Mathematical Methods for Images and Surfaces

The last two decades have witnessed a greatly increasing interest in mathematical methods for images and surfaces that are originated from biomedical and biological applications. The research problems have necessitated the development of mathematical theories of wavelets, frames, harmonic analysis, geometric flows, differential geometry, topology statistical methods, machine learning, and so forth. This theoretical development is enhanced by the related numerical algorithm development and modeling development. Currently, mean curvature flow, Willmore flow, level set, generalized Laplace-Beltrami operator are commonly used mathematical techniques for the analysis of biomedical images and for the generation of biomolecular surfaces. Additionally, wavelets, frames, compressive sensing, and harmonic analysis are popular tools for biomedical visualization and image processing. Moreover, differential geometry, topology, and geometric measure theory are powerful approaches for the multiscale modeling of biomolecular structure, dynamics, and transport. Finally, persistently stable manifold, topological invariant, Euler characteristic, Frenet frame, and machine learning are vital to the dimen-sionality reduction of extremely massive biomedical and biomolecular data. These ideas have been successfully paired with current investigation and discovery in biomedical and biological systems. However, many mathematical challenges remain in image and surface analysis, such as the well-posedness of mathematical models under physical and biological constraints, maximum-minimum principle for high-order geometric evolution equations, numerical analysis of multiply coupled partial differential equations, effectiveness of approximation theory, and the modeling of complex biomolecular phenomena. This special issue was called to address mathematical difficulties and challenges in image and surface analysis. It consists of seventeen research papers. The first paper presents octahedral symmetrical adapted functions (OSAFs) for three-dimensional (3D) reconstruction of macromolecule assembles. The authors employed two biomolecules, namely, heat shock protein Degp24 and the Red-cell L Ferritin, to verify the feasibility and advantages of their OSAF method. Their results indicate that the OSAF method is feasible and efficient for reconstructing structures of macromolecules. The second paper outlines statistical evaluations of the reproducibility and reliability of 3-Tesla high-resolution magnetization transfer brain images, which are very promising for early detection and monitoring of subtle brain changes. This paper analyzes 9 healthy subjects acquired from 12 brain regions of interest. It computes Spearman's correlation coefficient, coefficient of variation, and intraclass correlation coefficient. Multivariate mixed-effects regression models are built, and sensitivity analysis of various model specifications is carried out. The third paper investigates myocardial deformation and strain using suitably encoded cine MRI that admits disambiguation of material motion. The authors proposed a new method to …