Research On Projective Invariants In Computer Vision

One of the basic goals of computer vision is3D scene object recognition. That is to decide whether two images are the same scene object of different views. The main difficulty encountered in3D scene object recognition is that the measuring features between different views of the same object are different. One of the basic methods to solve this problem is using invariants of scene object for object recognition. An invariant of a scene object is a value that remains unchanged after some internal or external transformations applied to the object. These invariants are usually some functions of the observation features of a scene object. The change of the observation features of a scene object may caused by internal changes of the object or caused by the changes of the observation environment. The principal elements of an observation environment are lighting conditions and observation point and/or angle. The changes of the observation point and/or angle lead to the geometric changes of different views of the same object. But the images are related by some geometric transformations mathematically. This dissertation studies invariants of the images of scene objects under geometric transformations. The main geometric transformations considered are projective transformation and3D to2D projection. The underlying measuring features are pure geometric or geometric features plus grayscale measures. The topics of this dissertation include projective and permutation invariants,3D to2D projection invariants, projective invariance, and projective moment invariants.Projective invariant is classical in projective geometry. Yet the computation of projective invariant of points depends on the ordering of the underlying points. Different orderings of the underlying points usually result in different values of the invariant. This will cause problems in point pattern matching. Values that do not change under both projective transformations and permutation transformations are called projective and permutation invariants. Former construction methods of projective and permutation invariants are usually based on a high degree polynomial function and are of high computational complexity and are somewhat unstable. This dissertation presents two novel methods to construct projective and permutation invariants. The first method is based on the maximum function. The proposed projective and permutation invariants are raw values of the projective invariants. Experimental results indicate that this form of invariants are of low computational complexity and are more stable than previous constructions. The second construction is based on the enclosing degrees of a planar point set. These projective and permutation invariants have integral values. Experimental results validate the invariance of this construction.For3D to2D projection invariants, this dissertation makes four contributions. The first contribution is presenting a method to solve systems of nonlinear multivariable equations encountered in typical fundamental matrix estimation with rank-2constraint. The algorithm is based on the Lagrange multipliers method. After some careful transformations, the problem is reduced to the solution of a polynomial equation in a single variable. The second contribution is deriving an invariant relation between a set of six3D points and their projection images. This invariant relation can be used in model based scene object recognition. The third contribution is presenting an absolute3D to2D projection invariant of restricted geometric configuration. Experimental results indicate that the proposed invariant have satisfactory representation power. The fourth contribution is presenting a direct method for computing projective invariants of seven3D points from two of their2D projection images.Projective moment invariant is the key research topic of this dissertation. Since projective geometry is the basic model of computer vision, achieving projective invariance is the first goal of the vision problem. Yet it is proved that general projective moment invariants do not exist. Suk and Flusser proposed a method to construct approximate projective moment invariants through infinite series expansion. Unfortunately, their construction is not sound mathematically. Since there is no projective moment invariant, the problem can only be solved indirectly. That is to restrict the underlying transformations, to extend the definition of moment, to obtain approximate invariants, and to add assumptions. Based on the previous considerations, this dissertation makes three contributions. The first contribution is presenting a set of functions of generalized moments that are invariant under a type of restricted projective transformations. The second contribution is presenting a set of functions of generalized moments that are invariant under some more general restricted projective transformations. The third contribution is presenting a set of functions of generalized moments of image with respect to two reference points that are invariant under general projective transformations. Experimental results support the mathematical proof. These moment invariants have high discriminating power and are robust to slight noises. This dissertation presents test results of the performance of these invariants under dilation, erosion, blurring, and occlusion conditions.