Image Processing Reference
In-Depth Information
Chapter 7
Modeling of a Voxelated Surface
7.1
Voxelation and Approximation of 3-D Surface .................. 228
7.1.1
3-D Isothetic Covers ..................................... 230
7.1.2
Test Results .............................................. 232
7.2
Voxelation of Surface of Revolution ............................. 233
7.2.1
Various Techniques ...................................... 234
7.2.2
Digital Curves and Surfaces ............................. 236
7.2.3
Algorithm to Wheel-Throw a Single Piece .............. 238
7.2.4
Number-Theoretic Approach ............................ 241
7.2.4.1
Algorithm DCS (Digital Circle Using
Squares) .................................... 244
7.2.4.2
Run Length Properties of Digital Circles .. 244
7.2.5
Creating Realistic Potteries ............................. 248
7.2.6
Some Examples .......................................... 251
7.3
Summary ......................................................... 252
Exercises ......................................................... 253
Modeling of 3-D objects and surfaces is the prerequisite of many applica-
tions in computer graphics and computer vision. It should follow a scientific
or mathematical representation that conforms to optimal storage and e
-
cient computation. Modeling is primarily done in two broad categories: one
for regular geometric objects (polyhedra, platonic solids, etc.) or well-defined
mathematical surfaces, and another for real-world objects (sculptures, day-to-
day objects, living organisms, etc.). See Fig. 7.1 for some typical examples of
3-D models with triangulated surfaces.
In order to define the shape of a polyhedral object, a polygon or wire-
frame mesh is used. Such a mesh is a collection of vertices, edges, and faces.
The faces usually consist of triangles, but may also consist of quadrilaterals
or other simple convex polygons so as to simplify the rendering in 3-D com-
puter graphics, but depending on need and application, concave polygons,
orthogonal polygons, or polygons with holes are also sometimes used.
In the domain of digital geometry, a surface is expressed as a set of voxels.
To define the voxelated surface, in this chapter we expand some preliminary
concepts introduced in Chapter 1. A 3-D digital object A is defined as a fi-
nite subset of Z
3
, with all its constituent points (i.e., voxels) having integer
coordinates and connected in 26-neighborhood. Each voxel is equivalent to a
3-cell [115] centered at the concerned integer point (Fig. 7.2(a)). Note that a
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