Secure computing, copyright protection, and language grammars also rely on fundamental algebraic methods for implementation. In computer graphics and geometric design, the digital representation of fully three-dimensional objects has been the subject of intense research activity in the past decade, leading to powerful new algorithms. Algebraic geometry, splines, wavelets, optics and solid geometry are but a few of the mathematical tools brought to bear on this vital area. Applications include computer graphics, surface and solid rendering, animation, flight simulation, virtual reality, and scientific and medical imagery. Inevitable technological advances in graphics performance will continuously change the requirements for real-time rendering and the complexity of models to be visualized.
Galilean invariance plays an important role in movie restoration and recognition of moving objects.
In general, the reliable recognition of objects must form an integral part of any functional multimedia system. These include recognition of visual objects, target recognition, satellite images, speech and language recognition, optical character recognition, and language. Most of these are still in their infancy, with the underlying mathematical theory still in need of development. Applying symmetry groups to recognition problems in a natural fashion requires the adaptation of classical geometric theories of differential invariants and algebraic invariants, subjects that formed the core of pure mathematical research in the last century. Recent work on symmetry-incorporating numerical approximations hold significant promise for computation of invariants and object recognition in physical images.