of the graphs of ψk(x) andΨk(x) are similar to that of dist(x).

Proposition 3 We have

∆ϕk(x) = k(k + n)ϕk+2(x)

and, therefore, ψk(x) has no local minima inside D.

Note that the gradient and the other derivatives of ϕk(x)can be easily obtained by differentiating (2) by x as manytimes as needed.

4. Singular potentials for polylines and meshes

Let us start from the 2D case and consider the generalizeddouble-layer potentials generated by oriented segment ab.Given point x, γ = γ(x) is the angle between xa and xb, aand b are the distances from x to a and b, respectively, andρ(θ) is the distance from x to point y ∈ ab. See the rightimage of Fig. 1 for a visual illustration of the notations used.

The area of 4xab is equal to the sum of areas of 4xayand 4xyb. We have

ab sin γ = aρ(θ) sin θ + bρ(θ) sin(γ − θ),which gives an explicit expression of ρ(θ)

ρ(θ) =ab sin γ

a sin θ + b sin(γ − θ).

Then generalized double-layer potential ϕk(x) generatedby the segment is given by

ϕk(x) =∫ γ

0

dθ

ρ(θ)k=

1(ab sin γ)k

∫ γ

0

(a sin θ+b sin(γ−θ))kdθ.

For an arbitrary positive integer k, the value of this integralcan be evaluated symbolically (for example, one can useMaple or Mathematica for that). If k is odd, the integralcan be expressed as a polynomial of degree k of variable t,where t = tan(γ/2). In particular,

ϕ1(x) =(

1a

+1b

)t,

ϕ3(x) =13

(1a3

+1b3

)t+

16

(1a

+1b

)3

t(1 + t2

)Note that ϕ1(x) is the weight corresponding to ab andinduced by the mean value coordinates [10,15].

If k is even, then more complex expressions are obtainedfor ϕk(x), except the case when k = 0: ϕ0(x) = γ(x) is thestandard double-layer potential induced by ab.

As expected, in the 3D case, the situation is more com-plex. Let S be a triangulated surface and abc ∈ S be amesh triangle. While ϕ0(x) is just a solid angle at whichabc is seen from x and the formula for ϕ1(x) generated bytriangle abc is presented in [17], deriving a closed-form ex-pression for ϕk(x), k > 1, is a difficult and tedious task.Fortunately, a clear way to obtain analytical expressions fora family of singular integrals including ϕk(x) was proposedvery recently in [5].

Given a polyline in 2D (triangle mesh in 3D), potentialϕk(x) at point x is obtained by summing the contributionsof the polyline segments (mesh triangles).

5. Numerical experiments

In Fig. 3, we provide the reader with a visual comparisonof the level sets of dist(x) with those of Ψ1, Ψ3 and Ψ5 forsufficiently complex planar polylines. We observe that theapproximation quality improves quickly as k increases.

Fig. 4 visualizes the relative error between dist(x) andψk(x) and between dist(x) and Ψk(x) (normalized Lp-distance), k = 1, 3, 5, for a squared domain. As before, theerror goes down quickly as k increases. One can observethat the approximation error is relatively high near thecorners.

In Fig. 5, we investigate approximation properties of Lp-distance fields near a corner. As expected, the approxima-tion accuracy (as before, we calculate the relative error)at a small vicinity of a corner point is lower than near aninternal point of segment. Further, the relative error in-creases as the corner angle becomes sharper. However, evenfor a sharp corner, the approximation accuracy quickly im-proves, as k increases.

Finally, Fig. 6 illustrates our approach in 3D with therelative error between dist(x) and Ψ2(x) plotted on a cross-section of the object.

6. Discussion and conclusion

In this paper, we have introduced a family of gener-alized double-layer potentials and their corresponding

3

Fig. 3. Polygon approximation of two curves used for our experiments (left-most images). The polygon meshes consist in 201 and 29 segments

respectively. Contour plots for: the signed distance, Ψ1, Ψ3 and Ψ5 for a 2D polygon mesh (columns 2 to 5). Ψ5 (right-most images) provides

already a good approximation of the distance function (second column). Note that the holes in the curve for the letter A are properly handled.

Fig. 4. Relative errors for a square. The first row corresponds to therelative error between the exact signed distance and ψ1, ψ3 and ψ5

respectively. The second row uses the normalized functions: Ψ1, Ψ3

and Ψ5 instead.

Fig. 5. Relative errors between the exact signed distance and Ψ5 forvarious opening angles between two segments. The angles vary be-

tween 0 (top row, left column) and 45 (bottom row, right column).

Lp-distance fields. They possess interesting mathematicalproperties, and, in particular, the Lp-distance fields de-liver smooth and accurate approximations of the distancefunction.

This work is just a beginning of our study of generalizeddouble-layer potentials and their applications. A practical

Fig. 6. Relative errors between dist(x) and Ψ2(x) for a cube (left)

and the rocker-arm model (right).

computation of Lp-distance fields can be significantly ac-celerated via a hierarchical approach: an accurate evalua-tion of a Lp-distance field at a given point does not requirecareful calculations of contributions of surface parts distantfrom the point. For polylines in 2D and triangle meshes in3D, special correctors (boundary layers) should be addedto the Lp-distance fields in order to achieve a better ap-proximation accuracy at polyline/mesh vertices.

We hope that our approach can lead to improving andenhancing existing applications of approximate distancefunctions in image processing, computer vision, computergraphics, geometric modeling, and several other branches ofscientific computing [1,14,22]. In particular, we foresee po-tential applications of Lp-distance fields in computationalmechanics where some new promising computational tech-niques rely on constructing accurate and efficient approxi-mate smoothed distance functions and their derivatives [11,Section 3.2].

Appendix A: Proofs

Proof of Proposition 1. An integral version of Jensen’sinequality (see, for example, [29, Problem 7.5]) states that

F

(∫R

f(x)w(x) dx)≤∫R

F (f(x))w(x) dx,

where F (t) is convex and w(x) is a positive weight functionsatisfying ∫

R

w(x) dx = 1.

4

Let 1 < p < q <∞, F (t) = tq/p, R = [0, 2π], w ≡ 1/2π,and f(θ) = (1/ρ(θ))p. Since F (t) is strictly convex, we have[

12π

∫ 2π

0

dθ

ρ(θ)p

] 1p

<

[1

2π

∫ 2π

0

dθ

ρ(θ)q

] 1q

, ψp(x) > ψq(x),

where ψk(x) = (2π)1/kψk(x). The multidimensional case

n > 2 is considered similarly.Let x ∈ D is fixed and k → ∞. We have according to

Laplace’s method (see, for example, [3])

ϕk(x) =∫

Ω

dΩρk

=∫

Ω

expk ln

1ρ

dΩ

=C1 + o(1)k(n−1)/2

1ρkmin

, where ρmin = dist(x).

Thus

ψk(x) =[

1ϕk(x)

]1/k

= dist(x)[k(n−1)/2

C1 + o(1)

]1/k

= dist(x) [C2 + o(1)]1/k(k1/k

)(n−1)/2

, as k →∞,

where C1 and C2 are positive constants. It remains to notethat(k1/k

)n−12

= exp

(n− 1) ln k2k

= 1+

n− 12

ln kk

+O(

1k

),

as k →∞.

Proof of Proposition 2. First let us consider the 2Dcase and study an asymptotic behavior of potential ϕ1(x)generated by a single segment ab. See the left image ofFig. 7 for notations used. Assume that h, the distance fromx to ab, is small: h 1. We have γ = π − α− β,

a(h) = a+O(h2), α = arctan (h/a) = h/a+O(h2),

b(h) = b+O(h2), β = arctan (h/b) = h/b+O(h2),

tanγ

2=

2aba+ b

1h

+O(h), ϕ1(x) =2h

+O(h).

Fig. 7. Notations used for asymptotic analysis of potentials ϕk(x)

generated by a segment (left) and a circular arc (right).

In the 2D case, for a single straight segment, we have

ϕk(x) =∫

dθ

ρ(θ)k=∫

h dsyρ(θ)k+2

, ρ = |x− y|.

Note that

ρ =√h2 + z2,

∂ρ

∂h=h

ρ,

∂

∂h

(1

ρk+2

)= − (k + 2)h

ρk+4,

∂

∂h

(h

ρk+2

)=

1ρk+2

− (k + 2)h2

ρk+4

(see Fig. 7 for the notations used). Thus

∂ϕk∂h

=1hϕk − (k + 2)hϕk+2

or, equivalently,

(k + 2)ϕk+2 =1h2ϕk −

1h

∂ϕk∂h

. (7)

Note thatϕk(x) is odd w.r.t. h. Thus the expansion ofϕk(x)w.r.t. h 1 contains only odd degrees of h. For example,

ϕ1(x) =2h

+ a1h+ a3h3 + a5h

5 +O(h7), h→ 0.

Then (7) implies that

ϕ3(x) =43

1h3− 2a3h− 4a5h

3 +O(h5)

and we can observe that (7) kills the linear w.r.t. h term inthe expansion of ϕk(x). Now, applying the mathematicalinduction, we arrive at

ϕk(x) = ck/hk +O(h)

with ck defined by (5), where singular potential ϕk(x) isgenerated by segment ab and x approaches an internalpoint of ab. For the potential generated by a polygon con-taining segment ab, we have a slightly bigger remainder:

ϕk(x) = ck/hk +O(1), as h→ 0,

as the contribution of the other segments of the polygon isO(1). Thus

Ψk(x) =[

ckϕk(x)

]1/k

= h+O(hk+1

)as h→ 0.

In particular, if x is an internal point of ab, we have

∂sΨk(x)/ ∂ns = 0, s = 2, 3, . . . , k.

Now let us consider a domain D with a smooth bound-ary S. Without loss of generality we can assume that Dis convex (the case of non-convex domains can be treatedsimilarly to how it is done in [7]). Consider a point x ∈ Don a distance h 1 from S. Let ab be a circular arc os-culating S at the point closest to x such that the segmentab is perpendicular to the direction from x to its closestneighbor on S, as seen in the right image of Fig. 7. Let usalso consider another circular arc cd such that it is situatedinside D, ab and cd are collinear, and the direction from xto its closest neighbor on S is the bisector for cd.

Now we approximate ϕk(x) given by (3) by the sum ofintegrals of 1/ρ(θ)k over the circular arcs ab and cd. Notethat, as x approaches S, the osculating arc ab delivers abetter and better local approximation of S and the leadingterm in asymptotics ϕk(x), as h → 0, is the same as thatin the integral of 1/ρ(θ)k over the circular arc ab.

The law of cosines yields

ρ(θ) = −(R− h) cos θ +√R2 − (R− h)2 sin2 θ.

5

where R is the radius of the osculating circle. We can safelyassume thatR = 1. Our task now is to study the asymptoticbehavior of

Ik(h) =

π/2∫−π/2

[(h− 1) cos θ +

√1− (1− h)2 sin2 θ

]−kdθ.

It is easy to verify that

Ik(h) =ckhk

+O

(1

hk−1

), as h→ 0.

In the nD case, for a single (n − 1)-dimensional meshtetrahedron (a mesh triangle in 3D), we have

ϕk(x) =∫dΩyρk

=∫h dSyρk+n

, ρ = |x− y|.

So we arrive at a generalization of (7) for the multidimen-sional case

(k + n)ϕk+2(x) =1h2

ϕk(x)− 1h

∂ϕk(x)∂h

.

It is shown in [4] that

ϕ1(x) =vn−1

h+O(1)

which justifies that c1 = vn−1 in (5).

Proof of Proposition 3. Differentiating of (1) w.r.t xyields

∆φk(x) = k(k + n)φk+2(x).

For k = 1 and σ(y) ≡ 1 this formula coincides with that ob-tained in [4]. Obviously φk(x) is positive if surface ”dipole”density σ(y) is positive. Assume that φk(x) has a local max-imum at some point inside the domain bounded by S. Then∆φk(x) ≤ 0 at that point. We arrive at a contradictionwhich completes the proof.

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