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Belief rule base (BRB) systems are an extension of traditional IF-THEN rule based systems and capable of capturing complicated nonlinear causal relationships between antecedent attributes and consequents. In a BRB system, various types of information with uncertainties can be represented using belief structures, and a belief rule is designed with belief degrees embedded in its possible consequents. For a set of inputs to antecedent attributes, inference in BRB is implemented using the evidential reasoning (ER) approach. In this paper, the inference mechanism of the ER algorithm is analyzed first and its patterns of monotonic inference and nonlinear approximation are revealed. For a practical BRB system, it is difficult to determine its parameters accurately by using only experts' subjective knowledge. Moreover, the appropriate adjustment of the parameters of a BRB system using available historical data can lead to significant improvement on its prediction performance. In this paper, a training data selection scheme and an adaptive training method are developed for updating BRB parameters. Finally, numerical studies on a multimodal function and a practical pipeline leak detection problem are conducted to illustrate the functionality of BRB systems and validate the performance of the adaptive training technique.

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Fringe projector profilometry (FPP) is an important three-dimensional (3D) measurement technique, especially when high precision and speed are required. Thus, theoretical interrogation is critical to provide deep understanding and possible improvement of FPP. By dividing an FPP measurement process into four steps (system calibration, phase measurement, pixel correspondence, and 3D reconstruction), we give theoretical analysis on the entire process except for the extensively studied calibration step. Our study indeed reveals a series of important system properties, to the best of our knowledge, for the first time: (i) in phase measurement, the optimal and worst fringe angles are proven perpendicular and parallel to epipolar line, respectively, and can be considered as system parameters and can be directly made available during traditional calibration, highlighting the significance of the epipolar line; (ii) in correspondence, when two sets of fringes with different fringe orientations are projected, the highest correspondence precision can be achieved with arbitrary orientations as long as these two orientations are perpendicular to each other; (iii) in reconstruction, a higher reconstruction precision is given by the 4-equation methods, while we notice that the 3-equation methods are almost dominatingly used in literature. Based on these theoretical results, we propose a novel FPP measurement method which (i) only projects one set of fringes with optimal fringe angle to explicitly work together with the epipolar line for precise pixel correspondence; (ii) for the first time, the optimal fringe angle is determined directly from the calibration parameters, instead of being measured; (iii) uses 4 equations for precise 3D reconstruction but we can remove one equation which is equivalent to an epipolar line, making it the first algorithm that can use 3-equation solution to achieve 4-equation precision. Our method is efficient (only one set of fringe patterns is required in projection and the speed is doubled in reconstruction), precise (in both pixel correspondence and 3D reconstruction), and convenient (the computable optimal fringe angle and a closed-form 3-equation solution). We also believe that our work is insightful in revealing fundamental FPP properties, provides a more reasonable measurement for practice, and thus is beneficial to further FPP studies.