Time-dependent Affine Triangulation of Spatio-temporal Data

Transcription

1 Time-dependent Affine Triangulation of Spatio-temporal Data ABSTRACT Sofie Haesevoets Limburgs Universitair Centrum Department WNI 3590 Diepenbeek, Belgium In the geometric data model [6], spatio-temporal data are modelled as a finite collection of triangles that are transformed by time-dependent affinities. To facilitate querying and animation of spatio-temporal data, we present a normal form for data in the geometric data model. We propose an algorithm for constructing this normal form via a spatiotemporal triangulation of geometric data objects. This algorithm generates new geometric objects that form a partition both in space and in time. A particular property of the proposed partition is that it is invariant under time-dependent affine transformations, and hence independent of the coordinate system chosen when modelling the spatio-temporal data. We can show that our algorithm works correctly and has a polynomial time complexity (in the number of input triangles and the maximal degree of the transformation functions). We also discuss several possible applications of this spatio-temporal triangulation. Categories and Subject Descriptors H.2.1 [Database Management]: Logical Design; H.2.8 [Database Management]: Database Applications Spatial databases and GIS General Terms Algorithms, Theory, Design Keywords Spatio-temporal data models, Triangulations 1. INTRODUCTION AND SUMMARY At this moment, spatial databases are a well-established area of research. Since most natural and man-made phenomena have a temporal as well as a spatial extent a lot of attention has also been paid, during the last decade, to modelling and querying spatio-temporal data [3, 5, 6, 10, Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. GIS 04, November 12 13, 2004, Washington, DC, USA. Copyright 2004 ACM /04/ $5.00. Bart Kuijpers Limburgs Universitair Centrum Department WNI 3590 Diepenbeek, Belgium bart.kuijpers@luc.ac.be 12, 13, 17, 20]. Several data models for representing spatio-temporal data have been proposed already. In this article, we adopt the geometric data model that was introduced by Chomicki and Revesz [7] and later studied by them and the present authors [6]. In the geometric data model, spatio-temporal objects are finitely represented as geometric objects, which in turn are collections of atomic geometric objects. An atomic geometric object is given by its spatial reference object (which determines its shape), a time interval (which specifies its lifespan) and a transformation function (which determines the movement of the object during the time interval). This model is very natural and it would benefit from having a normal form. Atomic objects that constitute a geometric object do not need to have the same time domain and can have (partially) overlapping lifespans. Therefore, the querying of spatio-temporal objects or their animation requires a lot of computations that have to be performed at the time the result of a query or the animation are needed. We propose as a normal form a spatio-temporal triangulation that can be computed during a data preprocessing phase and that facilitates querying and animation in such a way that queries can be executed much more efficiently and require few additional computations The main reason for this is that in a triangulation the data is partitioned in space as well as in time. Hence, no objects overlap, thus reducing unnecessary computations. Our spatio-temporal triangulation is also invariant under time-dependent affinities. In the area of spatial and spatio-temporal database research, a lot of attention has been payed to affine invariance of both data and queries. The main purpose of studying affine invariance is to obtain methods and techniques that are not affected by affine transformations of the input spatial data. This means that a particular choice of origin or some artificial choice of unit of measure (e.g., inches versus centimeters) has no effect on the final result of the method, technique or query. Also in other areas, invariance under affinities is often relevant. In computer vision, the so-called weak perspective assumption [22] is widely adopted. When an object is repeatedly photographed under different camera angles, all the different images are assumed to be affine transformations of each other. This assumption led to the need for affineinvariant similarity measures between pairs of pictures [15, 16, 23]. In computer graphics, affine-invariant norms and triangulations have been studied [18]. In the field of spatial and spatio-temporal constraint databases [19, 21], affine-invariant query languages [11, 14] have been proposed.

2 The main contribution of this paper is a time-dependent affine triangulation algorithm that produces a unique and affine-invariant triangulation of a given geometric data object. We show that this algorithm runs in polynomial time in the size of the input, measured by the number of transformed triangles n in the input and by the maximal degree d of the transformation functions applied to them. The worstcase time complexity is of order n 11 d O(1) and the maximal number of atomic objects in the resulting triangulation is of order n 10 d 6. For static spatial data the time complexity of triangulating is of order n 4 and the number of returned triangles is of order n 3. We remark that such triangulations can also be computed via general purpose cell decomposition algorithms such as cylindrical algebraic decomposition [8] at a higher cost, however. Further, we also show some applications where the triangulation, when computed in a preprocessing stage, facilitates the computation of certain types of queries and operations. The outline of this paper is as follows. In Section 2, we explain the geometric data model and define spatial and spatio-temporal triangulations. We introduce an affine invariant spatial triangulation method in Section 3. Afterwards, in Section 4, we describe a novel affine-invariant triangulation of spatio-temporal data. We describe the algorithm in detail and give some properties. Then, we give some possible applications of the triangulation in Section 5 and we end with some concluding remarks in Section PRELIMINARIES AND DEFINITIONS We denote the set of real numbers by R and the twodimensional real space by R 2. We will use x and y (with or without subscripts) to denote spatial variables and t (with or without subscripts) to denote time variables. First, we give the definition of a spatial, a temporal and a spatio-temporal object. Next, we address the need of a normal form. Finally, we define affine triangulations of spatial and of spatio-temporal data. 2.1 Spatio-temporal Data in the Geometric Data Model In this section, we describe the geometric data model as introduced by Chomicki and Revesz [7, 6], in which geometric objects are finite collections of atomic (geometric) objects. But first, we define spatio-temporal data objects. Definition 1. A temporal object is a semi-algebraic 1 subset of R, a spatial object is a semi-algebraic subset of R 2 and a spatio-temporal object is a semi-algebraic subset of R 2 R. With the time domain of a spatio-temporal object, we mean its projection on the time axis. By well-known properties of semi-algebraic sets, this projection is a temporal object [2]. Example 1. The interval [0, 4] and the finite set {0, 1, 2, 3, 4} are examples of temporal objects. The unit circle in the plane is a spatial object, since it can be represented by the 1 A semi-algebraic set in R d is a Boolean combination of sets of the form {(x 1, x 2,..., x d ) R d p(x 1, x 2,..., x d ) > 0}, where p is a polynomial with integer coefficients in the real variables x 1, x 2,..., x d. Properties of semi-algebraic sets are well known [2]. -1 t=0 1 t=1 t=2 Figure 1: An example of a spatio-temporal object. polynomial inequalities (1 x 2 y 2 > 0) (x 2 +y 2 1 > 0), usually abbreviated by the formula x 2 + y 2 = 1. The set {(x, y; t) R 2 R x 2t x 2 + 2t y t 1 y t t 0 t 4} is a spatio-temporal object and it represents a translating rectangle during the time interval [0, 4]. At each moment t in this interval it has corner points (2t, t), (2 + 2t, t), (2 + 2t, t 1) and (2t, t 1), as illustrated in Figure 1. In the geometric data model [6, 7], spatio-temporal objects are finitely represented as geometric objects, which in turn are finite collections of atomic geometric objects. An atomic geometric object is given by its spatial reference object (which determines its shape), a time interval (which specifies its lifetime) and a transformation function (which determines the movement of the object during the time interval). Several classes of geometric objects were introduced, depending on the types of spatial reference objects and transformations [6, 7]. In this article, we consider spatio-temporal objects that can be represented as finite unions of triangles moved by time-dependent affine transformations (see also Definition 2). Other combinations have been studied [6] in which triangles, rectangles or polygons are transformed by time-dependent translations, scalings or affinities, that, in turn, are given by linear, polynomial or rational functions of time. The class of geometric objects that we consider is not only the most general of the classes that were previously studied, it is also one of the few classes that have the desirable property of being closed under the set operations union, intersection and difference [6]. In Section 4, we will rely on this closure property. Definition 2. An atomic geometric object O is a triple (S, I, f), where S R 2 is the spatial reference object of O, which is a (filled) triangle 2 ; I R is the time domain (a point or an interval) of O; and f : R 2 R R 2 is the transformation function of O, which is a time-dependent affinity of the form (x, y; t) a11(t) a12(t) a 21(t) a 22(t) t=3 t=4 x + y b1(t) b 2(t), where a ij(t) and b i(t) are rational functions of t (i.e., of the form p 1(t)/p 2(t), with p 1 and p 2 polynomials with rational coefficients in t) and the determinant of the matrix of the a ij s differs from zero for all t in I. 2 For technical reasons, we allow a triangle to degenerate into a line segment or a point.

3 are spatial objects that show what a spatio-temporal object looks like at a certain moment. t=0 t=1 t=2 t=3 t=4 t=5 Figure 2: A spatio-temporal object (a traffic sign) shown at six moments. We remark that this definition guarantees that there is a finite representation of atomic geometric objects by means of the polynomial constraint description of the time-interval, (the cornerpoints of) the reference triangle and the coefficients of the transformation matrices. An atomic geometric object O = (S, I, f) finitely represents the spatio-temporal object {(x, y; t) R 2 R t I ( x )( y ) ((x, y ) S (x, y) = f(x, y ; t))}, which we denote as st(o). Atomic geometric objects can be combined to more complex geometric objects. Definition 3. A geometric object is a set {O 1,..., O n} of atomic geometric objects. It represents the spatio-temporal object n i=1st(o i). By definition 3, the atomic geometric objects that compose a geometric object are allowed to overlap in time as well in space. This is a natural definition, but we will see in Section 2.2, that this flexibility in design leads to expensive computations when we want to query spatio-temporal objects represented this way. We define the time domain of a geometric object {O 1,..., O n} to be the smallest time interval that contains all the time intervals I i of the atomic geometric objects O i (this is the convex closure of these time intervals, denoted by n i=1 Ii). Remark that a spatio-temporal object is empty outside the time domain of the geometric object that defines it. Also, within the time domain, a spatio-temporal object is empty at any moment when no atomic object exists. Example 2. The spatio-temporal object of Example 1 can be represented by the geometric object {O 1, O 2}, where O 1 is represented by (S 1, [0, 4], f) and O 2 equals (S 2, [0, 4], f), with S 1 the triangle with corner points (0, 0), (2, 0), (2, 1), S 2 the triangle with corner points (0, 0), (0, 1), (2, 1), and f the transformation (x, y; t) (x + 2t + 2, y t 1). Example 3. Figure 2 shows a traffic sign at six moments (seen by an observer walking around it). This observation can be described by seven atomic geometric objects. During the interval [0, 3[, there exist three atomic objects, two triangles, and one line segment. At the time instant t = 3 there exists one atomic object that represents the shape of a line. During the interval ]3, 5] there exist three atomic objects, two triangles, and a line segment. To end this subsection, we define the snapshot of a spatio-temporal object at a certain moment in time. Snapshots Definition 4. Let O be an atomic object. Let τ 0 be a time moment in the time domain of O. The snapshot of O at time τ 0, denoted O τ 0, is the intersection of the spatio-temporal object st(o) with the plane in R 2 R defined by t = τ 0, i.e., the plane R 2 {τ 0}. Let {O 1,..., O n} be a geometric object. The snapshot of {O 1,..., O n} at time τ 0, denoted {O 1,..., O n} τ 0, is the union of the snapshots at τ 0 of all atomic objects that compose {O 1,..., O n}, i.e., n i=1o τ 0 i. As explained in Example 3, Figure 2 shows six snapshots of a geometric object representing a traffic sign seen by a moving observer. 2.2 The Benefits of a Normal Form As remarked after Definition 3, the atomic objects that compose a geometric object may overlap both in space as in time. As a consequence, it is impossible to answer some very basic queries about a geometric object without a lot of computations. To know the time domain of a geometric object, for example, one needs to check all atomic objects, sort the begin and end points of their time domains, and derive the union of all time domains. Also, there might be atomic objects that do not contribute at all to the shape of the geometric object as they are entirely overlapped by other atomic objects. These objects are taken along unnecessarily in computations. Furthermore, two geometric objects that represent the same spatio-temporal object can have a totally different representation by means of atomic objects. It is computationally expensive to derive from their different representations that they are the same. These drawbacks can only be solved by introducing a normal form for geometric objects, that makes their structure more transparent. This normal form should have the property that it is the same for all geometric objects that represent the same spatio-temporal object, independent of their initial representation by means of atomic objects. In this paper, we add the requirement that this normal form should be affine-invariant. If two geometric objects are the same up to an affine transformation, we also want their normal form representation to be the same up to this affine transformation. 2.3 Triangulations We end this section with some more definitions. We define the notions of spatial and spatio-temporal affine triangulation. Definition 5. Let {O 1,..., O n} be a geometric object and τ 0 be a time moment in the time domain of {O 1,..., O n}. A collection of triangles 3 {T 1,..., T m} in R 2 is a triangulation of the snapshot {O 1,..., O n} τ 0 if the interiors 4 of different T i are disjoint and the union m i=1t i equals {O 1,..., O n} τ 0. 3 Remark that we consider filled triangles and we allow a triangle to degenerate into a line segment or a point. 4 We define the interior as follows: the interior of a triangle is its topological interior; the interior of a line segment is the segment without endpoints; and the interior of a point is the point itself.

4 Figure 3: The triangulations of a snapshot (left) and of an affine transformation of the snapshot (right). A geometric object {T 1,..., T m} is a triangulation of a geometric object {O 1,..., O n} if for each τ 0 in the time domain of {O 1,..., O n}, {T 1,..., T m} τ 0 is a triangulation of the snapshot {O 1,..., O n} τ 0 and if st({o 1,..., O n}) = st({t 1,..., T m}). We remark that in the second part of Definition 5, at each moment τ 0, several of the T τ 0 i are allowed to be empty (i.e., τ 0 may be outside the time domain of T i). In Figure 3, two stars with their respective triangulations are shown. Triangulations, as defined here, are something like partitions of spatial objects into triangles. They are not partitions in the mathematical sense, as the elements of the partition may have common boundaries. For spatial data, it is customary to allow the elements of a partition to share boundaries (see for example [9]). A spatial triangulation method is a procedure that on input of a snapshot of a spatio-temporal object produces a triangulation of this snapshot. A spatio-temporal triangulation method is a procedure that on input a geometric object, produces a triangulation of this geometric object. Next, we define what it means for such methods to be affine-invariant. Definition 6. A spatial triangulation method T S is called affine invariant if and only if for any two snapshots A and B, for which there is an affinity α : R 2 R 2 such that α(a) = B, also α(t S(A)) = T S(B). A spatio-temporal triangulation method T ST is called affine invariant if and only if for any geometric objects {O 1,..., O n} and {O 1,..., O m} for which for each moment τ 0 in their time domains, there is an affinity α τ0 : R 2 R 2 such that if α τ0 ({O 1,..., O n} τ 0 ) = {O 1,..., O m} τ 0, also α τ0 (T ST ({O 1,..., O n}) τ 0 ) equals T ST ({O 1,..., O m}) τ AN AFFINE-INVARIANT SPATIAL TRIANGULATION METHOD We first propose an affine-invariant triangulation algorithm, called T S, that converts a snapshot of a geometric object into normal form. Furtheron, in Section 4, we will use this technique to construct an algorithm for converting geometric objects into normal form. We also give some properties of the output triangulation. The algorithm T S makes use of the fact that any set of (both ways unbounded) lines partitions the plane in a set of convex polygons and that this partition is invariant under affine transformations of the plane. Basically, the algorithm works as follows. Given a snapshot of a geometric object, we first compute its topological boundary and consider all lines that go through a segment of the boundary. As just remarked, these lines (also called the carriers of the snapshot) partition the plane in open convex polygons, open line segments and points. The points, as well as the closures (with endpoints) of the line segments that belong to the snapshot but are not adjacent to an open polygon are already returned as output. Next, the closures of the open polygons that belong to the given snapshot are triangulated by connecting their center of mass to each corner point. These triangles complete the output. Figure 3 shows the triangulation of a star-shaped snapshot. Remark that each bounded polygon is either part of at least one triangle of the original snapshot, or part of the convex closure of the original snapshot. Each line segment belongs to at most two polygons. In the algorithm, given below, we traverse each polygon in an a priori chosen direction such that each line segment is only traversed once in each direction. Algorithm 1 (or T S) gives the triangulation procedure more formally. The input of this triangulation algorithm is a snapshot A of a geometric object, which we assume to be given as a finite union of (possibly overlapping and possibly degenerated) triangles. To shorten and simplify the exposition, we assume that A is fully two-dimensional, or equivalently, that points and line segments that are not adjacent to a polygon belonging to A are already in the output (the description of the complete algorithm dealing with these cases becomes rather tedious since a lot of labelling cases have to be recorded). In Algorithm 1, we denote points in the plane by bold characters p, q,... and vectors between two points by pq, qr,.... The line segment between the points p and q will be represented pq. The unbounded line through the points p and q will be denoted by L p,q. We also refer to L p,q as the carrier of the line segment pq and of the vector pq. Algorithm 1. T S(input = A = { 1,..., k }; output = P = {T 1,..., T l }): /* The input and output are sets of triangles (representing a snapshot).*/ Step 1. Compute the boundary of A (e.g., using a planesweep algorithm [1]). Store all boundary line segments in a list B A. Step 2. Replace each set {p 1q 1, p 2q 2,...} of boundary segments in B A that are collinear by their convex closure. Step 3. Now we give each element pq of B A an orientation by replacing pq by θ(pq), where θ(pq) equals pq, if the oriented angle between (0, 0)(1, 0) and (0, 0)u, where u = q p, is not strictly negative 5 and is in the interval [0, π[, and θ(pq) equals qp otherwise. Step 4. Sort the vectors of B A by ascending positive angle with the vector (0, 0)(1, 0). /* In Step 5, we construct for each vector pq of B A a list points( pq) containing intersection points of pq with carri- 5 The orientation of the angle between the vectors pq and pr is the sign of the expression q 1r 2 q 2r 1 + p 2r 1 p 1r 2 + p 1q 2 p 2q 1. The function sign maps positive real numbers to +1, negative numbers to 1 and 0 to 0.

5 P Q b d a - h - c - i a - b g f - h - g - d e - i - d (A) (B) (C) (2) j - c - g - b b d (3) c 5 (1) i a i - c - h - a h 2 4 g (4) b - a 1 3 a c i f d - g - h - f h d - i - e 6 f e j e b - g - c - j j (D) (E) (F) (G) (H) Figure 4: The different steps of Algorithm 1 (A to G). Part H represents the list B A and, for each vector of B A (the vector labels correspond to those from D, where i is the inverse vector of i), the list of points on the line through that vector (the point labels correspond to those from E) ers of other vectors of B A together with their cross-ratios 6 relative to p and q.*/ Step 5. For every pair of vectors pq and rs of B A, compute the intersection point v of L pq and L rs. If v exists, compute the cross-ratios c 1 = Cr(p, v, q) and c 2 = Cr(r, v, s). If either 0 c 1 1 or 0 c 2 1, then add (v, c 1) to points( pq) and (v, c 2) to points( rs), such that the elements of points( pq) and points( rs) are sorted by increasing cross-ratio. Step 6. Traverse the list B A. For each element pq of B A, append a new element containing the vector qp to the end of B A. The list points( qp) consists of all elements of points( pq), but sorted in reverse order. /* In Step 7, we now define making one pass through B A, starting from the vector pq and the point v as follows: start with the element (v, c) in points( pq) and visit all later elements of points( pq), if there are any. Then, for all further elements rs of B A, visit all points in points( rs). If the end of B A is reached, restart at the beginning of B A, until the element pq is reached again.*/ Step 7. We now construct a set of cycles of vectors P as follows. For each element pq in B A and for each point p j of the list points( pq), make one pass through B A, starting from pq and p j and try to find a cycle of vectors p jq 0, q 0q 1,..., q l p j, where l 2. Each such vector q vq w consists of a pair of consecutive points in some list points( rs) and is the last unmarked line segment starting with q v and different from q vq v 1. If such a cycle of vectors is found, add it to P, and, in B A, place a mark on the first point of each vector in the cycle. Step 8. For each element Q of P, check if the polygon represented by Q is a part of A. If not 7, remove Q from P. /* In the last step, we output a set of triangles (which is the normalized representation of the input snapshot).*/ Step 9. For each element Q = ( p 0p 1, p 1p 2,..., p k p 0) of P do the following: If k = 2, then output (p 0, p 1, p 2), else compute the center of mass c = 1 k k+1 i=0 pi of the points 6 The cross-ratio Cr(p, q, r) of three collinear points p, q, r is pq, where pq is the length of the vector pq. 7 pr This step removes all polygons that do not belong to the original snapshot, but to its convex closure. p 0, p 1,..., p k and output all (p i, c, p (i+1)mod(k+1) ), for 1 i k. In the following, we will denote the result of applying algorithm T S to a snapshot A by T S(A). We give the following result without proof. Property 1. Let A = { 1,..., n} be a snapshot. Algorithm 1 returns on input A, a triangulation of A (in the sense of Definition 5). In the worst case, Algorithm 1 computes the triangulation of A in time of order n 4 and produces a triangulation of A consisting of order n 3 triangles. We remark that the worst-case number of triangles of order n 3 can actually be reached in examples. Example 4. Figure 4 illustrates the steps 1 7 of Algorithm 1 on an example snapshot, consisting of two triangles labelled P and Q (see (A) in Figure 4). First, the boundary of the union of P and Q computed (see (B)), and collinear segments are merged (see (C)). This corresponds to Step 1 and Step 2 of Algorithm 1. Then, the segments are oriented and sorted. The orientation is indicated in (D), and the labels of the vectors indicate their order in the list B A (see also the first five rows of (H)). In (E) the retained intersection points are labelled, and the first five rows of (H) show for each vector the list of points on the line through it. In Step 6 of the algorithm, the inverse vectors are added, this is shown in (H). Part (G) of the figure illustrates Step 7. For the vector hc, all adjacent vectors are indicated with dotted arrows. The numbers between brackets denote the order in which they are found in the list B A. The last vector before ch is chosen. This is cg, which is indeed the next edge of the polygon hc belongs to, if traversed in counterclockwise order. This triangulation is affine invariant, and produces the same output for all snapshots that represent the same set in R 2. We give the following property without proof. Property 2. Let A and B be snapshots given as sets of triangles such that there exists an affinity α : R 2 R 2 for which B = α(a). Then, for each triangle T of T S(A), it holds that the triangle α(t ) belongs to T S(B). We now define what is means for a snapshot to be in T S- normal form.

6 Definition 7. Let A be a snapshot given as the set of triangles { 1,..., n}. We say that A (or more precisely, this set of triangles) is in T S-normal form if T S on input A returns { 1,..., n} as output. In the next section, we will use the affine triangulation for snapshots to construct a triangulation of geometric objects. 4. AN AFFINE-INVARIANT SPATIO-TEM- PORAL TRIANGULATION METHOD In this section, we present an algorithm to convert geometric objects into normal form. We will adapt the technique of Algorithm 1 for time-dependent data. We start this section by defining isomorphic triangulations. Then we give an algorithm for computing an affineinvariant triangulation of geometric objects. We illustrate the algorithm with an example and end with some properties of the triangulation. Definition 8. Let S 1 and S 2 be two snapshots. We say that S 1 and S 2 are T S-isomorphic, denoted S 1 TS S 2, if there exists a bijective mapping h : R 2 R 2 with the following property: A triangle = (p 1, p 2, p 3) of T S(S 1) is bounded by 12, 23 and 31 (where each i,(i+1)mod 3 is either a triangle that shares the segment p ip (i+1)mod 3 with or is ɛ, which means that no triangle shares that boundary with ) if and only if, the triangle h( ) = (h(p 1), h(p 2), h(p 3)) belongs to T S(S 2) and is bounded by h( 12), h( 23) and h( 31), where each h( i,(i+1)mod 3 ) is either a triangle that shares the line segment h(p i)h(p (i+1)mod 3 ) with h( ) or is ɛ if i,(i+1)mod 3 equals ɛ. Example 5. The triangulations shown in Figure 3 are T S- isomorphic to each other. In Figure 2, all snapshots shown except the one at time moment t = 3 are T S-isomorphic. The snapshot at time moment t = 3 is clearly not isomorphic to the others, since it consists only of one line segment. Now, we introduce an algorithm T ST that constructs a time-dependent affine triangulation of spatio-temporal objects that are represented by geometric objects. Intuitively, this algorithm works as follows. The input is a geometric object O = {O 1,..., O n}. The algorithm first partitions the time domain I = n i=1ii of O into points and open intervals until each open interval is small enough such that all its snapshots are T S-isomorphic to each other. This partitioning is obtained by assuring that within one interval (or point), the positively oriented boundary segments have the same order when sorted by angle, and all intersection points exist and have the same order along the carriers they belong to (here, we use the terminology introduced in Algorithm 1). After the partition of the time domain is computed, the snapshot in the middle of each open interval, and at each time moment, is computed and triangulated using the spatial triangulation method T S given in Algorithm 1. The triangles returned by this algorithm will be reference objects for the atomic objects, returned by the spatio-temporal triangulation algorithm T ST. These atomic objects exist during the interval or moment under consideration. While computing the triangulation on the snapshot in the middle of an interval, using Algorithm 1, all actions on the structure B A are copied on a time-dependent structure similar to B A. This way, we know the functions that describe the movement of the corner points of the triangles that are returned by the spatial triangulation. Knowing these functions, together with the time interval and the reference object, we can deduce the transformation and construct atomic objects. Finally, for each point in the time partition, the snapshot of O at that time moment is computed and triangulated using Algorithm 1. After that, we merge again subsequent elements of the time partition that have an unnecessary partitioning, caused by atomic objects of O that do not influence the shape of the boundary of the spatio-temporal object. We decide whether two elements of the time partition can be merged, by testing whether their snapshot (in case of a time instant) or the snapshot at their middle point (in case of a time interval) are T S-isomorphic, and whether the atomic objects move further continuously across the intervals. This last step guarantees that the atomic objects live maximally and that the produced triangulation is the same for geometric objects that represent the same spatio-temporal object. In Algorithm 2, we will denote time-dependent points (p x(t), p y(t)), (q x(t), q y(t)),... in the plane by hatted bold ˆp, ˆq,... and we adapt all other notations of Algorithm 1 in the same way. Algorithm 2. T ST (input = O = {O 1,..., O n}; output = P = {T 1,..., T l }): /* The input and output are sets of atomic objects. */ Step 1. We create a list T of time moments and a list Seg with nodes of the form (ˆpˆq, t a, t b ). For each of the atomic objects O i = (S i, I i, f i), with 1 i n, add the endpoints t a and t b of I i to the list T and add three elements to Seg, each containing one of the three time-dependent boundary segments of O i and t a and t b (the moving boundary segments of the atomic geometric object O i = (S i, I i, f i) can be found by applying f i to the corner points of S i). Sort the list T according to <. Step 2. Let T = (τ 0, τ 1,..., τ k ). For each of the open intervals J j =]τ j, τ j+1[ (0 j k 1), we construct a list T j of time instants as follows. For each line segment ˆpˆq appearing in Seg, compute the function fˆpˆq (t) expressing the smallest angle between Lˆp,ˆq and L (0,0)(1,0). Now compute all time moments where two or more of the functions fˆpˆq (t) intersect. Add those time moments to T j. Add all points of the lists T j(0 j k 1) to T in a sorted way. /* In Step 3, we associate to each element (ˆpˆq, t a, t b ) of Seg a list MPoints(ˆpˆq) with nodes of the form (c(t), ˆv, ˆrŝ, K), where K is a list of time intervals.*/ Step 3. Let T = (τ 0, τ 1,..., τ k ). For each of the open intervals J j =]τ j, τ j + 1[ (0 j k 1), we construct a list T j of time instants as follows: for each pair of distinct elements (ˆpˆq, t a, t b ) and (ˆrŝ, t c, t d ) of Seg such that J j ]t a, t b [ and J j ]t c, t d [ do: Compute the time moments for which the (time-dependent) intersection ˆv between the two moving line segments exists and either 0 c pq(t) = Cr(ˆp, ˆv, ˆq) 1 or 0 c rs(t) = Cr(ˆr, ˆv, ŝ) To compute the cross-ratios, an orientation has to be chosen for the segments. This choice does not affect the further course of the algorithm, as 0 Cr(ˆp, ˆv, ˆq) 1 if and only if 0 Cr(ˆq, ˆv, ˆp) 1.

7 (A) (B) (C) (D) (E) (F) (G) (H) (I) Figure 5: The triangulations of the objects of Example 6 at time moments t = 0 (A), t = 1 (B), t = 1 (C), t = (D), t = 2 (E), t = 5 (F), t = 3 (G), t = 7 (H) and t = 4 (I). 2 2 If such time moments exist, add all intervals describing them to a list K pqrs and their end points to the list T j. Add the node (c pq(t), ˆv,ˆrŝ, K pqrs) to MPoints(ˆpˆq) and (c rs(t), ˆv, ˆpˆq, K pqrs) to MPoints(ˆrŝ). Add all points of the lists T j(0 j k 1) to T in a sorted way. Step 4. Let T = (τ 0, τ 1,..., τ k ). For each of the open intervals J j =]τ j, τ j + 1[ (0 j k 1), we construct a list T j of time instants as follows: for each element (ˆpˆq, t a, t b ) of Seg such that J j ]t a, t b [ do: Consider in the list MPoints(ˆpˆq) all elements (c i(t), ˆv i,ˆr iŝ i, K i) for which J j is contained in one of the elements of K i. Compute the time moments for which two or more of the functions c i(t) of those elements intersect (i.e. the intersection points change order ). Add those time moments to T j. Add all points of the lists T j (0 j k 1) to T in a sorted way. /* Now, we start constructing the output objects. They are collected in lists P I, where I either is an element of T or the open interval between two elements of T.*/ Step 5. Let T = (τ 0, τ 1,..., τ k ). For each of the J j = ]τ j, τ j + 1[ (0 j k 1): Compute the middle point of J j, we denote it by τ j. Compute the snapshot of O at time moment τ j. If O τ j is not empty, compute the boundary of O τ j (as in Step 1 of Algorithm 1). Store the boundary segments of O τ j in the list B τ j. If O τ j is empty, skip the next items and proceed with the next interval. Construct a copy Seg τ j of Seg only containing those elements (ˆpˆq, t a, t b ) for which J j ]t a, t b [ and the value of ˆpˆq at time moment τ j contains one of the segments of B τ j. For each element (ˆpˆq, t a, t b ) of Seg that is copied to Seg τ j, only those elements of MPoints(ˆpˆq) that contain an intersection point existing during J j are copied. Sort all intersection points by taking the value of their cross-ratio at time moment τ j. From now, compute the triangulation of O τ j using Algorithm 1, starting from Step 2, but copy each move in the structure B τ j on the list Seg τ j. Each operation on a line segment in B τ j is also performed on the corresponding line segment in Seg τ j. For each polygon that is returned after Step 8 of Algorithm 1, the formulas describing its moving corner points are extracted from Seg τ j. The formula for the time-dependent center of mass is computed from these corner points. For each triangle (p 1, p 2, p 3) in T S(O τ j ), and its corresponding moving corner points (ˆp 1, ˆp 2, ˆp 3 ) computed from Seg τ j, construct the atomic object O p1 p 2 p 3 = ((p 1, p 2, p 3), J j, f). Here, f is the unique transformation 9 that maps, for each time moment τ of J j, p i to the value of ˆp i at time moment τ. Add O p1 p 2 p 3 to P Jj. Step 6. Let T = (τ 0, τ 1,..., τ k ). For each time moment τ m in T, compute the snapshot of O τm. If O τm is not empty, triangulate it using Algorithm 1. For each triangle (p 1, p 2, p 3) in T S(O τm ), add the atomic object O p1 p 2 p 3 = ((p 1, p 2, p 3), {τ m}, Id) to P τm (Id is the identity transformation). Step 7. Let L be the sorted list that is the union of the list T = (τ 0, τ 1,..., τ k ) and all open intervals J j =]τ j, τ j + 1[ (0 j k 1). Repeat until no more changes can be made: For each pair of subsequent elements l I and l J of L, check whether they can be merged. We describe the case where l I = τ i and l J =]τ i, τ j[, the other possibilities can be dealt with in an analog way. We test whether the triangulation of the snapshot at moment τ i and the triangulation of the snapshot at the midpoint τ m of l J are T S-isomorphic. If they are, there exists a bijective mapping h between the triangles from both triangulations. For each pair of triangles T 1 and T 2 = h(t 1), being the reference object of the atomic objects (T 1, l I, f 1), respectively (T 2, I J, f 2), verify that f 2 is welldefined on l I and that T 1 equals (f 2(τ i))(t 2). If this is the case, replace l J by l I l J in L. For each atomic object in the spatio-temporal triangulation, existing during l J, replace its time interval by l I l J. Remove all atomic objects with time domain l I. /* Now output a set of atomic objects (providing the normalized representation of the input geometric object).*/ Step 8. Let L be the sorted list that is the union of the list T = (τ 0, τ 1,..., τ k ) and all open intervals J j =]τ j, τ j + 1[ (0 j k 1). For each element l of L, return P l. As before, we denote the output of algorithm T ST applied to input O by T ST (O). Example 6. Let O = {O 1, O 2} be a geometric object, where O 1 is given as ((( 1, 0), (1, 0), (0, 2)), [0, 4], Id) and 9 It is shown in [6] that such a unique transformation exists and the formula computing this transformation is given.

8 O 2 is given as ((( 3, 1), ( 1, 1), ( 2, 3)), [0, 4], f), where f : (x, y, t) (x + t, y). All atomic objects exist during the interval [0, 4], and the segments do not change their direction, so Step 1 through Step 4 of Algorithm 2 do not change anything. After Step 7, the list T equals (0, 1, 3, 5, 7, 4). As we can merge the objects for time moment t = 0 and t ]0, 1 [ and also for t 2 ] 7, 4[ and t = 4, the resulting spatio-temporal triangulation 2 partitions the interval [0, 4] in nine parts. The triangulation of each of these parts (for the intervals the snapshot at their midpoint is taken) is shown in Figure 5. We now list a few properties of Algorithm 2. Property 3. Algorithm 2 is affine-invariant spatio-temporal triangulation method. Recall that a spatio-temporal triangulation method is affine-invariant if it is invariant under time-varying affinities of the plane (see Definition 6). Of course, it is also invariant then under constant affinities: Corollary 1. Let O = {O 1,..., O n} and O = {O 1,..., O m} be two geometric objects such that there is an affinity α : R 2 R 2 such that, for each moment τ 0 in their time domains it holds that α(o τ 0 ) = O τ 0. Then, for each atomic element (S, I, f) of T ST (O), it holds that the element (α(s), I, f) belongs to T ST (O ). This shows that the partition is independent of the coordinate system used to represent the spatio-temporal object. The affine partitions of two spatio-temporal objects that are affine images of each other only differ in the coordinates of the spatial reference objects of the atomic objects. We now define what is means for a geometric object to be in T ST -normal form. Definition 9. Let O be a geometric object given as the set of atomic objects {O 1,..., O n}. We say that O (or more precisely, this set of atomic objects) is in T ST normal form, if T ST on input O returns {O 1,..., O n} as output. To end this section, we give, without proof, the complexity of computing the affine triangulation of geometric objects. Property 4. Let O = {O 1,..., O n} be a geometric object. If O = {O 1,..., O n} is the union of n atomic geometric objects, such that the maximum degree of the polynomials describing their transformation function is d, then the number of atomic objects in the triangulation of O, computed using Algorithm 2, is of order n 10 d 6. The complexity of computing the triangulation of O is of order n 11 d O(1). The dependency on the degree of the transformation functions follows from the fact that the complexity of computing the zeros of a polynomial depends on its degree. Note that in practice, however, the degree of the functions describing spatio-temporal objects is mostly low. In [4], for example, objects are assumed to move only by linear functions of time. Remark also that, in practice, either most of the original objects have the same time domain, making the number of intervals in the partition very small, or all different time domains, which greatly reduces the number of objects existing during each interval. So, in practice, the performance will be better than the worst case suggests. 5. APPLICATIONS We now describe some applications that we believe can benefit from the triangulation described in Algorithm 2. We first say what we mean by a spatio-temporal database. Definition 10. A spatio-temporal database is a set of geometric objects. For this section, we assume that each atomic object is labelled with the id of the geometric object it belongs to. 5.1 Efficient Rendering of Objects When a geometric object that is not in normal form has to be displayed to the user, there are two tasks to perform. First, the snapshots of the geometric object at each time moment in the time domain of the object have to be computed. This can be done in a brute force way by computing the snapshots of all atomic objects. Since some will be empty, this approach might lead to a lot of unnecessary computations. Another algorithm could keep track of the time domains of the individual atomic objects and keep a list of active ones at the moment under consideration, which has to be updated every instant. If the geometric object is in normal form, the atomic objects can be sorted by their time domains, and during each interval in the partition of time domain, the list of active atomic objects will remain the same. The second task is the rendering of the snapshot. If the geometric object is not in normal form, the snapshots of the atomic objects overlap, so pixels will be computed more than once. Another solution is computing the boundary, but this might take too long in real-time applications. When a geometric object is in normal form, no triangles overlap, so each pixel will be computed only once. 5.2 Moving Object Retrieval The triangulation provides a means of automatic affine invariant feature extraction for moving object recognition. Indeed, the number of intervals in the time domain indicates the complexity of the movement of the geometric object. This can be used as a first criterium for object matching. For objects having approximately the same number of intervals in their time domains, the snapshots at the middle of each time interval can be compared. If they are all similar, which can be, for example, defined as T S-isomorphic, the objects match. Or, if more exact comparison is needed, one can extract an affine-invariant description from the structure of the elements of the triangulation of the snapshot (see also Section 5.6). 5.3 Surveillance Systems In some applications, e.g., surveillance systems, it is important to know the time moments when something changed, when some discontinuity appeared. This could mean that an unauthorized person entered a restricted area, for example, or that a river has burst its banks. Triangulating the contours of the recorded images and reporting all single points and end points of intervals of the partition of the time domain indicates all moments when some discontinuity might have occurred. 5.4 Precomputing Queries If we do not triangulate each geometric object in a database separately, but use the contours of all geometric objects

9 together in the triangulation, the atomic objects in the result will have the following nice property. For each geometric object in the original database, an atomic object will either belong to (or be a subset of) it entirely, or not at all. This means that we can label each element of the spatio-temporal triangulation of the database with the set of id s of the geometric objects it belongs to. We illustrate this for the spatial case only in Figure 6. Suppose we have two triangles A and B. The set A is the union of the light grey and white parts of the figure, the set B is the union of the dark grey and white parts of the figure. After triangulation, we can label the light grey triangles with {A}, the white triangle with {A, B}, and the dark grey triangles with {B}. A Figure 6: The set operations between objects are pre-computed in the triangulation. Using this triangulation of databases in a preprocessing stage, means that the results of queries that ask for set operations between geometric objects are also pre-computed. Answering such a query boils down to checking labels of atomic objects. This means a lot of gain in speed at query time. Indeed, even to compute, for example, the intersection of two atomic objects, one has to compute first the intervals where the intersection exists, and then to consider all possible shapes the intersection can have, and again represent it by moving triangles. 5.5 Maintaining the triangulation. If a geometric object has to be inserted into or removed from a database (i.e., a collection of geometric objects), the triangulation has to be recomputed for the intervals in the partition of the time domain that contain the time domain of the object under consideration. This may require that the triangulation in total has to be recomputed. However, the nature of a lot of spatio-temporal applications is such that updates involve only the insertion of objects that exists later in time than the already present data. In that case, only the triangulation at the latest time interval of the partition should be recomputed together with the new object, to check whether the new data are a continuation of the previous. Also, data is removed only when it is outdated. In that case a whole time interval of data can be removed. Examples of such spatio-temporal applications are surveillance, traffic monitoring and cadastral information systems. B 5.6 Affine invariant Querying of Spatio-temporal Databases An interesting topic for further work is to compute a new, affine-invariant description of geometric objects in normal form, that does not involve coordinates of reference objects. The structure of the atomic objects in the spatio-temporal triangulation can be used for that. Once such a description is developed, a query language can be designed that asks for affine-invariant properties of objects only. 6. CONCLUDING REMARKS We adopted the hierarchical data model of Chomicki and Revesz [7] for moving objects, since it is natural and flexible. However, this model lacks a normal form, as different sets of objects might represent the same spatio-temporal set in R 2 R. Furthermore, we are interested in affine-invariant representation and querying of objects, as the choice of origin and unit of measure should not affect queries on spatiotemporal data. We first introduced a new affine-invariant triangulation method for spatial data. We then extended this method for spatio-temporal data in such a way that the time domain is partitioned in intervals for which the triangulations of all snapshots are isomorphic. The proposed affine-invariant triangulation is natural and can serve as an affine-invariant normal form for spatio-temporal data. 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