PDE-based Image Restoration: Efficient Numerical Algorithms and Software Engineering

As the field of image processing (IP) requires higher levels of reliability and efficiency, mathematical IP has become an important component. In particular, mathematical frameworks employing recent powerful tools of partial differential equations (PDEs) have been extensively studied to answer fundamental questions in IP. Such PDE-based methods turn out to allow researchers not only to introduce innovative mathematical models but also to analyze and improve traditional algorithms most of which have been developed heuristically. Developing appropriate numerical techniques for the PDE models is another important component for the PDE-based approaches. The proposal is concerned with the development of numerical algorithms for PDE-based image restoration and their applications to challenging problems. The models to be solved include nonlinear PDEs representing motion by mean curvature and Laplacian mean curvature flows, p-harmonic maps, and curvature-based total-variation minimization. The main goals are (a) to develop reliable and efficient numerical algorithms for image restoration, (b) to apply those algorithms for noise removal, image enhancement, and inpainting for real-life images such as medical imagery and satellite images, and (c) to construct related software packages. The newly developed numerical algorithms are expected to deliver large impact on mathematical image analysis; the resulting software packages must be applicable to various interesting problems dealing with images. Many applications in the modern digital age are based on images and therefore the resulting achievements must rely on their quality. Since images are not always in a good quality due to various types of noise, e.g., natural noise, defects in the sensors, and transmission problems, it is important to eliminate the noise automatically. Such image restoration is historically one of the oldest concerns and still a necessary processing step. As the field requires higher levels of reliability and efficiency for the last two decades, mathematical (PDE-based) image restoration has become an important component; it has been extensively studied to answer fundamental questions in image processing and to analyze/improve traditional methods. The proposal is concerned with the development of reliable and efficient computational algorithms for mathematical image restoration, their applications to real-life images, and the construction of software packages. The research emphasis will be on the computational aspects, trying to achieve truly practical algorithms based on the PDE models. Such reliable, efficient, practical algorithms will be implemented for software packages, which will be applicable for critically important problems including medical imaging, security control, crime scene investigation, and environmental watch. The proposed approaches and results can surely deliver benefits not only to research and education in academia but also to practitioners' image processing in industry. In particular, the software packages will provide convenient sources that are easy to maintain and modify for various applications. The investigators propose to explore mathematical frameworks and software engineering techniques to produce reliable and efficient numerical algorithms and related software packages which can support not only research-and-development but also classroom situations.