Digital Surfaces: the 3D Mesh

Surfaces can be represented digitally in several ways. In theoretical cases, 3D data can ideally be represented using mathematical equations such as Equation 1.2; for instance, any sphere can be represented using only four numbers, i.e., a radius and three numbers for the center. Such parametrized descriptions are great for expressing perfect planes, spheres, cones or tori, and with a little bit more effort these descriptions can describe combinations of (possibly transformed versions of) such primitives. However, for arbitrary surfaces, an exact representation quickly become cumbersome.

(a) B´ezier curve (b) Resulting surface

Figure 1.3: B´ezier curve and surface. This example shows how a bowling pin is modeled using a B´ezier curve. While this representation results in perfectly smooth curves, it is too

complex for real-time, interactive applications.

For approximating arbitrary surfaces, well-known approaches are built on B´ezier surfaces and B-splines. Such approximating surfaces can be designed arbitrarily accurate by defining sufficient control points. These surfaces are popular representations in CAD applications, but are overly complex for interactivevisualization. Figure 1.3 shows an example. A bowling pin can be represented by cubic B´ezier curves as shown in Figure 1.3a. By spinning this curve around a fixed center axis, the surface shown in Figure 1.3b is obtained.

Instead of relying on the curvature defined by the tangential vectors formed by control points, one can also consider a linear interpolation between control points, resulting in a polygon mesh. Such a mesh is defined by the control points or vertices, and the linear interpolation between these vertices which forms polygons.

Newell’s Utah teapot (Figure 1.4) is a famous 3D mesh from the early computer graphics days.

Figure 1.4: Example mesh:Utah teapot. This is an example of a 3D mesh. It represents the surface of a teapot, and approximates this surface by defining vertices, i.e., the points

where multiple lines cross, and faces, i.e., the triangles formed between the vertices.

Observe that, while the geometry information as such is given by the vertices, the connectivity information is equally important in defining the approximated surface.

Imagine what the surface would look like if vertices at the tip of the spout were connected with top of the lid; while the geometry information remains unchanged, a new surface

would be defined.

Similar to other multimedia domains, the distortion associated with a digital representation is related to the resolution determined by the sampling density on the one hand, and the quality of the samples on the other hand. Let us compare with image coding. Firstly, a digital image samples a picture, resulting for instance in a resolution of3 888 × 2 592 picture elements or pixels, meaning there are only 3 888 columns of pixels, over 2 592 rows. When reducing the sampling density to 128 × 86 pixels, fine detail is lost, as is illustrated by Figure 1.5a and its lower resolution approximation in Figure 1.5b. Secondly, the color value of each sample or pixel is obtained by mixing three color components, namely red, green and blue.

Each color value can be quantized to, for instance,8 bits per color component.

This allows for2^8×3≈ 17 million possible color values. Reducing the amount of quantization bits to3 bits per component as shown in Figure 1.5c shows banding artifacts: the colors can no longer vary smoothly, resulting in possibly large color jumps between neighboring pixels.

For meshes, the distortion is similarly determined by the vertex density. For instance, the brain surface depicted in Figure 1.2c is represented by a high-resolution mesh as shown in Figure 1.6a. The amount of vertices relates to the approximation accuracy. The denser sampling, as shown in Figure 1.6a, preserves

(a) Original image (b) Low resolution (c) Coarse quantization Figure 1.5: Image resolution and quantization. In (a), an original high-resolution and

finely-quantized image is displayed. (b) shows the resulting image at a much lower resolution, i.e., using fewer pixels, while (c) shows the resulting image at a much coarser

quantization, i.e., with the same number of pixels but with a lower color quality.

(a) High resolution:

294 012 vertices

(b) Low resolution:

3 941 vertices

(c) Coarse quantization:

294 012 vertices Figure 1.6: Mesh resolution and quantization. (a) shows an originally digitized version of

the brain isosurface depicted in Figure 1.2c. A lower-resolution mesh is shown in (b):

observe that the use of fewer vertices no longer preserves finer detail. (c) shows a high-resolution mesh where the vertex positions have a coarser quantization. Observe that

fine detail is again lost.

fine detail better compared to the sparser sampling shown in Figure 1.6b. This is referred to as the resolution of a mesh. Quantization additionally affects the accuracy of the vertex locations and results in banding artifacts similar to what is observed in image coding, where thex, y or z value of neighboring vertices either stays fixed, or jumps by a significant amount. This is depicted in Figure 1.6c.

Functions on 3D Surfaces At this point, the mesh approximates the geometry of an object, by combining vertices (i.e., geometry information which samples

the actual geometry) and faces (i.e., connectivity information which defines the remaining geometry by interpolation).

Taking these surfaces one step further, functions can be defined over a surface.

For instance, one could describe the average yearly temperature for each point on the surface of planet Earth. Useful values for describing the surface of an object often include appearance attributes, encompassing color information, reflectance, transparency, etc. After sampling, such information is typically provided either via specific vertex attributes in addition to the required geometric coordinates, or via several texture maps which are indexed by (u, v) coordinates given per vertex. Either the specific attributes or the(u, v) coordinates per vertex are then interpolated over the faces to obtain values over the entire surface.

In this dissertation, only geometric information per vertex is considered. The techniques proposed in this thesis can be extended to other attributes, which will require taking into consideration other specific attribute-related decision criteria. Additionally, only triangular meshes are considered, i.e., meshes for which all faces are triangles. For real-time rendering, this mesh representation is encountered most often; any polygon mesh can be converted to a triangle mesh without altering its geometry.