Seismic layer segmentation, oil-gas boundary surfaces recognition, and 3D volume data reconstruction are three important tasks in three-dimensional seismic image processing. Geophysical and geological parameters and properties have been known to exhibit progressive changes in a layer. However, there are also times when sudden changes can occur between two layers. A-connectedness was proposed to describe such a phenomenon. Based on graph theory, A-connectedness describes the relationship among pixels in an image. It is proved that A-connectedness is an equivalence relation. That is, it can be used to partition an image into different classes and hence can be used to perform image segmentation. Using the random graph theory and A-connectivity of the image, the length of the path in a A-connected set can be estimated. In addition to this, the normal Aconnected subsets preserve every path that is A-connected in the subsets. An O(nlogn) time algorithm is designed for the normal A-connected segmentation. Techniques developed are used to find objects in 2D/3D seismic images. Finding the interface between two layers or finding the boundary surfaces of an oil-gas reserve is often asked. This is equivalent to finding out whether a A-connected set is an interface or surface. The problem that is raised is how to recognize a surface in digital spaces. A-connectedness is a natural and intuitive way for describing digital surfaces and digital manifolds. Fast algorithms are designed to recognize whether an arbitrary set is a digital surface. Furthermore, the classification theorem of simple surface points is deduced: there are only six classes of simple surface points in 3D digital spaces. Our definition has been proved to be equivalent to Morgenthaler-Rosenfeld's definition of digital surfaces in direct adjacency. Reconstruction of a surface and data volume is important to the seismic data processing. Given a set of guiding pixels, the problem of generating a A-connected (subset of image) surface is an inverted problem of A-connected segmentation. In order to simplify the fitting algorithm, gradual variation, an equivalent concept of A-connectedness, is used to preserve the continuity of the fitted surface. The key theorem, the necessary and sufficient condition for the gradually varied interpolation, has been mathematically proven. A random gradually varied surface fitting is designed, and other theoretical aspects are investigated. The concepts are used to successfully reconstruct 3D seismic real data volumes. This thesis proposes A-connectedness and its applications as applied to seismic data processing. It is used for other problems such as ionogram scaling and object tracking. It has the potential to become a general technique in image processing and computer vision applications. Concepts and knowledge from several areas in mathematics such as Set Theory, Fuzzy Set Theory, Graph Theory, Numerical Analysis, Topology, Discrete Geometry, Computational Complexity, and Algorithm Design and Analysis have been applied to the work of this thesis.