Processing mesh animations: from static to dynamic geometry and back

Static triangle meshes are the representation of choice for artificial objects, as well as for digital replicas of real objects. They have proven themselves to be a solid foundation for further processing. Although triangle meshes are handy in general, it may seem that their discrete approximation of reality is a downside. But in fact, the opposite is true. The approximation of the real object's shape remains the same, even if we willfully change the vertex positions in the mesh, which allows us to optimize it in this way. Due to modern acquisition methods, such a step is always beneficial, often even required, prior to further processing of the acquired triangle mesh. Therefore, we present a general framework for optimizing surface meshes with respect to various target criteria. Because of the simplicity and efficiency of the setup it can be adapted to a variety of applications. Although this framework was initially designed for single static meshes, the application to a set of meshes is straightforward. For example, we convert a set of meshes into compatible ones and use them as basis for creating dynamic geometry. Consequently, we propose an interpolation method which is able to produce visually plausible interpolation results, even if the compatible input meshes differ by large rotations. The method can be applied to any number of input vertex configurations and due to the utilization of a hierarchical scheme, the approach is fast and can be used for very large meshes. Furthermore, we consider the opposite direction. Given an animation sequence, we propose a pre-processing algorithm that considerably reduces the number of meshes required to describe the sequence, thus yielding a compact representation. Our method is based on a clustering and classification approach, which can be utilized to automatically find the most prominent meshes of the sequence. The original meshes can then be expressed as linear combinations of these few representative meshes with only small approximation errors. Finally, we investigate the shape space spanned by those few meshes and show how to apply different interpolation schemes to create other shape spaces, which are not based on vertex coordinates. We conclude with a careful analysis of these shape spaces and their usability for a compact representation of an animation sequence

Od 3D počítačového modelování k realitě a zpět

Technologický pokrok neustále zrychluje a v jeho popředí se hřejí technologie spojené s 3D počítačovým modelováním, jako třeba 3D tisk, skenování a technologie spojené s virtuálně rozšířenou realitou. Tato práce dává nahlédnout za závoj tajemna, jež tyto technologie tak často obklopuje. Cílem je nabídnout krátký pohled na každou ze zmíněných oblastí. Čtenář bude mít možnost na vše nahlédnout z pohledu praktického, teoretického i trochu ajťáckého. To vše v naději, že se podaří tyto tři tak často vzdálené náhledy přivést blíž jak k sobě vzájemně, tak ke čtenáři.Technological advancements are faster than ever and on the frontier are applications and mechanisms entwined with 3D computer aided modelling, such as 3D printing, scan- ning and extended reality technologies. This work gives a peek behind the veil of mystery surrounding these technologies. We aim to give a brief look into each of the mentioned areas and let the reader experience them practically, mathematically and algorithmically in hope to bring these three so often separated views closer together and to the reader. 1Department of Mathematics EducationKatedra didaktiky matematikyMatematicko-fyzikální fakultaFaculty of Mathematics and Physic