Supercover model, digital straight line recognition and curve reconstruction on the irregular isothetic grids

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Supercover model, digital straight line recognition and

curve reconstruction on the irregular isothetic grids

David Coeurjolly, Loutfi Zerarga

To cite this version:

re ognition and urve re onstru tion on the irregular isotheti grids

David Coeurjolly and Lout Zerarga Laboratoire LIRIS

Université Claude Bernard Lyon 1 43 Bddu 11 novembre 1918 F-69622 Villeurbanne, Fran e

Abstra t

Onthe lassi aldis retegrid,theanalysisofdigitalstraightlines(DSLforshort)has beenintensivelystudiedfornearlyhalfa entury.Inthisarti le,weareinterestedina dis retegeometryonirregulargrids.Morepre isely,ourgoalistodenegeometri al properties on irregular isotheti grids that are tilings of the Eu lidean plane with dierent sized axis parallel re tangles. On these irregular isotheti grids, we dene digital straight lines with re ognition algorithms and a pro ess to re onstru t an invertible polygonal representation of anirregulardis rete urve.

1 Introdu tion

pro ess is alled aninvertiblere onstru tion of adis rete urve [4,26,12℄.The invertiblepropertyis animportantone indis retegeometrysin e itallowsto onvertdis retedatatoEu lidean onessu hthatnoinformationisaddednor lost.

In this arti le,we are interested in dening a geometry on irregular isotheti grids. More pre isely, we onsider grids dened by a tiling of the plane using axisparallelre tangles. Su ha gridmodel in ludes, for example,the lassi al dis retegrid, theelongatedgrids[25℄and thequadtreebasedgrids[17℄.In[8℄, a general framework has been proposed that denes elementary obje ts and a digitization framework, the super over model. An important aspe t of this general framework is the onsisten y with lassi al denitions if the dis rete spa e is onsidered.

Manyappli ationsmaybenetfromthesedevelopments.Forexample,we an ite the analysis of quadtree ompressed shapes, or the use of geometri al properties in obje ts represented by intervalor ane arithmeti s (see dis us-sionin[8℄).Basedonthisirregular model,wedenedigitalstraightlines with re ognition algorithms and a pro ess to re onstru t an invertible polygonal representation of anirregular dis rete urve.

Se tion2presentsmoreformal denitionsintheirregular grids:adja en y re-lations,obje ts,ar s, urves and the super overmodel. Based ona denition ofthe irregular isotheti digitalstraightlines, wepresentalgorithms to re og-nizemaximalirregulardis retestraightsegmentsandtore onstru tinvertible polygonal ar s and urves (Se tion 3). Experiments and results are shown in Se tion4.

2 Preliminary denitions

2.1 The irregular isotheti model

First of all, we dene an irregular isotheti grid, denoted

I

, as a tiling of the plane with isotheti re tangles. In this framework, the re tangles have not ne essarilythesame sizebut we an noti ethatthe lassi aldigitalspa eisa parti ular irregular isotheti grid. In that ase, all squares are entered in

Z

2

points and have a border size equal to 1. Figure 1 illustrates some examples of irregular isotheti grids. A re tangle of an isotheti grid is alled a pixel. Ea h pixel

P

is dened by its enter

(x

P

, y

P

) ∈ R

grid (

(x

P

, y

P

) ∈ Z

2

and

l

x

P

= l

y

P

= 1

), an elongated grid (

l

x

P

= λ

,

l

y

P

= µ

and

(x

P

, y

P

) = (λi, µj)

with

(i, j) ∈ Z

2

), a quadtree de omposition (for a ell of level

k

,

(x

P

, y

P

) = (

m

2

k

,

n

2

k

)

and

l

x

P

= l

y

P

=

2

k−1

1

for some

m, n

∈ Z

); a unilateral and equitransitivetiling bysquares:thesizeof thebiggestsquareisequalto thesum of thetwo othersquaresizes; nallya generalirregularisotheti grid.

Denition 1 (

ve−

adja en y,

e−

adja en y) Let

P

and

Q

betwo pixels.

P

and

Q

are ve-adja ent if:

|x

P

− x

Q

| =

l

x

P

+ l

Q

x

2

and

|y

P

− y

Q

| ≤

l

y

_P

+ l

y

_Q

2

,

or

|y

P

− y

Q

| =

l

_P

y

+ l

y

Q

2

and

|x

P

− x

Q

| ≤

l

x

P

+ l

x

Q

2

.

P

and

Q

are e-adja ent ifwe onsider anex lusiveor andstri t inequalities in the above ve-adja ent denition.

Inthefollowingdenitions,weusethenotationk-adja en yinordertoexpress either the ve-adja en yorthe e-adja en y. Using these adja en y denitions, several basi obje ts an be dened:

Denition 2 (

k−

path) Letus onsiderasetofpixels

E = {P

i

, i

∈ {1, . . . , n}}

and arelationof

k−

adja en y.

E

is a

k

− path

ifand only iffor ea h element

P

i

of

E

,

P

i

is

k−

adja ent to

P

i−1

.

Denition 3 (

k−

obje t) Let

E

beasetof pixels,

E

isa

k−

obje tifandonly if for ea h ouple of pixels

(P, Q)

belonging to

E × E

, there exists a

k−

path between

P

and

Q

in

E

.

Denition 4 (k-ar ) Let

E

bea set of pixels,

E

is a

k−

ar ifand onlyif for ea h the element of

E = {P

i

, i

∈ {1, . . . , n}}

,

P

i

has exa tly two

k−

adja ent pixels, ex ept

P

1

and

P

n

whi h are alled the extremities of the

k−

ar .

Denition 5 (k- urve) Let

E

be a set of pixels,

E

is a k- urve if and only if

E

is a k-ar and

P

1

= P

n

.

Ifwe onsiderpixelssu hthat

l

su h obje ts. A omplete topologi al analysis of

k−

urves and

k−

obje ts is not addressed here.

2.2 Super over model on the irregular isotheti grids

Before dening the digital straight lines on the irregular isotheti grids, we haveto onsideradigitizationmodel.Inthefollowing,we hoosetoextendthe super overmodel.Thismodelwasrstintrodu edbyCohen-OrandKaufman in[10℄ onthe lassi al dis retegrid and then widely used sin e itprovidesan analyti al hara terization of basi super over obje ts (e.g. lines, planes, 3D polygons, ...) [2,1℄.

Denition 6 (Super over on irregular isotheti grids) Let

F

bean Eu- lidean obje t in

R

2

. The super over

S

(F )

is dened on an irregular isotheti grid

I

by:

S

_{(F ) = {P ∈ I | B(P ) ∩ F 6= ∅}}

(1)

= {P ∈ I | ∃(x, y) ∈ F, |x

P

− x| ≤

l

x

P

2

and

|y

P

− y| ≤

l

_P

y

2

} .

(2)

where

B

(P )

is the re tangle entered in

(x

P

, y

P

)

of size

(l

x

P

, l

y

P

)

(if

l

x

P

= l

y

P

,

B(P )

is the ball entered in

(x

P

, y

P

)

of size

l

x

P

for the

L

∞

norm).

Properties of this model are dis ussed in[8℄.

Fig. 2.Illustration of thesuper overdigitization of a urve (left) and of a straight line (right).

Figure2illustrates someexamples ofthe super over digitizationofEu lidean obje ts. If

I

is the lassi al digital spa e (i.e.

(x

P

, y

P

) ∈ Z

2

and

l

x

P

= l

y

P

=

1

), many links exist between the super over of an Eu lidean straight line and lassi al digital straight line denitions [1,24℄. Sin e we have not any assumptiononthe irregulargrid,nostrongtopologi alproperty an bestated onthe super overof an Eu lidean straight line.

3.1 Denitions and IDSL Re ognition

Denition 7 (Irregular isotheti digital straight line) Let

S

be asetof pixelsin

I

,

S

is alledapie eof irregulardigitalstraightline(IDSLforshort) i there exists an Eu lidean straight line

l

su h that:

S

⊆ S(l) .

(3)

In other words,

S

isa pie e of IDSL i there exists

l

su h that for all

P

∈ S

,

B

∞

_{(P ) ∩ l 6= ∅}

.

Todete tif

B

∞

_{(P )∩l}

isemptyornot,weusethenotationspresentedinFigure 3. Hen e,

B

∞

_{(P ) ∩ l}

is not empty i

l

rosses either (or both) the diagonals

d

₁

or

d

2

of

P

.

l

(x

P

, y

P

)

d

1

d

2

Fig.3.Notationsusedto dete tif thepixelof enter

(x

P

, y

P

)

belongsto the super- overofa straight line

l

(

d

1

and

d

2

arethediagonalsof there tangle

P

).

Without loss of generality, we suppose that

l

is given by

y

= αx + β

with

(α, β) ∈ R

2

(an appropriate treatment an be design to handle the straight lines

x

= k

with

k

∈ R

).Tosolvethere ognitionproblem,weusethefollowing statement:

B

∞

_{(P ) ∩ l 6= ∅}

_{⇔ l ∩ d}

₁

_{6= ∅}

and

α

≥ 0

(4)

or

l

∩ d

2

6= ∅

and

α <

0

(5)

Given a pixel

P

, Equation (4) an be represented by two inequalities in the

(α, β)−

parameter spa e:

E

+

_{(P ) =}











α



x

P

−

l

x

P

2



+ β − y

P

−

l

y

_P

2

≤ 0

α



x

_P

+

l

P

x

2



+ β − y

P

+

l

y

_P

2

≥ 0

.

(6)

Details on the omputation of these inequalities an be found in [8℄. If we onsider Equation (5), we may obtain the following inequalities:

E

−

(P ) =











α



x

_P

−

l

x

P

2



+ β − y

P

+

l

y

_P

2

≥ 0

α



x

P

+

l

x

P

2



+ β − y

P

−

l

y

_P

2

≤ 0

.

(7)

E

+

_{(P )}

is dened for

α

≥ 0

and

E

−

_{(P )}

for

α <

0

. We an now dene the preimages of a pie eof IDSL:

Denition 8 (Preimages of an IDSL) Let

S

be a pie e of IDSL, the two preimages

P

+

and

P

−

of

S

are given by:

P

+

_{(S) =}

\

P

∈S

E

+

_{(P )}

and

P

−

_{(S) =}

\

P

∈S

E

−

(P ) .

(8)

Hen e, the re ognition pro ess an be des ribed asfollows:

Proposition 2 Let

S

be a set of pixels in a

I

-grid.

S

is a pie e of IDSL i

P

+

_{(S) 6= ∅}

or

P

−

_{(S) 6= ∅}

.

Using Proposition 2, the re ognition of a pie e IDSL leads to a linear pro-gramming problem: wehave tode ide whether alinear inequality system has a solution or not. To solve this problem, two dierent lasses of algorithms exist:the IDSL identi ation algorithmswhi h de ideif

S

is anIDSL ornot, and the IDSL re ognition algorithms whi h return the omplete preimages (maybe empty) of the re ognized IDSL. To solve the identi ation problem, in remental

O(n)

solutionsexist if

n

is the number of linear onstraints (i.e. the number of irregular pixels in our ase) [20,6℄. To ompletely des ribe the preimages,thein rementalPreparataandShamosalgorithm[21℄ maybeused whose omputational ost isoptimalin

O(n log n)

. In[8℄, analgorithm based onalinear programming pro edure isproposedto re ognizeIDSL givenaset of pixels. This algorithm an also be used to segment anirregular ar , i.e. to de omposethe ar intomaximal pie e of IDSL (see Figure 4).

Eu lidean straight lines are manually extra ted from the preimages asso iated to ea h IDSLsegment[8℄.

3.2 Invertible re onstru tion of irregular ar s and urves

In the following, we proposean algorithm to onstru t an Eu lidean polyline fromadis rete urvesu hthatitsdigitizationisequal totheoriginaldis rete urve. If we onsider the super over digitization model, a polyline

L

is an invertible re onstru tion of a dis rete urve

S

if it lies inside the dis rete urve. Morepre isely, for ea hEu lidean point

p

on

L

, thereexists apixel

P

in

S

su h that

p

belongsto

P

.

Usually, the re onstru tion task is a post-treatment of a DSL segmentation algorithm:rst wede omposethe dis rete urve intomaximal DSL,then, for ea hpie eofDSL,we omputearepresentativeEu lideansegment.The main drawba k of this approa h is that it is di ult to ensure the reversibility of the polyline verti es [4,12℄. In the lassi al dis rete grid, Sivignon et. al. [26℄ propose an invertible re onstru tion algorithm in whi h both the re ognition and the Eu lidean segment extra tion are performed at the same time. More pre isely, the authors redu e the problem for ing the rst extremity of the segments to be inside the dis rete urve. Then, they perform an analysis on the preimage of the segmentto ompute the se ond extremity.

In the following, we propose a similar algorithm without the omputation of the preimages that would have required omplex linear programming pro e-dures. The main idea is to use a visibility test te hnique ommonly used in omputational geometry tosolve shortestpath extra tion problems [7℄.

3.2.1 Visibility one based approa h

First,wedenethepredi ateTurnPositive

(a, b, c)

whi histrueifthepoints

Let

S

= {P

i

}

i=0..n

be a

k−

ar , we rst x the rst extremity

p

0

of the rst segmentsu hthat

p

0

∈ P

0

.Giventhe pixel

P

1

k−

adja entto

P

0

,wedenote

e

0

the Eu lidean segment shared by the two pixels

P

0

and

P

1

. We onsider the rst one

C

0

(p

0

, s, t)

with enter

p

0

anddenedbythetwopoints

s

and

t

su h that

{p

0

, t, s}

issorted ounter lo kwise(i.e. TurnPositive

(p

0

, t, s)

is true) and su hthat

s

and

t

oin ide with theextremitiesof

e

0

(see Figure5-(left)).

C

₀

is a visibility one sin e for ea h point

p

in the interse tion between

C

0

and the pixels of

S

, the segment

[p

0

p]

lies exa tly in

S

. In other words, the super over digitizationof

[p

0

p]

isa subset of

S

.

A ording to the previous denitions, the one

C

0

des ribes a subset of the preimages

P

+

_({P

0

, P

1

})

and

P

−

_({P

0

, P

1

})

in the parameter spa e. Indeed, ea hstraightline

(p

0

p

)

rossesthe pixels

P

0

and

P

1

.Morepre isely,the setof straight lines ontained inthe one

C

0

isthe segment inthe

(α, β)

-parameter spa e whi h orresponds to the interse tion between the preimages

P

+

and

P

−

and the straight line dened by the point

p

0

. Hen e, as proposed in[26℄, we ould have performed all omputations inthe parameter spa e

(α, β)

but the analysis using visibility ones leads toa more e ient algorithm.

Fig.5.Illustrationofthevisibility onebasedalgorithm:(fromlefttoright)therst one

C

0

,theupdate ofthe one onsideringthepixel

P

2

and an example whenthe visibilityfails.

The algorithm an be sket hed as follows: for ea h pixel

P

i

, we onsider the shared segment

e

i

between

P

i−1

and

P

i

. Then, we have a simple pro- edure to update the urrent one

C

j

(p

j

, s, t)

a ording to

e

i

(u, l)

(su h that TurnPositive

(p

0

, l, u

)

istrue).Thedierent ases arepresentedinFigure6. Notethat using the predi ateTurnPositive, Algorithm1 isvalidwhatever theorientationof the urveandthesegment

[ul]

isnotne essarilyverti alnor horizontal.

From the dierent ases presented in Figure 6, we an design a simple algo-rithm (Algorithm 1) with three possible outputs: the visibility fails, the one is updatedor the one remainsun hanged.

When the updatepro edurefails, itmeansthat thereis noeu lidean straight line going through

p

j

and rossingthe pixel

P

i

.In that ase, weneed tostart anew re ognitionpro ess. Hen e, wesetup anew one

C

j+1

(p

j+1

, s, t

)

where

s

and

t

are given by the edge

e

i

. To ompute the new enter of the one

p

s

t

u

l

p

s

t

u

l

p

s

t

u

l

p

s

u

l

t

p

s

t

u

l

Fig.6.Illustrationofthedierent ases whenweupdatea one:(fromleft toright) the oneisnotmodied,only thepoint

t

ismoved,onlythepoint

s

ismoved,both

s

and

t

aremoved, andnally,thevisibilityfails.

Algorithm 1 Visibility oneupdatepro edure

Let

C

j

(p

j

, s, t)

bethe urrent oneand

e

i

(u, l)

thesharedsegmentbetween

P

i

and

P

i−1

if (notTurnPositive

(p, t, u)

)orTurnPositive

(p, s, l)

then

return

∅

{thevisibilitytestfails} else

if TurnPositive

(p, t, l)

then

t

← l

return

C

j

(p

j

, s, t)

endif

if notTurnPositive

(p, s, u)

then

s

← u

return

C

j

(p

j

, s, t)

endif

endif

(dashed straight lines in Figure 5) and we dene

p

j+1

as the midpoint of the interse tion between the bise tor and the pixel

P

i−1

(this interse tion is not empty sin e

P

i−1

has already been onsidered).The idea ofthis strategy isto obtain a polyline as entered as possible in the dis rete urve. By denition of Algorithm 1, the segment

[p

j

, p

j+1

]

lies inside the irregular dis rete urve. Hen e, if we repeat the above pro ess for ea h pixel of the

k−

ar , the nal polyline isaninvertiblere onstru tion ofthe ar (see Figure5-(right)and 7).

3.2.2 Overall algorithm

Algorithm 2 Invertiblere onstru tion of a

k

-ar Let

S

= {P

i

}

i=0..n

bea

k−

ar and

p

0

therstpointin

P

0

Setj=0

Initializationofthe one

C

j

(p

j

, s, t)

using

P

0

and

P

1

su hthatTurnPositive

(p

j

, t, s)

istrue

p

j

istherstvertexofthenalpolyline for

i

from2to

n

do

Computethesharedsegment

e

i

between

P

i

and

P

i−1

C

′

_←

Updatethevisibility oneusingalgorithm1 if

C

′

_{= ∅}

then

Computethepoint

p

j+1

usingthebise torof

C

j

andthepixel

P

i−1

Initializationofanew one

C

j+1

with

p

j+1

and

e

i

Mark

p

j

asavertexofthenalpolyline else

C

j

← C

′

endif endfor

the preimages are onsidered.However,this restri tion allowsusto onstru t aninvertible polyline.

Fig.7.Illustrationofthere onstru tionalgorithm:(left)thesequen eof onesduring thevisibilitytest and (right),there onstru tedpolygonal urve.

3.2.3 Invertible re onstru tion of

k−

urves

Ifwe onsideranirregular

k−

urve

S

= {P

i

}

i=0..n

,the re onstru ted polyline mustbe losedandthusdenesasimplepolygon.Hen e,we anuseAlgorithm 2 for the pixels

P

0

to

P

n

and add a spe i analysis to handle the adja en y between

P

n

and

P

0

that reatesasfewaspossiblenewverti es.Let

C

j

(p

j

, s, t

)

bethe last visibility one su hthat the interse tionbetween

P

n

and this one is not empty. Several ases o ur (see Figure 8): for example, if

p

0

∈ C

j

, we lose the polyline using the segment

[p

j

p

0

]

. Otherwise, wemay move

p

0

along

(p

0

p

1

)

if there exists an interse tion between

C

j

and the straight line

(p

0

p

1

)

thatliesinside

P

0

(seeFigure8-

(b)

).Inthat ase,westill losethe urveusing

[p

j

p

′

₀

]

and the globalreversibility ofthe polygonal urve an be easilyproved. Other ases anbederived(forexampleusingthe visibilityfrom

p

0

to

P

n

)but additionnal verti es may be insertedto the polygonal urve (see Figure8).

Fig. 8. Dierent ases to end the re onstru tion of a

k−

urve:

(a)

and

(b)

we an losethe urve using

[p

0

p

j

]

or

[p

′

0

, p

j

]

,

(c)

a newvertex

p

j+1

must be inserted,and

Wehave onstru tedaC++librarytohandleelementaryirregularobje ts (ir-regularpixels,

k−

ar s and

k

- urves). Using thislibrary,wehaveimplemented the re onstru tion algorithm des ribed in the previous se tion (the ode is available on the following web page: http://liris. nrs.fr/~d oeurjo/ Code/Re onstru tion). Figure 10 presents the result of Algorithm 2 on an irregular

k−

ar . Sin e the lassi aldigital grid isa spe i irregular isotheti grid, Algorithm 2 an also be used to re onstru t a polygonal urve from a lassi al4- onne ted urve(see Figure10). In this ase, resultsare similarto [26℄.

Fig. 9. Resultof Algorithm 2 on an irregular

ve−

ar : the input

ve−

urve and the invertible re onstru tion usingAlgorithm2.

Fig.10.ResultofAlgorithm1ona lassi al4- onne ted urve:theinput4- onne ted urveand theinvertible re onstru tion usingAlg. 2.

5 Con lusion

inthenumberofsegments.Additionalpro essessimilarto[12℄inthe lassi al dis rete ase ouldbeinvestigated.

Sin e adaptive grids orQuadTree based de ompositions are spe i irregular isotheti models, an important future work isto use the proposed framework to provide geometri tools to hara terize obje t boundaries in su h grids. Furthermore, topologi al denitions and data stru ture to handle irregular obje ts is animportant ongoing resear h topi .

Referen es

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