Openness and Closedness

In addition to being permissible, we must require that a solid has a bounded maximum curvature and no singularities in a transition region of the bound-ary. This size of the transition region should be chosen in accordance with our

4The observation in the proof of Proposition 4.6 that the boundary mapping is continuous because it is composed of continuous functions is due to F.–E. Wolter (private communica-tions).

method of reconstruction. For the time being, we will simply assume that the transition region (see Section 3.4) is of widthr.

Intuitively, it seems that both things are ensured if we can roll a closed ball of radius r on either side of the boundary of the solid in such a way that it touches all points on the boundary from either side. This is illustrated in Figure 4.3. Since the ball touches the surface at all points, it seems certain that the

Figure 4.3: Balls rolling on either side of the boundary of a solid.

curvature of the boundary does not exceed the curvature of the ball. Likewise, the medial axis should not come closer to the boundary thanr. However these things need to be made more precise, and this can be done using Euclidean Morphology. The property that a ball can roll on the inside can be expressed formally by saying that the solid should be invariant with respect to opening with a closed ball of radius r. Likewise, we can use invariance with respect to closing to ensure that the ball can roll on the outside. These properties will be calledr–openness andr–closedness. r–openness is defined as follows

Definition 4.7 A solidS isr–open iff

S=O(S, b^r) (4.7)

where O(S, b^r) is the open operation on S using b^r as a structuring element.

r–openness simply means that we can represent the solid as the union of an (infinite) number of closed balls of radiusr. Hence, the sphere fits everywhere, and we are thus ensured that we can “roll” the ball on the interior side. Of course, the same thing must hold for the inverse solid, i.e. Sⁱ must also be r–open. To avoid using the inverse, we observe that it can be shown that

S=C(S, b^r) =⇒ Sⁱ=O(Sⁱ, b^r) (4.8) where C(S, b^r) is the close operation on S usingb^ras the structuring element.

In other words, ifS is closed with an open ball it implies thatSⁱ is open with

4.4 Openness and Closedness 71

a closed ball. This follows from (A.12) in Appendix A. Consequently, we can express the quality that we may roll the sphere on the exterior in the following way

Definition 4.8 A solidS isr–closed iff

S=C(S, b^r) (4.9)

whereb^ris an open ball of radiusr.

We can now state the main result of this chapter in the form of two propositions.

Both of these assume that we are dealing with permissible solids that are r–

open andr–closed. The first of these propositions is about the medial surfaces M(S) andM(Sⁱ).

Proposition 4.9 Given a permissible solid S that is r–open and r–closed, the shortest distance from any point p ∈ ∂S to any point belonging to M(S) or M(Sⁱ)isr.

b

b

0

b

p

S

Figure 4.4: Illustration of the proof that the medial axis is always at a distance of at leastr from∂S.

Proof: Due to symmetry, we only need to show that this holds for one of the medial surfaces. We chooseM(S) and give a proof by contradiction. Let there

be given a closed ball of radiusx < r at a pointp0,b^x_p0 ⊂S. Assume that this ball is maximal and touches ∂S at a point p. Because S is r–closed, there is an exterior ballb^r_p1 ⊆Sⁱ which also touchesp. Becauseb^r_p1 ⊆Sⁱ and b^x_p0 ⊂S they can only share points belonging to the boundary which means they must be tangent. Consequently, they share the tangent plane of∂S atp. BecauseS is r–open there is also a closed ball of radius r, b^r_p2 ⊆S, containing p. By a similar argument,b^r_p2shares the aforementioned tangent plane. However, ifb^r_p2 andb^x_p0 are tangent and both on the same side of∂S it follows thatb^x_p0 ⊂b^r_p2. This contradicts the assumption thatb^x_p0 is maximal.

Hence, no centre of a maximal ball is closer to∂Sthanr. LetCdenote the union of all centres of maximal balls. The arguments above imply that C ⊆ S⊖b^r since the erosion ofS byb^r includes all points that are at least a distance ofr from∂S.

We know thatS⊖b^ris a closed set⁵, and thatM(S) =C, i.e. M(S) isCplus the limit points ofC. Using (A.16), it now follows thatM(S) =C⊆S⊖b^r. In other words,M(S), is fully contained inS⊖b^rany point of which is at least a distance ofrfrom ∂S2

The next proposition regards maximum curvature:

Proposition 4.10 Given a permissible solidS that isr–open andr–closed and a point qso that |dS(q)|=σ where0≤σ < r

Kmax(q)≤ 1

r− |σ| (4.10)

if Kmax is defined at q.

Proof Again due to symmetry, only points inS need to be discussed. Letp be the point on∂S closest toq. Clearly,σ=kp−qk. Due to the fact thatS isr–open andr–closed, there are two tangent, closed balls of radiusrtouching pfrom either side of the boundary. Let these be b^r_p0 ⊆S and b^r_p1 ⊆Sⁱ. The configuration is shown in Figure 4.5.

Now, letb^r+σp1 be a sphere of radiusr+σwhich has the same centre asb^r_p1, and letb^rp⁻0^σ be a sphere of radiusr−σwith the same centre asb^r_p0.

For any point x in the interior of b^rp⁻0^σ, it is clear that d(x, ∂b^r_p0) > σ and

5Proposition III–27 in [151] states among other things that a closed set eroded by a compact set is a closed set.

4.4 Openness and Closedness 73

p q

b^rp₀

b^rp₁

b^r+σp₁

b^r-σp₀

p p1

0

S

Figure 4.5: Translated instances ofb^r touchqfrom either side.

consequently, thatd(x, ∂S)> σ. Hence,dS(x)<−σ. Similarly, for any point x∈b^r+σ_p1 ,d(x, ∂b^r_p1)< σ. Thus,d(x, ∂S)< σ and, therefore,dS(x)>−σ.

This means that any pointxnearqand on the same−σ–isosurface must lie on the boundary or between the two closed ballsb^r+σp1 andb^rp⁻0^σ.

Let there be given a normal section,α, which is formed as the intersection of the

−σ–isosurface and a plane containingqand the normal to the−σ–isosurface at q. The centre of the osculating circle to αat q lies in the line defined by the surface normal [80]. Since the closed balls b^rp⁻0^σ and b^r+σp1 share the tangent plane and, consequently, the normal (direction) of the −σ–isosurface at q, the intersection of the normal plane and these balls are circles of the same radiir−σ andr+σ. Because the osculating circle is defined as the limit position of three points that (as we know from the above) must be on the boundary or outside b^rp⁻0^σ and b^r+σp1 , we conclude that the minimum radius of the osculating circle to a normal section is r−σ. Finally, the radius of the osculating circle is the inverse of the curvature ofαatq. Consequently, the greatest normal curvature is _r−|¹_σ| 2

It should be noted that we are not guaranteed that the maximum curvature is defined at a given point. The signed distance function is only known to be C¹

and the principal curvatures are computed from the second order derivatives of the signed distance function. However, the continuity of the distance function reflects the continuity of the surface in the sense that it can be shown that for a C^k hypersurface where k ≥ 2, the signed distance function is C^k in a neighbourhood of the surface [94]. Since we are typically interested in solids whose surfaces are piecewise C^k where k ≥2 (e.g. a cube with filleted edges and corners) it is likely that the distance function is locally at least C² and hence that the maximum curvature is continuous in most of the volume and discontinuous only at points where the closest surface component changes.