Interpolation approach for 3D smooth reconstruction of subsurface objects

Abstract

The present paper assumes that 2D sections are cut from multiple 3D subsurface objects by means of a sparse set of finite planes. The problem considered is to develop an approach capable of creating a geodesic set by means of spatial 2D sections, which do not intersect after orthogonal projection on a plane. The geodesic set is to be used for generating intermediate 2D sections. Next, with the help of the generated 2D sections, together with the initial sections, 3D object reconstruction and visualization may be achieved. To fulfil this task, specific terms are introduced, such as extreme left (right) visible border points interpolated by means of a spatial polynomial curve. Then, each spatial couple of adjacent 2D sections are projected on a plane. If the projections cross each other, their union is considered to be a geodesic set. If the projections do not cross each other, they are linked at the extreme points by polynomial projections and again form a geodesic set. Each geodesic set is used for a morphological interpolation, and intermediate 2D sections are generated. Together with the initial sections, are used to build a 3D object surface. To validate the theory, a set of 2D sections is generated, using six cross sections (images) of an impermeable unit. Then, 3D unit visualizations are realized, using at first the initial section only, then the initial sections together with the generated set of 2D sections afterwards. The contributions of this approach to the problem of object visualization are discussed and compared with some other interpolation methods.