CPSC 695

Mesh OptimizationMarina L. Gavrilova

Brief Outline

Mesh Generation ApplicationsMesh OptimizationDynamic MeshesApplications

Terminology

DEM: Digital Elevation ModelViewer Dependent RefinementROAM: Realtime Optimally Adapting MeshCLOD: Continuous Level of Detail

What is mesh?It is often required to convert geometric objects in any other form to polygonal meshes of geometric simplexes (i.e. triangles) before renderingMeshes are used for computer modeling, biometric, GIS, cartographic, and animation applicationsMesh optimization techniques are used to reduce the mesh to an acceptable size (user defined), minimizing the loss of quality (subject to some form of error analysis).

Mesh definition

A mesh is defined as a pair (V,K) where V is the set of vertices and K is a simplicialcomplex specifying the connectivity of the mesh simplices (the adjacency of the vertices, edges and faces) [Hoppe].The 0-simplices are called vertices, the 1-simpices are called edges, and the 2-simplices are called faces. Triangulation is an example of mesh.

Example: 3D Terrain Rendering

• Uses DEM as input of the application

•Generates dynamic mesh to achieve frame coherent animated view in real-time

Mesh Optimization GoalsFor an arbitrary point (u,v) in the surface each goal contributes an

energy component to the energy function. The energy function E is defined as the weighted sum of all energy components where each corresponding weight pi is the priority of the energy component Ei:

In most optimization problems, the shape preservation is a majorgoal. Therefore, surface curvature index plays a significant role in the optimization process due to the fact that a curved surface requires greater number of simplexes compared to a planar (or less curved) surface to accommodate the same level of visual detail.

( , ) ( , )i iE u v p E u v= ∑

Mesh optimization problemsThere are three major categories for mesh optimization

problems:A fully detailed geometric mesh exists and the task is to

find an optimized mesh under the specified energy function and satisfying a user-defined level of detail (i.e. specified number of vertices).

A fully detailed geometric mesh exists and one has to find a hierarchical representation for rendering in dynamic multiple levels of detail (MLOD).

A surface is defined as a parametric function. One has to find an approximate mesh representation of that surface under specified goals and constraints in a framework for dynamic real-time continuous level of detail (CLOD).

Mesh optimization

Mesh optimization is the process of reducing a mesh M to a mesh M0, where M0contains less number of vertices and geometric primitives (i.e. triangles) than M. The goal is to find such mesh M0 that the no other mesh exists, which represent a better approximation of the mesh M0.

Methods for mesh optimization

Mesh optimization can be considered as is a compression method for geometric models with large details. In computer graphics, various processes are utilized such as:

regular tessellations,polygonized spline surfaces, polygonized parametric surfaces, polygonized implicit surfaces,range scanned surfaces and subdivision surfaces.

Basic ApproachThe entire optimization process can be viewed as an optimal spatial subdivision problem. There exist only two legal moves to achieve optimality: subdivide and merge. Models that are based on this concept do not require computationally extensive optimization process. Instead, the mesh progressively adapts to an optimal solution. These models usually define a multi-criteria based error metric to assess the energy of the mesh. Error metrics are more flexible than energy functions in a sense that they are capable of assessing an intermediate mesh both at a local and a global level.

Dynamic mesh conceptAt any given stage, one of the two following courses of action is chosen: subdivide or merge. If the number of vertices reaches the upper limit, details are reduced on the regions assessed with the lowest error contribution. At the same time, whenever the vertex count becomes lower than the upper limit, details are expanded on the region assessed with the highest error contribution. This greedy approach allows the intermediate mesh to optimally adapt to the minimal energy configuration, satisfying the specified set of constraints. This is the fundamental basis for a dynamic mesh algorithm.

Real-time terrain renderingUses a graphics Engine/LibraryCentral focus on efficient mesh representationView coherence and frame rate constancyLimited/Variable Level of DetailSpeed optimizationRepresenting layer data as textures or particles

Advantages of static modelsThere are various static mesh models used in GIS applications and computer games. They are regular tessellation, DelaunayTriangulation (DT) [ref], Triangulated Irregular Network (TIN). They provide the following advantages:

Well-known and well understood modelsEasy to implementOptimization is performed only once, so rendering application is simple to designEfficient for small scale objects

Disadvantages of static modelsDuring rendering of massive terrain or other GIS data:

Most of the geometry of a terrain is outside the view frustum due to its massive size. Static methods waste a lot of CPU cycles trying render the entire object. Although the enormous amount of detail is stored in memory for the entire object the visible area might constitute a tiny fraction of that detail. Consequently, static models are highly inefficient for terrain modeling.Viewpoint dependent refinement is not possible with static meshes. Have a bad triangle to screen pixel ration.Unable to obtain a steady performance.

Understanding Height FieldsDEM: Digital Elevation Model

• Contains only relative Height

• Regular interval

• Pixel color determine height

•Discrete resolution

Dynamic meshDynamic mesh generation is based on real-time adaptive models that allow local transformation of a given mesh to different levels of details and adapt progressively to the changes in the rendering environment. Dynamic models use some form of hierarchical data structure to store a mesh in multiple LOD.There are many different versions of dynamic mesh currently under research and development. Several patented technologies have been released in the industry (i.e. Intels MRM). The two most prominent approaches are Progressive Meshes (PM) and Real-time Optimally Adapting Mesh (ROAM).

PM vs. ROAM

PM can be seen as a bottom up approach where as ROAM is a top-down approach. PM starts with an arbitrary mesh in maximum detail. It requires a massive amount of computation for the generation of the multi resolution hierarchical data structure representing the mesh at different LOD.ROAM starts with a simple mesh and refines detail in real-time during the rendering cycle.

Converting Height field data into 3D topological mesh

200 255 150 100

100 255 255 200

200 150 200 100

• Pixel value (z) is used as Height Map

• Vertices are generated as points in 3D

• A Mesh is triangulated

Mesh Representation

Goals:

Speed

Quality

Constancy

Representations:

Progressive Meshes

ROAM: Real-time Optimally Adapting Mesh

Progressive MeshesDeveloped by Hugues Hoppe, Microsoft Research Inc. Published first in SIGGRAPH 1996.

How PM works…

ROAM :•Can Subdivide when more details necessary

•Merge & Split Queue

•Tree Structured

ROAM advantagesReal-time Optimally Adapting Mesh is an efficient top-down dynamic mesh model. In this model, real-time display of complex surfaces can be provided by dynamically computing a multi-resolution triangular mesh for each view.One can minimize geometric distortions on the screen while maintaining a fixed triangle count. Visual discontinuities could be minimized in several ways, and efficient mesh corrections can ensure selected lines of site, or object proximity are correctly represented.An incremental priority-queue based approach in ROAM allows exploiting frame-to-frame coherence for fast efficient computation of the adapting mesh.

Geometric subdivision

Problems with Geometric Subdivisions

ROAM principle

The basic operating principle of ROAM

Quadtree and Bintree for ROAM

Comparison between Quadtree (top) and Bintree (bottom)

ROAM Implementation and Improvements

Pre-process DEMReduce overhead for maintaining Split queueUse degenerated split queueReplace priority merge queue (HEAP) with a simple FIFO queue.Merge arbitrary obscured regions (outside visible frustum) only

Preprocessing of DEM input

Convolute the surface (2nd Difference)Efficient LOD distributionUsed in Advanced Shading (feature preservation)

Differentiating the Surface: Convolution

Original DEM of Kluane Park First differentiation. Blue area Shows steep mountains.

Final LOD distribution image Shows rocky areas require more details

Second Differentiation. Blue areas indicate steep mountain peaks.

Differentiating the Surface (cont’d)…

Take Weighted Average of neighboring pixel Gradients

2 passes gives us the rate of gradient changes

Statistical Analysis of ROAM

Bottleneck: Split and Merge Operation and visibility testOnly 2.7% triangles are merged within the visible frustum75% of split operations occurs on closest 5% region of the visible frustum

Proposed Priority Function for Split QueueDetermines RefinementDistribute LOD to preferred areaCan be changed

f=face, C=Camera U=Uniformity F=Frustum priorityD=Curvature priorityP=Distance priorityDET=Curvature (preprocessed)Lev(f)=Subdivision level of face f

( )UfLevCfCenterPfCenterDETDCfAngularityF

Cfiority)(1

)())((),(),(Pr

+

−×+×+×=

Speeding up the processAmortizing refinement

Updating the Split queue over multiple framesAllow split queue (heap) to degenerateRefine integrity using error threshold testRe-generate queue after constant number of framesDramatic gain in speed

Maintaining Degenerated S-Queue

Understanding Geo-MorphSmooth interpolation of vertex over subdivision

Requires maintenance of one additional queue (G-queue)

SmoothingCompute face normal Compute vertex normalLocal computation of normal during refinement (Geo-Morph)Apply vertex normal when renderingAdvanced shading: Linear combination of face and vertex normal (Hybrid)

Example of smoothing

Texturing

Compute uv coordinate during creation of each vertex (planar transform)

Load texture bitmap in the pipeline

Apply texture handle

Result with Texturing

Results (Output obtained)

Results (Runtime)Rendered Frame VS Time

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1 18 35 52 69 86 103

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Results (Runtime) (Cont’d)Frames Per Second

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Results (Runtime) (Cont’d)Mesh Face Counts

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ROAM Face Rendered Faces

Results (Runtime) (Cont’d)

Geo Morph Load

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Summary of Results

High Speed (45 FPS average)Continuity is achieved (Geo-Morph) Visual appeal (smoothing & Texture)Less variant Frame rateEfficient Understructure (amortized update)

Other Applications

Motion PlanningSpatial-Temporal AnalysisMassive Terrain RenderingComputer Games

Next: Photo Realistic 3D Terrain