Browsing by Author "Daubechies, Ingrid"

In this dissertation, we investigate several problems in shape analysis. We start by discussing the shape matching problem. Given that homeomorphisms of shapes are computed in practice by interpolating sparse correspondence, we give an algorithm to refine pairwise mappings in a collection by employing a simple metric condition to obtain partial correspondences of points chosen in a manner that outlines the shapes of interest in a relatively small number of points. We then use this mapping algorithm in two separate applications. First, we investigate the extent to which classical assumptions and methods in statistical shape analysis hold for near continuous discretizations of surfaces spanning different species and genus groups. We find that these methods yield biologically meaningful information, and that resulting operations with these correspondences, including averaging and linear interpolation, yield biologically meaningful surfaces even for distinct geometries. As a second application, we discuss the problem of dictionary learning on shapes in an effort to find sparse decompositions of shapes in a collection. To this end, we define a construction of wavelet-like and ridgelet-like objects that are easily computable at the level of the discretization, both of which have natural interpretation in the smooth case. We then use these in tandem with feature points to create a sparse dictionary, and show that standard sparsification practices still retain biological information.

The thesis is divided into two parts: the moving anchor parameterization method and the comparative morphology through Willmore-type functionals. In the first part we show that parametrizing points on curves or surfaces or their higher dimensional analogs in Euclidean space by their distances to well-chosen anchor points can lead to representations that are much less curved. We then use this feature to construct approximation methods that are very simple and that achieve results comparable in quality to standard higher-order methods.\\ The second part is motivated by the observation that the Dirichlet normal energy of a 2-dimensional surface embedded in 3D measures, in some sense, the deviation of the surface from the minimal surface fore the Willmore functional. A midified version of this functional with different parameters, is also of interest in applications; this suggests that their minimal surfaces (with respect to which these modified ``Willmore-like''-functionals measure the deviation) are of interest as well. The second part of the thesis concerns the construction of such examples.

In this thesis we explore the efficiency of signal representations and their robustness in signal reconstruction in three subfields of signal and image processing.

The first result concerns regularity limitation in the construction of directional wavelet bases due to redundancy constraint on the scheme, in an effort to construct “optimal” directional bases with multiresolution and perfect reconstruction proper- ties. We showed that for orthonormal and biorthogonal bases with dilated quincunx downsampling, the wavelets cannot be well localized; however, this regularity limit can be circumvented in a tight frame with dyadic downsampling and a redundancy factor smaller than 2.

The second result introduces a novel framework for patch-based image models combining local structure of patches and nonlocal information in image domain. In particular, we built convolution framelets from local and nonlocal bases, which form a tight frame of the image space and has energy concentration when the local and nonlocal bases are coherent. We applied this framework to reinterpret and improve state-of-the-art low dimensional manifold model.

The final result proposes a new paradigm of phase retrieval, considering signal reconstruction up to a larger equivalence class than a uniform phase shift. It is known that in the classical setting, phase retrieval in infinite or high dimension is inherently unstable. We showed that stability can be achieved, however, for frames of Gabor wavelets or Cauchy wavelets in this new paradigm.

We introduce Hypoelliptic Diffusion Maps (HDM), a novel semi-supervised machine learning framework for the analysis of collections of anatomical surfaces. Triangular meshes obtained from discretizing these surfaces are high-dimensional, noisy, and unorganized, which makes it difficult to consistently extract robust geometric features for the whole collection. Traditionally, biologists put equal numbers of ``landmarks'' on each mesh, and study the ``shape space'' with this fixed number of landmarks to understand patterns of shape variation in the collection of surfaces; we propose here a correspondence-based, landmark-free approach that automates this process while maintaining morphological interpretability. Our methodology avoids explicit feature extraction and is thus related to the kernel methods, but the equivalent notion of ``kernel function'' takes value in pairwise correspondences between triangular meshes in the collection. Under the assumption that the data set is sampled from a fibre bundle, we show that the new graph Laplacian defined in the HDM framework is the discrete counterpart of a class of hypoelliptic partial differential operators.

This thesis is organized as follows: Chapter 1 is the introduction; Chapter 2 describes the correspondences between anatomical surfaces used in this research; Chapter 3 and 4 discuss the HDM framework in detail; Chapter 5 illustrates some interesting applications of this framework in geometric morphometrics.

In this thesis, we propose probabilistic models on fibre bundles for learning the generative process of data. The main tool we use is the diffusion kernel and we use it in two ways. First, we build from the diffusion kernel on a fibre bundle a projected kernel that generates robust representations of the data, and we test that it outperforms regular diffusion maps under noise. Second, this diffusion kernel gives rise to a natural covariance function when defining Gaussian processes (GP) on the fibre bundle. To demonstrate the uses of GP on a fibre bundle, we apply it to simulated data on a Mobius strip for the problem of prediction and regression. Parameter tuning can also be guided by a novel semi-group test arising from the geometric properties of diffusion kernel. For an example of real-world application, we use probabilistic models on fibre bundles to study evolutionary process on anatomical surfaces. In a separate chapter, we propose a robust algorithm (ariaDNE) for computing curvature on each individual surface. The proposed machinery, relating diffusion processes to probabilistic models on fibre bundles, provides a unified framework for ideas from a variety of different topics such as geometric operators, dimension reduction, regression and Bayesian statistics.

This dissertation introduces algorithms that analyze oscillatory signals adaptively. It consists of three chapters. The first chapter reviews the adaptive time-frequency analysis of 1-dimensional signals. It introduces models that capture the time-varying behavior of oscillatory signals. Then it explains two state-of-the-art algorithms, named the SynchroSqueezed Transform (SST) and the Concentration of Frequency and Time (ConceFT), that extract the instantaneous information of signals; this chapter ends with a discussion of some of the shortcomings of SST and ConceFT, which will be remedied by the new methods introduced in the remainder of this thesis. The second chapter introduces the Ramanujan DeShape Algorithm (RDS); it incorporates the periodicity transform to extract adaptively the fundamental frequency of a non-harmonic signal. The third part proposes an algorithm that rotates the time-frequency content of an oscillatory signal to obtain a time-frequency representation that has fewer artifacts. Numerical results illustrate the theoretical analysis.