Geometry processing

Polygon Mesh Processing by Mario Botsch et al. is a textbook on the topic of Geometry Processing.

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.

R3). 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 Möbius 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 c is the number of connected components, χ=|V|−|E|+|F|.

This image shows a mesh of a pair of pants, with Euler characteristic -1. This is explained by the equation to compute the characteristic: 2c - 2h - b. The mesh has 1 connected component, 0 topological holes, and 3 boundaries (the waist hole and each leg hole): 2 - 0 - 3 = -1. A triangle mesh is constructed out of a point cloud. Sometimes shapes are initialized only as "point clouds," a collection of sampled points from the shape's surface. Often, these point clouds need to be converted to meshes.

Depending on how a shape is initialized or "birthed," the shape might exist only as a nebula of sampled points that represent its surface in space. To transform the surface points into a mesh, the Poisson reconstruction strategy can be employed. This method states that the indicator function, a function that determines which points in space belong to the surface of the shape, can actually be computed from the sampled points. The key concept is that gradient of the indicator function is 0 everywhere, except at the sampled points, where it is equal to the inward surface normal. More formally, suppose the collection of sampled points from the surface is denoted by ni. Then the gradient of the indicator function is defined as:

‖▽χ−𝐕‖ is minimized, where (x,y,z) with X with surface X onto surface t that minimize the following objective function:

X is small. An iterative solution such as Iterative Closest Point (ICP) is therefore employed to solve for small transformations iteratively, instead of solving for the potentially large transformation in one go. In ICP, n random sample points from λ:

ℒ emits the necessary condition

∇2 is chosen to be M−1 term is to map the image of the Laplacian from areas to points. Because the variation is free, this results in a self-adjoint linear problem to solve with a parameter L is through analyzing the geometry of connected triangles on the mesh.

βij are the angles opposite the edge minU∑ij∈E||ui−uj||2

Where minu,v∫S||∇u||2+||∇v||2dA

There are a few other things to consider. We would like to minimize the angle distortion to preserve orthogonality. That means we would like minu,v∫S12||∇u||2+12||∇v||2−∇u⋅∇v⊥

To avoid the problem of having all the vertices mapped to a single point, we also require that the solution to the optimization problem must have a non-zero norm and that it is orthogonal to the trivial solution.

An example of as-rigid-as-possible deformation

Deformation is concerned with transforming some rest shape to a new shape. Typically, these transformations are continuous and do not alter the topology of the shape. Modern mesh-based shape deformation methods satisfy user deformation constraints at handles (selected vertices or regions on the mesh) and propagate these handle deformations to the rest of shape smoothly and without removing or distorting details. Some common forms of interactive deformations are point-based, skeleton-based, and cage-based. In point-based deformation, a user can apply transformations to small set of points, called handles, on the shape. Skeleton-based deformation defines a skeleton for the shape, which allows a user to move the bones and rotate the joints. Cage-based deformation requires a cage to be drawn around all or part of a shape so that, when the user manipulates points on the cage, the volume it encloses changes accordingly.

Handles provide a sparse set of constraints for the deformation: as the user moves one point, the others must stay in place.

A rest surface x̂:Ω→ℝ3, where S. Ideally, the transformed shape adds as little distortion as possible to the original. One way to model this distortion is in terms of displacements xi=Rxî+t to each handle i, where 𝐑:

isInside(q) which will return S, and r in any direction from a query point, and count the number of times S then the ray must either not pass through S it must pass through twice, because S is bounded, so any ray entering it must exit. So if q is inside, the same logic applies to the previous case, but the ray must intersect isInsider(q)={1countr is odd0countr is even

Now, oftentimes we cannot guarantee that the S an odd number of times. To quantify this, let us say we cast rayTest(q)=1k∑i=1kisInsideri(q) which is the average value of q of each triangle in the mesh to determine if wn(q) is proportional to the sum of the solid angle contribution from each triangle in the mesh:

S. Therefore, we say:

1. 1 2 Botsch, Mario; Kobbelt, Leif; Pauly, Mark; Alliez, Pierre (2010). Polygon Mesh Processing. CRC Press. ISBN 9781568814261.
2. ↑ Szymon Rusinkiewicz, Marc Levoy. "Efficient Variants of the ICP Algorithm" (PDF).
3. ↑ Levy, Bruno (2002). "Least squares conformal maps for automatic texture atlas generation" (PDF). ACM Transactions on Graphics. 21 (3): 362–371. doi:10.1145/566654.566590. Archived from the original (PDF) on 2017-03-15. Retrieved 2017-03-14.
4. ↑ Sorkine, Olga; Alexa, Marc (2007). "As-Rigid-As-Possible Surface Modeling" (PDF). Proceedings of EUROGRAPHICS/ACM SIGGRAPH Symposium on Geometry Processing: 109–116.
5. ↑