4.6.2 QEM Simplification Summary

Fig. 4.20 Simplification results using the Quadric Error Metric. From left to right: original mesh (172,974 vertices), simplified by 80 % (34,595 vertices) and simplified by 90 % (17,296 vertices)

[29, 54] with applications in statistics and applications to scientific and medical datasets. Visualizing 3D data in particular has attracted considerable research attention, specifically where the aim is to create realistic renderings. This has been the target of computer graphics for many decades [51]. However, visualization in general encompasses a broader set of aims beyond only simulating the physics of real world image formation. For example, we may wish to navigate massive and complex datasets [71] where only a small fraction can be observed at any one time. Alternatively, our dataset may be volumetric and acquired in slices [37]. Such data can either be viewed as 2D images, formed by sampling over a plane that intersects the dataset, or rendered as 3D surfaces by applying surface fitting techniques to the raw data.

Many free tools exist for the visualization of data. In the domain of mesh data, these include MeshLab and Graphite, while for volumetric data include MicroView. In Fig. 4.21 we demonstrate some of the most common means of visualizing 3D mesh data. We begin by plotting vertices as 2D points, i.e. a point cloud. Although difficult to interpret, such visualizations can be used with raw data and give an overview of sampling density. In (b) and (c) we draw only edges, in the latter case only drawing edges which are not occluded by other faces. This gives an impression of a solid model and is easier to interpret than the raw wireframe. In (d) we show the object projected to a 2D depth map where darker indicates further away. Note that the depth data appears smooth and it is difficult to visualize small surface features when rendered in this way. In (e) we plot the vertex normals at each vertex as a blue line. Note that we use a simplified version of the surface here in order to be able to draw all of the vertex normals. In (f) we plot the principal curvature directions for the same simplified mesh as blue and red vectors. In (g) we show a smooth-shaded rendering of the original high resolution data. By varying material properties and illumination, this is the classical rendering approach used in computer graphics. In (h) we show a flat shaded rendering of the low resolution version of the mesh. In this case, triangles are shaded with a constant color which destroys the effect of a smooth surface. However, this view can be useful for inspecting the geometry of the surface without the deceptive effect of interpolation shading. Finally, in (i) we show an example of coloring the surface according to the output of a function, in

(e.g. the resolution of medical imaging devices) and we begin to process large sequences of dynamic data (acquired through 4D capture) or databases of meshes. Large databases arise in applications such as 3D face recognition where fine surface detail is required to distinguish faces, yet many millions of subjects may be enrolled in the database. Not only must the database be stored efficiently but it must be possible to perform queries in real-time to be of use in real-world applications. The progressive mesh representations described above allow progressive transmission of 3D data, but this must be extended to dynamic data. Moreover, the transformation of a mesh into a progressive mesh is expensive and may not be viable for huge datasets. The growing trend to outsource storage and processing of data to “the cloud” necessitates 3D representations that can be stored, interacted with and edited in a distributed manner.

However, on the other hand, there is a growing wish to be able to access such data in a resource limited environment. For example, over bandwidth-limited network connections or on mobile devices with limited computational power. The advances in rendering technology exhibited in computer generated movie sequences will begin to find its way into consumer products such as game consoles and mobile devices. These platforms are highly resource limited in terms of both processing power and storage. Hence, the efficiency of the data structures used to store and process 3D data will be of critical importance in determining performance.

Statistical representations of 3D surfaces (such as the 3D morphable model [6]) have proven extremely useful in vision applications, where their compactness and robustness allow them to constrain vision problems in an efficient way. However, their application is limited because of model dominance whereby low frequency, global modes of variation dominate appearance. Extensions to such statistical representations that can capture high frequency local detail while retaining the compact storage requirements are a significant future challenge. A related challenge is how such models can be learnt from a sparse set of samples within a very high dimensional space, so that the model’s generalization capabilities can be improved.

4.9 Concluding Remarks

Representations of 3D data provide the interface between acquisition or sensing of 3D data and ultimately processing the data in a way that enables the development of useful applications. Anyone involved in 3D imaging and processing must be aware of the native representations used by sensing devices and the various advantages and limitations of higher level surface and volume based representations. In many cases, data in a raw representation must be preprocessed before conversion to the desired higher level format, for example through surface fitting, smoothing or resampling. Subsequently, in choosing the most suitable representation for a particular application, it must be established what operations need to be performed on the data and what requirements exist in terms of computational and storage efficiency. Whether involved with the low level design of 3D sensors or interpreting and processing high