The Feasibility of Motion and Structure Computations

Abstract

We address the problem of interpreting image velocity fields generated by a moving monocular observer viewing a stationary environment under perspective projection to obtain 3-D information about the relative motion of the observer (egomotion) and the relative depth of environmental surface points (environmental layout). The algorithm presented in this paper involves computing motion and structure from a spatio-temporal distribution of image velocities that are hypothesized to belong to the same 3-D planar surface. However, the main result of this paper is not just another motion and structure algorithm that exhibits some novel features but rather an extensive error analysis of the algorithm’s preformance for various types of noise in the image velocities. Waxman and Ullman [83] have devised an algorithm for computing motion and structure using image velocity and its 1st and 2d order spatial derivatives at one image point. We generalize this result to include derivative information in time as well. Further, we show the approximate equivalence of reconstruction algorithms that use only image velocities and those that use one image velocity and its 1st and/or 2”d spatio-temporal derivatives at one image point. The main question addressed in this paper is: “How accurate do the input image velocities have to be?’ or equivalently, “How accurate does the input image velocity and its I~ and 2& order derivatives have to be?“. The answer to this question involves worst case error analysis. We end the paper by drawing some conclusions about the feasibility of motion and structure calculations in general. I.1 Introduction In this paper, we present a algorithm for computing the motion and strncture parameters that describe egomotion and environmental layout from image velocity fields generated by a moving monocular observer viewing a stationary environment. Egomotion is defined as the motion of the observer relative to his environment and can be described by 6 parameters; 3 dvth-scaled translational parameters, Z and 3 rotation parameters, o. Environmental layout refers to the 3-D shape and location of objects in the environment. For monocular image sequences, en$ronmental layout is described by the normalized surface gradient, a, at each image point. To determine these motion and structure parameters we derive nonlinear equations relating image velocity at some image int ?(?*,t ‘) to the underlying motion and structure parameters at (P,c). The computaP tion of egomotion and environmental layout from image velocity is sometimes called the reconstruction problem; we reconstruct the observer’s motion, and the layout of his environment, from (timevarying) image velocity. A lot of research has been devoted to