A novel scheme for the detection of object boundaries is presented. The technique is based on active contours deforming according to intrinsic geometric measures of the image. The evolving contours naturally split and merge, allowing the simultaneous detection of several objects and both interior and exterior boundaries. The proposed approach is based on the relation between active contours and the computation of geodesics or minimal distance curves. The minimal distance curve lays in a Riemannian space whose metric as defined by the image content. This geodesic approach for object segmentation allows to connect classical "snakes" based on energy minimization and geometric active contours based on the theory of curve evolution. Previous models of geometric active contours are improved as showed by a number of examples. Formal results concerning existence, uniqueness, stability, and correctness of the evolution are presented as well. >

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Geodesic active contours

A novel scheme for the detection of object boundaries is presented. The technique is based on active contours evolving in time according to intrinsic geometric measures of the image. The evolving contours naturally split and merge, allowing the simultaneous detection of several objects and both interior and exterior boundaries. The proposed approach is based on the relation between active contours and the computation of geodesics or minimal distance curves. The minimal distance curve lays in a Riemannian space whose metric is defined by the image content. This geodesic approach for object segmentation allows to connect classical “snakes” based on energy minimization and geometric active contours based on the theory of curve evolution. Previous models of geometric active contours are improved, allowing stable boundary detection when their gradients suffer from large variations, including gaps. Formal results concerning existence, uniqueness, stability, and correctness of the evolution are presented as well. The scheme was implemented using an efficient algorithm for curve evolution. Experimental results of applying the scheme to real images including objects with holes and medical data imagery demonstrate its power. The results may be extended to 3D object segmentation as well.

Geodesic Active Contours

We introduce flat systems, which are equivalent to linear ones via a special type of feedback called endogenous. Their physical properties are subsumed by a linearizing output and they might be regarded as providing another nonlinear extension of Kalman's controllability. The distance to flatness is measured by a non-negative integer, the defect. We utilize differential algebra where flatness- and defect are best defined without distinguishing between input, state, output and other variables. Many realistic classes of examples are flat. We treat two popular ones: the crane and the car with n trailers, the motion planning of which is obtained via elementary properties of plane curves. The three non-flat examples, the simple, double and variable length pendulums, are borrowed from non-linear physics. A high frequency control strategy is proposed such that the averaged systems become flat.

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Flatness and defect of non-linear systems: introductory theory and examples

to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

Mirror symmetry (MS) was discovered several years ago in string theory as a duality between families of 3-dimensional Calabi-Yau manifolds (more precisely, complex algebraic manifolds possessing holomorphic volume elements without zeros). The name comes from the symmetry among Hodge numbers. For dual Calabi-Yau manifolds V, W of dimension n (not necessarily equal to 3) one has 

$$\dim {H^p}(V,{\Omega ^q}) = \dim {H^{n - p}}(W,{\Omega ^q}).$$

.

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Homological Algebra of Mirror Symmetry

These lecture notes are devoted to the theory of equations of associativity describing geometry of moduli spaces of 2D topological field theories. Introduction. Lecture 1. WDVV equations and Frobenius manifolds. {Appendix A.} Polynomial solutions of WDVV. {Appendix B.} Symmetriies of WDVV. Twisted Frobenius manifolds. {Appendix C.} WDVV and Chazy equation. Affine connections on curves with projective structure. Lecture 2. Topological conformal field theories and their moduli. Lecture 3. Spaces of isomonodromy deformations as Frobenius manifolds. {Appendix D.} Geometry of flat pencils of metrics. {Appendix E.} WDVV and Painlev\'e-VI. {Appendix F.} Branching of solutions of the equations of isomonodromic deformations and braid group. {Appendix G.} Monodromy group of a Frobenius manifold. {Appendix H.} Generalized hypergeometric equation associated to a Frobenius manifold and its monodromy. {Appendix I.} Determination of a superpotential of a Frobenius manifold. Lecture 4. Frobenius structure on the space of orbits of a Coxeter group. {Appendix J.} Extended complex crystallographic groups and twisted Frobenius manifolds. Lecture 5. Differential geometry of Hurwitz spaces. Lecture 6. Frobenius manifolds and integrable hierarchies. Coupling to topological gravity.

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Geometry of 2D topological field theories

The first € price and the £ and $ price are net prices, subject to local VAT. Prices indicated with * include VAT for books; the €(D) includes 7% for Germany, the €(A) includes 10% for Austria. Prices indicated with ** include VAT for electronic products; 19% for Germany, 20% for Austria. All prices exclusive of carriage charges. Prices and other details are subject to change without notice. All errors and omissions excepted. B.A. Dubrovin, A.T. Fomenko, S.P. Novikov Modern Geometry—Methods and Applications

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Modern geometry--methods and applications

The basic content of this survey is an exposition of a recently developed method of constructing a broad class of periodic and almost-periodic solutions of non-linear equations of mathematical physics to which (in the rapidly decreasing case) the method of the inverse scattering problem is applicable. These solutions are such that the spectrum of their associated linear differential operators has a finite-zone structure. The set of linear operators with a given finite-zone spectrum is the Jacobian variety of a Riemann surface, which is determined by the structure of the spectrum. We give an explicit solution of the corresponding non-linear equations in the language of the theory of Abelian functions.

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Non-linear equations of korteweg-de vries type, finite-zone linear operators, and abelian varieties

CONTENTS Introduction Chapter I. Theta functions. General information § 1. Definition of theta functions and their simplest properties § 2. Theta functions of a single variable § 3. On Abelian tori § 4. Addition theorems for theta functions Chapter II. Theta functions of Riemann surfaces. The Jacobi inversion problem § 1. Periods of Abelian differentials on Riemann surfaces. Jacobi varieties § 2. Abel's theorem § 3. Some remarks on divisors on a Riemann surface § 4. The Jacobi inversion problem. Examples Chapter III. The Baker-Akhiezer function. Applications to non-linear equations § 1. The Baker-Akhiezer one-point function. The Kadomtsev-Petviashvili equation and equations associated with it § 2. The Baker-Akhiezer two-point function. The Schrodinger equation in a magnetic field Chapter IV. Effectivization of the formulae for the solution of KdV and KP equations. Recovery of a Riemann surface from its Jacobi variety. The problem of Riemann and the conjecture of Novikov § 1. The KdV equation. Genus g = 1 or 2 § 2. The KP equation. Genus 2 and 3 § 3. The KP equation. Genus g ≥ 2. Canonical equations of Riemann surfaces § 4. The problem of Riemann on relations between the periods of holomorphic differentials on a Riemann surface and the conjecture of Novikov Chapter V. Examples of Hamiltonian systems that are integrable in terms of two-dimensional theta functions § 1. Two-zone potentials § 2. The problem of Sophie Kovalevskaya § 3. The problems of Neumann and Jacobi. The general Garnier system § 4. Movement of a solid in an ideal fluid. Integration of the Clebsch case. A multi-dimensional solid Appendix. I.M. Krichever. The periodic non-Abelian Toda chain and its two-dimensional generalization References

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Theta functions and non-linear equations

CONTENTS Introduction Chapter I. Hamiltonian theory of systems of hydrodynamic type § 1. General properties of Poisson brackets § 2. Hamiltonian formalism of systems of hydrodynamic type and Riemannian geometry § 3. Generalizations: differential-geometric Poisson brackets of higher orders, differential-geometric Poisson brackets on a lattice, and the Yang-Baxter equation § 4. Riemann invariants and the Hamiltonian formalism of diagonal systems of hydrodynamic type. Novikov's conjecture. Tsarev's theorem. The generalized hodograph method Chapter II. Equations of hydrodynamics of soliton lattices § 5. The Bogolyubov-Whitham averaging method for field-theoretic systems and soliton lattices. The results of Whitham and Hayes for Lagrangian systems § 6. The Whitham equations of hydrodynamics of weakly deformed soliton lattices for Hamiltonian field-theoretic systems. The principle of conservation of the Hamiltonian structure under averaging § 7. Modulations of soliton lattices of completely integrable evolutionary systems. Krichever's method. The analytic solution of the Gurevich-Pitaevskii problem on the dispersive analogue of a shock wave § 8. Evolution of the oscillatory zone in the KdV theory. Multi-valued functions in the hydrodynamics of soliton lattices. Numerical studies § 9. Influence of small viscosity on the evolution of the oscillatory zone References

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Boris Dubrovin

Papers

Geometry of 2D topological field theories

Modern geometry--methods and applications

Non-linear equations of korteweg-de vries type, finite-zone linear operators, and abelian varieties

Theta functions and non-linear equations

Hydrodynamics of weakly deformed soliton lattices. Differential geometry and Hamiltonian theory