Reconstruction of binary geological images using analytical edge and object models

Reconstruction of fields using partial measurements is of vital importance in different applications in geosciences. Solving such an ill-posed problem requires a well-chosen model. In recent years, training images (TI) are widely employed as strong prior models for solving these problems. However, in the absence of enough evidence it is difficult to find an adequate TI which is capable of describing the field behavior properly. In this paper a very simple and general model is introduced which is applicable to a fairly wide range of binary images without any modifications. The model is motivated by the fact that nearly all binary images are composed of simple linear edges in micro-scale. The analytic essence of this model allows us to formulate the template matching problem as a convex optimization problem having efficient and fast solutions. The model has the potential to incorporate the qualitative and quantitative information provided by geologists. The image reconstruction problem is also formulated as an optimization problem and solved using an iterative greedy approach. The proposed method is capable of recovering the image unknown values with accuracies about 90% given samples representing as few as 2% of the original image.