Motion Segmentation and Face Clustering Based on Low-Rank Representation

Low-Rank Representation (LRR) is a mathematical tool widely used in subspace clustering, particularly suitable for capturing structural information in high-dimensional data. In computer vision, LRR effectively addresses two classical problems: motion segmentation and face clustering. The core concept involves representing data as linear combinations of low-rank matrices to uncover underlying subspace structures. In implementation, this typically involves constructing a data matrix where columns represent feature vectors, then solving an optimization problem using nuclear norm minimization.

Motion Segmentation The objective of motion segmentation is to separate differently moving objects in video sequences. When applying LRR, features from video frames (such as trajectories or optical flow) are stacked into a data matrix. The LRR model decomposes this matrix into a low-rank component (representing common motion patterns) and sparse noise (capturing individual motion variations), enabling automatic segmentation of moving objects. Code implementation often requires trajectory extraction using algorithms like Kanade-Lucas-Tomasi (KLT) tracker, followed by matrix construction and optimization using Alternating Direction Method of Multipliers (ADMM) solver.

Face Clustering Face clustering aims to group face images according to identity. Since face images of the same identity typically lie in the same linear subspace, LRR constructs a global low-rank affinity matrix to characterize similarities between samples. Compared to traditional methods like K-means, LRR demonstrates greater robustness to noise and occlusions, effectively handling non-rigid variations such as facial expressions and lighting changes. Implementation typically involves face alignment and feature extraction (using techniques like PCA or deep features) before applying LRR optimization.

Key Implementation Points Data Preprocessing: Normalize motion trajectories or facial features to eliminate scale differences using z-score normalization or min-max scaling. Low-Rank Optimization: Solve for the low-rank matrix through nuclear norm minimization using algorithms like Singular Value Thresholding (SVT) or Iterative Thresholding. Clustering Enhancement: Apply spectral clustering or hierarchical clustering to the generated affinity matrix to improve grouping performance. MATLAB implementations commonly use built-in functions like 'svd' for matrix decomposition and 'spectralcluster' for final grouping.

MATLAB implementations of LRR typically rely on optimization toolboxes (such as ADMM solvers), with special attention required for computational efficiency when handling large-scale datasets. While LRR offers theoretical rigor, parameter selection (particularly regularization coefficients) significantly impacts results and requires task-specific tuning. Practical implementation should include cross-validation for parameter optimization and consider dimensionality reduction techniques for computational feasibility.