Affine Invariant Detection: Edge Maps, Anisotropic Diffusion, and Active Contours

In this paper we undertake a systematic investigation of affine invariant object detection and image denoising. Edge detection is first presented from the point of view of the affine invariant scale-space obtained by curvature based motion of the image level-sets. In this case, affine invariant maps are derived as a weighted difference of images at different scales. We then introduce the affine gradient as an affine invariant differential function of lowest possible order with qualitative behavior similar to the Euclidean gradient magnitude. These edge detectors are the basis for the extension of the affine invariant scale-space to a complete affine flow for image denoising and simplification, and to define affine invariant active contours for object detection and edge integration. The active contours are obtained as a gradient flow in a conformally Euclidean space defined by the image on which the object is to be detected. That is, we show that objects can be segmented in an affine invariant manner by computing a path of minimal weighted affine distance, the weight being given by functions of the affine edge detectors. The gradient path is computed via an algorithm which allows to simultaneously detect any number of objects independently of the initial curve topology. Based on the same theory of affine invariant gradient flows we show that the affine geometric heat flow is minimizing, in an affine invariant form, the area enclosed by the curve.     

Recognition of centre symmetry set singularities

Let M be a smooth surface in real affine 3-space. Consider the pairs of points of this surface at which the tangent planes are parallel and in particular the chords (infinite lines) joining these pairs. We study in detail and classify the singularities of the envelope of these chords, that is a (singular) surface tangent to all of them. This is called the Centre Symmetry Set (CSS) of M. The study is local in character and is based upon a more general investigation by the authors of the n-dimensional case. The construction of the CSS is affinely invariant and generalises the focal set of a surface in euclidean space and the affine focal set of a surface in affine space. Many standard and some unusual singularities occur in a natural way as the singularities of the CSS. There are illustrations of the various cases by means of concrete examples.      

Singularities of Centre Symmetry Sets

The center symmetry set (CSS) of a smooth hypersurface S inan affine space Rⁿ is the envelope of lines joining pairs ofpoints where S has parallel tangent hyperplanes. The idea stemsfrom a definition of Janeczko, in an alternative version dueto Giblin and Holtom. For n = 2 the envelope is always real,while for n > 3 the existence of a real envelope dependson the geometry of the hypersurface. In this paper we make alocal study of the CSS, some results applying to n 5 and othersto the cases n = 2,3. The method is to construct a generatingfunction whose bifurcation set contains the CSS and possiblysome other redundant components. Focal sets of smooth hypersurfacesare a special case of the construction, but the CSS is an affineand not a euclidean invariant. Besides the familiar local formsof focal sets there are other local forms corresponding to boundarysingularities, and yet others which do not appear to have arisenelsewhere in a geometrical context. There are connections withFinsler geometry. This paper concentrates on the theory andthe proof of the local normal forms for the CSS. 2000 MathematicsSubject Classification 57R45, 58K40, 32S25, 58B20.     

New results on affine invariant points

We prove a conjecture of B. Grünbaum stating that the set of affine invariant points of a convex body equals the set of points invariant under all affine linear symmetries of the convex body. As a consequence we give a short proof of the fact that the affine space of affine linear points is infinite dimensional. In particular, we show that the set of affine invariant points with no dual is of the second category. We investigate extremal cases for a class of symmetry measures. We show that the centers of the John and Löwner ellipsoids can be far apart and we give the optimal order for the extremal distance between the two centers.     

Singularities of Families of Chords

A chord is a straight line joining two points of a pair of hypersurfaces in an affine space such that the tangent hyperplanes at these points are parallel. We classify the singularities of envelopes of the families of chords determined by generic pairs of plane curves and surfaces in three-space. The list contains all bifurcation diagrams of simple boundary singularities (of the corresponding multiplicity).     

An analytic invariant of ordinary curve singularities

It is well known that analytically equivalent ordinary plane curve singularities have projectively equivalent tangent cones. In this note we introduce an analytic invariant in order to show two non analytically equivalent ordinary 5-fold points with projectively equivalent (or equal) tangent cones.

     

Focal loci of algebraic varieties i

The focal locus ∑_x of an affine variety X is roughly speaking the (projective) closure of the set of points O for which there is a smooth point x ∈X and a circle with centre O passing through x which osculates X inx. Algebraic geometry interprets the focal locus as the branching locus of the endpoint map ∈ between the Euclidean normal bundle N_x and the projective ambient space (∈ sends the normal vector O - x to its endpoint O), and in this paper we address two general problems:.

1)Characterize the"degenerate"case where the focal locus is not a hyper surface.

2)Calculate, in the case where ∑x is a hypersurface, its degree (with multiplicity).     

Properties of Affinely Regular Polygons

Intuitively, one might consider an affinely regular polygon of the Eucidean plane to be the result of applying an affine transformation to a regular polygon. These affinely regular polygons, and their kindred that go by the same name in the Euclidean plane as well as in more general affine planes, have been onjects of investigations at all levels of sophistication and in a remarkable variety of contexts. For example, they arise in linear algebra as a set of vectors that are cyclically permuted by a unimodular matrix. Our purpose is to describe this concept and its attributes in a general setting. The main result is Theorem 1 where we present seven equivalent definitions of affine regularity, one of which appears for the first time. We are careful to distinguish these definitions from the weaker intuitive definition. Our work also features an application of Chebyshev polynomials to describe parameters associated with these polygons.     

Complex affine distributions

Geometry of affine immersions is the study of hypersurfaces that are invariant under affine transformations. As with the hypersurface theory on the Euclidean space, an affine immersion can induce a torsion-free affine connection and a (pseudo)-Riemannian metric on the hypersurface. Moreover, an affine immersion can induce a statistical manifold, which plays a central role in information geometry. Recently, a statistical manifold with a complex structure is actively studied since it connects information geometry and Kähler geometry. However, a holomorphic complex affine immersion cannot induce such a statistical manifold with a Kähler structure. In this paper, we introduce complex affine distributions, which are non-integrable generalizations of complex affine immersions. We then present the fundamental theorem for a complex affine distribution, and show that a complex affine distribution can induce a statistical manifold with a Kähler structure.     

The deformation of flat connections and affine manifolds

Geodesically complete affine manifolds are quotients of the Euclidean space through a properly discontinuous action of a subgroup of affine Euclidean transformations. An equivalent definition is that the tangent bundle of such a manifold admits a flat, symmetric and complete connection. If the completeness assumption is dropped, the manifold is not necessarily obtained as the quotient of the Euclidean space through a properly discontinuous group of affine transformations. In fact the universal cover may no longer be the Euclidean space. The main result of this paper states that if a flat connection of a bundle can be properly deformed into a metric connection then its Euler class vanishes. This is a partial result toward an old question of Chern.