We present a new compressed representation of free trajectories of moving objects. It combines a partial-sums-based structure that retrieves in constant time the position of the object at any instant, with a hierarchical minimum-bounding-boxes representation that allows determining if the object is seen in a certain rectangular area during a time period. Combined with spatial snapshots at regular intervals, the representation is shown to outperform classical ones by orders of magnitude in space, and also to outperform previous compressed representations in time performance, when using the same amount of space.
A lossy way to reduce size is to generate a new trajectory that approximates the original one, by keeping the most representative points. The best known method of this type is the Douglas-Peucker algorithm @cite_11 . Other strategies record speed and direction, discarding points that can be reasonably predicted with this data @cite_19 . A lossless way to reduce space is to use differential encodings of the consecutive values @math , @math , and time @cite_12 @cite_14 @cite_1 .
@cite_14 @cite_1 @cite_19 @cite_12 @cite_11
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The need to store vast amounts of trajectory data becomes more problematic as GPS-based tracking devices become increasingly prevalent. There are two commonly used approaches for compressing trajectory data. The first is the line generalisation approach which aims to fit the trajectory using a series of line segments. The second is to store the initial data point and then store the remaining data points as a sequence of successive deltas. The line generalisation approach is only effective when given a large error margin, and existing delta compression algorithms do not permit lossy compression. Consequently there is an uncovered gap in which users expect a good compression ratio by giving away only a small error margin. This paper fills this gap by extending the delta compression approach to allow users to trade a small maximum error margin for large improvements to the compression ratio. In addition, alternative techniques are extensively studied for the following two key components of any delta-based approach: predicting the value of the next data point and encoding leading zeros. We propose a new trajectory compression system called Trajic based on the results of the study. Experimental results show that Trajic produces 1.5 times smaller compressed data than a straight-forward delta compression algorithm for lossless compression and produces 9.4 times smaller compressed data than a state-of-the-art line generalisation algorithm when using a small maximum error bound of 1 meter. Spanners, emulators, and approximate distance oracles can be viewed as lossy compression schemes that represent an unweighted graph metric in small space, say O(n1+Δ) bits. There is an inherent tradeoff between the sparsity parameter Δ and the stretch function f of the compression scheme, but the qualitative nature of this tradeoff has remained a persistent open problem. It has been known for some time that when Δ ≥ 1 3 there are schemes with constant additive stretch (distance d is stretched to at most f(d) = d + O(1)), and recent results of Abboud and Bodwin show that when Δ In this paper we show that the lower bound of Abboud and Bodwin is just the first step in a hierarchy of lower bounds that characterize the asymptotic behavior of the optimal stretch function f for sparsity parameter Δ ∈ (0,1 3). Specifically, for any integer k ≥ 2, any compression scheme with size [EQUATION] has a sublinear additive stretch function f: f(d) = d + Ω(d1−1 k). This lower bound matches Thorup and Zwick's (2006) construction of sublinear additive emulators. It also shows that Elkin and Peleg's (1 + ϵ, β)-spanners have an essentially optimal tradeoff between Δ, ϵ, and β, and that the sublinear additive spanners of Pettie (2009) and Chechik (2013) are not too far from optimal. To complement these lower bounds we present a new construction of (1 + ϵ, O(k ϵ)k−1)-spanners with size [EQUATION], where hk Our lower bound technique exhibits several interesting degrees of freedom in the framework of Abboud and Bodwin. By carefully exploiting these freedoms, we are able to obtain lower bounds for several related combinatorial objects. We get lower bounds on the size of (β, ϵ)-hopsets, matching Elkin and Neiman's construction (2016), and lower bounds on shortcutting sets for digraphs that preserve the transitive closure. Our lower bound simplifies Hesse's (2003) refutation of Thorup's conjecture (1992), which stated that adding a linear number of shortcuts suffices to reduce the diameter to polylogarithmic. Finally, we show matching upper and lower bounds for graph compression schemes that work for graph metrics with girth at least 2γ + 1. One consequence is that 's (2010) additive O(γ)-spanners with size [EQUATION] cannot be improved in the exponent. Predictive lossy compression has been shown to represent a very flexible framework for lossless and lossy onboard compression of multispectral and hyperspectral images with quality and rate control. In this paper, we improve predictive lossy compression in several ways, using a standard issued by the Consultative Committee on Space Data Systems, namely CCSDS-123, as an example of application. First, exploiting the flexibility in the error control process, we propose a constant-signal-to-noise-ratio algorithm that bounds the maximum relative error between each pixel of the reconstructed image and the corresponding pixel of the original image. This is very useful to avoid low-energy areas of the image being affected by large errors. Second, we propose a new rate control algorithm that has very low complexity and provides performance equal to or better than existing work. Third, we investigate several entropy coding schemes that can speed up the hardware implementation of the algorithm and, at the same time, improve coding efficiency. These advances make predictive lossy compression an extremely appealing framework for onboard systems due to its simplicity, flexibility, and coding efficiency. This paper presents a way to embed watermarks into 2D vectordata. The watermarking system provides a high capacity and is robust against the following attacks: polyline simplifications like the Douglas-Peucker algorithm [1], moving and cropping of data and addition of small amounts of random noise. The system is designed for adding information to digital maps. The attacks mentioned above can happen during the daily work with these maps, so the watermark will not be destroyed by working with the data. Nevertheless it is possible to destroy it on purpose by using other attacks. The information is embedded by changing the x y-coordinates of datapoints within the tolerance of the data. A block code is used to reconstruct missing data when not enough datapoints are available and a synchronization provides a way to detect the information even when the map is cropped.