Multiresolution Mesh Morphing

1
Multiresolution Mesh Morphing

• By Aaron Lee et al.

http//cm.bell-labs.com/who/wim/papers/morphing/
Presented By Boris Lipshitz Visualization and
Animation - winter 2002
2
Given Problem and Goal of the work

• Build User controlled morphing of two triangle
  meshes of arbitrary topology.
• Morphing (Metamorphosis) - process of gradually
  changing a source object through intermediate
  objects into a target object.
• Any two irregular connectivity meshes with the
  only requirement that they be topologically
  equivalent.
• The large size of the meshes makes it difficult
  to manipulate them efficiently - dealt using
  Multiresolution representation.

3
Overview

• Establishing correspondence map between source
  and target meshes with suitable user controls.
• MAPS (Multiresolution Adaptive Parameterization
  of Surfaces) algorithm to parameterize both
  meshes over simple base domains.
• Bring base domains to correspondence.
• User control of mapping by specifying any number
  of feature pairs and allowing the user to tune
  the mapping between the base domains.

4
Overview of the correspondence map computation

• The user specifies pairs of feature points(red)
  and lines (yellow) in the original mesh.

5
Hierarchical Surface Representation

• The approach is inspired by hierarchy proposed by
  Dobkin-Kirkpatrick (DK) , which guarantees that
  the number of levels L is O(logN). While the
  original DK hierarchy is built for convex
  polyhedra this approach can deal with general
  polyhedra.
• The basic idea is vertex
• removing with all his adjacent
• faces and then retriangulation
• of the hole.

6
Vertex Removal

• Instead of DK algorithm approach based only on
  topological information with random selection of
  vertices , this one uses priority queue based on
  both geometric and topological information.
• Vertices with small and flat 1-ring neighborhoods
  will be chosen first.
• At level l every vertex of outdegree less than 12
  will be assigned a priority inversely
  proportional to combination of area and curvature

7
Example of mesh hierarchy
8
Flattening and Retriangulation

• After vertex removal hole should be
  retriangulated. The easiest way to do it is
  flattening it , and then retriangulating using ,
  for example , constrained Delaunay triangulation
  (CDT).
• One possibility is to find a plane into which to
  project the hole without overlaping triangles ,
  but its very expansive.
• Conformal map is used which is always exist, its
  easy to compute, and its a bijection and thus
  never maps two triangles on top of each other

9
Flattening and Retriangulation -Cont
10
Parameterization

• Constracting a bijection from original mesh to
  base domain.
• The mapping of a point through is a point
  which can be written as
• where is a face of the base domain
  and are barycentric coordinates, i.e.,
• is computed successively between original mesh
  and intermediate one until we reach base
  domain.

11
Parameterization - Cont.

• After retriangulation of a hole in the plane, the
  just removed vertex gets assigned barycentric
  coordinates with respect to the containing
  triangle on the coarser level. Similarly, all the
  finest level vertices that were mapped to a
  triangle of the hole now need to be reassigned to
  a triangle of the coarser level.

12
Parameterization - Cont.

• Visualization of computed mapping. For each point
  from the original mesh, its mapping is
  shown with a dot on the base domain.

13
The Base Domain Correspondence Map

• Construction of the base domain correspondence
  map consists of the following steps
• globally align the source and destination base
  domains and project the source base domain to the
  target base domain.
• apply an iterative relaxation procedure to
  improve the mapping.
• user adjustment of the coarse correspondence to
  produce the final mapping.

14
Global alignment of base domains

• Feature points are guaranteed to be in the base
  domain, then their correspondence map is

15
Relaxation on a Mesh

• The initial projection is improved through an
  iterative relaxation procedure.

The vertex is moved in a direction computed as a
weighted average of the directions given by the
shortest path (bold arrows)
16
Additional Controls

• The user can exercise further control in the base
  domain correspondence mapping
• Vertex on the source base domain onto any point
  on the target base domain.

User allowed to map a vertex on one domain to a
vertex , point on an edge, or a point in a
triangle on the other base domain.
17
Extending

• How to compute the map for any point (not vertex)
  to the source base domain?

The source base domain triangle maps to a
triangular shaped region (shaded) on the target
base domain. Compute piecewise linear harmonic
mapping (Eck et al.) of this region to a triangle
so as to place the target base domain vertices on
the source base domain.
18
The final Correspondence Map

• Our goal

What we get so far
which are the source vertices placed on the
target mesh with the source connectivity
19
The Metamesh

• The purpose of the metamesh is to combine the
  source connectivity and target connectivity.
• To define target connectivity vertices, edges and
  faces needed to be found.

Vertices defined as follows
20
Tracing Edge Segments

• To find the connectivity of the metamesh the
  edges are drawn on the target mesh between the
  points

• If two vertices of some finest level source edge
  belong to the same target triangle - connect them
  with a line.
• Otherwise - connect them with a segmented line.

21
Tracing Edge Segments -Cont
In practice its easiest to find the intersection
points in the target base domain and map them
through

• Finally these segments have cut the target
  triangles into polygons. Retriangulate those
  polygons and add new edges and triangles to
  metamesh connectivity.

22
Properties of the Metamesh

• Finding intersection points is easier if the
  source mesh contains triangles which are smaller
  than the target mesh triangles.
• By using metamesh same triangles can be lined up
  to form source triangles and then target
  triangles.
• The positions of intermediate meshes needed in
  the morph are given by
• The user can specify how will vary with time
  and also spatial control is provided by letting
  depend on location

23
Results
24
Conclusions and Future Work

• The main restriction of this method is the
  requirement that source and target share the same
  genus. Thus fundamental work on extending MAPS to
  deal with genus changes is needed.
• Once a dense correspondence map is established
  the morph is done by spatially varying linear
  interpolation. But more sophisticated controls
  would be desirable in production work.
• When source and target are quite dissimilar , the
  user needs to spend more time to guide the
  correspondence map. More tools which naturally
  combine user control with automated computation
  are needed