Affine invariant scale-space

  title={Affine invariant scale-space},
  author={Guillermo Sapiro and Allen R. Tannenbaum},
  journal={International Journal of Computer Vision},
A newaffine invariant scale-space for planar curves is presented in this work. The scale-space is obtained from the solution of a novel nonlinear curve evolution equation which admits affine invariant solutions. This flow was proved to be the affine analogue of the well knownEuclidean shortening flow. The evolution also satisfies properties such ascausality, which makes it useful in defining a scale-space. Using an efficient numerical algorithm for curve evolution, this continuous affine flow… 

On projective plane curve evolutionOlivier

In this paper, we investigate the evolution of curves of the projective plane according to a family of projective invariant intrinsic equations. This is motivated by previous work for the Euclidean

Affine-invariant curvature estimators for implicit surfaces

Affine-Invariant Estimators for Implicit Surfaces 1

Affine Differential Geometry provides a set of measures invariant under a set of transformations that is more general than rigid motions. This may provide robust bases to several applications for

Plane curve matching under affine transformations

It is common to use an affine transformation to approximate the plane curve matching problem under a projective transformation. The plane curve itself can be used as an identity to solve the

Invariant image descriptors and affine morphological scale-space

This research report proposes a novel approach to build interest points and descriptors which are invariant to a subclass of affine transformations, based on the so-called Affine Morphological Scale-Space, a non-linear filtering which has been proved to be the natural equivalent of the classic linear scale-space when affine invariance is required.

Numerical Invariantization for Morphological PDE Schemes

This paper focuses on the Hamilton-Jacobi equation governing dilation and erosion processes which displays morphological symmetry, i.e. is invariant under strictly monotonically increasing transformations of gray-values.

Morphological Counterparts of Linear Shift-Invariant Scale-Spaces

A simple and fairly general dictionary that allows to translate any linear shift-invariant evolution equation into its morphological counterpart and vice versa is presented, based on a scale-space representation by means of the symbol of its (pseudo)differential operator.

Affine Invariant Flows in the Beltrami Framework

  • N. Sochen
  • Mathematics
    Journal of Mathematical Imaging and Vision
  • 2004
The geometric Beltrami framework is shown to incorporate and explain some of the known invariant flows e.g. the equi-affine invariant flow for hypersurfaces, and it is demonstrated that the new concepts put forward enable us to suggest new invariants namely the case where the codimension is greater than one.

Multiscale, curvature-based shape representation for surfaces

This paper presents a multiscale, curvature-based shape representation technique for general genus zero closed surfaces. The method is invariant under rotation, translation, scaling, or general

On Affine Invariant Descriptors Related to SIFT

This paper presents some basic geometric affine invariant quantities, and uses them to construct some distinctive descriptors for object detection, and suggests that these descriptors behave more robustly than SIFT with respect to affine deformations.



On affine plane curve evolution

Abstract An affine invariant curve evolution process is presented in this work. The evolution studied is the affine analogue of the Euclidean Curve Shortening flow. Evolution equations, for both

A Theory of Multiscale, Curvature-Based Shape Representation for Planar Curves

A shape representation technique suitable for tasks that call for recognition of a noisy curve of arbitrary shape at an arbitrary scale or orientation is presented and several evolution and arc length evolution properties of planar curves are discussed.

Representations Based on Zero-Crossing in Scale-Space-M


The purpose of this paper is to remove regularity assumptions on the class of convex bodies for which equality is obtained in three closely related affine extremal problems. These three problems are

The Curve Shortening Flow

This is an expository paper describing the recent progress in the study of the curve shortening equation $${X_{{t\,}}} = \,kN $$ (0.1) Here X is an immersed curve in ℝ2, k the geodesic

The normalized curve shortening flow and homothetic solutions

The curve shortening problem, by now widely known, is to understand the evolution of regular closed curves γ: R/Z -> M moving according to the curvature normal vector: dy/dt = kN = -"the ZΛgradient

Parabolic equations for curves on surfaces Part I. Curves with $p$-integrable curvature

This is the first of a two-part paper in which we develop a theory of parabolic equations for curves on surfaces which can be applied to the so-called curve shortening of flow-by-mean-curvature

On the evolution of curves via a function of curvature

Parabolic equations for curves on surfaces Part II. Intersections, blow-up and generalized solutions

We describe a theory for parabolic equations for immersed curves on surfaces, which generalizes the curve shortening or flow-by-mean-curvature problem, as well as several models in the theory of

Scale and the differential structure of images