Augmented reality (AR) applications typically extend the user's view of the real world with virtual objects. In recent years, AR has gained increasing popularity and attention, which has led to improvements in the required technologies. AR has become available to almost everyone. Researchers have made great progress towards the goal of believable AR, in which the real and virtual worlds are combined seamlessly. They mainly focus on issues like tracking, display technologies and user interaction, and give little attention to visual and physical coherence when real and virtual objects are combined. For example, virtual objects should not only respond to the user's input; they should also interact with real objects. Generally, AR becomes more believable and realistic if virtual objects appear fixed or anchored in the real scene, appear indistinguishable from the real scene, and response to any changes within it. This thesis examines on three challenges in the field of computer vision to meet the goal of a believable combined world in which virtual objects appear and behave like real objects. Firstly, the thesis concentrates on the well-known tracking and registration problem. The tracking and registration challenge is discussed and an approach is presented to estimate the position and viewpoint of the user so that virtual objects appear fixed in the real world. Appearance-based line models, which keep only relevant edges for tracking purposes, enable absolute registration in the real world and provide robust tracking. On the one hand, there is no need to spend much time creating suitable models manually. On the other hand, the tracking can deal with changes within the object or the scene to be tracked. Experiments have shown that the use of appearance-based line models improves the robustness, accuracy and re-initialization speed of the tracking process. Secondly, the thesis deals with the subject of reconstructing the surface of a real environment and presents an algorithm to optimize an ongoing surface reconstruction. A complete 3D surface reconstruction of the target scene offers new possibilities for creating more realistic AR applications. Several interactions between real and virtual objects, such as collision and occlusions, can be handled with physical correctness. Whereas previous methods focused on improving surface reconstructions offline after a capturing step, the presented method de-noises, extends and fills holes during the capturing process. Thus, users can explore an unknown environment without any preparation tasks such as moving around and scanning the scene, and without having to deal with the underlying technology in advance. In experiments, the approach provided realistic results where known surfaces were extended and filled in plausibly for different surface types. Finally, the thesis focuses on handling occlusions between the real and virtual worlds more realistically, by re-interpreting the occlusion challenge as an alpha matting problem. The presented method overcomes limitations in state-of-the-art methods by estimating a blending coefficient per pixel of the rendered virtual scene, instead of calculating only their visibility. In several experiments and comparisons with other methods, occlusion handling through alpha matting worked robustly and overcame limitations of low-cost sensor data; it also outperformed previous work in terms of quality, realism and practical applicability. The method can deal with noisy depth data and yields realistic results in regions where foreground and background are not strictly separable (e.g. caused by fuzzy objects or motion blur).