Optimising Spatial and Tonal Data for PDE-based Inpainting

Some recent methods for lossy signal and image compression store only a few selected pixels and fill in the missing structures by inpainting with a partial differential equation (PDE). Suitable operators include the Laplacian, the biharmonic operator, and edge-enhancing anisotropic diffusion (EED). The quality of such approaches depends substantially on the selection of the data that is kept. Optimising this data in the domain and codomain gives rise to challenging mathematical problems that shall be addressed in our work. In the 1D case, we prove results that provide insights into the difficulty of this problem, and we give evidence that a splitting into spatial and tonal (i.e. function value) optimisation does hardly deteriorate the results. In the 2D setting, we present generic algorithms that achieve a high reconstruction quality even if the specified data is very sparse. To optimise the spatial data, we use a probabilistic sparsification, followed by a nonlocal pixel exchange that avoids getting trapped in bad local optima. After this spatial optimisation we perform a tonal optimisation that modifies the function values in order to reduce the global reconstruction error. For homogeneous diffusion inpainting, this comes down to a least squares problem for which we prove that it has a unique solution. We demonstrate that it can be found efficiently with a gradient descent approach that is accelerated with fast explicit diffusion (FED) cycles. Our framework allows to specify the desired density of the inpainting mask a priori. Moreover, is more generic than other data optimisation approaches for the sparse inpainting problem, since it can also be extended to nonlinear inpainting operators such as EED. This is exploited to achieve reconstructions with state-of-the-art quality. We also give an extensive literature survey on PDE-based image compression methods

Sensing and Compression Techniques for Environmental and Human Sensing Applications

In this doctoral thesis, we devise and evaluate a variety of lossy compression schemes for Internet of Things (IoT) devices such as those utilized in environmental wireless sensor networks (WSNs) and Body Sensor Networks (BSNs). We are especially concerned with the efficient acquisition of the data sensed by these systems and to this end we advocate the use of joint (lossy) compression and transmission techniques. Environmental WSNs are considered first. For these, we present an original compressive sensing (CS) approach for the spatio-temporal compression of data. In detail, we consider temporal compression schemes based on linear approximations as well as Fourier transforms, whereas spatial and/or temporal dynamics are exploited through compression algorithms based on distributed source coding (DSC) and several algorithms based on compressive sensing (CS). To the best of our knowledge, this is the first work presenting a systematic performance evaluation of these (different) lossy compression approaches. The selected algorithms are framed within the same system model, and a comparative performance assessment is carried out, evaluating their energy consumption vs the attainable compression ratio. Hence, as a further main contribution of this thesis, we design and validate a novel CS-based compression scheme, termed covariogram-based compressive sensing (CB-CS), which combines a new sampling mechanism along with an original covariogram-based approach for the online estimation of the covariance structure of the signal. As a second main research topic, we focus on modern wearable IoT devices which enable the monitoring of vital parameters such as heart or respiratory rates (RESP), electrocardiography (ECG), and photo-plethysmographic (PPG) signals within e-health applications. These devices are battery operated and communicate the vital signs they gather through a wireless communication interface. A common issue of this technology is that signal transmission is often power-demanding and this poses serious limitations to the continuous monitoring of biometric signals. To ameliorate this, we advocate the use of lossy signal compression at the source: this considerably reduces the size of the data that has to be sent to the acquisition point by, in turn, boosting the battery life of the wearables and allowing for fine-grained and long-term monitoring. Considering one dimensional biosignals such as ECG, RESP and PPG, which are often available from commercial wearable devices, we first provide a throughout review of existing compression algorithms. Hence, we present novel approaches based on online dictionaries, elucidating their operating principles and providing a quantitative assessment of compression, reconstruction and energy consumption performance of all schemes. As part of this first investigation, dictionaries are built using a suboptimal but lightweight, online and best effort algorithm. Surprisingly, the obtained compression scheme is found to be very effective both in terms of compression efficiencies and reconstruction accuracy at the receiver. This approach is however not yet amenable to its practical implementation as its memory usage is rather high. Also, our systematic performance assessment reveals that the most efficient compression algorithms allow reductions in the signal size of up to 100 times, which entail similar reductions in the energy demand, by still keeping the reconstruction error within 4 % of the peak-to-peak signal amplitude. Based on what we have learned from this first comparison, we finally propose a new subject-specific compression technique called SURF Subject-adpative Unsupervised ecg compressor for weaRable Fitness monitors. In SURF, dictionaries are learned and maintained using suitable neural network structures. Specifically, learning is achieve through the use of neural maps such as self organizing maps and growing neural gas networks, in a totally unsupervised manner and adapting the dictionaries to the signal statistics of the wearer. As our results show, SURF: i) reaches high compression efficiencies (reduction in the signal size of up to 96 times), ii) allows for reconstruction errors well below 4 % (peak-to-peak RMSE, errors of 2 % are generally achievable), iii) gracefully adapts to changing signal statistics due to switching to a new subject or changing their activity, iv) has low memory requirements (lower than 50 kbytes) and v) allows for further reduction in the total energy consumption (processing plus transmission). These facts makes SURF a very promising algorithm, delivering the best performance among all the solutions proposed so far

Nearly Consistent Finite Particle Estimates in Streaming Importance Sampling

In Bayesian inference, we seek to compute information about random variables such as moments or quantiles on the basis of {available data} and prior information. When the distribution of random variables is {intractable}, Monte Carlo (MC) sampling is usually required. {Importance sampling is a standard MC tool that approximates this unavailable distribution with a set of weighted samples.} This procedure is asymptotically consistent as the number of MC samples (particles) go to infinity. However, retaining infinitely many particles is intractable. Thus, we propose a way to only keep a \emph{finite representative subset} of particles and their augmented importance weights that is \emph{nearly consistent}. To do so in {an online manner}, we (1) embed the posterior density estimate in a reproducing kernel Hilbert space (RKHS) through its kernel mean embedding; and (2) sequentially project this RKHS element onto a lower-dimensional subspace in RKHS using the maximum mean discrepancy, an integral probability metric. Theoretically, we establish that this scheme results in a bias determined by a compression parameter, which yields a tunable tradeoff between consistency and memory. In experiments, we observe the compressed estimates achieve comparable performance to the dense ones with substantial reductions in representational complexity

pBWT: Achieving succinct data structures for parameterized pattern matching and related problems

The fields of succinct data structures and compressed text indexing have seen quite a bit of progress over the last two decades. An important achievement, primarily using techniques based on the Burrows-Wheeler Transform (BWT), was obtaining the full functionality of the suffix tree in the optimal number of bits. A crucial property that allows the use of BWT for designing compressed indexes is order-preserving suffix links. Specifically, the relative order between two suffixes in the subtree of an internal node is same as that of the suffixes obtained by truncating the furst character of the two suffixes. Unfortunately, in many variants of the text-indexing problem, for e.g., parameterized pattern matching, 2D pattern matching, and order-isomorphic pattern matching, this property does not hold. Consequently, the compressed indexes based on BWT do not directly apply. Furthermore, a compressed index for any of these variants has been elusive throughout the advancement of the field of succinct data structures. We achieve a positive breakthrough on one such problem, namely the Parameterized Pattern Matching problem. Let T be a text that contains n characters from an alphabet , which is the union of two disjoint sets: containing static characters (s-characters) and containing parameterized characters (p-characters). A pattern P (also over ) matches an equal-length substring S of T i the s-characters match exactly, and there exists a one-to-one function that renames the p-characters in S to that in P. The task is to find the starting positions (occurrences) of all such substrings S. Previous index [Baker, STOC 1993], known as Parameterized Suffix Tree, requires (n log n) bits of space, and can find all occ occurrences in time O(jPj log +occ), where = jj. We introduce an n log +O(n)-bit index with O(jPj log +occlog n log ) query time. At the core, lies a new BWT-like transform, which we call the Parame- terized Burrows-Wheeler Transform (pBWT). The techniques are extended to obtain a succinct index for the Parameterized Dictionary Matching problem of Idury and Schaer [CPM, 1994]