Chapter 2 Trajectory Indexing and Retrival

This 26 pages paper is a bit short as a survey but a little too long for me the first day to write a “papaer analysis”. But I will finish it rather than give up in front of my first piece of work.

As a survey, I can learn some full-scale knowledge on the topic of Trajectory Indexing and Retrival.

If an paper has a smooth section setting, it’s convenient for readers to read and understand, this paper did.

Query Tricks:

1) Construct indecies;

2) when tranverse the index, Depth-first or width-first can be used. Then, the filter-and-refine paradigm is necessary.

1 Three Query

The central theme of the paper is three kinds of trajectory query:

1.1 P-Query

Two kinds:

Given trajectories, ask for POI;

Given POI(1 or more), ask for trajectories.

1.1.1 Ask for trajectories

1) Construct index: for line segment or sampled position

2) Filter-and-refine: remember to mark visited

1.1.2 Ask for POI

Fantastic Idea!



Figure 1. ask for POI

If there is a segment of (s, e), and a lot of POIs around it.

1) Search the POI nearest to s —- a;

2) Draw circles of (s, a), (a, e), and all POI out of circles will not be candidates;

3) Now, turn to e to determine which POI is nearest in its current circle —- c;

4) Draw the perpendicular bisector of (a, c) and get point s1;

5) Repeat the above steps on (a, s1), (s1, e);

The last result graph:



Figure 2. Result of above example

1.2 R-Query

Two kinds:

Given a region, ask for trajectories passed it;

Find out regions that some trajectory all passed in a period of time.

MR-tree, HR+-tree, 3D R-tree are often used here.

1.3 T-Query

Two kinds:

ask for similar trajectories;

ask for trajectories within a distance threshold.

The distance between a point qi and a trajectory pj:



The distance between a set of query point Q and a trajectory R:



By induce both the lower and upper bound, a filter-and-refine can be adapted.

2 Trajectory Similarity Measures

Point to Trajectory: if one point, take the min distance to the trajectory(measure can be Lp-norm or network distance); if multiple point, use e^x!!!

Trajectory to trajectory:

1) Closest-Pair-Distance;

2) Sum-of-Pairs Distance;

3) Dynamic Time Warping (DTW);

4) Longest Common Subsequence (LCSS);

5) Edit Distance with Real Pelnaty (EPR);

6) Edit Distance on Real Sequences (EDR);

All the six methods were originally proposed for time series data. And they all are related, the latter one makes improvement on the former ones. For example, 1) and 2) are both based on Euclidean distance which need the two trajectories have the same length, and 3) break the limitation first by using repeating. And LCSS conquer the sensitivity to noise and so on.

An obvious drawback here is that some variants or formulas did not well explain. Maybe they were evolved in other chapters in this book.

3 Trajectory Indexes

Mainly three index approaches:

Multidimensional access method: 3D R-tree, STR-tree;

Multiversion structures: MR-tree, HR-tree, HR+-tree;

Grid based: SETI, MTSB-tree;

It can be seen that most of these methods are sons of R-tree. The powerful R-tree have 4 uses in general:

Retrieve all spatial objects within a specified query region;

Search for a certain point;

Process a nearest neighbor query

Spatial join queries.

3.1 3D R-tree

Figure 2. 3D R-tree

At first, 3D R-tree just extend out a temporal dimension like Figure 2.(a), but 1) a big box to hold just a small line segment results to a big space waste; 2) Which trajectory does each segment belongs to is unknow. Then, 3D R-tree was modified into Figure 2(b). Just use (id, trajecory#, MBB, orientation ) to stand for each segment.

3.2 STR-Tree

Figure 3. STR-Tree

The difference between 3D R-tree and STR-Tree is just the insertion and deletion.

3.3 TB-Tree

Figure 4. TB-Tree

As shown in Figure 4, all segments of a trajetory are linked by a linked list.

In fact, the paper didn’t say what are the other leaf nodes stands for. Confused…………….

3.4 HR-Tree

Figure 5. HR-Tree

One tree for one timestamp. To save the space, reuse the nodes.

3.5 HR+-Tree

Similar to HR-Tree, but more space economic by allowing entries of different timestamps to be placed in the same node.

HR+-Tree is more adaptive to support interval query.

Both HR-Tree and HR+-Tree are efficient in timestamp queries.

3.6 MV3R-Tree

Figure 6. MV3R-Tree

It consists of two structures: a multiversion R-tree(MVR-tree) and a 3D R-tree built on the leaves of the MVR-tree in order to process interval query.

4 Applications

This paper also provide abundant applications mainly include:

Traffice flow patterns;

Behavior mining;

Path Planning;