3D Projective Invariants Research Articles

Invariants of the Space Point Element Structure and Their Applications

In this paper, a new geometric structure of projective invariants is proposed. Compared with the traditional invariant calculation method based on 3D reconstruction, this method is comparable in the reliability of invariant calculation. According to this method, the only thing needed to find out is the geometric relationship between 3D points and 2D points, and the invariant can be obtained by using a single frame image. In the method based on 3D reconstruction, the basic matrix of two images is estimated first, and then, the 3D projective invariants are calculated according to the basic matrix. Therefore, in terms of algorithm complexity, the method proposed in this paper is superior to the traditional method. In this paper, we also study the projection transformation from a 3D point to a 2D point in space. According to this relationship, the geometric invariant relationships of other point structures can be easily derived, which have important applications in model-based object recognition. At the same time, the experimental results show that the eight-point structure invariants proposed in this paper can effectively describe the essential characteristics of the 3D structure of the target, without the influence of view, scaling, lighting, and other link factors, and have good stability and reliability.

A Linear Method to Derive 3D Projective Invariants from 4 Uncalibrated Images

A well-known method proposed by Quan to compute projective invariants of 3D points uses six points in three 2D images. The method is nonlinear and complicated. It usually produces three possible solutions. It is noted previously that the problem can be solved directly and linearly using six points in five images. This paper presents a method to compute projective invariants of 3D points from four uncalibrated images directly. For a set of six 3D points in general position, we choose four of them as the reference basis and represent the other two points under this basis. It is known that the cross ratios of the coefficients of these representations are projective invariant. After a series of linear transformations, a system of four bilinear equations in the three unknown projective invariants is derived. Systems of nonlinear multivariable equations are usually hard to solve. We show that this form of equations can be solved linearly and uniquely. This finding is remarkable. It means that the natural configuration of the projective reconstruction problem might be six points and four images. The solutions are given in explicit formulas.

GEOMETRIC INVARIANTS CONSTRUCTION FROM MULTIPLE VIEWS

Geometric invariants have wide applications in computer vision and their precision has long been a hot topic. In most of the existing methods, three-dimensional (3D) invariants have been obtained by reconstruction of the object structure, where fundamental matrices between image pairs should be first established. Consequently, there are additional errors introduced during invariants construction and could be very time consuming. In this paper, a novel algorithm to calculate 3D projective invariants from multiple images has been proposed, without reconstructing the object structures explicitly. We have employed the geometric configuration of points and lines in general position to deduce the formulation of 3D invariants. It has been verified in our experiments that our proposed method is considerably accurate when compared with the ground truth, and more efficient when compared with reconstruction based methods.

A Geometric Approach for the Theory and Applications of 3D Projective Invariants

A central task of computer vision is to automatically recognize objects in real-world scenes. The parameters defining image and object spaces can vary due to lighting conditions, camera calibration and viewing position. It is therefore desirable to look for geometric properties of the object which remain invariant under such changes in the observation parameters. The study of such geometric invariance is a field of active research. This paper presents the theory and computation of projective invariants formed from points and lines using the geometric algebra framework. This work shows that geometric algebra is a very elegant language for expressing projective invariants using n views. The paper compares projective invariants involving two and three cameras using simulated and real images. Illustrations of the application of such projective invariants in visual guided grasping, camera self-localization and reconstruction of shape and motion complement the experimental part.

ANALYSIS AND COMPUTATION OF PROJECTIVE INVARIANTS FROM MULTIPLE VIEWS IN THE GEOMETRIC ALGEBRA FRAMEWORKS

A central task of computer vision is to automatically recognize objects in real-world scenes. The parameters defining image and object spaces can vary due to lighting conditions, camera calibration and viewing positions. It is therefore desirable to look for geometric properties of the object which remain invariant under such changes. In this paper we present geometric algebra as a complete framework for the theory and computation of projective invariants formed from points and lines in computer vision. We will look at the formation of 3D projective invariants from multiple images, show how they can be formed from image coordinates and estimated tensors (F, fundamental matrix and T, trilinear tensor) and give results on simulated and real data.

The unique solution of projective invariants of six points from four uncalibrated images

In this paper, we show that four uncalibrated images are sufficient to uniquely determine the 3D projective invariants of a set of six points in general position in space. An algorithm for computing the unique solution is proposed. It computes by solving a set of linear equations for its two linear independent solutions and is simpler than other algorithms which compute the three possible solutions of 3D projective invariants by solving a set of nonlinear equations. However, a fourth image is needed to get the linear unique solution. Finally, experimental results have shown the feasibility of this algorithm.