about digital level layers
DESCRIPTION
About Digital Level Layers. GT Géométrie Discrète, 03/12/2010. Yan Gerard & Laurent Provot ISIT, Clermont Universités. [email protected] [email protected]. Outline. I Linear Primitives. II Unlinear Primitives. III Some Applications of DLL. - PowerPoint PPT PresentationTRANSCRIPT
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About Digital Level Layers
Yan Gerard & Laurent Provot
ISIT, Clermont Universités
GT Géométrie Discrète, 03/12/2010
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Outline
I Linear Primitives
II Unlinear Primitives
III Some Applications of DLL
IV Algorithms
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ILinear Primitives
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digital straight line
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digital plane
and more generally digital hyperplanes of Zd
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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
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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
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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
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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
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Digital hyperplanes of Zd have at least 3 definitions
Topology Morphology Algebra
Neighborhood Structuring element value h’-h
Parameters
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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 ?
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IIUnlinear Primitives
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Let S be a continuous level set of equation f(x)=0
Problem: define a digital primitive for S.
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Problem: define a digital primitive for S.
Three approaches
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Topology Morphology Algebra
Three approaches
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Topology Morphology Algebra
Structuring element
Three approaches
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Topology Morphology Algebra
We consider the lattice points between two ellipses f(x)=h et f(x)=h’
Three approaches
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Topology Morphology Algebra
Three approaches
The three approaches are equivalent for linear structure
but not for unlinear shapes
Advantages and drawbacks ?
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Topology Morphology Algebra
Three approaches
Topology
Morphology
Recognitionalgorithm
Properties
Advantages and drawbacks ?
Algebraic characterization
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Topology Morphology Algebra
Three approaches
Topology
Morphology
Algebraic characterization
Recognitionalgorithm
Properties
SVM
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Algebra
Topology
Morphology
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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).
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IIISome Applications of DLL
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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
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Estimation of the kth derivative of a digital function
Principle :
Input: Points
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Estimation of the kth derivative of a digital function
Principle :
+ Vertical thickness (or maximal roughness)>1Input: Points
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Estimation of the kth derivative of a digital function
Principle :
+ Vertical thickness (or maximal roughness)>1Input: Points + order k
Polynomial of degree ≤ k
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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
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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)…
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Second derivative
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Second derivative
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Vectorization of Digital Shapes
Principle :
Lattice set SInput: Recognition
DLL containing S
Alternative ?
Digitization Undesired neighbors
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Vectorization of Digital Shapes
Principle :
Lattice set SInput: Recognition
DLL containing SDigitization
Undesired neighbors
Forbidden neighbors+ Recognition
DLL between the inliers and outliers
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IVAlgorithms
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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
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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