In computer graphics a projection describes the mapping of scene geometry to the screen. While linear projections such as perspective and orthographic projection are common, increasing applications are being found for nonlinear projections, which do not necessarily map straight lines in the scene to straight lines on the screen. Nonlinear projections occur in reflections and refractions on curved surfaces, in art, and in visualisation. This thesis presents a new nonlinear projection technique called a trilinear projection that is based on the trilinear interpolation of surface normals used in Phong shading. Trilinear projections can be combined to represent more complicated nonlinear projections. Nonlinear projections have previously been implemented with ray tracing, where rays are generated by the nonlinear projections and traced into the scene. However for performance reasons, most current graphics software uses scanline rendering, where a scene point is imaged on a screen as a function of the projection parameters. The techniques developed in this thesis are of this nature. This thesis presents several algorithms used in trilinear projection: 1. An algorithm to analytically determine which screen locations image a given scene point. 2. An algorithm that correctly connects projected vertices. Each scene point may be imaged multiple times, which means a projected scene triangle may form from one to four different shapes of from two to nine vertices. Once connected, the projected shapes may be rendered with standard scanline algorithms. 3. An algorithm to more accurately render the curved edges between projected vertices. 4. A scene-space edge-clipping algorithm that handles continuity issues for projected shapes across composite projections. The trilinear projection technique is demonstrated in two different application areas: visualisation, and reflections and refractions. Specifically, various nonlinear projections that are congruent with pre-existing visualisation techniques are implemented with trilinear projections and a method for approximating the reflections and refractions on curved surfaces with trilinear projections is presented. Finally, the performance characteristics of the trilinear projection is explored over various parameter ranges and compared with a naive ray tracing approach.

Abstract Training a visual inspection device is not straightforward but suffers from the high variation in material to be inspected. This variation causes major difficulties for a human, and this is directly reflected in classifier training. Many inspection devices utilize rule-based classifiers the building and training of which rely mainly on human expertise. While designing such a classifier, a human tries to find the questions that would provide proper categorization. In training, an operator tunes the classifier parameters, aiming to achieve as good classification accuracy as possible. Such classifiers require lot of time and expertise before they can be fully utilized. Supervised classifiers form another common category. These learn automatically from training material, but rely on labels that a human has set for it. However, these labels tend to be inconsistent and thus reduce the classification accuracy achieved. Furthermore, as class boundaries are learnt from training samples, they cannot in practise be later adjusted if needed. In this thesis, a visual based training method is presented. It avoids the problems related to traditional training methods by combining a classifier and a user interface. The method relies on unsupervised projection and provides an intuitive way to directly set and tune the class boundaries of high-dimensional data. As the method groups the data only by the similarities of its features, it is not affected by erroneous and inconsistent labelling made for training samples. Furthermore, it does not require knowledge of the internal structure of the classifier or iterative parameter tuning, where a combination of parameter values leading to the desired class boundaries are sought. On the contrary, the class boundaries can be set directly, changing the classification parameters. The time need to take such a classifier into use is small and tuning the class boundaries can happen even on-line, if needed. The proposed method is tested with various experiments in this thesis. Different projection methods are evaluated from the point of view of visual based training. The method is further evaluated using a self-organizing map (SOM) as the projection method and wood as the test material. Parameters such as accuracy, map size, and speed are measured and discussed, and overall the method is found to be an advantageous training and classification scheme.