Loading in 2 Seconds...

Regular and Irregular Multi-resolution Terrain Models: a Comparison

Loading in 2 Seconds...

- By
**tudor** - Follow User

- 123 Views
- Uploaded on

Download Presentation
## PowerPoint Slideshow about 'Regular and Irregular Multi-resolution Terrain Models: a Comparison' - tudor

**An Image/Link below is provided (as is) to download presentation**

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

### Regular and Irregular Multi-resolution Terrain Models: a Comparison

Leila De Floriani * Paola Magillo

Department of Computer Science

University of Genova, Genova (Italy)

* currently at the University of Maryland, College Park, MD

Outline

- Motivations
- Related Work
- Regular and Irregular Multi-Triangulations (MTs)
- Vertex-Based Multi-Triangulation (Vertex-based MT)
- Hierarchy of Right Triangles (HRT)
- Level-Of-Detail (LOD) Queries for Terrain Modeling
- Data Structures
- Experimental Results and Comparisons
- Summary and Future Work

Why Multi-resolution Terrain Models?

- Large-size terrain data sets
- high storage requirements and high processing times
- Multi-resolution terrain models
- encompass a collection of terrain representations at different levels of resolutions

Why Multi-resolution Terrain Models?

- Multi-resolution terrain models
- allow extracting terrain representations at a variable resolution

Regular and Irregular Multi-Resolution Models

- Data on a grid versus scattered data
- Regular versus irregular multi-resolution models:
- Regular models: based on a nested domain decomposition
- Irregular models: compact way of encoding the steps performed by a simplification process applied to an irregular triangle mesh

Regular and Irregular Multi-Resolution Models

- Both are instances of a common framework: the Multi-Triangulation
- Our objective: compare and analyze regular and irregular multi-resolution models
- Comparison:
- size of the models
- space requirements of their encoding structures
- efficiency in extracting meshes at variable resolution: Level-Of-Detail (LOD) queries

Related Work

- Regular multi-resolution models:
- Triangle quadtrees (Gomez and Guzman, 1979; Dutton, 1983)
- Restricted quadtrees (Von Herzen and Barr, 1987; Sivan and Samet, 1992)
- Hierarchies of right triangles (Duchaineau et a., 1997; Evans et al., 2001; Lindstrom et al., 1996; Pajarola, 1998)
- Irregular multi-resolution models:
- Nested meshes (De Floriani and Puppo, 1995; Scarlatos and Pavlidis, 1994)
- Pyramidal triangle meshes (De Berg and Dobrindt, 1995; De Floriani, 1989)
- Progressive meshes (Hoppe, 1996; Taubin et al., 1998)
- Continuous LOD models (Hoppe, 1998; Xia et al., 1997; Maheshvari et al.,1997; El Sana and Varshney 1999: De Floriani et al., 1998)

Basic Concepts: Modifications

- Modification of a triangle mesh: replace a connected set of triangles with other triangles covering the same region

Basic Concepts: Dependencies

- A modification M2 depends on a modification M1 iff M2 changes some triangles that have been changed by M1

M2 depends on M1

M2 does not depend on M1

The Multi-Triangulation (MT)

- A base mesh
- A set of modifications
- A partial order (dependency relation)

The Multi-Triangulation (MT)

- Vertex-Based Multi-Triangulation (Vertex-based MT):
- Scattered data
- Modification: vertex insertion
- Hierarchy of Right Triangles (HRT):
- Data on a grid
- Modification: simultaneous bisection of two adjacent right triangles

LOD Queries

- A set of basic queries for analysis and visualization of a terrain at different levels of detail
- Instances of selective refinement:

extract from a Multi-triangulation a mesh with the smallest possible number of triangles satisfying some user-defined criterion based on LOD

- LOD based on approximation error
- LOD can be uniform on the whole domain,or variable at each point of the domain.

Data Structures for Multi-Triangulations

- They must support efficiently:
- do/undo modifications on the extracted mesh
- test dependency links (to decide whether a modification can be applied)
- Data Structures for Vertex-Based MTs:
- Procedural encoding of modifications
- Partial order represented as a Directed Acyclic Graph (DAG)
- Approximation error associated either with triangles or with modifications

Procedural Encoding of Modifications in a Vertex-based MT

- To insert a vertex:
- Recognize the triangles to be deleted – hard
- Create triangles incident in the new vertex – easy
- To remove a vertex:
- Delete triangles incident in the vertex – easy
- Reconstruct the triangles inside the hole – hard

Encoding a Triangulated Polygon

- Encode an anchor edge
- Perform a depth-first traversal as a tree
- Encode the traversal as a bit stream

10 00 11 11

Building a Vertex-based MT

- A vertex-based MT is built through error-driven techniques based on
- coarsening an initial mesh through vertex insertion (VI)
- decimating the full-resolution mesh through vertex removal
- iterative removal of a single vertex (VR)
- removal of a set of independent vertices (IVR):
- Shape of a vertex-based MT (number of triangles, size of modifications) depends on its construction strategy
- Size of the modifications with different strategies:
- MT-VI : each modification creates 5 triangles on average
- MT-IVR: each modification creates 5.5 triangles on average
- MT-VR : each modification creates 6 triangles on average

Encoding a Hierarchy of Right Triangles

- Modifications and dependency links are implicitly represented
- Each triangles is uniquely identified by a binary location code
- From the location code of a triangle t we can retrieve:
- vertex coordinates and height values
- modifications involving t
- dependency links for such modifications
- Only height values and errors (associated with triangles) are stored

Comparison:Storage Costs of the Data Structures

- Full-resolution mesh (encoded in a standard triangle- based data structure) : 54n bytes
- Vertex-based MT: 27n (error on modifications)

33n bytes (error on triangles)

- between 1/2 and 3/5 of the space required by mesh at full resolution
- Hierarchy of right triangles:6n bytes
- 1/5 of the space required by a vertex-based MT
- 1/9 of the space required by mesh at full resolution

Comparison:Level-Of-Detail (LOD) Queries

- Uniform LOD across the domain
- Variable LOD:
- Domain-based LOD: max resolution inside a window
- Field-based LOD: max resolution for selected contour values
- Best solution = fewer extracted triangles for the same error value

Comparison: Uniform LOD

- MT-VI is the best one
- MT-VR, MT- IVR, HRT are comparable
- Motivation: error-driven construction strategy

HRT -

VI -

IVR -

VR -

HRT -

VI -

IVR -

VR -

Mount Marcy

Devil Peak

Comparison: Variable LOD

- HRT is the best one
- MT-VI, MT-IVR are comparable (MT-IVR slightly better)
- MT-VR is the worst one
- Motivation: smaller modifications, fewer dependency links

(HRT = each modification creates 4 triangles)

HRT -

VI -

IVR -

VR -

HRT -

VI -

IVR -

VR -

Window focus

Mount Marcy

Devil Peak

Comparison: Variable LOD

- HRT gives the best results
- MT-VI, MT-IVR are comparable (MT-VI slightly better)
- MT-VR give the worst results
- Motivation: smaller modifications, fewer dependency links

(HRT = each modification creates 4 triangles)

HRT -

VI -

IVR -

VR -

HRT -

VI -

IVR -

VR -

Field

focus

Mount Marcy

Devil Peak

Comparison: Variable LOD

error = 1.3% of height range, focused inside a window

HRT 1614 triangles MT-VI 2072 triangles

Comparison: Variable LOD

error = 1.3% of height range, focused on a field value

HRT 6697 triangles MT-VI 7138 triangles

Summary

- Both can generate meshes with topology (triangle-triangle adjacency links)

Current and Future Work

- Out-of-core techniques for a vertex-based MT: data structures, simplification methods, query algorithms
- 3D extension for volume data visualization:
- Efficient neighbor-finding techniques for a hierarchy of tetrahedra (Lee, De Floriani and Samet, 2001)
- Compact data structures for irregular 3D MTs (De Floriani et al., 2002)
- Analysis and comparison of regular and irregular 3D MTs on large volume data sets

Generating a Vertex-based MT

- Error-driven vertex insertion (VI)top-down
- At each step insert the data point corresponding to the maximum error (maximum vertical distance from the existing surface)
- Error-driven vertex removal (VR) bottom-up
- Start with a full-resolution mesh
- At each step remove the vertex whose removal causes the least error increase (minimum vertical distance from the new surface)
- Error-driven independent vertex removal (IVR) bottom-up
- Start with a full-resolution mesh
- At each step remove an independent set of vertices selected as the ones causing the least error increase

Generating a Vertex-based MT

- Error-driven vertex insertion (VI)top-down
- At each step insert the data point corresponding to the maximum error (maximum vertical distance from the existing surface)

Generating a Vertex-based MT

- Error-driven vertex removal (VR) bottom-up
- Start with a full-resolution mesh
- At each step remove the vertex whose removal causes the least error increase (minimum vertical distance from the new surface)

Generating a Vertex-based MT

- Error-driven independent vertex removal (IVR) bottom-up
- Start with a full-resolution mesh
- At each step remove an independent set of vertices selected as the ones causing the least error increase

Space Requirements for a Vertex-based MT

- Storage cost:
- 27n bytes (errors associated with modifications)
- 33n bytes (errors associated with triangles)

where n = number of vertices in the data set

by assuming (pessimistic experimental estimates)

- number of triangles = 4n
- number of arcs in the DAG = 3n
- Size (number of triangles) depends on the error-driven MT construction strategy:
- vertex insertion (VI)
- vertex removal (VR)
- removal of a set of independent vertices (IVR)

Download Presentation

Connecting to Server..