Small-Determinant Directional Dilation Matrices for Anisotropic Multiresolution Analysis

Dilation matrices are important in multiple subdivision and multiple multiresolution analysis, as they govern the process of data refinement and play a crucial role in capturing directional features. One common limitation in the existing methods is the relatively large determinant of their dilation matrices, leading to high computational and storage costs. To address this issue, this paper proposes a novel family of pairs of directional dilation matrices with determinant 3. Such dilation matrices satisfy the joint expansion property and directional sensitivity. The joint expansion property is verified via the joint spectral radius, while by connecting the action of the matrices to certain elliptic elements of PSL(2,R), their directional adaptability can be established. Compared to most of the existing dilation matrices, the proposed ones achieve a balance between determinant and directional adaptability and provide a new insight into the construction of directional dilation matrices. This makes them suitable for addressing practical anisotropic problems.