This book considers specific inferential issues arising from the analysis of dynamic shapes with the attempt to solve the problems at hand using probability models and nonparametric tests. The models are simple to understand and interpret and provide a useful tool to describe the global dynamics of the landmark configurations. However, because of the non-Euclidean nature of shape spaces, distributions in shape spaces are not straightforward to obtain.
The book explores the use of the Gaussian distribution in the configuration space, with similarity transformations integrated out. Specifically, it works with the offset-normal shape distribution as a probability model for statistical inference on a sample of a temporal sequence of landmark configurations. This enables inference for Gaussian processes from configurations onto the shape space.
The book is divided in two parts, with the first three chapters covering material on the offset-normal shape distribution, and the remaining chapters covering the theory of NonParametric Combination (NPC) tests. The chapters offer a collection of applications which are bound together by the theme of this book.
They refer to the analysis of data from the FG-NET (Face and Gesture Recognition Research Network) database with facial expressions. For these data, it may be desirable to provide a description of the dynamics of the expressions, or testing whether there is a difference between the dynamics of two facial expressions or testing which of the landmarks are more informative in explaining the pattern of an expression.
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Part I Offset Normal Distribution for Dynamic Shapes |
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1 Basic Concepts and Definitions |
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3 | (12) |
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1.1 Landmark Coordinates and the Configuration Space |
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3 | (2) |
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5 | (2) |
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1.3 Coordinate Systems in Two Dimensions |
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7 | (8) |
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1.3.1 Kendall's Shape Coordinates |
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7 | (1) |
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1.3.2 Bookstein Coordinates |
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7 | (2) |
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1.3.3 Procrustes Coordinates |
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9 | (4) |
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13 | (2) |
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2 Shape Inference and the Offset-Normal Distribution |
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15 | (18) |
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15 | (1) |
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2.2 The Gaussian Distribution in the Configuration Space |
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16 | (3) |
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2.3 The Gaussian Distribution in the Pre-form Space |
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19 | (1) |
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2.4 The Offset-Normal Shape Distribution |
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19 | (3) |
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2.5 EM Algorithm for Estimating μ and Σ |
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22 | (4) |
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2.5.1 EM for Complex Covariance |
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24 | (2) |
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2.5.2 Cyclic Markov and Isotropic Covariances |
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26 | (1) |
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2.6 Data Analysis: The FG-NET Data |
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26 | (7) |
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30 | (3) |
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3 Dynamic Shape Analysis Through the Offset-Normal Distribution |
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33 | (26) |
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3.1 The Offset-Normal Distribution in a Dynamic Setting |
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34 | (3) |
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3.1.1 The Probability Density Function |
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35 | (2) |
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3.2 EM Algorithm for Estimating μ and Σ |
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37 | (4) |
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3.3 Separable Covariance Structure |
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41 | (7) |
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3.3.1 EM for the Offset Shape Distribution of a Matrix-Variate Normal Distribution |
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42 | (2) |
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3.3.2 Complex Landmark Covariance Structures |
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44 | (1) |
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45 | (2) |
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3.3.4 Temporal Independence |
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47 | (1) |
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3.4 Offset Normal Distribution and Shape Polynomial Regression for Complex Covariance Structure |
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48 | (3) |
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3.4.1 Modeling the Dynamics of Facial Expressions by Shape Regression |
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50 | (1) |
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51 | (2) |
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3.6 Mixture Models for Classification |
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53 | (6) |
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55 | (4) |
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Part II Combination-Based Permutation Tests for Shape Analysis |
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4 Parametric and Non-parametric Testing of Mean Shapes |
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59 | (14) |
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4.1 Inferential Procedures for the Analysis of Shapes |
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59 | (2) |
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4.2 NPC Approach in Shape Analysis |
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61 | (8) |
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4.2.1 Brief Description of the Nonparametric Methodology |
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61 | (3) |
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4.2.2 A Two Independent Sample Problem with Landmark Data |
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64 | (2) |
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4.2.3 A Suitable Algorithm |
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66 | (3) |
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4.3 General Framework for Longitudinal Data Analysis in NPC Framework |
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69 | (4) |
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71 | (2) |
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5 Applications of NPC Methodology |
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73 | (32) |
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73 | (2) |
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75 | (1) |
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5.3 NPC Methodology for Longitudinal Data |
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76 | (18) |
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5.4 Introduction on Paired Landmark Data |
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94 | (2) |
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5.5 Evaluating Symmetry Within Happy Facial Expression: Object Symmetry |
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96 | (9) |
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102 | (3) |
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Appendix A Shape Inference and the Offset-Normal Distribution |
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105 | (8) |
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A.1 EM Algorithm for Estimating μ and Σ |
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105 | (2) |
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A.2 EM for Complex Covariance |
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107 | (6) |
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111 | (2) |
| Index |
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113 | |
Chiara Brombin is Assistant Professor in Statistics at the Faculty of Psychology (University Vita-Salute San Raffaele, Milano) and national coordinator of the research project FIRB 2012 (RBFR12VHR7) "Interpreting emotions: a computational tool integrating facial expressions and biosignals based on shape analysis and Bayesian networks". Her research interests focus on applied statistics and include nonparametric permutation tests, statistical shape analysis, multivariate statistics, linear mixed-effect models, joint models for longitudinal and time-to-event data.
Luigi Salmaso is Full Professor of Statistics at the Department of Management and Engineering at University of Padova. His research interests include biostatistics, statistical methods for marketing research, design of experiments, nonparametric statistics and agricultural statistics. Specific topics of interests include permutation tests, resampling techniques and ranking and selection methods.
Luigi Ippoliti is an Associate Professor in Statistics at the University "G. d'Annunzio"of Chieti Pescara, Italy. His research activity is mainly focused on the analysis of multivariate processes with temporal, spatial and spatio-temporal structures with interests in economic, environmental and Neuro-Physiological applications.
Specific topics of interests include hierarchical spatio-temporal models, image processing, functional data analysis and dynamic shape analysis.
Lara Fontanella is a Researcher in Statistics at the University G. d'Annunzio of Chieti-Pescara, Italy. Her research interests focus mainly on Latent Variable models and Statistical Analysis of Dynamic Shapes, with applications to environmental, neuro-physiological, social and economic data.
Caterina Fusilli holds a Bachelor's Degree in Statistics and Information Technologies and a Master Degree in Statistics for Biomedicine, Environment and Technology from the University "La Sapienza" of Rome. She also received the Ph.D degree in Economics and Statistics from the University "G. d'Annunzio" of Chieti - Pescara. She is a postdoctoral research fellow in the Bioinformatic unit at the IRCCS Casa Sollievo della Sofferenza - Mendel Institute (Rome). Her research interests include the Next-Generation Sequencing, Bioinformatics, Shape Analysis, Cluster Analysis and Finite Mixture Models.
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