Irina A. Kogan

Rational invariants of a group action. Construction and rewriting

Geometric constructions applied to a rational action of an algebraic group lead to a new algorithm for computing rational invariants. A finite generating set of invariants appears as the coefficients of a reduced Grobner basis. The algorithm comes in two variants. In the first construction the ideal of the graph of the action is considered. In the second one the ideal of a cross-section is added to the ideal of the graph. Zero-dimensionality of the resulting ideal brings a computational advantage. In both cases, reduction with respect to the computed Grobner basis allows us to express any rational invariant in terms of the generators.

Smooth and Algebraic Invariants of a Group Action: Local and Global Constructions

We provide an algebraic formulation of the moving frame method for constructing local smooth invariants on a manifold under an action of a Lie group. This formulation gives rise to algorithms for constructing rational and replacement invariants. The latter are algebraic over the field of rational invariants and play a role analogous to Cartans normalized invariants in the smooth theory. The algebraic algorithms can be used for computing fundamental sets of differential invariants.

3D Face Recognition using Euclidean Integral Invariants Signature

A novel 3D face representation and recognition approach is presented in this paper. We represent a 3D face by a set of level curves of geodesic function starting from the nose tip, which is invariant under isometric transformation of the surfaces. A pose change induces a special Euclidean transformation (a composition of a rotation and a translation) of the surface that represents a face and leads to the Euclidean transformation of the iso-geodesic curves. A change of facial expression induces isometric transformation of the iso-geodesic curves. Although the set of isometric transformations of a surface is larger than the set of Euclidean transformations in 3D, we assume that iso-geodesic curves undergo piecewise Euclidean transformations, i.e. the transformation of relatively small segments of the level curves is Euclidean. A Euclidean invariant integral signature for curves in 3D is presented in this paper. Euclidean invariant integral signature provides a classification of spatial curves which is independent of their position in 3D space and parameterization, and is not sensitive to noise. A recognition procedure based on comparing face feature in the invariant signature space is proposed. Substantiating examples are provided with an achieved classification accuracy of 95% for faces with various poses and facial expressions.

Rotation invariant topology coding of 2D and 3D objects using Morse theory

In this paper, we propose a numerical algorithm for extracting the topology of a three-dimensional object (2 dimensional surface) embedded in a three-dimensional space /spl Ropf//sup 3/. The method is based on capturing the topology of a modified Reeb graph by tracking the critical points of a distance function. As such, the approach employs Morse theory in the study of translation, rotation, and scale invariant skeletal graphs. The latter are useful in the representation and classification of objects in /spl Ropf//sup 3/.

Object-image correspondence for curves under central and parallel projections

We present a novel algorithm for deciding whether a given planar curve is an image of a given spatial curve, obtained by a central or a parallel projection with unknown parameters. A straightforward approach to this problem consists of setting up a system of conditions on the projection parameters and then checking whether or not this system has a solution. The computational advantage of the algorithm presented here, in comparison to algorithms based on the straightforward approach, lies in a significant reduction of a number of real parameters that need to be eliminated in order to establish existence or non-existence of a projection that maps a given spatial curve to a given planar curve. Our algorithm is based on projection criteria that reduce the projection problem to a certain modification of the equivalence problem of planar curves under affine and projective transformations. The latter problem is then solved using a separating set of rational differential invariants. A similar approach can be used to solve the projection problem for finite lists of points. The motivation comes from the problem of establishing a correspondence between an object and an image, taken by a camera with unknown position and parameters.

SYSTEMS OF HYPERBOLIC CONSERVATION LAWS WITH PRESCRIBED EIGENCURVES

We study the problem of constructing systems of hyperbolic conservation laws in one space dimension with prescribed eigencurves, i.e. the eigenvector fields of the Jacobian of the flux are given. We formulate this as a typically overdetermined system of equations for the eigenvalues-to-be. Equivalent formulations in terms of differential and algebraic-differential equations are considered. The resulting equations are then analyzed using appropriate integrability theorems (Frobenius, Darboux and Cartan-Kahler). We give a complete analysis of the possible scenarios, including examples, for systems of three equations. As an application we characterize conservative systems with the same eigencurves as the Euler system for 1-dimensional compressible gas dynamics. The case of general rich systems of any size (i.e. when the given eigenvector fields are pairwise in involution; this includes all systems of two equations) is completely resolved and we consider various examples in this class.

Invariants of objects and their images under surjective maps

We examine the relationships between the differential invariants of objects and of their images under a surjective maps. We analyze both the case when the underlying transformation group is projectable and hence induces an action on the image, and the case when only a proper subgroup of the entire group acts projectably. In the former case, we establish a constructible isomorphism between the algebra of differential invariants of the images and the algebra of fiber-wise constant (gauge) differential invariants of the objects. In the latter case, we describe residual effects of the full transformation group on the image invariants. Our motivation comes from the problem of reconstruction of an object from multiple-view images, with central and parallel projections of curves from three-dimensional space to the two-dimensional plane serving as our main examples.

Object-Image Correspondence for Algebraic Curves under Projections

We present a novel algorithm for deciding whether a given planar curve is an image of a given spatial curve, obtained by a central or a parallel projection with unknown parameters. The motivation comes from the problem of establishing a correspondence be- tween an object and an image, taken by a camera with unknown position and parameters. A straightforward approach to this problem consists of setting up a system of conditions on the projection parameters and then checking whether or not this system has a solution. The computational advantage of the algorithm presented here, in comparison to algorithms based on the straightforward approach, lies in a significant reduction of a number of real parameters that need to be eliminated in order to establish existence or non-existence of a projection that maps a given spatial curve to a given planar curve. Our algorithm is based on projection criteria that reduce the projection problem to a certain modification of the equivalence problem of planar curves under affine and projective transformations. To solve the latter problem we make an algebraic adaptation of signature construction that has been used to solve the equivalence problems for smooth curves. We introduce a notion of a classifying set of rational differential invariants and produce explicit formulas for such invariants for the actions of the projective and the affine groups on the plane.

Extensions for Systems of Conservation Laws

Extensions are central in the theory of conservation laws by providing intrinsic selection criteria for weak solutions. Given a system u t + f(u) x = 0 the extensions solve certain second order PDEs which are typically overdetermined. Determining the size of the set of extensions can be a challenging task. Instead of analyzing this second order system directly, we consider the equations satisfied by the lengths β i of the eigenvectors r i of the Jacobian matrix Df, measured with the inner product defined by an extension. For a given eigen-frame {r i } the extensions are determined uniquely, up to trivial affine parts, by these lengths. The β i solve a first order algebraic-differential system (the β-system) to which standard integrability theorems can be applied. The number of extensions is determined by determining the number of free constants and functions in the general solution to the β-system. We provide a complete breakdown for 3 × 3-systems, and for rich frames whose β-system has trivial algebraic part. Our framework is motivated by the work [16] where the authors consider conservative systems with prescribed eigen-frames. It is natural to ask how many extensions the resulting systems have, and the answer depends in an essential manner on the prescribed frame.

3D Mixed Invariant and its Application on Object Classification

A new integro-differential invariant for curves in 3D transformed by affine group action is presented in this paper. The derivatives involved are of the first order, and therefore this invariant is significantly less sensitive to noise than classical affine differential invariants, the simplest of which involves derivatives of order 5. A classification procedure based on characteristic curves of an object surface is considered using our proposed mixed invariants. Substantiating examples are provided to verify efficiency and discriminant power of the characteristic spatial curve based 3D object classification.