Local Geometry Processing for Deformations of Non-Rigid 3D Shapes

Geometry processing and in particular spectral geometry processing deal with many different deformations that complicate shape analysis problems for non-rigid 3D objects. Furthermore, pointwise description of surfaces has increased relevance for several applications such as shape correspondences and matching, shape representation, shape modelling and many others. In this thesis we propose four local approaches to face the problems generated by the deformations of real objects and improving the pointwise characterization of surfaces. Differently from global approaches that work simultaneously on the entire shape we focus on the properties of each point and its local neighborhood. Global analysis of shapes is not negative in itself. However, having to deal with local variations, distortions and deformations, it is often challenging to relate two real objects globally. For this reason, in the last decades, several instruments have been introduced for the local analysis of images, graphs, shapes and surfaces. Starting from this idea of localized analysis, we propose both theoretical insights and application tools within the local geometry processing domain. In more detail, we extend the windowed Fourier transform from the standard Euclidean signal processing to different versions specifically designed for spectral geometry processing. Moreover, from the spectral geometry processing perspective, we define a new family of localized basis for the functional space defined on surfaces that improve the spatial localization for standard applications in this field. Finally, we introduce the discrete time evolution process as a framework that characterizes a point through its pairwise relationship with the other points on the surface in an increasing scale of locality. The main contribute of this thesis is a set of tools for local geometry processing and local spectral geometry processing that could be used in standard useful applications. The overall observation of our analysis is that localization around points could factually improve the geometry processing in many different applications

Clustering under prior knowledge with application to image segmentation

This paper proposes a new approach to model-based clustering under prior knowledge. The proposed formulation can be interpreted from two different angles: as penalized logistic regression, where the class labels are only indirectly observed (via the probability density of each class); as finite mixture learning under a grouping prior. To estimate the parameters of the proposed model, we derive a (generalized) EM algorithm with a closed-form E-step, in contrast with other recent approaches to semi-supervised probabilistic clustering which require Gibbs sampling or suboptimal shortcuts. We show that our approach is ideally suited for image segmentation: it avoids the combinatorial nature Markov random field priors, and opens the door to more sophisticated spatial priors (e.g., wavelet-based) in a simple and computationally efficient way. Finally, we extend our formulation to work in unsupervised, semi-supervised, or discriminative modes

Clustering under prior knowledge with application to image segmentation

This paper proposes a new approach to model-based clustering under prior knowledge. The proposed formulation can be interpreted from two different angles: as penalized logistic regression, where the class labels are only indirectly observed (via the probability density of each class); as finite mixture learning under a grouping prior. To estimate the parameters of the proposed model, we derive a (generalized) EM algorithm with a closed-form E-step, in contrast with other recent approaches to semi-supervised probabilistic clustering which require Gibbs sampling or suboptimal shortcuts. We show that our approach is ideally suited for image segmentation: it avoids the combinatorial nature Markov random field priors, and opens the door to more sophisticated spatial priors (e.g., wavelet-based) in a simple and computationally efficient way. Finally, we extend our formulation to work in unsupervised, semi-supervised, or discriminative modes

TIME SERIES ANALYSIS AND CLUSTERING TO CHARACTERIZE CARDIORESPIRATORY INSTABILITY PATTERNS IN STEP-DOWN UNIT PATIENTS

Background: Cardiorespiratory instability (CRI) in noninvasively monitored step-down unit (SDU) patients has a variety of etiologies, and therefore likely manifests in different patterns of vital signs (VS) changes. Objective: We sought to describe differences in admission characteristics and outcomes between patients with and without CRI. We explored use of clustering techniques to identify VS patterns within initial CRI epoch (CRI1) and assessed inter-cluster differences in admission characteristics, outcomes and medications. Methods: Admission characteristics and continuous monitoring data (frequency 1/20 Hz) were recorded in 307 patients. Vital sign (VS) deviations beyond local instability trigger criteria for 3 consecutive minutes or for 4 out of a 5 minute moving window were classified as CRI events. We identified CRI1 in 133 patients, derived statistical features of CRI1 epoch and employed hierarchical and k-means clustering techniques. We tested several clustering solutions and used 10-fold cross validation and ANOVA to establish best solution. Inter-cluster differences in admission characteristics, outcomes and medications were assessed. Main Results: Patients transferred to the SDU from units with higher monitoring capability were more likely to develop CRI (n=133, CRI 44% vs no CRI n=174, 31%, p=.042). Patients with at least one event of CRI had longer hospital length of stay (CRI 11.3 + 10.2 days vs no CRI 7.8 + 9.2, p=.001) and SDU unit stay (CRI 6.1 + 4.9 days vs no CRI 3.5 + 2.9, p< .001). Four main clusters(C) were derived. Clusters were significantly different based on age (p=0.001; younger patients in C1 and older in C2), number of comorbidities (p<0.01; more C2 patients had ≥2), and admission source (p=0.008; more C1 and C4 patients transferred in from a higher intensity monitoring unit). Patients with CRI differed significantly (p<.05) from those without CRI based on medication categories. Conclusions: CRI1 was associated with prolonged hospital and SDU length of stay. Patients transferred from a higher level of care were more likely to develop CRI, suggesting that they are sicker. Future study will be needed to determine if there are common physiologic underpinnings of VS clusters which might inform monitoring practices and clinical decision-making when CRI first manifests

Clustering under prior knowledge with application to image segmentation

This paper proposes a new approach to model-based clustering under prior knowledge. The proposed formulation can be interpreted from two different angles: as penalized logistic regression, where the class labels are only indirectly observed (via the probability density of each class); as finite mixture learning under a grouping prior. To estimate the parameters of the proposed model, we derive a (generalized) EM algorithm with a closed-form E-step, in contrast with other recent approaches to semi-supervised probabilistic clustering which require Gibbs sampling or suboptimal shortcuts. We show that our approach is ideally suited for image segmentation: it avoids the combinatorial nature Markov random field priors, and opens the door to more sophisticated spatial priors (e.g., wavelet-based) in a simple and computationally efficient way. Finally, we extend our formulation to work in unsupervised, semi-supervised, or discriminative modes.