Geometry processing is an area of research that uses concepts from applied mathematics, computer science and engineering to design efficient algorithms for the acquisition, reconstruction, analysis, manipulation, simulation and transmission of complex 3D models. As the name implies, many of the concepts, data structures, and algorithms are directly analogous to signal processing and image processing. For example, where image smoothing might convolve an intensity signal with a blur kernel formed using the Laplace operator, geometric smoothing might be achieved by convolving a surface geometry with a blur kernel formed using the Laplace-Beltrami operator. Applications of geometry processing algorithms already cover a wide range of areas from multimedia, entertainment and classical computer-aided design, to biomedical computing, reverse engineering, and scientific computing. Geometry processing is a common research topic at SIGGRAPH, the premier computer graphics academic conference, and the main topic of the annual Symposium on Geometry Processing.
Geometry processing as a life cycle
Geometry processing involves working with a shape, usually in 2D or 3D, although the shape can live in a space of arbitrary dimensions. The processing of a shape involves three stages, which is known as its life cycle. At its "birth," a shape can be instantiated through one of three methods: a model, a mathematical representation, or a scan. After a shape is born, it can be analyzed and edited repeatedly in a cycle. This usually involves acquiring different measurements, such as the distances between the points of the shape, the smoothness of the shape, or its Euler characteristic. Editing may involve denoising, deforming, or performing rigid transformations. At the final stage of the shape's "life," it is consumed. This can mean it is consumed by a viewer as a rendered asset in a game or movie, for instance. The end of a shape's life can also be defined by a decision about the shape, like whether or not it satisfies some criteria. Or it can even be fabricated in the real world, through a method such as 3D printing or laser cutting.
Discrete Representation of a Shape Like any other shape, the shapes used in geometry processing have properties pertaining to their geometry and topology. The geometry of a shape concerns the position of the shape's points in space, tangents, normals, and curvature. It also includes the dimension in which the shape lives (ex. R 2 {\displaystyle R^{2}} or R 3 {\displaystyle R^{3}} ). The topology of a shape is a collection of properties that do not change even after smooth transformations have been applied to the shape. It concerns dimensions such as the number of holes and boundaries, as well as the orientability of the shape. One example of a non-orientable shape is the Mobius strip. In computers, everything must be discretized. Shapes in geometry processing are usually represented as triangle meshes, which can be seen as a graph. Each node in the graph is a vertex (usually in R 3 {\displaystyle R^{3}} ), which has a position. This encodes the geometry of the shape. Directed edges connect these vertices into triangles, which by the right hand rule, then have a direction called the normal. Each triangle forms a face of the mesh. These are combinatoric in nature and encode the topology of the shape. In addition to triangles, a more general class of polygon meshes can also be used to represent a shape. More advanced representations like progressive meshes encode a coarse representation along with a sequence of transformations, which produce a fine or high resolution representation of the shape once applied. These meshes are useful in a variety of applications, including geomorphs, progressive transmission, mesh compression, and selective refinement.
Properties of a shape
Euler Characteristic One particularly important property of a 3D shape is its Euler characteristic, which can alternatively be defined in terms of its genus. The formula for this in the continuous sense is χ = 2 c − 2 h − b {\displaystyle \chi =2c-2h-b} , where c {\displaystyle c} is the number of connected components, h {\displaystyle h} is number of holes (as in donut holes, see torus), and b {\displaystyle b} is the number of connected components of the boundary of the surface. A concrete example of this is a mesh of a pair of pants. There is one connected component, 0 holes, and 3 connected components of the boundary (the waist and two leg holes). So in this case, the Euler characteristic is -1. To bring this into the discrete world, the Euler characteristic of a mesh is computed in terms of its vertices, edges, and faces. χ = | V | − | E | + | F | {\displaystyle \chi =|V|-|E|+|F|} .
Surface reconstruction
Poisson reconstruction from surface points to mesh
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