2.4 Properties of the Volume of the Coverage of Multiple Cameras
2.4.1 Continuity and Di ff erentiability
In two dimensions, the authors of [58] prove that the volume of the field of view of one camera with unlimited opening angle in a (two-dimensional) polygonal environment is locally Lipschitz. Further-more, the authors prove that the non-differentiable points of the volume lie on a set of lines in the polygon. When changing the camera position, a point within these lines is a camera position where two vertices of the field of view coincide. Neither a limited opening angle, nor several cameras, nor the three-dimensional case, nor a 3DBGS method have been considered in their publication, which will be covered in the following paragraphs.
With Equation (2.7), the volume is a polynomial of the components of the vertices. So, as long as these components are continuous or differentiable, the property is inherited by the volume. But the components of the vertices are neither always differentiable nor continuous. In this section, the continuity of vertices and faces is investigated. A visual eventis a point of network parameters at which a vertex or face is either not differentiable or not continuous.
Let V ∈ Cx(k,S) be such a vertex of the coverage with a fixedS ⊂ S, threshold k ∈ N, and sensor network parametersx=(a1, . . . ,aN)∈P1×. . .×PN. It is an intersection of at least three faces classified in Section 2.3.3. A face is defined by a plane (two-dimensional affine subspace) and by its boundary (half-spaces), cf. Definition 2.1.1. When adjusting the parameters of a network the vertex moves and the planes and boundaries could change. Thus, visual events can be caused by the following issues:
Visual event 1 The dimension of them-dimensional affine subspace assigned to the face (or the bound-ary) is reduced or increased or the subspace undefined.
Visual event 2 The intersection of the faces is outside the boundaries of one/several face(s).
Furthermore, the two-dimensional affine subspace of a face is determined by a normal ni ∈ ∂B31(0) as well as the distance to the origindi ∈Rof the facei=1,2,3. SinceVis an intersection of these planes
of the normals of the three planes is regular and withd=
d1 d2 d3T
the equation systemN · V=d holds. With the rule of Cramer the components of the vertexVare rational functions ofni,i = 1,2,3 and the distances di. The numerator is a polynomial of the parameters and the denominator is detN. Thus, the components of the vertex are inC∞ as long as the above matrix N remains regular and the components ofN anddare inC∞. The components of the normal of the faces are not inC∞if and only if Visual event 1 occurs. The additional visual event is the following:
Visual event 3 The normals of the faces are linearly dependent.
The repositioning of a camera is illustrated in Figure 2.18. The images depict examples for the Visual events 1–3: The top, left pictogram shows a projection face which vanishes after the movement of the camera. The image to the right in the same row displays the rotation of the camera to a point where the normal of an opening face and the normal of an environmental face are linearly dependent. Both pictures below depict a vertex which meets the boundary of an environmental face FE. In the left image the boundary is a vertex defined by a projection faceFPandFE. In the right image the boundary is defined by an opening faceFOandFE.
In the following paragraphs, it is shown that Visual event 1 only occurs if one of the vertices meets another face of the coverage or if one of the positions of the cameras are in a subspace assigned to an
2.4. PROPERTIES OF THE VOLUME OF THE COVERAGE OF MULTIPLE CAMERAS 57
anchor defined in Definition 2.1.1. In order to show this, assume (2.1) for all cameras and define the following set:
K:=n
A× ∂B31(0)
×Pπ(2)× · · · ×Pπ(N) |A⊂Rnsubspace assigned to an anchorE ⊂ ∂E, π∈SN
o
(2.10) Its elements are the subsets of network parameters where at least one camera position is in a 1-dimensional affine subspace of an anchor of the environment.
A σ
−1
(E,a)({detectable})
A σ
−1
(E,a)({detectable})
A σ
−1(E,a)({detectable})
FP FE
A
σ− (E,a1)({
detect able})
FE
FO
Figure 2.18: Illustration of Visual events 1–3 with the field of view (yellow) of one single camera (green) in 2D; The movement (green arrow) of the camera changes the faces (magenta arrows or crosses) of the camera coverage σ−1(E,a)({detectable}) (yellow) and thereby changes a vertex (red dot/arrow, blue cross). The orange line and dot illustrate the critical position or orientation of the camera where the visual events arise. Top left: Event 1; Top, right: Event 3; Bottom: Event 2 with an environmental face FE and differing boundaries toFE. The boundary is a vertex defined by a projection faceFP andFE (bottom, left), and defined by an opening faceFOandFE (bottom, right).
The sets Kof Equation (2.10) andI of Definition 2.3.8 are used in both the next lemmas. The latter addresses Visual event 3.
Lemma 2.4.1
Let E be an environment. Let N,k∈Nwith k≤N and S ⊂S. LetIbe as in Definition 2.3.8 andKas in
Equation(2.10). Let the following set be connected: coverage of one camera (Equation (2.8)), w.l.o.g. camera with parameters (p,o)∈E×∂B31(0), or it could be a constant face (Fa).
The affine subspace of constant faces and opening faces always exists, thus, only the existence of the following faces of the coverage is critical:
(FP): The two-dimensional affine subspace assigned to F is completely defined by p and an anchor E ⊂ ∂Eof the environment, cf. Lemma 2.3.7, unless the position is in the subspaceAof the edge.
This case is excluded, however, sinceA× ∂B31(0)∈K.
(FS): The affine subspace is defined bypand an anchorE ⊂ ∂Eof the dynamic objects of the environ-ment, cf. Lemma 2.3.7. This is the same case as (FP), however: The face is replaced by another silhouette face if the position is in the subspace of either one of the polygons adjacent to theE, which are the vertex incidence surfaces of Table 2.3 type CPP.
The normal and distance to the origin must be inC∞:
(FP) and (FS): The normal of a plane can be calculated by three points in the affine subspace: Let the linearly independent pointsz1,z2,z3∈ F be ordered counterclockwise when viewed from a point inside the environment from whichz1,z2,z3are visible. The normalnof a hyperplane satisfies
ZT·n=d·1 withZ =
z1 z2 z3
and1T =
1 1 1
We choose the position of the camera pand two points of the anchor as the pointszr,r = 1,2,3.
With the rule of Cramer, the components of the normals are inC∞ unless detZT = 0, which are the above two cases.
(FO): The frustum of a camera is defined as an intersection of two theta-spaces, again defined by the position p, directiond, openingv, and opening angleθ. Its boundary is{x ∈R3 | ](ρ(d,v)(x),d) =
±2θ}. The normaln∈ ∂B31(0) ofF lies in the subspace ofdandvwithn=λdd+λvvand its angle todis known. Unlessθ=πin which case the normal holdsn:=d, it can be constructed with the ratio tan(π−θ2 )=±λλv
d.
In the first case, the components of the normals are inC∞unless a vertex meets an affine subspace of an anchor or a vertex incidence surface. In the second case the normals are inC∞. Additionally, the distance to the origindholdsd= zT ·nfor any pointz∈ F, w.l.o.g. this can be the position of the camerap, cf.
Lemma 2.3.7. This settles the proof.
2.4. PROPERTIES OF THE VOLUME OF THE COVERAGE OF MULTIPLE CAMERAS 59
We have seen that the normals of the faces are continuously differentiable and the affine subspaces of the faces will keep their dimensionality unless a vertex incidence surface or an affine subspace of an anchor is met. In the following lemma, the normals of the faces are proven to be linearly independent under similar conditions:
Lemma 2.4.2
Let E be a bounded environment. Let the variables be declared as in Lemma 2.4.1. Let x ∈ D. Let Cx(k,S) be the k-reliable coverage of Section 2.1. Let Vbe a vertex of Cx(k,S). LetN be as in of Equation(2.9)for the normals of the vertexV.
Then,N is regular for all x∈D.
Proof. Network parametersx ∈ Dexists whereVis a vertex ofCx(k,S). At this point,N = N(x) is regular and constitutes the case c.ii) in Figure 2.19. The matrix of any three normalsn1,n2,n3is singular if the normals can be linear combined, which is true if the planes are arranged like the illustrations a), b), c.i), or c.iii). We will give an intuitive reason why the latter cases cannot emerge from case c.ii) with the continuity of the normals (Lemma 2.4.1) inD:
First, we motivate that V meets an additional face before the cases c.i) and c.iii) emerge, since the polyhedral area is bounded byE: In general, all edges of a bounded polyhedral area have two vertices as boundary points,VandV. Since the normals are continuous with the network parameters, c.i) emerges from c.ii) if the edge [V,V] between two planes (intersection of two planes) levels up with the third plane. When leveling up, theVmeets another face of the coverage which can only happen if xis in a vertex incidence surface. These have been excluded fromD.
The three edgesE1, ...,E3in c.iii) as well as the two edges in b) are parallel with a distance bigger than 0. The three planes in a) are also distanced by a scalar bigger than 0. Before a),b), and c.iii) can emerge from c.ii) the edges/planes need to open up. Thereby, the vertexV=E1∩ · · · ∩ E3slides to the boundary of the edges since these are bounded (since the environment is bounded). The vertex therefore meets another vertex defined by some of the original faces and at least one additional face (or else it would be the same vertex). But again the parameter vector of the network is in a vertex incidence surface and holds x<D.
a) b) c.i)
c.ii) c.iii)
Figure 2.19: Illustration of the arrangement of the non-coinciding affine subspaces of three faces that define a vertex. They can be distinguished by how many planes are parallel, first: a) All planes are parallel (first picture); b) Two planes are parallel (second picture), the third intersects both (if it intersects one it intersects a parallel one too); and c) No two planes are parallel, instead all planes intersect. We know that two planes intersect in a line. The intersection lines of three planes can then c.i) be identical, c.ii) meet in one point or c.iii) be parallel (the intersecting lines cannot be skew since they are situated at the same three hyperplanes). Picture from [146].
With these lemmas, the way is cleared for the final result of the observations of polyhedral areas and their vertices. The volume of the fused coverage is continuously differentiable except for a small set of network parameters, since the vertices are continuously differentiableµ-almost everywhere:
Theorem 2.4.3
Let the variables be declared as in Lemma 2.4.2. The components of the vertex are in C∞ µ-almost everywhere. In particular, they are in C∞ for all x∈D.
Proof. The components of the vertex are inC∞ forx ∈Dsince: Lemma 2.4.2 and 2.4.1 show the non-existence of the Visual events 1 and 3 if x∈D. As a last condition, the vertex must not pass through the boundaries of the face, an edge of the fused coverage. This edge is an intersection of two faces of the coverage. When a vertex meets such an edge, it also meets an additional face, which can only happen if x<D. A set inKcan be denoted as
A× ∂B31(0)
×Pπ(2)× · · · ×Pπ(N)with a suitable affine subspace of an anchorA⊂Rnand permutationπ∈SN, which is a null set inPπ(1)×Pπ(2)× · · · ×Pπ(N). With Theorem 2.3.9 it is known that the vertex incidence surfaces are null sets.
Thus, the volume of the coverage is continuously differentiableµ-almost everywhere. The continuity is relevant for the solver that is proposed in the next chapter. The solver also has to be chosen by the number of local optima of the objective function. More than one local optimum of the volume of the fused coverage is shown in the next section.