Search

This paper is devoted to a new paradigm for the invariant recognition of visual objects through introducing ‘the generating invariant’ of the underlying visual geometry which allows to numerically calculate differential signature curves in a fully group-invariant manner. Then, we utilize the results to work on the unsolved problem of approximating similarity differential invariant signatures based on suitable combination of joint invariants of the underlying group action. We also illustrate that, compared to the traditional and current schemes, the new paradigm is reliable, stable, significantly minimizes the effects of noise and indeterminacy, and Signature-inverse Theorem is correct in terms of it. Besides we demonstrate that the ratio of elapsed time and inaccuracy of the outcomes resulted from the Calabi et al.’s scheme to our formulation follow an exponential decay model. And finally, application to real images are discussed.

Let M be a closed smooth manifold. In 1999, Friedlander and Nadirashvili introduced a new differential invariant $$I_1(M)$$ using the first normalized nonzero eigenvalue of the Lalpace–Beltrami operator $$\Delta _g$$ of a Riemannian metric g. They defined it taking the supremum of this quantity over all Riemannian metrics in each conformal class, and then taking the infimum over all conformal classes. By analogy we use k-th eigenvalues of $$\Delta _g$$ to define the invariants $$I_k(M)$$ indexed by positive integers k. In the present paper the values of these invariants on surfaces are investigated. We show that $$I_k(M)=I_k({\mathbb {S}}^2)$$ unless M is a non-orientable surface of even genus. For orientable surfaces and $$k=1$$ this was earlier shown by Petrides. In fact Friedlander and Nadirashvili suggested that $$I_1(M)=I_1({\mathbb {S}}^2)$$ for any surface M different from $${\mathbb {RP}}^2$$ . We show that, surprisingly enough, this is not true for non-orientable surfaces of even genus, for such surfaces one has $$I_k(M)>I_k({\mathbb {S}}^2)$$ . We also discuss the connection between the Friedlander–Nadirashvili invariants and the theory of cobordisms, and conjecture that $$I_k(M)$$ is a cobordism invariant.

In this paper, a variable-coefficient cubic–quintic nonlinear Schrodinger equation involving five arbitrary real functions of space and time is analyzed from the point of view of symmetry analysis by using Lie’s invariance infinitesimal criterion. The infinitesimal generators of corresponding equivalence transformations are presented. The first-order differential invariants are constructed to identify when the equation can be mapped to a constant-coefficient cubic–quintic nonlinear Schrodinger equation. The constrained conditions on the variable coefficients we arrived here extend the cases discussed before and present more general results. Some brightlike and darklike solitary wave solutions for special potentials and cubic–quintic nonlinearities are obtained.

Single-View Geometry (SVG) studies the world-to-image mapping or warp, which is the relationship that exists between a body’s model and its image. For a rigid body observed by a projective camera, the warp is described by the usual camera matrix and its properties. However, it is clear that for a body whose deformation state changes between the body’s model and its image, the ‘simple,’ globally parameterized warp described solely by the camera matrix, breaks down. Existing work has exploited deformation to reconstruct the deformed body from its image, but did not establish the properties of the deformable warp. Studying these properties is part of deformable SVG and forms a recent research topic. Because deformations may take place anywhere in the object’s body, and because they may be uncorrelated, the warp is local in nature. Using a differential framework is thus an obvious choice. We propose a differential–algebraic projective framework based on modeling the body’s surface by a locally rational projective embedding and on the 1D projective camera. We show that this leads, via the study of univariate rational functions, to differential invariants that the warp must satisfy. It may seem surprising, given the generic hypothesis made on the observed body, hardly stronger than mere local smoothness, that constraints can still be found. Our framework generalizes the Schwarzian derivative, the first-order projective differential invariant, which holds under the assumption that the body’s shape is locally linear. Our invariants may be used to construct regularizers to be used in warp estimation. We report experimental results of two types on simulated and real data. The first type shows that the proposed invariants hold well for an independently estimated warp. The second type shows that the proposed regularizers improve warp estimation from point correspondences compared to the classical derivative-penalizing regularizers.

In Grebenev, Waclawczyk, Oberlack (2019 Phys A: Math. Theor. 52, 33), the conformal invariance (CI) of the characteristic $${\varvec{X}}_{1}(t)$$ (the zero-vorticity Lagrangian path) of the first equation (i.e. for the evolution of the 1-point PDF $$f_1({\varvec{x}}_{1},\omega _{1},t)$$ , $${\varvec{x}}_{1} \in D_1 \subset {\mathbb {R}}^2)$$ of the inviscid Lundgren–Monin–Novikov (LMN) equations for 2d vorticity fields was derived. The infinitesimal operator admitted by the characteristics equation generates an infinite-dimensional Lie pseudo-group G which conformally acts on $$D_1$$ . We define the conformal invariant differential form $$\mathrm{d}s^2 = f_1\cdot \left( {\mathrm{d}X_{1}^{ 1}}^2 + {\mathrm{d}X_{1}^{ 2}}^2\right) $$ along the characteristic $$\left. {\varvec{X}}_{1}(t)\right| _{\omega _{1} = 0}$$ together with the simple action functional $${\mathcal F}({\varvec{X}}_{1},\mathrm{d}s^2)$$ . We demonstrate that $$G_{{\mathcal {Y}}}$$ , which is a subgroup of the group G restricted on the variables $${\varvec{x}}_{1}$$ and $$f_1$$ , gives rise to a symmetry transformations of $${\mathcal {F}}({\varvec{X}}_{1},\mathrm{d}s^2)$$ . With this, we calculate the second-order universal differential invariant $$J_2^{{\mathcal {Y}}}$$ (or the multiscale representation of the invariants) of $$G_{{\mathcal {Y}}}$$ under the action on the zero-vorticity characteristics. We show that $${{\mathcal {F}}}({\varvec{X}}_{1},\mathrm{d}s^2)$$ is a scalar invariant and generates all differential invariants, which look like the quantities of different scales, from $$J_2^{{\mathcal {Y}}}$$ by the operators of invariant differentiation. It gives insight into the geometry of a flow domain nearby point $${\varvec{x}}_{1}$$ in the sense of Cartan.

We construct differential-invariant solutions to automorphic systems and propose a method for classifying differential-invariant submodels.

A point classification of ordinary differential equations of the form y″ = F(x, y) is considered. The algebra of differential invariants of the action of the point symmetry pseudogroup on the right-hand sides of equations of the form y″ = F(x, y) is calculated, and Lie’s problem on the point equivalence of such equations is solved.

Algebras of symmetries and the corresponding algebras of differential invariants for plane flows of inviscid fluids are given. Their dependence on thermodynamical states of media are studied and a classification of thermodynamical states is given.

In this paper (cf. Lychagin and Yumaguzhin, in Anal Math Phys, 2016) a class of totally geodesics solutions for the vacuum Einstein equations is introduced. It consists of Einstein metrics of signature (1,3) such that 2-dimensional distributions, defined by the Weyl tensor, are completely integrable and totally geodesic. The complete and explicit description of metrics from these class is given. It is shown that these metrics depend on two functions in one variable and one harmonic function.

We construct explicit solutions of the Einstein–Maxwell equations in the case when distributions defined by the Faraday tensor are completely integrable and totally geodesic.

This paper introduces a new, fully recursive algorithm for computing moving frames and differential invariants of Lie pseudo-group actions. The recursive method avoids unwieldy symbolic expressions that complicate the treatment of large scale applications of the equivariant moving frame method. The development leads to novel results on partial moving frames, structure equations, and new differential operators underlying the moving frame construction. In particular, our methods produce a streamlined computational algorithm for determining moving frames and differential invariants of finite-dimensional Lie group actions.

We study the field of rational differential invariants of the left \(\mathbf {PSL}_{2}\left( \mathbb {R}\right) \)-action on the homogeneous space \(\mathbf { Diff(S}^{1})/\mathbf {PSL}_{2}\left( \mathbb {R}\right) \) and use them to give an alternative discription of the shape space and fingerprints, introduced by Mumford and Sharon (Int J Comput Vis 70(1):55–75, 2006).

The action of the pseudogroup of point transformations on the set of smooth functions on the contact 1-jet space J1ℝ is considered. Differential invariants of this action are found, and the problem of local point equivalence of regular smooth functions is solved.

We apply nonconjugate subgroups of the Poіncare group P(1,4) to perform a partial preliminary group classification of the nonlinear five-dimensional d’Alembert equation. We also present a brief survey of the results obtained.

We present a method for automatically solving apictorial jigsaw puzzles that is based on an extension of the method of differential invariant signatures. Our algorithms are designed to solve challenging puzzles, without having to impose any restrictive assumptions on the shape of the puzzle, the shapes of the individual pieces, or their intrinsic arrangement. As a demonstration, the method was successfully used to solve two commercially available puzzles. Finally we perform some preliminary investigations into scalability of the algorithm for even larger puzzles.

The authors construct Maurer-Cartan equation, the generating set of the differential invariant algebra and their syzygies for the symmetry groups of a (2+1)-dimensional Burgers equation, based on the theory of equivariant moving frames of infinite-dimensional Lie pseudo-groups.

Scale-invariant interest points have found several highly successful applications in computer vision, in particular for image-based matching and recognition.This paper presents a theoretical analysis of the scale selection properties of a generalized framework for detecting interest points from scale-space features presented in Lindeberg (Int. J. Comput. Vis. 2010, under revision) and comprising: an enriched set of differential interest operators at a fixed scale including the Laplacian operator, the determinant of the Hessian, the new Hessian feature strength measures I and II and the rescaled level curve curvature operator, as well as an enriched set of scale selection mechanisms including scale selection based on local extrema over scale, complementary post-smoothing after the computation of non-linear differential invariants and scale selection based on weighted averaging of scale values along feature trajectories over scale. It is shown how the selected scales of different linear and non-linear interest point detectors can be analyzed for Gaussian blob models. Specifically it is shown that for a rotationally symmetric Gaussian blob model, the scale estimates obtained by weighted scale selection will be similar to the scale estimates obtained from local extrema over scale of scale normalized derivatives for each one of the pure second-order operators. In this respect, no scale compensation is needed between the two types of scale selection approaches. When using post-smoothing, the scale estimates may, however, be different between different types of interest point operators, and it is shown how relative calibration factors can be derived to enable comparable scale estimates for each purely second-order operator and for different amounts of self-similar post-smoothing.A theoretical analysis of the sensitivity to affine image deformations is presented, and it is shown that the scale estimates obtained from the determinant of the Hessian operator are affine covariant for an anisotropic Gaussian blob model. Among the other purely second-order operators, the Hessian feature strength measure I has the lowest sensitivity to non-uniform scaling transformations, followed by the Laplacian operator and the Hessian feature strength measure II. The predictions from this theoretical analysis agree with experimental results of the repeatability properties of the different interest point detectors under affine and perspective transformations of real image data. A number of less complete results are derived for the level curve curvature operator.

We formulate and prove a criterion of the equivalence of functional bases of differential invariants of an arbitrary finite order k for nonconjugate subgroups of the local Lie groups of point transformations.

This paper is devoted to differential invariants of equations $$y''=a^{3}(x,y)y^{\prime3}+a^{2}(x,y)y^{\prime2}+a^{1}(x,y)y'+a^{0}(x,y).$$ w.r.t. point transformations. The natural bundle of these equations and its bundles of k-jets of sections, k=0,1,2,…, are considered. The action of the pseudogroup of all point transformations on these bundles is investigated. Tensor differential invariants distinguishing orbits of this action on jet bundles of second and third orders are constructed. A complete collection of generators and their differential syzygies is obtained for the algebra of all scalar differential invariants of a wide class of considered equations containing generic equations.

The problem of feedback equivalence for control systems is considered. An algebra of differential invariants and criteria for the feedback equivalence for regular control systems are found.

A symmetry-based method for constructing solutions to systems of differential equations founded on the reduction of exterior differential systems invariant under the action of an infinite dimensional pseudogroup is proposed. One can associate to any system of differential equations Δ=0 with a symmetry group \(\mathcal{G}\) an exterior differential system \(\mathcal{I}\) invariant under \(\mathcal{G}\) so that solutions of Δ=0 correspond to integral manifolds \(\mathcal{I}\). The \(\mathcal{G}\)-invariant exterior differential system gives rise to a reduced system \(\overline{\mathcal{I}}\) specified on a cross section to the pseudogroup orbits, and it is shown that solutions to Δ=0 can be reconstructed from integral manifolds \(\overline{\mathcal{I}}\) by solving an equation of generalized Lie type for the jets of pseudogroup transformations. In particular, as opposed to the classical method of symmetry reduction, every solution to the system of differential equations can, under some mild regularity assumptions, be constructed by the present algorithm.

We demonstrate that the invariant operators on a homogeneous space generate differential invariants and invariant differentiation operators. The coordinate-free method of this article makes it possible to simply the computations essentially, namely to reduce them to operations of linear algebra. Some examples are exhibited.

This paper discusses non-negativity and positivity concepts and related properties for the state and output trajectory solutions of dynamic linear time-invariant systems described by functional differential equations subject to point time-delays. The various non-negativity and positivity introduced hierarchically from the weakest one to the strongest one while separating the corresponding properties when applied to the state space or to the output space as well as for the zero-initial state or zero-input responses. The formulation is developed by defining cones for the input, state and output spaces of the dynamic system.

We develop new computational algorithms, based on the method of equivariant moving frames, for classifying the differential invariants of Lie symmetry pseudo-groups of differential equations and analyzing the structure of the induced differential invariant algebra. The Korteweg-deVries (KdV) and Kadomtsev-Petviashvili (KP) equations serve to illustrate examples. In particular, we deduce the first complete classification of the differential invariants and their syzygies of the KP symmetry pseudo-group.

For comparison of shapes under subgroups of the projective group, we can use a lot of invariants and especially differential invariants coming from multiscale analysis. But such invariants, as we have to compute curvature, are very sensitive to the noise induced by the dicretization grid. In order to resolve this problem we use size functions which can recognize the ``qualitative similarity" between graphs of functions that should be theorically coinciding but, unfortunately, change their values due to the presence of noise. Moreover, we focus this study on a projective differential invariant which allows to decide if one shape can be considered as the deformation of another one by a rotation of the camera.

In this paper, we investigate the action of the pseudogroup of all point transformations on the bundle of equations y″=u0(x,y)+u1(x,y)y′+u2(x,y)(y′)2+u3(x,y)(y′)3. We calculate the 1st nontrivial differential invariant of this action. It is a horizontal differential 2-form with values in some algebra, it is defined on the bundle of 2-jets of sections of the bundle under consideration. We prove that this form is a unique obstruction to linearizability of these equations by point transformations.

Calculation of differential invariants and invariant differentiation operators of a subalgebra of the Lie algebra admitted by a system of differential equations enables us to construct differential invariant submodels. We classify submodels for every subalgebra of an optimal system of subalgebras. Classification includes the invariant submodels and partially invariant submodels considered earlier. We give examples of classification for three-dimensional subalgebras admitted by the equations of gas dynamics.

In this paper, we derive an explicit group-invariant formula for the Euler- Lagrange equations associated with an invariant variational problem. The method relies on a group-invariant version of the variational bicomplex induced by a general equivariant moving frame construction, and is of independent interest.

In this paper, an extension of all Lie group actions on R 2to coordinates defined by potentials is given. This provides a new solution to the equivalence problems of curves under the projective group and two of its subgroups. The potentials correspond to integrals of higher and higher-order producing an infinite number of independent integral invariants. Applications to computer vision are discussed.

This paper outlines a new general construction, named “multi-space”, that forms the proper geometrical foundation for the numerical analysis of differential equations – in direct analogy with the role played by jet space as the basic object underlying the geometry of differential equations. Application of the theory of moving frames leads to a general framework for constructing symmetry-preserving numerical approximations to differential invariants and invariant differential equations.

Corrected versions of the numerically invariant expressions for the affine and Euclidean signature of a planar curve introduced by Calabi et al. in (Int. J. Comput. Vision, 26: 107–135, 1998) are presented. The new formulas are valid for fine but otherwise arbitrary partitions of the curve. We also give numerically invariant expressions for the four differential invariants parameterizing the three dimensional version of the Euclidean signature curve, namely the curvature, the torsion and their derivatives with respect to arc length.

The primary goal of this paper is to provide a rigorous theoretical justification of Cartan’s method of moving frames for arbitrary finite-dimensional Lie group actions on manifolds. The general theorems are based a new regularized version of the moving frame algorithm, which is of both theoretical and practical use. Applications include a new approach to the construction and classification of differential invariants and invariant differential operators on jet bundles, as well as equivalence, symmetry, and rigidity theorems for submanifolds under general transformation groups. The method also leads to complete classifications of generating systems of differential invariants, explicit commutation formulae for the associated invariant differential operators, and a general classification theorem for syzygies of the higher order differentiated differential invariants. A variety of illustrative examples demonstrate how the method can be directly applied to practical problems arising in geometry, invariant theory, and differential equations.

We introduce a new paradigm, the differential invariant signature curve or manifold, for the invariant recognition of visual objects. A general theorem of E. Cartan implies that two curves are related by a group transformation if and only if their signature curves are identical. The important examples of the Euclidean and equi-affine groups are discussed in detail. Secondly, we show how a new approach to the numerical approximation of differential invariants, based on suitable combination of joint invariants of the underlying group action, allows one to numerically compute differential invariant signatures in a fully group-invariant manner. Applications to a variety of fundamental issues in vision, including detection of symmetries, visual tracking, and reconstruction of occlusions, are discussed.

This is the first in a series of papers devoted to the development and applications of a new general theory of moving frames. In this paper, we formulate a practical and easy to implement explicit method to compute moving frames, invariant differential forms, differential invariants and invariant differential operators, and solve general equivalence problems for both finite-dimensional Lie group actions and infinite Lie pseudo-groups. A wide variety of applications, ranging from differential equations to differential geometry to computer vision are presented. The theoretical justifications for the moving coframe algorithm will appear in the next paper in this series.