# Hole Filling

Document Sample

Atomic Volumes for
Mesh Completion

Joshua Podolak
Szymon Rusinkiewicz

Princeton University
Outline
   Problem Description

   Background

   Algorithm

   Results
Motivation
   Scanners usually need two unobstructed lines of sight
between the cameras and the model
   Even with multiple scans, there are areas containing
no depth information that need to be filled
Filling Holes
Holes bounded by ring of half-edges.
Triangulate?
Holes bounded by ring of half-edges.

In simple cases, it is sufficient to create a patch by
triangulating the half-edge loops around the holes.
Challenge

Triangulation is not enough.
Challenge

Triangulation is not enough.
Talk Overview
   Problem Description

   Background

   Algorithm

   Results
Related Work

   Point cloud reconstruction
[Amenta01, Kolluri04]

   Half-edge boundary triangulation
[Berg97, Liepa03]

   Implicit hole filling
[Curless96, Davis02, Masuda04, Ju04, Sharf04]

   Volumetric hole filling
[Murali97]
Related Work

   Point cloud reconstruction
[Amenta01, Kolluri04]

   Half-edge boundary triangulation
[Berg97, Liepa03]

   Implicit hole filling
[Curless96, Davis02, Masuda04, Ju04, Sharf04]

   Volumetric hole filling
[Murali97]
Related Work

   Point cloud reconstruction
[Amenta01, Kolluri04]

   Half-edge boundary triangulation
[Berg97, Liepa03]

   Implicit hole filling
[Curless96, Davis02, Masuda04, Ju04, Sharf04]

   Volumetric hole filling
[Murali97]
Related Work

   Point cloud reconstruction
[Amenta01, Kolluri04]

   Half-edge boundary triangulation
[Berg97, Liepa03]

   Implicit hole filling
[Curless96, Davis02, Masuda04, Ju04, Sharf04]

   Volumetric hole filling
[Murali97]
Talk Overview
   Problem Description

   Background

   Algorithm

   Results
Definition: Atomic Volume
A volume is atomic if it cannot be intersected by the
surface of the model.

An atomic volume must be either completely inside or
completely outside the model.

2 atomic volumes
Overview of Approach
1.   Divide all of space into atomic volumes.
   Regions of space that will either be wholly in or wholly out of
the reconstructed solid.
2.   For each atomic volume, decide whether it is inside
the mesh or outside.
3.   The boundary between interior volumes and exterior
volumes is the new surface.
Step 1: Spatial Partitioning
Step 1: Spatial Partitioning
Blank cubes do not contain any part of the input model.

Blank Region

Blank cubes
Step 1: Spatial Partitioning
Inside/Outside (IO): Cubes containing elements of the surface
away from the holes.

Blank Region

In Region

Out Region

IO cubes
Step 1: Spatial Partitioning
Hole cubes are subdivided until they can be
trivially triangulated [Mitchell92] .

Blank Region

In Region

Out Region

Hole Region
Hole cubes
Step 2: Label Assignment
   Most of the volumes can be labeled as inside or outside based on
the normals of the input mesh.
   Filling the hole requires labeling the remaining volumes as inside
and outside.

Inside

Outside
Step 2: Label Assignment
   Atomic volumes correspond to nodes in the graph with edges
between neighboring volumes.
Step 2: Label Assignment
   Atomic volumes correspond to nodes in the graph with edges
between neighboring volumes.

   Atomic volumes on either side of a surface are not connected.
Step 2: Label Assignment
   Atomic volumes corresponds nodes in the graph with edges
between neighboring volumes.

   Atomic volumes on either side of a surface are not connected.

   The mesh graph can be split into two disjoint sub graphs.
Mesh Graph

Blank Region

In Region

Out Region

Hole Region

Blank cube
Mesh Graph

Blank Region

In Region

Out Region

Hole Region

I/O cube
Mesh Graph

Blank Region

In Region

Out Region

Hole Region

Hole cube
Mesh Graph

Blank Region

In Region

Out Region

   In a watertight surface, there can be no   Hole Region
path between inside and outside nodes.
Step 2: Label Assignment
   Splitting the graph into two sub graphs is equivalent to
labeling each of the atomic volumes.
   Initially labeled atomic volumes must retain original
labeling.
Labeling Volumes
   Minimum Cut
–   Add source and sink nodes.
–   Constraint edges have a weight of infinity.

Sink
(outside)

Source
Constraint
edges                              (inside)
Creating Patch
   Every edge cut in the graph corresponds to a surface

Blank Region

In Region

Out Region

Hole Region
Topological Control
other sources of information such as Space Carving,
Shadow Carving, or direct user input.

   Adding an edge with weight infinity between an atomic
volume to the sink node constrains that volume to be
outside the model.

   Atomic volumes containing points with contradicting
constraints may be subdivided.
Topological Control
   In the figure on the right, points are added manually to
indicate that the space between the two toes is outside.

Initial mesh      Unconstrained         Constrained
solution              solution
Smoothing
   Big atomic volumes cause faceting of the surface.
   We introduce a smoothing step to address faceting.

Blank Region

In Region

Out Region

Hole Region
Smoothing
   Big atomic volumes cause faceting of the surface.
   We introduce a smoothing step to address faceting.
Smoothing
   Big atomic volumes cause faceting of the surface.
   We introduce a smoothing step to address faceting.

Unsmoothed         Smoothed
Smoothing
Coarse smoothing will changes the labeling of entire
atomic volumes.

Fine smoothing keeps the topology of the mesh graph
constant, but allows the boundary between adjacent
atomic volumes to change.

Unsmoothed                    Smoothed
Smoothing
Constrained Laplacian Smoothing.
We do not move vertices that cause the mesh graph to change.

Result after five iterations
Talk Overview
   Problem Description

   Background
–   Related Work
–   Volumetric Representation

   Algorithm

   Results
Results

Half-torus filled with two   Cross section of spiral torus.
different patches.
Results

Bunny: 70k faces. 78 sec.
Angel: 350k faces. 15 min.
Conclusions
   Atomic Volumes is a discrete volumetric method for
mesh completion.

   The use of an adaptive subdivision of space allows the
algorithm to focus on target areas.

   Using a mesh-graph provides flexibility in determining
the patch.
Future Work
   Soft constraints.

–   Flexible atomic volumes.
–   Thin plate energy.
–   Texture Synthesis.

   Signal processing.
–   Localized filtering.
–   Deformation.
Thank you
Mesh Graph
   A mesh graph contains a node for each atomic volume.

   An edge between neighboring atomic volumes represents the
strength of the belief that they will be labeled similarly.

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 views: 28 posted: 4/17/2010 language: English pages: 44