About Digital Level Layers Yan Gerard & Laurent Provot ISIT, Clermont Universités GT Géométrie Discrète, 03/12/2010

Outline I Linear Primitives II Unlinear Primitives III Some Applications of DLL IV Algorithms

I Linear Primitives

digital straight line

digital plane and more generally digital hyperplanes of Z d

The boundary of the lattice points in the half-space of equation a.x<h Digital hyperplanes of Z d have at least 3 definitions TopologyMorphology Algebra

Digital hyperplanes of Z d have at least 3 definitions TopologyMorphology Algebra The track on Z d of a Minskowski sum H+Structuring Element Structuring element

Digital hyperplanes of Z d have at least 3 definitions TopologyMorphology Algebra The track on Z d of a Minskowski sum H+Structuring Element Structuring element ball N 0 ball N 1 ball N 2 segments

The lattice points in an affine strip of double equation h< a.x <h’ Digital hyperplanes of Z d have at least 3 definitions TopologyMorphology Algebra

Digital hyperplanes of Z d have at least 3 definitions TopologyMorphology Algebra NeighborhoodStructuring elementvalue h’-h Parameters

Digital hyperplanes of Z d have at least 3 definitions TopologyMorphology Algebra More generally NeighborhoodStructuring elementvalue h’-h Ball N 8 Ball N 1 h’-h=N (a) 8 Ball N 1 Ball N 8 h’-h=N 1 (a) Ball N ? Ball N h’-h=N* (a) The three definitions collapse But what about unlinear primitives ?

II Unlinear Primitives

Let S be a continuous level set of equation f(x)=0 Problem: define a digital primitive for S.

Three approaches

TopologyMorphology Algebra Three approaches

TopologyMorphology Algebra Structuring element Three approaches

TopologyMorphology Algebra We consider the lattice points between two ellipses f(x)=h et f(x)=h’ Three approaches

TopologyMorphology Algebra Three approaches The three approaches are equivalent for linear structure but not for unlinear shapes Advantages and drawbacks ?

TopologyMorphologyAlgebra Three approaches Topology Morphology Recognition algorithm Properties Advantages and drawbacks ? Algebraic characterization

TopologyMorphologyAlgebra Three approaches Topology Morphology Algebraic characterization Recognition algorithm Properties SVM

Algebra Topology Morphology

Algebra Definition: Topology Morphology This kind of primitives is not a surface!!!!!! The lattice set characterized by a double-inequality h<f(x)<h’ is called a Digital Level Layer (DLL for short).

III Some Applications of DLL

Estimation of the k th derivative of a digital function Previous works : A. Vialard, J-O Lachaud, F De Vieilleville An approximation based on maximal straight segments S. Fourey, F. Brunet, A. Esbelin, R. Malgouyres An approximation based on convolutions Error Bounding O(h 1/3 ) for k=1 O(h (2/3) ) for k k An approximation based on DLL Recognition L. Provot, Y. G O(h (1/(k+1)) ) for k

Estimation of the k th derivative of a digital function Principle : Input:Points

Estimation of the k th derivative of a digital function Principle : + Vertical thickness (or maximal roughness)>1 Input:Points

Estimation of the k th derivative of a digital function Principle : + Vertical thickness (or maximal roughness)>1 Input:Points + order k Polynomial of degree ≤ k

Estimation of the k th derivative of a digital function Principle : DLL of double-inequation -roughness ≤ y-P(x) ≤ +roughness containing S Output: Polynomial of degree ≤ k the derivative of P(x) as digital derivative

Estimation of the k th derivative of a digital function Previous works : A. Vialard, J-O Lachaud, F De Vieilleville An approximation based on maximal straight segments S. Fourey, F. Brunet, A. Esbelin, R. Malgouyres An approximation based on convolutions Error Bounding O(h 1/3 ) for k=1 O(h (2/3) ) for k k An approximation based on DLL Recognition L. Provot, Y. G O(h (1/(k+1)) ) for k Increase the degree Relax the maximal vertical thickness Different general algorithms (chords or GJK)…

Second derivative

Vectorization of Digital Shapes Principle : Lattice set S Input: Recognition DLL containing S Alternative ? Digitization Undesired neighbors

Vectorization of Digital Shapes Principle : Lattice set S Input: Recognition DLL containing S Digitization Undesired neighbors Forbidden neighbors + Recognition DLL between the inliers and outliers

IV Algorithms

Problem of separation by a level set f(x)=0 with f in a given linear space Problem of linear separability in a descriptive space well-known in the framework of Support Vector Machine (Kernel trick: Aizerman et al. 1964) or Computational Geometry GJK computes the closest pair of points from the two convex hulls Recognition of topological surfaces

Problem of separation by two level sets f(x)=h and f(x)=h’ with f in a given linear space Problem of linear separability by two parallel hyperplanes We introduce a variant of GJK in nD Recognition of DLL with forbidden points Thank you for your attention