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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
ILinear Primitives
digital straight line
digital plane
and more generally digital hyperplanes of Zd
The boundary of the lattice points in the half-space of equation a.x<h
Digital hyperplanes of Zd have at least 3 definitions
Topology Morphology Algebra
Digital hyperplanes of Zd have at least 3 definitions
Topology Morphology Algebra
The track on Zd of a Minskowski sum H+Structuring Element
Structuring element
Digital hyperplanes of Zd have at least 3 definitions
Topology Morphology Algebra
The track on Zd of a Minskowski sum H+Structuring Element
Structuring element
ball N0
ball N1
ball N2
segments
The lattice points in an affine strip of double equation h< a.x <h’
Digital hyperplanes of Zd have at least 3 definitions
Topology Morphology Algebra
Digital hyperplanes of Zd have at least 3 definitions
Topology Morphology Algebra
Neighborhood Structuring element value h’-h
Parameters
Digital hyperplanes of Zd have at least 3 definitions
Topology Morphology Algebra
More generally
Neighborhood Structuring element value h’-h
Ball N 8 Ball N1h’-h=N (a)8
Ball N1Ball N 8 h’-h=N1 (a)
Ball N ? Ball N h’-h=N* (a)
The three definitions collapse
But what about unlinear primitives ?
IIUnlinear Primitives
Let S be a continuous level set of equation f(x)=0
Problem: define a digital primitive for S.
Problem: define a digital primitive for S.
Three approaches
Topology Morphology Algebra
Three approaches
Topology Morphology Algebra
Structuring element
Three approaches
Topology Morphology Algebra
We consider the lattice points between two ellipses f(x)=h et f(x)=h’
Three approaches
Topology Morphology Algebra
Three approaches
The three approaches are equivalent for linear structure
but not for unlinear shapes
Advantages and drawbacks ?
Topology Morphology Algebra
Three approaches
Topology
Morphology
Recognitionalgorithm
Properties
Advantages and drawbacks ?
Algebraic characterization
Topology Morphology Algebra
Three approaches
Topology
Morphology
Algebraic characterization
Recognitionalgorithm
Properties
SVM
Algebra
Topology
Morphology
Algebra
Definition:
Topology
MorphologyThis kind of primitives is not a surface!!!!!!
The lattice set characterized by a double-inequality h<f(x)<h’ is called aDigital Level Layer (DLL for short).
IIISome Applications of DLL
Estimation of the kth 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(h1/3) for k=1
O(h(2/3) ) for kk
An approximation based on DLL Recognition
L. Provot, Y. GO(h(1/(k+1)) ) for k
Estimation of the kth derivative of a digital function
Principle :
Input: Points
Estimation of the kth derivative of a digital function
Principle :
+ Vertical thickness (or maximal roughness)>1Input: Points
Estimation of the kth derivative of a digital function
Principle :
+ Vertical thickness (or maximal roughness)>1Input: Points + order k
Polynomial of degree ≤ k
Estimation of the kth derivative of a digital function
Principle :
DLL of double-inequation -roughness ≤ y-P(x) ≤ +roughness containing SOutput:
Polynomial of degree ≤ k
the derivative of P(x) as digital derivative
Estimation of the kth 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(h1/3) for k=1
O(h(2/3) ) for kk
An approximation based on DLL Recognition
L. Provot, Y. GO(h(1/(k+1)) ) for k
Increase the
degree
Relax the maximal vertical
thickness
Different general algorithms (chords or GJK)…
Second derivative
Second derivative
Vectorization of Digital Shapes
Principle :
Lattice set SInput: Recognition
DLL containing S
Alternative ?
Digitization Undesired neighbors
Vectorization of Digital Shapes
Principle :
Lattice set SInput: Recognition
DLL containing SDigitization
Undesired neighbors
Forbidden neighbors+ Recognition
DLL between the inliers and outliers
IVAlgorithms
Problem of separation by a level set f(x)=0
with f in a given linear space
Problem of linear separabilityin 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 separabilityby two parallel hyperplanes
We introduce a variant of GJK in nD
Recognition of DLL with forbidden points
Thank you
for
your attention
