image: More and more abstract mathematics has been applied in the research of computer graphics, geometric modeling and computer vision. The scope covers from tensor algebra, exterior algebra, geometric algebra, projective geometry, differential geometry, Riemannian geometry to topology, differentiable manifolds and Hilbert spaces. view more
Credit: World Scientific Publishing, 2014
Professor Hongyu Guo has published his latest book "Modern Mathematics and Applications in Computer Graphics and Vision" with World Scientific, which makes significant contribution to the literature and research community of computer graphics and computer vision.
Unlike other areas of computer science, where discrete mathematics is mostly applied, computer graphics and vision, as well as game programming, utilize many areas of continuous mathematics. For example, ray tracing in computer graphics rendering is purely the computational simulation of the optical process using laws in physics.
The regular university curriculum does not provide proper preparation of the continuous mathematics needed for these students. Researchers and professionals in these areas have their struggles too. More and more modern arsenal of abstract mathematics is utilized in research literature, for example, the concepts of tensors and differentiable manifolds. Prof. Guo's book is intended to help them to better understand and apply the concepts and theories needed in their study and research. Prof. Guo's book also explains many state of the art research applications in computer graphics and machine learning, ranging from color spaces, support vector machines and algorithms in manifold learning.
Prof. Guo's book also provides results in perspective analysis and perspective depth inference in his recent research. As early as Renaissance, in the first treaties on perspective—On Painting, Leon B. Alberti raised questions like what are the relationships of the images of the same scene observed from different positions. When the scenes are planar, the answers were well understood in the context of projective geometry in the nineteenth century. However, there have not been explicit discussions in literature when the scene is 3-dimensional. Prof. Guo explores these problems and provides his research results.
Perspective depth inference from a single image is also investigated. Some new concepts---perspective diminution factor and the perspective foreshortening factor---are proposed and defined in a rigorous manner and applied to the analysis of perspective distortion.
Photo forensic analysis is one of the applications of perspective depth inference.
There have been cases in the court, in which both the plaintiff and the defendant provide photo evidences of the scene of interest or dispute. An epigraph in the book has the following quote: "Photographs do not lie … too often." Indeed, different photographs of the same scene can give people quite different perceptions. In the court case, the plaintiff and the defendant may provide different photographs, being shot with a wide-angle lens, or with a telephoto lens from different distances. Both the plaintiff and the defendant try to use the photographs to their own advantage to show either the distance in the scene is far or close. Both photos are real and unaltered. However, the spatial perception from the photographs are quite different. The distance in a wide-angle image looks much bigger than in the one with telephoto lens. With perspective depth inference on the images, the spatial distance can be calculated and inferred through the measurements on the photograph. Furthermore, the camera position when the photograph was shot can be inferred through measurements on the photograph as well. This may provide key forensic evidences in many court cases.
More and more abstract mathematics has been applied in the research of computer graphics, geometric modeling and computer vision. The scope of this book covers from tensor algebra, exterior algebra, geometric algebra, projective geometry, differential geometry, Riemannian geometry to topology, differentiable manifolds and Hilbert spaces. More information on the book can be found at http://www.worldscientific.com/worldscibooks/10.1142/8703.