# Browsing by Author "Daubechies, Ingrid"

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Item Open Access A new fully automated approach for aligning and comparing shapes.(Anatomical record (Hoboken, N.J. : 2007), 2015-01) Boyer, Doug M; Puente, Jesus; Gladman, Justin T; Glynn, Chris; Mukherjee, Sayan; Yapuncich, Gabriel S; Daubechies, IngridThree-dimensional geometric morphometric (3DGM) methods for placing landmarks on digitized bones have become increasingly sophisticated in the last 20 years, including greater degrees of automation. One aspect shared by all 3DGM methods is that the researcher must designate initial landmarks. Thus, researcher interpretations of homology and correspondence are required for and influence representations of shape. We present an algorithm allowing fully automatic placement of correspondence points on samples of 3D digital models representing bones of different individuals/species, which can then be input into standard 3DGM software and analyzed with dimension reduction techniques. We test this algorithm against several samples, primarily a dataset of 106 primate calcanei represented by 1,024 correspondence points per bone. Results of our automated analysis of these samples are compared to a published study using a traditional 3DGM approach with 27 landmarks on each bone. Data were analyzed with morphologika(2.5) and PAST. Our analyses returned strong correlations between principal component scores, similar variance partitioning among components, and similarities between the shape spaces generated by the automatic and traditional methods. While cluster analyses of both automatically generated and traditional datasets produced broadly similar patterns, there were also differences. Overall these results suggest to us that automatic quantifications can lead to shape spaces that are as meaningful as those based on observer landmarks, thereby presenting potential to save time in data collection, increase completeness of morphological quantification, eliminate observer error, and allow comparisons of shape diversity between different types of bones. We provide an R package for implementing this analysis.Item Open Access Algorithms with Applications to Anthropology(2018) Ravier, Robert JamesIn 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.

Item Open Access Development and Assessment of Fully Automated and Globally Transitive Geometric Morphometric Methods, With Application to a Biological Comparative Dataset With High Interspecific Variation.(Anatomical record (Hoboken, N.J. : 2007), 2018-04) Gao, Tingran; Yapuncich, Gabriel S; Daubechies, Ingrid; Mukherjee, Sayan; Boyer, Doug MAutomated geometric morphometric methods are promising tools for shape analysis in comparative biology, improving researchers' abilities to quantify variation extensively (by permitting more specimens to be analyzed) and intensively (by characterizing shapes with greater fidelity). Although use of these methods has increased, published automated methods have some notable limitations: pairwise correspondences are frequently inaccurate and pairwise mappings are not globally consistent (i.e., they lack transitivity across the full sample). Here, we reassess the accuracy of published automated methods-cPDist (Boyer et al. Proc Nat Acad Sci 108 () 18221-18226) and auto3Dgm (Boyer et al.: Anat Rec 298 () 249-276)-and evaluate several modifications to these methods. We show that a substantial percentage of alignments and pairwise maps between specimens of dissimilar geometries were inaccurate in the study of Boyer et al. (Proc Nat Acad Sci 108 () 18221-18226), despite a taxonomically partitioned variance structure of continuous Procrustes distances. We show these inaccuracies are remedied using a globally informed methodology within a collection of shapes, rather than relying on pairwise comparisons (c.f. Boyer et al.: Anat Rec 298 () 249-276). Unfortunately, while global information generally enhances maps between dissimilar objects, it can degrade the quality of correspondences between similar objects due to the accumulation of numerical error. We explore a number of approaches to mitigate this degradation, quantify their performance, and compare the generated pairwise maps (and the shape space characterized by these maps) to a "ground truth" obtained from landmarks manually collected by geometric morphometricians. Novel methods both improve the quality of the pairwise correspondences relative to cPDist and achieve a taxonomic distinctiveness comparable to auto3Dgm. Anat Rec, 301:636-658, 2018. © 2017 Wiley Periodicals, Inc.Item Open Access Differential Geometry Tools for Data Analysis(2021) Nag, PanchaliThe 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.

Item Open Access Electrocardiographic J Wave and Cardiovascular Outcomes in the General Population (from the Atherosclerosis Risk In Communities Study).(Am J Cardiol, 2016-09-15) O'Neal, Wesley T; Wang, Yi Grace; Wu, Hau-Tieng; Zhang, Zhu-Ming; Li, Yabing; Tereshchenko, Larisa G; Estes, E Harvey; Daubechies, Ingrid; Soliman, Elsayed ZThe association between the J wave, a key component of the early repolarization pattern, and adverse cardiovascular outcomes remains unclear. Inconsistencies have stemmed from the different methods used to measure the J wave. We examined the association between the J wave, detected by an automated method, and adverse cardiovascular outcomes in 14,592 (mean age = 54 ± 5.8 years; 56% women; 26% black) participants from the Atherosclerosis Risk In Communities (ARIC) study. The J wave was detected at baseline (1987 to 1989) and during follow-up study visits (1990 to 1992, 1993 to 1995, and 1996 to 1998) using a fully automated method. Sudden cardiac death, coronary heart disease death, and cardiovascular mortality were ascertained from hospital discharge records, death certificates, and autopsy data through December 31, 2010. A total of 278 participants (1.9%) had evidence of a J wave. Over a median follow-up of 22 years, 4,376 of the participants (30%) died. In a multivariable Cox regression analysis adjusted for demographics, cardiovascular risk factors, and potential confounders, the J wave was not associated with an increased risk of sudden cardiac death (hazard ratio [HR] 0.74, 95% CI 0.36 to 1.50), coronary heart disease death (HR 0.72, 95% CI 0.40 to 1.32), or cardiovascular mortality (HR 1.16, 95% CI 0.87 to 1.56). An interaction was detected for cardiovascular mortality by gender with men (HR 1.54, 95% CI 1.09 to 2.19) having a stronger association than women (HR 0.74, 95% CI 0.43 to 1.25; P-interaction = 0.030). In conclusion, our findings suggest that the J wave is a benign entity that is not associated with an increased risk for sudden cardiac arrest in middle-aged adults in the United States.Item Open Access High Dimensional Signal Representation(2017) Yin, RujieIn 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.

Item Open Access Hypoelliptic Diffusion Maps and Their Applications in Automated Geometric Morphometrics(2015) Gao, TingranWe 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.

Item Open Access Neural Network Approximation of Refinable Functions.(CoRR, 2021) Daubechies, Ingrid; DeVore, Ronald; Dym, Nadav; Faigenbaum-Golovin, Shira; Kovalsky, Shahar Z; Lin, Kung-Ching; Park, Josiah; Petrova, Guergana; Sober, BarakIn the desire to quantify the success of neural networks in deep learning and other applications, there is a great interest in understanding which functions are efficiently approximated by the outputs of neural networks. By now, there exists a variety of results which show that a wide range of functions can be approximated with sometimes surprising accuracy by these outputs. For example, it is known that the set of functions that can be approximated with exponential accuracy (in terms of the number of parameters used) includes, on one hand, very smooth functions such as polynomials and analytic functions (see e.g. \cite{E,S,Y}) and, on the other hand, very rough functions such as the Weierstrass function (see e.g. \cite{EPGB,DDFHP}), which is nowhere differentiable. In this paper, we add to the latter class of rough functions by showing that it also includes refinable functions. Namely, we show that refinable functions are approximated by the outputs of deep ReLU networks with a fixed width and increasing depth with accuracy exponential in terms of their number of parameters. Our results apply to functions used in the standard construction of wavelets as well as to functions constructed via subdivision algorithms in Computer Aided Geometric Design.Item Open Access Non-Convex Planar Harmonic MapsKovalsky, Shahar Z; Aigerman, Noam; Daubechies, Ingrid; Kazhdan, Michael; Lu, Jianfeng; Steinerberger, StefanWe formulate a novel characterization of a family of invertible maps between two-dimensional domains. Our work follows two classic results: The Rad\'o-Kneser-Choquet (RKC) theorem, which establishes the invertibility of harmonic maps into a convex planer domain; and Tutte's embedding theorem for planar graphs - RKC's discrete counterpart - which proves the invertibility of piecewise linear maps of triangulated domains satisfying a discrete-harmonic principle, into a convex planar polygon. In both theorems, the convexity of the target domain is essential for ensuring invertibility. We extend these characterizations, in both the continuous and discrete cases, by replacing convexity with a less restrictive condition. In the continuous case, Alessandrini and Nesi provide a characterization of invertible harmonic maps into non-convex domains with a smooth boundary by adding additional conditions on orientation preservation along the boundary. We extend their results by defining a condition on the normal derivatives along the boundary, which we call the cone condition; this condition is tractable and geometrically intuitive, encoding a weak notion of local invertibility. The cone condition enables us to extend Alessandrini and Nesi to the case of harmonic maps into non-convex domains with a piecewise-smooth boundary. In the discrete case, we use an analog of the cone condition to characterize invertible discrete-harmonic piecewise-linear maps of triangulations. This gives an analog of our continuous results and characterizes invertible discrete-harmonic maps in terms of the orientation of triangles incident on the boundary.Item Open Access Probabilistic Models on Fibre Bundles(2019) Shan, ShanIn 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.

Item Open Access Stop memorizing: A data-dependent regularization framework for intrinsic pattern learningZhu, Wei; Qiu, Qiang; Wang, Bao; Lu, Jianfeng; Sapiro, Guillermo; Daubechies, IngridDeep neural networks (DNNs) typically have enough capacity to fit random data by brute force even when conventional data-dependent regularizations focusing on the geometry of the features are imposed. We find out that the reason for this is the inconsistency between the enforced geometry and the standard softmax cross entropy loss. To resolve this, we propose a new framework for data-dependent DNN regularization, the Geometrically-Regularized-Self-Validating neural Networks (GRSVNet). During training, the geometry enforced on one batch of features is simultaneously validated on a separate batch using a validation loss consistent with the geometry. We study a particular case of GRSVNet, the Orthogonal-Low-rank Embedding (OLE)-GRSVNet, which is capable of producing highly discriminative features residing in orthogonal low-rank subspaces. Numerical experiments show that OLE-GRSVNet outperforms DNNs with conventional regularization when trained on real data. More importantly, unlike conventional DNNs, OLE-GRSVNet refuses to memorize random data or random labels, suggesting it only learns intrinsic patterns by reducing the memorizing capacity of the baseline DNN.Item Open Access Two New Methods to Improve Adaptive Time-Frequency Localization(2021) Chen, ZiyuThis 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.