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1. Automatic Feature Selection for Shape Registration in Additive Manufacturing

   Lin, Weizhi; Dai, Peng; Huang, Qiang (April 2020, IISE ANNUAL CONFERENCE & EXPO)

   null (Ed.)

   There is a growing importance in characterizing 3D shape quality in additive manufacturing (a.k.a. 3D printing). To accurately define the shape deviation between the designed product and actual build, shape registration of scanned point cloud data serves as a prerequisite for a reliable measurement. However, manual registration is currently heavily involved, for example, in obtaining initial matching of the design and the scanned product based on landmark features. The procedure can be inefficient, and more importantly, introduce potentially large operator-to-operator variations for complex geometries and deformation. Finding a sparse shape correspondence before refined registration would be meaningful to address this problem. In that case, automatic landmark selection has been a challenging issue, particularly for complicate geometric shapes like teeth. In this work we present an automatic landmark selection method for complicated 3D shapes. By incorporating subject matter knowledge (e.g., dental biometric information), a 3D shape will be first segmented through a new density-based clustering method. The geodesic distance is proposed as the distance metric in the revised clustering procedure. Geometrically informative features in each segment are automatically selected through the principal component analysis and Hotelling's T2 statistic. The proposed method is demonstrated in dental 3D printing application and could serve as a basis of sparse shape correspondence. 

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2. Scalar-on-Shape Regression Models for Functional Data Analysis

   Bhadra, Sayan; Srivastava, Anuj (November 2024, arXiv.org)

   Functional data contains two components: shape (or amplitude) and phase. This paper focuses on a branch of functional data analysis (FDA), namely Shape-Based FDA, that isolates and focuses on shapes of functions. Specifically, this paper focuses on Scalar-on-Shape (ScoSh) regression models that incorporate the shapes of predictor functions and discard their phases. This aspect sets ScoSh models apart from the traditional Scalar-on-Function (ScoF) regression models that incorporate full predictor functions. ScoSh is motivated by object data analysis, {\it, e.g.}, for neuro-anatomical objects, where object morphologies are relevant and their parameterizations are arbitrary. ScoSh also differs from methods that arbitrarily pre-register data and uses it in subsequent analysis. In contrast, ScoSh models perform registration during regression, using the (non-parametric) Fisher-Rao inner product and nonlinear index functions to capture complex predictor-response relationships. This formulation results in novel concepts of {\it regression phase} and {\it regression mean} of functions. Regression phases are time-warpings of predictor functions that optimize prediction errors, and regression means are optimal regression coefficients. We demonstrate practical applications of the ScoSh model using extensive simulated and real-data examples, including predicting COVID outcomes when daily rate curves are predictors. 

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3. SrvfRegNet: Elastic Function Registration Using Deep Neural Networks

   https://doi.org/10.1109/CVPRW53098.2021.00503  

   Chen, Chao; Srivastava, Anuj (June 2021, 2021 IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops (CVPRW))

   Registering functions (curves) using time warpings (re-parameterizations) is central to many computer vision and shape analysis solutions. While traditional registration methods minimize penalized-L2 norm, the elastic Riemannian metric and square-root velocity functions (SRVFs) have resulted in significant improvements in terms of theory and practical performance. This solution uses the dynamic programming algorithm to minimize the L2 norm between SRVFs of given functions. However, the computational cost of this elastic dynamic programming framework – O(nT 2 k) – where T is the number of time samples along curves, n is the number of curves, and k < T is a parameter – limits its use in applications involving big data. This paper introduces a deep-learning approach, named SRVF Registration Net or SrvfRegNet to overcome these limitations. SrvfRegNet architecture trains by optimizing the elastic metric-based objective function on the training data and then applies this trained network to the test data to perform fast registration. In case the training and the test data are from different classes, it generalizes to the test data using transfer learning, i.e., retraining of only the last few layers of the network. It achieves the state-of-the-art alignment performance albeit at much reduced computational cost. We demonstrate the efficiency and efficacy of this framework using several standard curve datasets. 

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4. Uncovering Specific-Shape Graph Anomalies in Attributed Graphs

   https://doi.org/10.1609/aaai.v33i01.33015433  

   Wu, Nannan; Wang, Weijun; Chen, Feng; Li, Jianxin; Li, Bo; Huai, Jinpeng (July 2019, The Thirty-Third AAAI Conference on Artificial Intelligence (AAAI-19))

   As networks are ubiquitous in the modern era, point anomalies have been changed to graph anomalies in terms of anomaly shapes. However, the specific-shape priors about anomalous subgraphs of interest are seldom considered by the traditional approaches when detecting the subgraphs in attributed graphs (e.g., computer networks, Bitcoin networks, and etc.). This paper proposes a nonlinear approach to specific-shape graph anomaly detection. The nonlinear approach focuses on optimizing a broad class of nonlinear cost functions via specific-shape constraints in attributed graphs. Our approach can be used in many different graph anomaly settings. The traditional approaches can only support linear cost functions (e.g., an aggregation function for the summation of node weights). However, our approach can employ more powerful nonlinear cost functions and enjoys a rigorous theoretical guarantee on the near-optimal solution with the geometrical convergence rate. 

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5. Shape Analysis of Functional Data with Elastic Partial Matching

   https://doi.org/10.1109/TPAMI.2021.3130535  

   Bryner, Darshan; Srivastava, Anuj (November 2021, IEEE Transactions on Pattern Analysis and Machine Intelligence)

   Elastic Riemannian metrics have been used successfully for statistical treatments of functional and curve shape data. However, this usage suffers from a significant restriction: the function boundaries are assumed to be fixed and matched. Functional data often comes with unmatched boundaries, {\it e.g.}, in dynamical systems with variable evolution rates, such as COVID-19 infection rate curves associated with different geographical regions. Here, we develop a Riemannian framework that allows for partial matching, comparing, and clustering functions under phase variability {\it and} uncertain boundaries. We extend past work by (1) Defining a new diffeomorphism group G over the positive reals that is the semidirect product of a time-warping group and a time-scaling group; (2) Introducing a metric that is invariant to the action of G; (3) Imposing a Riemannian Lie group structure on G to allow for an efficient gradient-based optimization for elastic partial matching; and (4) Presenting a modification that, while losing the metric property, allows one to control the amount of boundary disparity in the registration. We illustrate this framework by registering and clustering shapes of COVID-19 rate curves, identifying basic patterns, minimizing mismatch errors, and reducing variability within clusters compared to previous methods. 

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