Mathematical Methods for Objects Reconstruction

The volume collects several contribution to the INDAM workshop "Mathematical Methods for Objects Reconstruction: from 3D Vision to 3D Printing" held in Rome, February 10-12, 2021.

The 3D reconstruction and data acquisition has grown considerably in recent years with a wide spectrum of applications. However, there are still a lot of limitations in actual techniques motivating new efforts on mathematical modeling and numerical methods, often in real applications the presence of big data coming from cameras and sensors increases the complexity of the problem. This is particularly relevant for medical applications, where calibration of the system and accuracy are really important for the success of surgery. Modern 3D reconstruction techniques are expected to provide the theoretical grounds for new technological developments and for substantial advances in real-life applications. To this end, efficiency and reliability of numerical methods are crucial issues. In general, we aim at comparing and (possibly) integrating a number of techniques of heterogeneous origin, such as optimization problem via least-squares resolution, variational approaches or differential formulations, photometric stereo techniques and multi-view data. Potential interactions include exploring the relation between new mathematical models in computer vision as well as the development of numerical methods developed in different communities, from mathematics to engineering, from computer science to medicine. It is therefore important to compare these different classes of problems and methods adopted, analyzing advantages and drawbacks of them.

The goal of the workshop was precisely to discuss new methods and conceptual structures for managing these challenging problems.  The chapters reflect this goal and the authors are academic researchers and some experts from industry working in the areas of 3D modeling, computer vision, 3D printing and/or developing new mathematical methods for these problems.  The contributions present methodologies and challenges raised by the emergence of large-scale 3D reconstruction applications and low-cost 3D printers. The volume collects complementary knowledges from different areas of mathematics, computer science and engineering on research topics related to 3D printing, which are, so far, widely unexplored.

Young researchers and future scientific leaders in the field of 3D data acquisition, 3D scene reconstruction, and 3D printing software development will find a excellent introduction to these problems and  to the mathematical techniques necessary to solve them.