Mathematical analysis of image matching techniques

Image matching is a fundamental problem in Computer Vision with direct applications in robotics, remote sensing and geospatial data analysis. The present paper reports an analytical and experimental evaluation of classical local feature-based image matching algorithms in the context of satellite imagery, specifically, the Scale-Invariant Feature Transform (SIFT) and the Oriented FAST and Rotated BRIEF (ORB) that is built on the Features from Accelerated Segment Test (FAST) keypoint detector and the Binary Robust Independent Elementary Features (BRIEF) descriptor. Each method is decomposed into a common matching pipeline consisting of the keypoint detection, local descriptor extraction, descriptor matching, and geometric verification. Formally, an image with \( c \) color channels is modeled as a function \( I: \mathbb{R}^2 \rightarrow \mathbb{R}^c\). Keypoints (features) are locally distinctive points in the image, defined as spatial locations \( \mathbf{x} \in \mathbb{R}^2 \). Each keypoint is associated with a compact descriptor vector \( f(\mathbf{x}) \in \mathbb{R}^D \), where $D$ denotes the descriptor dimensionality. The descriptor matching stage attempts to establish a correspondence \( f(\mathbf{x}_i) \approx f(\mathbf{x}'_i) \) between sets of points according to a predefined distance metric, such that the pair \( (\mathbf{x}_i,\mathbf{x}'_i) \) represents the same physical location. To ensure geometric consistency, we incorporate a verification employing the Random Sample Consensus (RANSAC) algorithm, where a ho\-mog\-ra\-phy matrix \( \mathbf{H} \in R^{3 \times 3} \) is estimated to model a projective transformation between the images. A match is denoted as an inlier (or valid match) if it satisfies the condition \( \| \mathbf{x}'_i - \mathbf{H} \mathbf{x}_i \| < \epsilon \) for a predefined threshold $\epsilon$. The proportion of correspondences satisfying this constraint is referred to as the Inlier Ratio, is used as a measure of matching confidence and serves as the primary metric in the evaluation. The study utilizes a manually constructed dataset of satellite images. The dataset includes GPS-annotated map tiles with intentional partial overlaps between adjacent images enabling a reliable evaluation via a pairwise image matching. We examine the impact of varying the number of extracted keypoints on the resulting Inlier Ratio.