Multi-scale techniques for multi-dimensional data analysis

Large datasets of geometric data of various nature are becoming more and more available as sensors become cheaper and more widely used. Due to both their size and their noisy nature, special techniques must be employed to deal with them correctly. In order to efficiently handle this amount of data and to tackle the technical challenges they pose, we propose techniques that analyze a scalar signal by means of its critical points (i.e. maxima and minima), ranking them on a scale of importance, by which we can extrapolate important information of the input signal separating it from noise, thus dramatically reducing the complexity of the problem. In order to obtain a ranking of critical points we employ multi-scale techniques. The standard scale-space approach, however, is not sufficient when trying to track critical points across various scales. We start from an implementation of the scale-space which computes a linear interpolation between scales in order to make tracking of critical points easier. The linear interpolation of a process which is not itself linear, though, does not fulfill some theoretical properties of scale-space, thus making the tracking of critical points much harder. We propose an extension of this piecewiselinear scale-space implementation, which recovers the theoretical properties (e.g., to avoid the generation of new critical points as the scale increases) and keeps the tracking consistent. Next we combine the scale-space with another technique that comes from the topology theory: the classification of critical points based on their persistence value. While the scale-space applies a filtering in the frequency domain, by progressively smoothing the input signal with low-pass filters of increasing size, the computation of the persistence can be seen as a filtering applied in the amplitude domain, which progressively removes pairs of critical points based on their difference in amplitude. The two techniques, while being both relevant to the concept of scale, express different qualities of the critical points of the input signal; depending on the application domain we can use either of them, or, since they both have non-zero values only at critical points, they can be used together with a linear combination. The thesis will be structured as follows: In Chapter 1 we will present an overview on the problem of analyzing huge geometric datasets, focusing on the problem of dealing with their size and noise, and of reducing the problem to a subset of relevant samples. The Chapter 2 will contain a study of the state of the art in scale-space algorithms, followed by a more in-depth analysis of the virtually continuous framework used as base technique will be presented. In its last part, we will propose methods to extend these techniques in order to satisfy the axioms present in the continuous version of the scale-space and to have a stronger and more reliable tracking of critical points across scales, and the extraction of the persistence of critical points of a signal as a variant to the standard scale-space approach; we will show the differences between the two and discuss how to combine them. The Chapter 3 will introduce an ever growing source of data, the motion capture systems; we will motivate its importance by discussing the many applications in which it has been used for the past two decades. We will briefly summarize the different systems existing and then we will focus on a particular one, discussing its peculiarities and its output data. In Chapter 4, we will discuss the problem of studying intra-personal synchronization computed on data coming from such motion-capture systems. We will show how multi-scale approaches can be used to identify relevant instants in the motion and how these instants can be used to precisely study synchronization between the different parts of the body from which they are extracted. We will apply these techniques to the problem of generating a classifier to discriminate between martial artists of different skills who have been recorded doing karate’s movements. In Chapter 5 will present a work on the automatic detection of relevant points of the human face from 3D data. We will show that the Gaussian curvature of the 3D surface is a good feature to distinguish the so-called fiducial points, but also that multi-scale techniques must be used to extract only relevant points and get rid of the noise. In closing, Chapter 6 will discuss an ongoing work about motion segmentation; after an introduction about the meaning and different possibilities of motion segmentation we will present the data we work with, the approach used to identify segments and some preliminary tools and results.