## Compressed Sensing applied to the Reconstruction of Permeability Fields

This work explores new approaches for the problem of multichannel facies image recovery from the theory and algorithmic solutions provided in the mark of compressed sensing. In this task, the geological field is recovered from pixel-based linear measurements without the use of any prior information from a statistical model. $\ell_1$−minimization algorithms are explored, and a performance guaranteed results are adopted to evaluate their reconstruction performances. From this analysis, we formulate the problem of basis selection, where it is shown that for unstructured pixel-based measurements the Discrete Cosine Transform is the best choice for the problem. In the experimental side, signal-to-noise ratios and similarity perceptual indicators are used to evaluate the quality of the reconstructions. The potential of this new approach is demonstrated in under-sampled scenario of 2–4% of direct data, which is known to be very challenging in the absence of prior knowledge from a training image.

Contributions:

1. Hernan Calderon, Jorge F. Silva, Julin M. Ortiz, and Alvaro Egana, “Reconstruc- tion of Multichannel Facies based on RIPless Compressed Sensing,” Computers and Geosciences, vol. 77, pp. 54-65, April, 2015.

## Well Placement Strategies for Uncertainty Reduction in Categorical Random Fields: Formulation, Mathematical Analysis and Application to Multiple-Point Simulations

In this research, the problem of optimal well-placement (or sensor placement) is addressed from the perspective of minimizing the posterior uncertainty of non-sensed positions of a discrete random field given the information of the sensed positions. Information theoretic quantities are adopted to formalize the problem of optimal well-placement for field characterization, where concepts like information of the measurements, average posterior uncertainty, and the resolvability capacity of the field are introduced. We study the implications of information-driven sensing strategies on the characterization of a field --- using information theoretic measures for characterization--- and the inference of non-sensed variables from the sensed ones. On the application, we explore a simple Markov chain context where the statistics of the random object is known. We also consider the practical case where multiple points simulations (MPS)  are used to simulate channelized facies fields adopting a training image.

## Analysis and Classification of Natural Rock Textures based on New Transform-based Features

This work develops a mathematical method to extract relevant information about natural rock textures to address the problem of automatic classification. Classical methods of texture analysis cannot be directly applied in this context, since rock tex- tures are typically characterized by both stationary patterns (a classic kind of texture) and geometric forms, which are not properly captured with conventional methods. Due to the presence of these two phenomena, a new classification approach is pro- posed in which each rock texture class is individually analyzed developing a specific low-dimensional discriminative feature. For this task, multi-scale transform domain representations are adopted, allowing the analysis of the images at several levels of scale and orientation. The proposed method is applied to a database of digital photographs acquired in a porphyry copper mining project, showing better performance than state-of-the-art techniques, and additionally presenting a low computational cost.

Contributions:

1. Rodrigo Lobos, Jorge F. Silva, Julian M. Ortiz, Gonzalo Diaz, Alvaro Egana, “Analysis and Classification of Natural Rock Textures based on New Transform- based Features,” Mathematical Geosciences, July, vol. 48, pp. 835 - 870, 2016.