On mesh

Research On Mesh
Thenraja Vettivelraj
Department Of Computer Science
Swansea University
M.Sc, Department of Computer Science, Department of Computer Science, Swansea University

Topics
 Introduction
 “Mutual Tessellation”
 “Mesh Optimization”
 “Progressive Mesh”
 Energy Function
 Applications and Conclusion.
 References
M.Sc, Department of Computer Science, Swansea University

Introduction
 Graphics
• Graphics has its necessity in all the fields and nowadays the people want
the graphics to be so realistic.
• Since it helps to analyze the data and also interact with the people it
plays a vital role.
• Examples in video games, cinemas, Technology
 Note: The centered equations and images are copied from the
source file for time consumption.

Aim
 Aim
• Our Ultimate aim is to construct a Mesh M from the given set of data
points Mesh M0 scattered in three dimensions, where the topology of the
mesh M0 remains the same in Mesh M. The result Mesh M has to fit the
data well and also we have to reduce the number of vertices.
• Concentrating on the geometry and topology of the image alone not
sufficient henceforth we have to take care of the overall appearance.
• In [5] they have given that how to construct a surface from the given
unorganized points. An algorithm to get the topology from the
constructed surface and from that how to get the rough geometrical
surface has been discussed in [6].

Research On Mesh
 Initially Greg Turk [1] introduced the concept “Mutual
tessellation”. By using this procedure we can represent the
same object in different level of detail which is very much
helpful for achieving higher frame rates.
 The paper "Mesh optimization" by Hugues Hoppe et al [2]
talks about minimizing the number of triangles from a initial
dense triangular mesh.
 Goal(Mesh Optimization):
Efficient, Lossless and Continuous resolution of the image
with minimum number of triangles.
 Challenges(Mesh Optimization):
Complex triangle meshes, Transmission bandwidth,
Storing capacities and Rendering performance.

Research On Mesh
 “Hugues Hoppe” [3] introducing the concept of "Progressive meshes".
In addition to the mesh optimization here we are also taking care of the
overall appearance by the scalar (colour, normal and texture
coordinates) and discrete attributes (material identifier).
 Challenges(Progressive Mesh)
• Mesh Simplification-Created meshes optimized for rendering efficiency.
• Level-of-detail (LOD) approximation-To improve efficiency it is
necessary to define the different versions of the model in different
details
• Progressive transmission-Mesh M0 is first transferred followed by
increments . Needs additional time for sending successive LOD.
• Mesh Compression-Mesh Simplification and Minimize the space to
store.
• Selective refinement-Details added in only desired areas.

Research On Mesh
 Solution
• For these drawbacks we have two solutions.
First send coarser form M0 followed by n levels of detail. In
order to refine into original mesh M=Mn
.
Hence the PM contains the information from M0
, M1
, M2
,…Mn
.
• Preserving overall appearance by its discrete (material identifier)
and scalar attributes such as colour, normal and texture coordinates

Mesh Optimization
 Definition of a Mesh
• According to [4] "A graphics object that is composed entirely of polygons that
have common vertices and edges".
 Mesh Optimization
• Mesh simplification algorithm can also be considered here because it also speaks
about the reducing in number of faces in a congested mesh while slightly disturbing
the shape with reference to Turk [7] and Schroeder et al. [8].
• The exchange between geometric fit and compact representation is controlled via a
user controlled parameter Crep. Increase in the value of Crep results in fewer
triangles.
Figure1:mesh simplification.

Research On Mesh
 Meshes in computer graphics
• In computer graphics we are representing the objects as triangle
meshes.
• Expression for mesh M= (K, V, D, S) where K specifies the connectivity
of vertices, edges, faces and where V denotes the set of vertices {V1,V2,
….Vm} in the shape of the mesh R3
, scalar attributes are colour (r, g, b),
normal(nx, ny, nz) and texture coordinates (u, v). And discrete attributes.
A corner is represented as (v,f ).
Figure 2. Example of mesh representation with a single face

Mesh Representation
 Progressive Mesh Representation
• Edge collapse is sufficient in simplifying the edges. Edge swap and the Edge split
is only in surface reconstruction and so here the Edge collapse is enough.
• The Vsplit contains information about how to split the edges. By transferring into
ecol we can compress the mesh and can store it in an efficient manner. Below the
description how to make the edge collapse.
Figure 4.a, b: Sequence of edge collapse and its result

Research On Mesh
 Geomorphs
• In a progressive mesh representation the geomorphs can be formed
between any two meshes.
• Consider the finer Mesh Mf
and a coarser Mesh Mc
and this case the
geomorphs lies between 0 <= C < f <= n, where each Mf
will be having a
unique mapping in Mc
and this is called Surjective map Ac
.
• For proper resolution we will get the image slowly. Imagine an image,
if we focus it then the image will keep on changing slowly until it gets
the desired image, it won’t change in quick frames.
 Progressive Transmission and Mesh Compression
• In the progressive transmission first the compact Mesh M0 is transmitted
first followed by vspliti records until the original mesh M is recovered.
• Instead, of storing all the index vertices (si,li,ri) of vspliti just store the si
alone and in the remaining 5 bits store the remaining.

Research On Mesh
 Selective refinement
• Supported by progressive Mesh
• Details will be added only in desired areas.
• Can travel in low level bandwidth.
• Consider the application supply a callback function REFINE (v) that returns a
Boolean function that what the neighbour mesh V supposed to do. The intial
mesh Mc
is refined by iterating the list (Vsplitc,…….Vsplitn-1)but only Vspliti (si,li,ri,Ai) if
1) All three vertices present in the mesh {Vsi,Vli,Vri} and
2) REFINE (Vsi) is true
Figure 5. Selective refinement for terrain(Using conditions 1 and 2)

Energy Function
 Energy Function
• The energy metric E (M) is defined as M= {K,V,D,S} with respect to the original M
• E(M)=Edist(M)+Espring(M)+Escalar(M)+Edisc(M)
 Preserving surface geometry
• Mesh M= (K,V) minimizes the energy function
E(K,V)=E dist (K,V) + E rep (K) + E spring (K,V)
• The distance energy E dist is as follows
• This E dist + E rep does not produce the desired result, because it produces some
spike regions.
Figure 6. c) Out of phase I(M0) d)Optimization without Espring.

Energy function
 Scalar attributes
 The scalar attribute Escalar for vertices is defined as
• the range constraints the (r,g,b) lies between 0 and 1.
 Discontinuity Curves
• That is the discrete face attributes like material identifier and the scalar
attributes have the problems in identifying the shadow boundaries. They
would not be sharp in some cases which are supposed to be sharp and
vice versa and we come to know that because the edge collapse we are
doing will change the topology of the mesh. Hence forth we are
introducing the new energy term Edisc which will overcome the
problems that we are facing.

Applications and Conclusion
 Applications
This mesh optimization, retiling polygonal model and
progressive meshes technique has given new dimensions in
medical areas, mainly in scans. And also in the graphics field it
has reduced the manual work.
 Conclusion
And hence by minimizing the energy formula we are achieving
our goals.

References
• [1] Greg Turk. Re-tiling polygonal surfaces. Computer Graphics (SIGGRAPH ’92
Proceedings), 26(2):55–64, July 1992.
• [2] H. Hoppe, T. DeRose, T. Duchamp, J. McDonald, and W. Stuetzle. Surface
reconstruction from unorganized points. Computer Graphics (SIGGRAPH ’92
• [3] Hugues Hoppe, Progressive Meshes, Microsoft Research.
• [4]Computer graphics dictionary By Roger T. Stevens.
• [5] T. DeRose, H. Hoppe, T. Duchamp, J. McDonald, and W.Stuetzle.Fitting of
surfaces to scattered data. SPIE, 1830:212–220, 1992
• [6] H. Hoppe, T. DeRose, T. Duchamp, J. McDonald, and W. Stuetzle. Surface
reconstruction from unorganized points. Computer Graphics (SIGGRAPH ’92
• [7] Greg Turk. Re-tiling polygonal surfaces. Computer Graphics (SIGGRAPH ’92
• [8] William Schroeder, Jonathan Zarge, and William Lorensen. Decimation of
triangle meshes. Computer Graphics (SIGGRAPH ’92 Proceedings), 26(2):65–70,
July 1992.

Thank You

On mesh

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