Image Projective Invariants

In machine vision, objects are observed subject to an unknown projective transformation, and it is usual to use projective invariants for either testing for a false alarm or for classifying an object. For four collinear points, the cross-ratio is the simplest statistic which is invariant under projective transformations. We obtain the distribution of the cross-ratio under the Gaussian error model with different means. The case of identical means, which has appeared previously in the literature, is derived as a particular case. Various alternative forms of the cross-ratio density are obtained, e.g. under the Casey arccos transformation, and under an arctan transformation from the real projective line of cross-ratios to the unit circle. The cross-ratio distributions are novel to the probability literature; surprisingly various types of Cauchy distribution appear. To gain some analytical insight into the distribution, a simple linear-ratio is also introduced. We also give some results for the projective invariants of five coplanar points. We discuss the general moment properties of the cross-ratio, and consider some inference problems, including maximum likelihood estimation of the parameters.

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Projective invariants of submanifolds

Projective Differential Geometry Old and New ◽

10.1017/cbo9780511543142.006 ◽

2004 ◽

pp. 103-152

Keyword(s):

Projective Invariants

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Image-Space Marker Detection and Recognition Using Projective Invariants

2011 International Conference on 3D Imaging, Modeling, Processing, Visualization and Transmission ◽

10.1109/3dimpvt.2011.55 ◽

2011 ◽

Cited By ~ 7

Author(s):

Filippo Bergamasco ◽

Andrea Albarelli ◽

Andrea Torsello

Keyword(s):

Image Space ◽

Projective Invariants ◽

Marker Detection ◽

Space Marker ◽

Detection And Recognition

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XII.—Clebsch-Aronhold Symbols and the Theory of Symmetric Functions

Proceedings of the Royal Society of Edinburgh Section A Mathematical and Physical Sciences ◽

10.1017/s0080454100007068 ◽

1951 ◽

Vol 63 (2) ◽

pp. 155-173

Author(s):

H. W. Turnbull ◽

A. H. Wallace

Keyword(s):

Bilinear Form ◽

Fundamental Theorem ◽

Symmetric Functions ◽

Projective Invariants ◽

Invariant Matrices ◽

Latent Roots ◽

Square Matrix

SynopsisA square matrix A = (aij) is expressed symbolically in terms of Clebsch-Aronhold equivalent symbols aij = aiaj = βibj = …, and the symbolic expressions for symmetric functions of the latent roots of A are considered, the relation between these functions and projective invariants of the bilinear form uAx being noted. The Newton and Brioschi relations between the symmetric functions are obtained by reduction of symbolic determinants and permanents respectively, and the Wronskian relations are shown to be equivalent to certain identities between determinants and permanents due to Muir. Also the fundamental theorem of symmetric functions is obtained symbolically as a consequence of the first fundamental theorem of invariants. The paper concludes with a note on the symbolization of the h-bialternants, that is of the traces of irreducible invariant matrices of A.

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Invariants of the Space Point Element Structure and Their Applications

Mathematical Problems in Engineering ◽

10.1155/2020/3295492 ◽

2020 ◽

Vol 2020 ◽

pp. 1-13

Author(s):

Yanping Mui ◽

Youzheng Zhang ◽

Guitao Cao

Keyword(s):

Object Recognition ◽

3D Reconstruction ◽

Traditional Method ◽

3D Structure ◽

Geometric Invariant ◽

Basic Matrix ◽

Projective Invariants ◽

Single Frame ◽

Space Point ◽

Geometric Relationship

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.

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GEOMETRIC INVARIANTS CONSTRUCTION FROM MULTIPLE VIEWS

International Journal of Modeling Simulation and Scientific Computing ◽

10.1142/s1793962311000402 ◽

2011 ◽

Vol 02 (02) ◽

pp. 195-206

Author(s):

CHENG JIN

Keyword(s):

Three Dimensional ◽

Ground Truth ◽

Multiple Views ◽

Multiple Images ◽

Object Structure ◽

Projective Invariants ◽

Geometric Invariants ◽

Image Pairs ◽

Fundamental Matrices ◽

Novel Algorithm

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.

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