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Optimal covering of a straight line

application to discrete convexity Jean-Marc Chassery, Isabelle Sivignon gipsa-lab, CNRS, Grenoble, France

Continuous convexity

Optimal covering of a straight line applied to discrete convexity — JM Chassery, I Sivignon -- DGCI 2013 2

P Q

From Continuous space to discrete space

Optimal covering of a straight line applied to discrete convexity — JM Chassery, I Sivignon -- DGCI 2013 3

P Q

From discrete space to continuous space

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Tessel P Q

Convexity on dilated covering balls

Optimal covering of a straight line applied to discrete convexity — JM Chassery, I Sivignon -- DGCI 2013 5

ε ? ; ½ ≤ ε < 1 P Q

Discretization of a straight line using OBQ quantization process

{ (x,y) € R2 ; 3x − 8y + µ = 0} { (x,y) € Z

2 ; 0 ≤ 3x − 8y + µ < 8} Optimal covering of a straight line applied to discrete convexity — JM Chassery, I Sivignon -- DGCI 2013 6

ax − byc + µ = 0 ax − byd + µ = r yd - yc = r/b

Upper and lower leaning points

ax − by + µ = 0 ax−by+µ=b−1

Optimal covering of a straight line applied to discrete convexity — JM Chassery, I Sivignon -- DGCI 2013 7

The vertical distance between a pixel of remainder r of the DSL and the line L is equal to r/b, with r varying from 0 to b−1. The value r = b−1 is obtained for a lower leaning point

Covering by balls with radius = (b-1)/b

Not covered Not covered

Optimal covering of a straight line applied to discrete convexity — JM Chassery, I Sivignon -- DGCI 2013 8

Local configuration: D = U [proj(Pi), proj(Pi+1)] Pi and Pi+1 are 4-connected

Optimal covering of a straight line applied to discrete convexity — JM Chassery, I Sivignon -- DGCI 2013 9

Local configuration: D = U [proj(Pi), proj(Pi+1)] Pi and Pi+1 are 8-connected

Optimal covering of a straight line applied to discrete convexity — JM Chassery, I Sivignon -- DGCI 2013 10

Optimal covering of a straight line applied to discrete convexity — JM Chassery, I Sivignon -- DGCI 2013 11

Using Thales Theorem We prove: ρ(r)=a(b-r)/b(a+b) and ε(r)= ρ(r) + r/b = (a+r) / (a+b)

Theorem

Let L be a straight line of equation ax−by+µ = 0 ,

Let L its digitization with the OBQ scheme. The union of balls B(Pi,ε) centered on pixels Pi of the DSL L with radius ε = max(1/2, (|a|+|b|−1)/(|a|+|b|) ) covers the straight line L. This set doesn’t contain any other digital pixels excepted those of the DSL.

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Optimal covering of the straight line

3x − 8y + µ = 0

Optimal covering of a straight line applied to discrete convexity — JM Chassery, I Sivignon -- DGCI 2013 13

{ (x,y) € R2 ; 3x − 8y + µ = 0} { (x,y) € Z

2 ; 0 ≤ 3x − 8y + µ < 8}

ε = 10/11

Application to discrete convexity

Optimal covering of a straight line applied to discrete convexity — JM Chassery, I Sivignon -- DGCI 2013 14

Initial connected component

Optimal covering of a straight line applied to discrete convexity — JM Chassery, I Sivignon -- DGCI 2013 15

Delimitation by minimal and maximal columns

Discrete convex hull component

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ε = 1/2 ε = 1/2 ε = 1/2 ε = 1/2 ε = 2/3 ε = 2/3 ε = 1/2 The discrete convex hull is ε-convex with ε = 2/3

Conclusion

• Discrete convexity is defined in perfect compatibility with

continuous convexity.

• When sampling value tends to zero, discrete convexity

definition is similar to continuous convexity definition.

Optimal covering of a straight line applied to discrete convexity — JM Chassery, I Sivignon -- DGCI 2013 17