MULTIGRID INTERPOLATION AND DISCONTINUITY DETECTION IN HIERARCHICAL SURFACE RECONSTRUCTION Raid Al-Tahir Department of Surveying and Land Information The University of the West Indies Trinidad and Tobago Commission III KEY WORDS: Surface Reconstruction, Surface Interpolation, Discontinuity Detection. ABSTRACT Working within the framework of hierarchical surface reconstruction first introduced by (Schenk et al., 1991), this research focuses on the subsystem of surface interpolation and defines the tasks for interpolation and analysis of a surface. The main objective of this research is to develop an algorithm that produces a dense representation of a surface from a sparse set of observations and, at the same time, facilitates preliminary labeling of discontinuities in the surface. Additionally, this study establishes the instrument and the mechanism for the communication between the levels of the image pyramid as well as the subsystems of the hierarchical surface reconstruction paradigm. The suggested approach is the simultaneous interpolation of the surface and detection of its discontinuities by adopting the weak membrane model. The proposal is a new implementation of the Graduate Non-Convexity (GNC) algorithm based on multigrid techniques. The GNC was originally developed by (Blake and Zisserman, 1987) for a single grid, for applications in computer vision. As it was initially proposed, GNC depends a great deal on the optimal setting of the control parameters. In this work, a specific procedure has been developed that determines the control parameters automatically and less subjectively. 1 INTRODUCTION In digital photogrammetry, surface reconstruction is the frame that enclaves all the tasks for modeling the portion of the real world that has been photographed. Automatic surface reconstruction entails two major problems: determining conjugate points or features (matching) and densifying the matched points in object space (interpolation). The two tasks are usually performed sequentially in a hierarchical approach, without interacting with one another except for providing an approximations to the next level of hierarchy. In order to improve the success rate and the reliability of automated surface reconstruction the results from densifying and analyzing the surface in any level must be considered by the matching on subsequent levels. The tasks of interpolation and surface analysis may give clues about surface discontinuities and occlusions - a vital feedback for the matching process. The conceptual framework developed in (Schenk et al., 1990; Schenk and Toth, 1991) for surface reconstruction possesses several appealing features. The most distinguished features, as related to this study, are its modularity and hierarchy. For these reasons, this study adopts this framework to define the type of input, output and the expected tasks for surface interpolation subsystem. This subsystem is of particular interest for reconstructing surfaces from large-scale aerial scenes of urban areas. Because of man-made features and buildings, aerial imagery of urban areas is usually associated with a higher percentage of occlusion, repeated patterns or similar features, and many surface discontinuities. All these problems affect the success rate of any matching algorithm. Besides, there is no straightforward answer to the selection of a specific surface interpolation method that would accommodate such characteristics. The objective for this study is then to develop an algorithm that produces a dense representation for a surface from a sparse set of observations and, at the same time, facilitates preliminary labeling of discontinuities in the surface. Moreover, this study caters the base for the flow of information between different modules in the hierarchical surface reconstruction paradigm. 2 SURFACE INTERPOLATION In digital photogrammetry, surface reconstruction is the frame that incorporates all the tasks for modeling the portion of the real world that has been photographed. Automatic surface reconstruction entails two major problems: determining conjugate points or features (matching) and densifying the matched points in object space (interpolation). While there are several approaches for reconstructing surfaces from digital images, the operation and success of all of these approaches depend upon having a dense representation for the surface. A scheme for surface interpolation, hence, is required for furnishing the needed representation, and facilitating further analysis of the surface. It is clear that the objectives of surface interpolation are • Constructing as realistic a surface representation as possible. • Preserving essential surface characteristics implied by the observations. A violation to this may occur when a smooth surface is interpolated over breaklines.