Deeper Understanding on Solution Ambiguity in Estimating 3D Motion Parameters by Homography Decomposition and its Improvement

3D pose estimation or 3D motion estimation is one of the most important problems in the computer vision field. For this problem, a lot of methods for fixing the shape restoration problem from consecutive images have been proposed up until now, including non-linear methods and linear methods. When the object is a plane, the camera displacement parameters can be extracted (assuming that the intrinsic camera parameters are known) from the homography matrix that can be measured from two views. This process is called homography decomposition. The standard algorithms for homography decomposition give numerical solutions using the singular value decomposition (SVD) of the matrix, proposed by Faugeras and Zhang. There are generally at least two possible solutions only under the visual reality constraint by using only two pieces of images. The ambiguity of the two solutions cannot be resolved if there is no further a priori information available. This is the solution ambiguity problem in estimating the 3D motion parameters by homography decomposition. From then on, the inevitability of the ambiguity problem to estimate solutions has been widely accepted for nearly 30 years. It is considered to be one of the basic properties and mathematical defects in homography decomposition, even today. My research focuses on this solution ambiguity problem of homography decomposition. The different results have been found in my research: solution ambiguity is not inevitable, and the unique solution can be obtained conditionally. Two kinds of dependences of the solution ambiguity problem in estimating the 3D motion parameters by homography decomposition have been clarified for the first time: (1) My research pointed out the dependence of solution ambiguity on 3D motion parameters for the fixed object size, and defined a criterion about the 3D motion parameters to obtain the unique estimated solution: “If all distances between the iv feature points and the camera do not become closer after 3D motion, the solution ambiguity problem can be avoided”. This conclusion has been geometrically and theoretically derived and it is the first time this dependence has been mentioned. However, because the constraint for the theoretical proof is more general than that in the previous work, this theory should be more reliable. (2) My research pointed out the dependence of solution ambiguity on object size for the fixed 3D motion parameters and explained it with geometry. Meanwhile, a criterion about the object size to obtain the unique estimated solution has been defined: “If the feature region is large enough, or the range of 3D movements is relatively small compared to the size of the feature region, the solution ambiguity problem can be avoided by using homography decomposition”. This dependence and its geometrical proof have been presented for the first time until now. Besides the two kinds of dependences of the solution ambiguity problem in estimating the 3D motion parameters by homography decomposition, my research also proposes a kind of constraint condition of joined two planes, in order to guarantee the unique real estimated solution. The effectiveness of the constraint condition has been tested by the simulation experiments. The occurrence ratio of the ambiguous solutions is smaller than 0.3‰. My research clarified the mechanism for solution ambiguity obtained by homography decomposition in the estimation of 3D motion parameters. Both dependencies on 3D motion parameters and object size have been theoretically and experimentally verified. All of the theories and the constraint condition have been verified by simulation experiments used the huge database. In my research, I do not have any intention of proposing a new approach to solving the solution ambiguity problem of homography decomposition more effectively in my research. The only purpose of my research is to give an accurate interpretation and a complete description of facts that have been misunderstood for a long time. Therefore, the simulation results should have been sufficiently accurate, so that a demonstration with real images was not necessary in my research. v Chapter