A cornerstone of geometric reconstruction, rotation averaging seeks the set of absolute rotations that optimally explains a set of measured relative orientations between them. In addition to being an integral part of bundle adjustment and structure-from-motion, the problem of synchronizing rotations also finds applications in visual simultaneous localization and mapping, where it is used as an initialization for iterative solvers, and camera network calibration. Nevertheless, this optimization problem is both non-convex and high-dimensional. In this paper, we address it from a maximum likelihood estimation standpoint and make a twofold contribution. Firstly, we set forth a novel primal-dual method, motivated by the widely accepted spectral initialization. Further, we characterize stationary points of rotation averaging in cycle graphs topologies and contextualize this result within spectral graph theory. We benchmark the proposed method in multiple settings and certify our solution via duality theory, achieving a significant gain in precision and performance.
One of the first decisions to be made when building a mineral resource model is the definition of geological/geostatistical domains. Cluster analysis is a set of techniques in machine learning that can be especially suited for this matter. In order to compare different approaches, two clustering algorithms were investigated in this study: k-means and the dual-space clustering algorithm. Choosing the most appropriate method and the number of clusters can be challenging and some metrics are needed to support these decisions, including the validation of the spatial distribution of the clusters, which is not always appropriately discussed in the literature. We introduce the use of correlograms of the indicators for that matter. Although clustering techniques can be robust for an application in resource modelling, expert knowledge is still necessary when applying cluster analysis to resource modeling, since final decisions should not be based solely on statistical indexes, but also on experience. In this paper, the proposed methodology was tested in a threedimensional dataset related to a phosphate/titanium deposit.
Machine learning is a broad field of study that can be applied in many areas of science. In mining, it has already been used in many cases, for example, in the mineral sorting process, in resource modeling, and for the prediction of metallurgical variables. In this paper, we use for defining estimation domains, which is one of the first and most important steps to be taken in the entire modeling process. In unsupervised learning, cluster analysis can provide some interesting solutions for dealing with the stationarity in defining domains. However, choosing the most appropriate technique and validating the results can be challenging when performing cluster analysis because there are no predefined labels for reference. Several methods must be used simultaneously to make the conclusions more reliable. When applying cluster analysis to the modeling of mineral resources, geological information is crucial and must also be used to validate the results. Mining is a dynamic activity, and new information is constantly added to the database. Repeating the whole clustering process each time new samples are collected would be impractical, so we propose using supervised learning algorithms for the automatic classification of new samples. As an illustration, a dataset from a phosphate and titanium deposit is used to demonstrate the proposed workflow. Automating methods and procedures can significantly increase the reproducibility of the modeling process, an essential condition in evaluating mineral resources, especially for auditing purposes. However, although very effective in the decision-making process, the methods herein presented are not yet fully automated, requiring prior knowledge and good judgment.
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