[notes] polygon mesh processing learning notes (10)

Spherical parametrizations

Definition

• hold genus-0 The mesh of is mapped to the sphere

Spherical parameterization , It is required that the genus of the grid is 0, Closed

• application

  • Corresponding ,correspondence
  • deformation ,morphing
  • remeshing

• Problem analysis

  • Spherical constraints
  • Bijective constraints （no foldover）
  • low distortion

• Challenge

  • No Tutte’s embedding method
  • Non-linear, non-convex optimization problem

Hierarchical method

A hierarchical approach

• Pipeline
  • mesh do decimation, Then map to the sphere , then refinement, The final output triangular sphere
• Decimation
  • curvature error metric （CEM）
  • According to measurement , Each time, we simplify the parts with high curvature
• Refinement
  • Insert a new vertex on the ball
  • Insert a point , Check for distortion , To optimize

Two hemispheres

Two hemispheres

• Pipeline
  • Divide the grid into two sub grids
  • Map each sub mesh onto a flat disk
  • Map the disk onto the hemispherical surface
  • Get spherical mapping

Curvature flow

Directional Field

Definition

• Direction information of spatial transformation , Assign to each point

  • Defined on a domain , Each store gives one direction

• Magnitude + direction

• Multi-valued field

  • Multiple directions （ It may be rotationally symmetric , For example, rotation 90 degree ）
  • You can think of a set that defines a direction
  • such as ,Rotationally-symmetric direction fields （RoSy fields）
  • The number of directions can be defined $N=1,2,4,6$
  • Yes RoSy Come on , Yes 4 A direction , Apart, 90 degree
  • It can also be two pairs of independent directions （ Total or 4 A direction , Every pair The difference between 180 degree , But the two one. pair There is no angular relationship between them ）

• Common fields ：1-vector field （ There is size and direction ）,2-direction field （ Only the direction has no size , Also called line field,2-RoSy field）,$1^3$-vector field（ Three independent vectors , This 3 Express 3 Independent ）;4-vector field（4 Vector field of direction , There is only one independent direction ）;4-direction field（ No length , Also called unit cross field,4-RoSy field）,$2^2$-vector field（ Two pairs of good villages , Also called frame field）,$2^2$-direction field ( Two pairs of directions , Also called non-ortho. cross field)…

• Some examples

  • Direction of principal curvature （2-direction field）
  • Stress or strain tensors ( Stress )
  • Gradient field （ One or more directions ）
  • The advection field of a flow
  • Diffusion data from MRI

• Generate or design

  • User constraints

  • Alignment conditions

  • Fairness objectives

  • Physical requirements

• application

  • Mesh generation（ Quadrilateral mesh generation ）
  • All-hex meshing（ Hexahedral mesh generation ）
  • Deformation
  • Texture mapping and synthesis
  • Architectural geometry （ Conjugate direction ）

Discretization

Tangent spaces Tangent space

• The tangent plane can be defined on the patch 、 edge 、 On vertex
• Construct a tangent plane for a point , Count him surface normal vector
  • On the other hand , Is the normal direction
  • Opposite edge or vertex , Is the local average

Discrete connections

• Given two tangent planes $i$ and $j$, Need one connection, To compare two directional objects defined on them .
• The most common method is to parameterize the two tangent planes , Compare in the parameter plane .Levi-Civita

Vector field topology

• Singular points of vector fields , At this point the vector disappears or does not well-defined
• 2D case

Discrete field topology

• Period jump

Matching: multi-valued field

• $N>1$ directionals per tangent space
• An additional degree of freedom:
  • the correspondence between the individual directionals in tangent space $i$ To those in the adjacent tangent space $j$;
• A matching between two $N-$sets of directional is a bijective map $f$ between them (or their indices).
  • It preserves order: $f(u_r)=v_s\leftarrow \to f(u_{r+1})=v_{s+1}$

Effort

• Based on a matching $f$, the notions of rotation and principal rotation can be generalized to multi-valued fields
• ${\delta }_{ij}^r$: rotation between $u_r$ and $f(u_r)$
• Effort of the matching : ${\sum }_{r=1}^N{\delta }_{ij}^r$

Representation

Angle-based

Pros: direcitons, period jumps, are represented explicitly

Cons: k

Cartesian and Complex

Complex Polynomials

• Consider the field as the root of a complex polynomial .
• Compare the coefficients of complex polynomials on two triangles

Objectives and Constraints

Different applications have different requirements .

Objective function

• smooth
• Parallelism ——As parallel as possible. Give Way effort As small as possible .
• Orthogonality

constraint

• alignment ：fit certain prescribed directions
• Symmetry
• Surface mapping

Integrable field