Exploration of Method for Defining Metrics between Symmetric Positive Semi-definite Matrices = 대칭 양의 준정부호 행렬 간의 거리 척도 정의 방법에 대한 탐구

다국어 초록 (Multilingual Abstract)

This paper studies geometry-aware dissimilarities between symmetric positive semidefinite (PSD) matrices arising from covariance descriptors of images. The motivating application is unsupervised clustering of material/texture images under nuisance factors such as scale, illumination, and pose. Each image is mapped to a covariance descriptor computed from dense local features consisting of RGB values and magnitudes of a bank of 2D Gabor wavelet responses, yielding a symmetric positive definite (SPD) matrix per image. Then we explore fixed-rank modeling by adopting the framework of [1]. Specifically, we map each descriptor to S+(p, n)
with p < n via eigenvalue pruning, and define pairwise dissimilarities as the length functional proposed by [1] and several standard distance metrics between SPD matrices. The resulting dissimilarity matrices are used for multidimensional scaling (MDS) and clustering. We compare against standard SPD dissimilarities (Frobenius, Log-Euclidean, and Affine-invariant Riemannian metric) under controlled experimental protocols, and analyze the effects of hyperparameters such as the target rank p and the weighting parameter k.

목차 (Table of Contents)

• Abstrarct i
• Contents ii
• List of Figures iv
• List of Tables v
• List of Algorithms vi
• Chapter 1 Introduction 1
• 1.1 Motivation and problem statement 1
• 1.2 Image-based clustering 1
• 1.3 Why geometry-aware dissimilarities 2
• 1.4 Related perspectives: SPD learning and applications 2
• 1.5 Challenges 2
• 1.6 Paper organization 3
• Chapter 2 Preliminaries 4
• 2.1 Notation 4
• 2.2 Local image features 4
• 2.2.1 RGB intensity 4
• 2.2.2 Complex-valued Gabor wavelet response 5
• 2.2.3 Local feature vector 5
• 2.3 Covariance descriptors 5
• 2.4 Geometry of SPD matrices and limitations of Euclidean distances 6
• 2.5 Fixed-rank PSD manifold (S(p, n), g) 6
• Chapter 3 Method 8
• 3.1 Covariance descriptor (CovD) construction 8
• 3.1.1 Sampling protocol and preprocessing 8
• 3.1.2 Remarks 8
• 3.2 Fixed-rank PSD representation and dissimilarity matrix construction 9
• 3.2.1 Rank pruning of covariance descriptors 9
• 3.2.2 Representation in S(p, n) via (U, R²) 9
• 3.2.3 Length-based dissimilarity on S(p, n) 10
• 3.2.4 Dissimilarity matrix for clustering and MDS 11
• 3.2.5 Practical considerations: choosing p and k 11
• Chapter 4 Experiments 12
• 4.1 Dataset 12
• 4.1.1 Data explanation 12
• 4.1.2 Dataset selection 12
• 4.2 Experimental protocol and evaluation setup 13
• 4.3 Baseline dissimilarities 13
• 4.4 Clustering and visualization method 14
• 4.4.1 Partitioning Around Medoids (PAM) 14
• 4.4.2 MDS (Multidimensional Scaling) 14
• 4.5 Quantitative clustering performance measures 14
• 4.5.1 Adjusted Rand Index (ARI) 14
• 4.5.2 Silhouette score 15
• 4.5.3 Evaluation Framework: From Matrix to Metrics 15
• 4.6 Experimental results and ablation studies 16
• 4.6.1 Qualitative Visualization via MDS 17
• 4.6.2 Quantitative Analysis 18
• Chapter 5 Discussion and Contributions 22
• 5.1 Summary of empirical findings 22
• 5.2 Contributions 23
• 5.3 Limitations 23
• 5.4 Future research directions 24
• Bibliography 25