Optimal Methods for Discrete Configurations of
some Continuous Shapes.
Dr. Adama Arouna KONÉ
Faculté des Sciences et Techniques de l’USTT-Bamako,
Département de Mathématiques et Informatique, Mali.
First network meeting for Sida-and ISP-funded PhD students in mathematics
Stockholm 7-8 March 2017
7 mars 2017
Dr. Adama Arouna KONÉ (FST)
Digital geometry and optimization
7 mars 2017
1 / 25
Outline
1
Overview of our first paper
2
PhD thesis
3
Recent works
4
Conclusion
Dr. Adama Arouna KONÉ (FST)
Digital geometry and optimization
7 mars 2017
2 / 25
Overview of our first paper
Outline
1
Overview of our first paper
2
PhD thesis
3
Recent works
4
Conclusion
Dr. Adama Arouna KONÉ (FST)
Digital geometry and optimization
7 mars 2017
3 / 25
Overview of our first paper
Covering a Euclidean Line or Hyperplane by
Dilations of its Discretization
Paper accepted for publication in Vietnam Journal of Mathematics 2015-12-28.
Online First : 2016-06-03
DOI 10.1007/s10013-016-0202-2
Dr. Adama Arouna KONÉ (FST)
Digital geometry and optimization
7 mars 2017
4 / 25
Overview of our first paper
Covering a Euclidean straight line
Chord property in the sense of Azriel Rosenfeld (1974)
Chord property in the sense of Azriel Rosenfeld (1974)
A subset L of R2 has the chord property in the sense of Rosenfeld
(1974) if
chord(L) ⊂ L + B< (0, 1),
where L + B< (0, 1) = δB< (0,1) (L) is the dilation L by B< (0, 1)
Dr. Adama Arouna KONÉ (FST)
Digital geometry and optimization
7 mars 2017
5 / 25
Overview of our first paper
Covering a Euclidean straight line
Chord property in the sense of Azriel Rosenfeld (1974)
Consequently
(x + B< (0, 1)) ∩ Z2 = {x}, ∀x ∈ Z2
⇒ ∀M ⊂ Z2 having chord property, we have
chord(M ) ∩ Z2 = M
Dr. Adama Arouna KONÉ (FST)
Digital geometry and optimization
7 mars 2017
5 / 25
Overview of our first paper
Covering a Euclidean straight line
Euclidean straight line and its discretization
.M =
.L =
n
(x, F (x)) ∈ R2 , F (x) =
α1 x+µ
α2
o
n
(x, f (x)) ∈ Z2 , f (x) =
j
α1 x+µ
α2
ko
is a
Euclidean Line
.M =
n
(x, f (x)) ∈ Z2 , f (x) =
j
α1 x+µ
α2
+ 1
2
ko
L : α1 x − α2 y + µ = 0 où (x, y) ∈ R2
Two kinds of discretization M of L
Dr. Adama Arouna KONÉ (FST)
Digital geometry and optimization
7 mars 2017
6 / 25
Overview of our first paper
Covering a Euclidean straight line
Structuring element
The structuring element U is a subset of R2 defined by :
U (r1 , r2 ) = {(x1 , x2 ) ∈ R × R; |x1 | 6 r1 , |x2 | 6 r2 } = [−r1 , r1 ] × [−r2 , r2 ]
Where r1 and r2 are defined for the chosen floor function as discretization by
r2 =
α −1
α
1 (1 − r ) + 2
,
if r1 ∈ [ 1
, 1] ;
1
α2
α2
2
1 − α1 ,
if 1 6 r1 ,
2


 r = max 1 −
1
1 ,
, 2
if r1 = r2 = r.
α +α




1
2
and for the chosen closest integer as discretization by
r2 =





α1
α2
1,
2
(1 − r1 ) +
bα2 /2c
,
α2
2bα /2c−α
1,1 +
if r1 ∈ [ 2
2bα2 /2c−α2
2α1
];
2
2 6 r ,
if 1 +
1
2α1



 r = max 1 − dα2 /2e , 1 ,
if r1 = r2 = r.
α1 +α2 2
Dr. Adama Arouna KONÉ (FST)
Digital geometry and optimization
7 mars 2017
7 / 25
Overview of our first paper
Covering a Euclidean straight line
Different forms of U in the plane.
Dr. Adama Arouna KONÉ (FST)
Digital geometry and optimization
7 mars 2017
8 / 25
Overview of our first paper
Covering a Euclidean straight line
Problem of covering a Euclidean straight line by dilations of its discretization
. Whether the dilation of M by a structuring element U contains the straight
line L?
. In other words, whether L ⊂ δU (M ) = M + U ?
Example of covering a Euclidean straight line
Covering by means of the floor function.
Covering by means of the closest integer.
Example of covering a portion of L by M + U (r1 , r2 )
Dr. Adama Arouna KONÉ (FST)
Digital geometry and optimization
7 mars 2017
9 / 25
Overview of our first paper
Covering a Euclidean straight line
Counterexample
Example of partial covering a Euclidean straight line
12
12
10
10
8
8
6
6
4
4
2
Floor function
Closest integer
Euclidean straight line
2
0
-2
Euclidean straight line
0
2
4
6
8
10
12
14
16
18
Covering by means of the floor function.
0
0
2
4
6
8
10
12
14
16
18
Covering by means of the closest integer.
Example of covering a portion of L by M + U ( 1
, 1)
2 2
These examples confirm that the structuring element U 12 , 12 or the closed ball
centered at origin and radius r = 12 is not always suitable for optimal covering.
Dr. Adama Arouna KONÉ (FST)
Digital geometry and optimization
7 mars 2017
10 / 25
Overview of our first paper
Extension of the concept of Covering
Covering a Euclidean hyperplane
Let L be a Euclidean hyperplane of R3 define by
α3 x3 = α1 x1 + α2 x2 + µ
and given by a function
F (x) =
α1 x1 + α2 x2 + µ
, x ∈ R2 , α = (α1 , α2 , α3 ) ∈ R3
α3
and µ ∈ R, with L = graph(F ),
and D = graph(f ) is the graph of a function f : Z2 → Z, thus a subset of Z3 .
f = bF |Z3 c or f = F |Z3 + 12
Dr. Adama Arouna KONÉ (FST)
Digital geometry and optimization
7 mars 2017
11 / 25
Overview of our first paper
Extension of the concept of Covering
Euclidean Hyperplane and its Discretization
4
3
2
1
0
-1
-2
0
10
2
8
4
6
6
4
8
2
10
0
Figure –
Figure –
D ⊂ Z3
L ⊂ R3
Dr. Adama Arouna KONÉ (FST)
Digital geometry and optimization
7 mars 2017
12 / 25
Overview of our first paper
Extension of the concept of Covering
The structuring element U is a subset of R3 defined by :
U (r) = {x ∈ R3 ; |xj | 6 rj , j = 1, . . . , 3} =
3
Y
[−rj , rj ] ⊂ R3 .
j=1
Where r1 , r2 and r3 are defined for the floor function as discretization by
 P
αj
α3 −1
α3 (1−θ)−1
2
1

, j = 1, 2 ;

j=1 α3 (1 − rj ) + α3 , rj ∈ 2 , 1 +
2αj






α (1−θ)−1
1 ,
r3 =
1 + 3 2α
6 rj , j = 1, 2 ;
θ = max 1 − α +α1 +α , 2

1
2
3
j





P

α

α3 −1
j
2
1 6 r , j = 1, 2.
max
,θ ,
j
j=1 α (1 − rj ) + α
2
3
3
and for the closest integer as discretization by
jα k
lα m
"
#

3
3 −α
2
αj
P2

3

2
2
1,1 +

(1
−
r
)
+
,
r
∈
, j = 1, 2 ;

j
j
j=1
α
α
2
4α

3
3
j






l
m

α3

2
−α3
2
1
r3 =
1+
6 rj , j = 1, 2 ;
 2,
4αj





jα k
!


3

αj
P2


2
1 6 r , j = 1, 2.

, 1
,
 max
j
j=1 α (1 − rj ) + α
2
2
3
Dr. Adama Arouna KONÉ (FST)
3
Digital geometry and optimization
7 mars 2017
13 / 25
Overview of our first paper
Extension of the concept of Covering
Structuring element
Figure – Geometrical shape of the structuring element U ∈ R3 i.e. in 3-D .
Dr. Adama Arouna KONÉ (FST)
Digital geometry and optimization
7 mars 2017
14 / 25
Overview of our first paper
Extension of the concept of Covering
Example of full covering
Figure – Example of covering
Df l + U ( 89 , 89 , 89 ) ⊃ L.
Figure – Example of covering
Figure – Example of covering
Df l + U ( 12 , 12 , 11
) ⊃ L.
8
Figure – Example of covering
Dr. Adama Arouna KONÉ (FST)
Dcl + U ( 79 , 97 , 79 ) ⊃ L.
Dcl + U ( 12 , 21 , 98 ) ⊃ L.
Digital geometry and optimization
7 mars 2017
15 / 25
Overview of our first paper
Extension of the concept of Covering
Counterexample
Figure – counterexample
Mf l + U ( 21 , 12 , 12 ) 6⊃ L.
Figure – counterexample
Mcl + U ( 12 , 12 , 12 ) 6⊃ L.
Same remark as in the plan as in 3-D, these counterexamples also confirm that a
choice for any arbitrary size of the structuring element without take into account
the local characteristics of the plane (the slope of the plane) risks to provide
unappropriate discrete transcription of this plan.
Dr. Adama Arouna KONÉ (FST)
Digital geometry and optimization
7 mars 2017
16 / 25
PhD thesis
Outline
1
Overview of our first paper
2
PhD thesis
3
Recent works
4
Conclusion
Dr. Adama Arouna KONÉ (FST)
Digital geometry and optimization
7 mars 2017
17 / 25
PhD thesis
Géométrie Digitale Utilisée pour la Discrétisation et le
Recouvrement Optimal des Objets Euclidiens
114 pages, Presented 2016 January 14 at the Faculté des Sciences Techniques (FST) de
l’Université des Sciences, des Techniques et des Technologies de Bamako (USTTB)-Mali.
Thesis realized in co-supervision with USTTB-FST and Uppsala University.
Scientific advisors
Professor Emeritus Christer Oscar Kiselman and Professor Gunilla Borgefors in Swedwn
Professor Ouatenu Diallo and Dr DibyDiarra in Mali.
Thesis funded by
International Science Programme (ISP-IPMS) managed by Dr. Leif Abrahamsson throught
it Network Partial Differential Equation (EDP) managed by Professor Hamidou Touré of
Burkina Faso.
Dr. Adama Arouna KONÉ (FST)
Digital geometry and optimization
7 mars 2017
18 / 25
Recent works
Outline
1
Overview of our first paper
2
PhD thesis
3
Recent works
4
Conclusion
Dr. Adama Arouna KONÉ (FST)
Digital geometry and optimization
7 mars 2017
19 / 25
Recent works
Recent works
Pr Christer Oscar Kiselman invited me to work on :
project abstract : Digital hyperplanes
Digital planes in all dimensions are studied. The general goal is to generalize to any
dimension the results of Kiselman’s 2011 paper in Mathematika(11-1)
project abstract : Digital Geometry, Mathematical Morphology and
Discrete Optimization
An important relation was found between the following properties :
Strong property cvxh(A) ⊂ A + B(0, 1),
A ⊂ Zn ,
A ⊂ Zn ,
Generalization of chord property chord(A) ⊂ A + B(0, 1),
Zn -cnvexity cvxh(A) ∩ Zn = A,
A ⊂ Zn .
wher B(0, 1) is the open unit ball of Rn for l∞ metric.
Dr. Adama Arouna KONÉ (FST)
Digital geometry and optimization
7 mars 2017
20 / 25
Recent works
Relations between theses properties
The Zn -convexity allow us to have a digital caracterization of hyperplan. The Zn -convexity is mainly
focused on the slabs concept.
Dr. Adama Arouna KONÉ (FST)
Digital geometry and optimization
7 mars 2017
21 / 25
Recent works
I invited Dr. Diby Diarra to Work on :
project abstract : Optimal covering of a Euclidean hyperplane by means
of Hausdorff distance
The main purpose is to find an optimal covering of a Euclidean hyperplanes by
means of Hausdorff distance applied between a shape and its convex hull as a
maximal deviation.
We approach this covering concept by take into account any discretization.
Dr. Adama Arouna KONÉ (FST)
Digital geometry and optimization
7 mars 2017
22 / 25
Conclusion
Outline
1
Overview of our first paper
2
PhD thesis
3
Recent works
4
Conclusion
Dr. Adama Arouna KONÉ (FST)
Digital geometry and optimization
7 mars 2017
23 / 25
Conclusion
Conclusion
. These studies show that the need to improve the actual methods of
transcription from continuous shape into discrete shape is more than
necessary.
. However, the results obtained from the methods we have proposed
subsequently require the experimental interpretation of their applicability
and adaptability to real problems.
. These methods whose we present ensure the preservation of information
about continuous shapes into digital shapes.
. They stress the importance of the choice of discretization.
Dr. Adama Arouna KONÉ (FST)
Digital geometry and optimization
7 mars 2017
24 / 25
Conclusion
for your attention !
Dr. Adama Arouna KONÉ (FST)
Digital geometry and optimization
7 mars 2017
25 / 25