Geometric Constructions in the Digital Plane

??, ??, 1–36 (??) c ?? Kluwer Academic Publishers, Boston. Manufactured in The Netherlands. °

Geometric Constructions in the Digital Plane PETER VEELAERT [email protected]

Hogeschool Gent, Dept. INWE, Schoonmeersstraat 52, B-9000 Ghent, Belgium Received ??; Revised ?? Editors: ??

Abstract.

We adapt several important properties from affine geometry so that they become applicable

in the digital plane. Each affine property is first reformulated as a property about line transversals. Known results about transversals are then used to derive Helly type theorems for the digital plane. The main characteristic of a Helly type theorem is that it expresses a relation holding for a collection of geometric objects in terms of simpler relations holding for some of the subcollections. For example, we show that in the digital plane a collection of digital lines is parallel if and only if each of its 2-membered subcollections consists of two parallel digital lines. The derived Helly type theorems lead to many applications in digital image processing. For example, they provide an appropriate setting for verifying whether lines detected in a digital image satisfy the constraints imposed by a perspective projection. The results can be extended to higher dimensions or to other geometric systems, such as projective geometry. Keywords:

1.

digital geometry, digital straight line, incidence relation

Introduction

cepts and methods of classical continuous geometry so that they become applicable in the digital

Many papers in digital image processing deal with the problem of how to translate or adapt the con-

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plane, that is, the plane consisting of all points that have integral coordinates. The primary goal

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of these papers is not to develop a “digital ge-

cuss could also be reformulated within a frame-

ometry” that stands completely on its own, but

work aimed at the digitization of either Euclidean

rather to examine in how far classical properties,

or projective geometry.

such as straightness and convexity, are left intact by a digitization process. Intact here means that we can reconstruct the geometric relations in the original image from the relations that are discovered in the digitized image. Fig.

A.1 shows a

typical application, where after an edge detection and clustering process, subsets of the digital plane have been recognized as being “digital representa-

When we consider the plane R2 , affine geometry provides a strict and very elegant scheme of axioms and properties with which we can describe and verify geometrical structure. For example, two lines are either parallel or they intersect in a unique point. With rules like this the verification of, say, perspectivity constraints becomes a

tions” of straight line segments.

logical and straightforward process. If the lines To detect more structure, we can now verify

detected in Fig.

A.1 would not be digital but

whether these subsets satisfy additional geomet-

affine lines, verifying whether three of these lines

ric relations, and in particular, whether they rep-

pass through a common point is a process that

resent lines that are either parallel, collinear, or

seems almost self-evident: first choose two non-

concurrent. For example, we must be able to ver-

parallel lines and locate their unique point of in-

ify whether the lines detected in the image of 3D

tersection (the axioms tell us such a point must

scene satisfy the constraints imposed by a per-

exist); then find out whether this point lies on the

spective projection, and if so, locate their points

third line. In the digital plane this kind of ele-

of intersection, i.e., the vanishing points of the

gant and simple framework is not available. The

projection [7, 15, 26]. The goal of this paper is to

intersection of two digitized lines is not necessar-

provide a mathematical framework in which such

ily a digital point, and two digital points do not

problems can be solved in a rigourous way. In par-

define a unique digital straight line, unless we in-

ticular, we will give the basic definitions for paral-

troduce additional criteria to select such a line. In

lelism, collinearity, and concurrency in the digital

fact, progress only seems possible if we sacrifice at

plane. Since these three properties are basic con-

least some of the classical notions, and concen-

stituents of affine geometry, we refer to them as

trate on the preservation of a more restricted set

affine properties, although much of what we dis-

of fundamental concepts.

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Although some sacrifice seems necessary, we will

transitivity: we cannot reduce the number of line

show that in the digital plane we can still pre-

pairs that must be verified. There is an example

serve a substantial part of affine geometry, and

of three digital lines S1 , S2 , S3 in the digital plane,

we prove that the “digitized” properties can be

where S1 is digitally parallel with S2 , S2 is digi-

verified by constructions that are still purely ge-

tally parallel with S3 , but where S1 is not digitally

ometric, though slightly more complicated than

parallel with S3 . In contrast, in the affine plane

what would be required by affine or Euclidean ge-

parallelism is an equivalence property, and we do

ometry. We must allow, however, two important

not have to verify the parallelism of all pairs. It

deviations. First, points and lines are no longer

suffices to select one line from a collection and to

the sole basic geometric objects. We need a richer

verify whether all the other lines are parallel with

class of objects. We must characterize, for ex-

it.

ample, the lines passing through two points by a new kind of object, called the preimage of the two points, and we must describe the intersection of two digital straight lines as an intersection of two preimages. Second, and even more importantly, each incidence relation is replaced by a Helly type property. To be precise, let k be a positive integer, let P be some (geometric) property, and let F be a collection of m geometric objects. Then P is called a “Helly type property” if the following theorem is always true: The collection F has property P if and only if each of its k-membered subcollections has property P . For example, we introduce parallelism in the digital plane, and we prove that a collection of digital lines is parallel if and only if each pair of digital lines is parallel.

Some results in this paper are extensions of the properties and concepts introduced by other authors. The first appearance of a Helly type property in the digital plane, although at the time not recognized as such, was Rosenfeld’s chord property for digital straight line segments [22]. Ronse later showed that the chord property has a straightforward proof based on Santalo’s Theorem, which is one of the primary examples of a Helly type theorem [19, 20, 21]. Extending the work of Rosenfeld, Kim defined digital straight lines in 3D space, and Stojmenovi´c and Toˇsi´c introduced a general digitization scheme for digital m-flats of arbitrary dimension [12, 24]. It was then shown that digital hyperplanes also possess chordal properties of the Helly type [28].

Typical for a Helly type property is the lack of To characterize straight digital sets, we introduce a new definition for the thickness, domain

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and preimage of a set. Our definition of thick-

of the objects and properties that we introduce

ness is very similar but not completely identical to

(e.g., domain, preimage, parallelism, collinearity

the arithmetical thickness introduced by Reveill`es

and concurrency) can be computed or verified

and later generalized by Andres et al for discrete

by standard algorithms from computational ge-

analytical hyperplanes [3, 17]. Our definition of

ometry [16].

domain extends the previous definition used by

tle transversal problems, one can apply the algo-

Dorst and Smeulders and by McIlroy [8, 9, 14]. In

rithms given by Avis and Doskas, as well as Atal-

fact, according to our more general definition, a

lah and Bajaj [4, 5]. Transversal algorithms are

domain is not necessarily a quadrangle or trian-

also discussed by Amenta in the context of Gen-

gle. We also generalize the concept of preimage,

eralized Linear Programming [1].

which was introduced for digital straight lines by Anderson and Kim [2]. In this paper we show, however, that thickness, domain and preimage all possess Helly type properties. Furthermore, we introduce parallelism and concurrency for digital point sets, and show that also these properties are of the Helly type.

To verify some of the more sub-

This paper is organized as follows. In Section 2 we consider digital straightness and show how it naturally leads to Helly type properties. Next, in Sections 3 and 4, we discuss several important tools that are needed later on, i.e., the thickness, domain, preimage and collinearity region of a set. We show that each of these concepts has Helly

Although Helly’s original theorem dates from 1923 ([11]), until this day it has been the subject of further extensions and variations. Danzer et al and Hadwiger et al give extensive overviews of what was known in 1963 [6, 10]. More recent advances can be found in [1, 4, 5, 13, 25, 30]. One

type properties. In Section 5 three important geometric concepts are adapted to the digital plane: collinearity, parallelism, and concurrency. In Section 6 we discuss how all the foregoing concepts can be made digitization scheme independent. We conclude the paper in Section 7.

relatively recent result by Tverberg is used in this 2.

paper [25].

Digital straightness

Although this paper focuses mainly on theoret-

The digital plane is the subset of the real plane

ical aspects, the translation of the given proper-

R2 formed by all points that have integral coor-

ties into algorithms is often straightforward. Most

dinates. The points in the digital plane are called

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digital points. A set of digital points is called a digital set.

5

One of the basic problems in the digital plane is to find all properties of its digitally straight subsets. One such property follows almost im-

A common technique used to digitize curves is the Grid Intersect Quantization scheme (GIQ) [8, 22]. Let τ be a positive real number. To each digital point (x, y) we assign a cell C((x, y); τ ) ⊂ R2 that consists of two parts: (i) the points (a, y) where a satisfies x−τ /2 ≤ a < x+τ /2, and (ii) the points (x, b) where b satisfies y−τ /2 < b ≤ y+τ /2. In the sections that follow we consider arbitrary values for τ . In the GIQ scheme, however, one chooses τ = 1. Thus, as illustrated in Fig. A.2(a), each cell is the union of two line segments, i.e. a horizontal segment parallel to the x-axis, and a vertical segment parallel to the y-axis. Let B be a straight line in the plane defined by the equation

mediately if we formulate digital straightness as a transversal problem. First note that the scheme illustrated in Fig.

A.2(a) is equivalent with a

second scheme, where each cell is replaced by its convex hull, as shown in Fig. A.2(b). In fact, it is an elementary property of 2-dimensional geometry that a straight line intersects a connected set, i.e. a cell, if and only if it intersects the convex hull of the set. According to this second scheme, a set of points is digitally straight if the convex hull of their cells can be transversed by a straight line. Hence we can apply the following result on transversals, which was proved by Tverberg, after being conjectured by Gr¨ unbaum [25].

αx + β − y = 0, where we assume that α 6= −1. (α = −1 can be handled as a special case, but will not be discussed in this paper). According to the GIQ scheme, the digitization dig(B) of B consists of all digital points (x, y) whose cell C((x, y); 1) is

Theorem 1. (Tverberg)

A finite family F

of disjoint translates of a single convex set C in R2 admits a line transversal if each 5-membered subfamily of F admits a line transversal.

intersected by the line B: Thus we arrive at a first Helly type property about straightness simply by rephrasing Theorem dig(B) = {(x, y) ∈ Z2 : C((x, y); 1) ∩ B 6= ∅}.

1. Note that the cells of two distinct digital points are in fact disjoint when τ ≤ 1.

Accordingly, a digital set S is called digitally straight if there exists a straight line B such that

Theorem 2.

S ⊆ dig(B).

points of S lie on a common digital straight line if

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Let S be a finite digital set. The

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and only if the points of each 5-membered subset

digy (B) denote the digitization according to the

of S lie on a common digital straight line.

cells Cy ((x, y); 1) shown in Fig.

A.2(d), and we

call this set digitally straight with respect to the As for the size of the subcollections, this result can be further optimized if we take into account that a cell consists of a vertical and a horizontal segment, and that depending on the slope of

y-axis. Recall that α = −1 is a special case not discussed here. Ronse was the first to show that to these anisotropic definitions of straightness we can apply the following Helly type theorem [6, 19]:

the line B we can predict which segment intersects B. If we have −1 < α ≤ 1, it is sufficient to consider only the intersection with the vertical segment. In fact, whenever such a line intersects the horizontal segment of a cell, it always inter-

Theorem 3. (Santalo)

A finite family F of

vertical line segments in R2 admits a line transversal if and only if each 3-membered subfamily of F admits a line transversal.

sects the vertical segment too [22]. Hence, for lines with slope −1 < α ≤ 1, we can replace each cell by a cell that contains only the vertical segment, as illustrated in Fig. A.2(c). For each digital point (x, y), we let Cx ((x, y); 1) be the vertical line segment comprising all points (x, b) ∈ R2 with y−1/2 ≤ b < y+1/2, as illustrated in Fig. A.2(c). We use digx (B) to denote the digitization of B us-

Consider, for example, the points of the finite digital set S shown in Fig.

A.3(a).

Clearly,

the set S is digitally straight if and only if the cells Cx ((x, y); 1) can be transversed by a single straight line. Thus another Helly type property follows as an immediate corollary.

ing the cells Cx ((x, y); 1), thus Theorem 4. digx (B) = {(x, y) ∈ Z2 : Cx ((x, y); 1) ∩ B 6= ∅}.

A finite digital set S is digitally

straight with respect to the x-axis if and only if each of its 3-point subsets is digitally straight with

Hence, for lines with slope −1 < α ≤ 1, we

respect to the x-axis.

have dig(B) = digx (B). In accordance with this anisotropic digitization scheme, a digital set S is

We can derive another, more quantitative, ver-

called digitally straight with respect to the x-axis

sion of this result. Let (x1 , y1 ), . . . , (xm , ym ) be

if there exists a line B such that S ⊂ digx (B).

the points of S. Then the cells Cx ((xi , yi ); 1) have

Likewise, for a line with slope |α| > 1, we let

a common line transversal if and only if the system

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??    −1/2 ≤ α + βx1 − y1 < 1/2    S : ...      −1/2 ≤ α + βx − y < 1/2 m m

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form

Sijk

   |α + βxi − yi | < 1/2    : |α + βxj − yj | < 1/2      |α + βx − y | < 1/2 k k

has a solution for the indeterminates α, β. The indeterminates of this system can be eliminated if we use Helly’s original theorem [18, 23].

has a solution, for any 3 distinct points of the set S. It remains to determine under what conditions a subsystem has a solution. Let Di , Dj , Dk denote

Theorem 5. (Helly)

Let F be a family of

the cofactors of the last column of the matrix

convex subsets of Rd with at least d + 1 elements. If F satisfies the following two conditions:





 1 xi yi  1 x y j j   1 xk ym

  ,  

1. the intersection of any d + 1 sets in F is nonwhich contains the coefficients of the subsystem

empty, 2. F is finite or all elements of F are compact,

Sijk . Using standard linear algebra one can easily prove (see [28]) that Sijk has a solution if and only

then the intersection of all the elements in F is non-empty.

|Di yi + Dj yj + Dk yk | < (|Di | + |Dj | + |Dk |) /2.

In fact, each relation −1/2 ≤ αxi + β − yi < 1/2 in S defines a convex set in the αβ parameter space, that is, a convex strip bounded by two parallel lines. For example, Fig.

if the coordinates of the three points satisfy

A.3(b) shows

Hence, by expanding the cofactors, we obtain, as a slight generalization of Rosenfeld’s chord property [22], a second, more detailed Helly type characterization of digital straightness [28].

the strips that correspond with the set shown in Fig.

A.3(a). The gray region shows the (non-

Theorem 6.

Let S be a finite digital set that

empty) intersection of the convex sets. According

contains at least two points with distinct x-

to Helly’s Theorem the intersection of the convex

coordinates. Then S is digitally straight with re-

strips defined by S is non-empty if and only if the

spect to the x-axis if and only if each of its 3-

intersection of any 3 strips is non-empty. In other

point subsets (xi , yi ), (xj , yj ), (xk , yk ) satisfies ei-

words, S has a solution if each subsystem of the

ther xi = xj = xk or

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|(xk − xj )yi + (xi − xk )yj + (xj − xi )yk |
αx+β}.

a collinearity region is a preimage to which two

Similarly, let B − be the open halfplane of points

strips of vertical height τ /2 have been added.)

trated in Fig.

A.5(b).

A.6. The boundaries of the region corre-

0 that lie below B. The complement of dig−1 x (S ; τ )

then consists of two parts, where each part is the

4.2.

Shape of the preimage

intersection of three halfplanes. To be precise, T + is the intersection of the three halfplanes cl(B1+ ), cl(B2+ ) and cl(B3+ ), where B1 passes through the points (x1 , y1 +τ /2) and (x2 , y2 −τ /2), B2 through the points (x1 , y1 −τ /2) and (x2 , y2 +τ /2), and B3 through the points (x1 , y1 +τ /2) and (x2 , y2 +τ /2). Likewise, T − = cl(B1− )∩cl(B2− )∩B4− , where B4 is the line passing through the points (x1 , y1 − τ /2) and (x2 , y2 −τ /2). (Notice that we do not take the closure of B4− , since B4 lies within dig−1 x (S; τ ).) Since both sets T + and T − are the intersection of three halfplanes, both are convex. Finally, it is clear that any line B that lies in dig−1 x (S; τ ) 0 + is also in dig−1 x (S ; τ ) and therefore separates T

and T − .

We have seen that the preimage of a set S can be written as the intersection of the preimages of point pairs. We now show that this intersection also inherits some of the shape properties of a point pair preimage. As a byproduct, we obtain an efficient method to compute a preimage from the domain of a set, and conversely, a domain from a preimage. Although this shape analysis is very relevant from the computational viewpoint, we note that it is not strictly needed when we adapt the incidence relations of affine geometry in Section 5. From Lemma 1 it is clear that the complement of the preimage of a point pair consists of two convex regions. Surprisingly, we can prove that

Fig. A.6 shows the collinearity region of a small

the complement of the preimage of an arbitrary

set. In accordance with Theorem 7, the collinear-

set also consists of two convex sets. If we regard a

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B (B

+

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preimage simply as an intersection of a collection

Since p is not in

), there must be a line

of preimages of point pairs, as in Theorem 7, this result is not obvious. Although the complement

B1 such that p ∈ B1− . Likewise, since p is not T in B (B − ), there must be a line B2 such that

of a preimage consists of two disjoint connected

p ∈ B2+ . By Lemma 2 of the Appendix, it fol-

components, and although each component is a

lows that p ∈ dig−1 x (S; τ ). Hence, p cannot lie

union of convex sets, this does not imply that the union itself is convex. However, if we look at a

in the complement of the preimage, and therefore T T + − dig−1 x (S; τ ) = ( B (B )) ∪ ( B (B )). Thus the

preimage as the region that is being swept over

complement of the preimage is the union of two

by a set of lines B whose parameters belong to

disjoint sets. Since each of these sets is the inter-

domx (S; τ ), we can prove the following result:

section of convex sets, they are convex themselves.

Let S be a finite subset of R2

Theorem 7 and Proposition 3 yield sufficient in-

and let σx (S) < τ . Then the preimage of S is the

formation about the composition and shape of a

complement of two disjoint convex regions.

preimage to derive an efficient method for com-

Proposition 3.

puting preimages from domains. The following Proof:

By definition, dig−1 x (S; τ ) consists of

all point that lie on the lines B : y = αx + β for which (α, β) ∈ domx (S; τ ). For such a line B, we let B + denote the open halfplane of points (x, y) ∈ R2 that satisfy y − αx − β > 0. That is, B

+

comprises all points above B, but not

on B. Likewise, let B − denote the open halfdig−1 x (S; τ )

= plane of points below B. Hence, T + ∪ B − ), where the intersection is taken B (B over all lines B in the domain domx (S; τ ). It T immediately follows that B (B + ) ⊂ dig−1 x (S; τ ) T and B (B − ) ⊂ dig−1 x (S; τ ). We now claim that T T + − dig−1 x (S; τ ) = ( B (B )) ∪ ( B (B )). Let p be T T a point in R2 outside ( B (B + )) ∪ ( B (B − )).

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proposition is proven in [29].

Proposition 4.

Let S be a finite set, and let

σx (S) < τ . If a line y = αx + β is contained in dig−1 x (S; τ ) then (α, β) ∈ domx (S; τ ). In particular, the parameter point (α, β) is a vertex of cl(domx (S; τ )) if and only if the line y = αx + β contains a boundary segment of cl(digx−1 (S; τ )).

4.3.

Efficient computation of domain, preimage and collinearity region

In the sections that follow domains, preimages and collinearity regions will be used extensively to introduce and verify new properties. We have al-

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ready shown how these objects can be computed

U that contains S. Since P ⊆ S, the points of P

by standard algorithms from computational geom-

must be boundary points of the convex hull of S.

etry. Before proceeding, however, we examine how this computation can be made more efficient. In

Also the domain of a set can be computed after

particular, we show that all points of S that are

excluding the points of S that lie in the interior

not part of the boundary of its convex hull can be

of its convex hull. The following proposition is

discarded during a calculation. A first theorem

proven in [29].

refers to the computation of thickness. Proposition 6.

Let S be a finite subset of

R2 , and let σx (S) < τ . Proposition 5.

Let S be a finite subset of R2

domx (H; τ ).

such that all its points have distinct x-coordinates. Let H be the subset of S that contains those points of S that are also boundary points of the convex hull of S. Then σx (S) = σx (H). In particular, a

Then domx (S; τ ) =

Furthermore, by Proposition 4, it follows that also for the preimage and the collinearity region −1 we must have dig−1 x (S; τ ) = digx (H; τ ), and

collx (S; τ ) = collx (H; τ ).

finite digital set S, consisting of points with distinct x-coordinates, is digitally straight if and only

5.

Incidence relations in the digital plane

if its subset that only includes boundary points of After introducing the necessary tools such as the

its convex hull is digitally straight.

domain of a set, we now adapt the basic properties of geometry so that they can be applied to Proof:

By Proposition 2, S contains a 3-point

arbitrary finite points sets, and in particular, to

subset P whose points lie on the unique pair of

subsets of the digital plane. It is important to re-

enclosing lines of S and for which σx (P ) = σx (S).

mark that from now on basic properties such as

Therefore it suffices to prove that P ⊆ H, since

parallelism will get a more general meaning than

we would then also have σx (P ) = σx (H). Let

the one that is usually assigned to them in affine

U denote the convex set that includes the points

geometry. When we refer to the original affine

on each of the two parallel enclosing lines of S

concepts, we will always state this explicitly, as in

and the region in between those lines. Then each

“affine parallelism”. Furthermore, the term “digi-

point in P lies on the boundary of a convex set

tal,” as in digitally collinear, is reserved for digital

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sets and when τ = 1, to comply with current lit-

ness such that σx (Si ) < τ .

erature.

are collinear relative to the x-axis if and only if

Each geometric property will be translated such

Then S1 and S2

S1 ⊂ collx (S2 ; τ ) and S2 ⊂ collx (S1 ; τ ).

that it is preserved by a digitization process. Thus if we have a set of lines that are either collinear, parallel, or concurrent according to affine geometry, the digitizations of these sets must also be digitally collinear, parallel, or concurrent in the

Proof:

According to the definition, S1 and S2

are collinear if and only if the thickness of S1 ∪ S2 is smaller than τ . Since the thicknesses of both S1 and S2 are smaller than τ by choice, it follows from Proposition 1 that it is sufficient to compute

digital plane.

the thickness of the following 3-point subsets: (i) 5.1.

the 3-point subsets that contain a pair of points in

Collinearity

S1 , and a single point of S2 ; (ii) the 3-point subDefinition 6.

Let S1 , . . . , Sn be finite subsets

sets that contain a pair of points in S2 , and a sin-

of R2 , and let σx (Si ) < τ . We say that these sets

gle point of S1 . The subsets mentioned in (i) are

are collinear relative to the x-axis if and only if

straight if S1 ⊂ collx (S2 ; τ ). Similarly, the subsets

∩i domx (Si ; τ ) is non-empty.

mentioned in (ii) are straight if S2 ⊂ collx (S1 ; τ ).

Hence, in terms of preimages, the sets Si are called collinear if the intersection of the preimages of the sets contains an (affine) line B. Note that B is one of the lines whose parameters lie in ∩i domx (Si ; τ ). In particular, when the sets Si are

Thus the collinearity of two sets can be verified by a geometric construction. For a collection of more than two sets, we have the following Helly type result.

digitally straight sets and when we choose τ = 1,

Theorem 8.

then according to the above definition the sets Si

tion of finite subsets of R2 , and let σx (Si ) < τ .

are collinear if and only if there is an affine line B

Then this collection is collinear if and only if each

in the real plane such that (S1 ∪. . .∪Sn ) ⊂ digx B.

of its 3-membered subcollections is collinear.

Let S1 , . . . , Sn be a finite collec-

For two sets S1 and S2 , the following result interprets collinearity in terms of collinearity regions.

Proof:

The thickness of the set S1 ∪ . . . ∪ Sn

is smaller than τ if the thickness of each of its 3Let S1 and S2 be finite subsets

points subsets is smaller than τ . Since the points

of R2 , and let τ be a maximal acceptable thick-

of each 3-point subset can only be selected from at

Proposition 7.

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most three sets, it suffices to verify whether each

the projection πα (domx (S1 ; τ )) of the domain

subcollection with 3 sets is collinear.

domx (S1 ; τ ). Clearly, πα (domx (S; τ )) is convex,

Proposition 7 and Theorem 8 completely characterize the collinearity of collections of sets. According to Theorem 8, a collection of sets is collinear if each of its 3-membered subcollections is collinear It is not sufficient that each 2membered subcollection is collinear, as shown in Fig. A.7 with a simple counterexample. Fig. A.7 shows three distinct digital sets together with their collinearity regions and preimages, for τ = 1. To exemplify the shape of these preimages, the boundaries of the preimage of S2 have been highlighted. Although each pair of sets is digitally collinear, as can be seen from the collinearity re-

since it is the projection of a convex set. The projection πα (domx (S; τ )) has the following geometrical meaning: it is equal to the open interval ]α1 , α2 [, where α1 is the infimum of the slopes of 2 the lines B ⊂ dig−1 x (S; τ ), and where α is the

supremum.

Definition 7.

Let S1 , . . . , Sn be a finite collec-

tion of finite subsets of R2 such that all sets have a thickness smaller than the maximal acceptable thickness, that is, maxi (σx (Si )) < τ . Then these sets are parallel relative to the x-axis if and only T if i πα (domx (Si ; τ )) is non-empty.

gions, the collection of three sets is not digitally

Hence, a collection of sets Si is parallel if and

collinear. To be precise, we have Si ⊂ collx (Sj ; τ ),

only if there exist parallel, affine lines Bi : y =

for each pair of sets Si , Sj , and by Proposition

αx + βi for which (α, βi ) ∈ domx (Si ; τ ). Or, in

7 this means that each pair of sets is collinear.

terms of preimages, the Si are parallel if there

Nevertheless, the intersection ∩i dig−1 x (Si ; τ ) of

exists a collection of affine parallel lines Bi such

the three preimages, shown as a gray region in

that Bi lies in the preimage of Si .

Fig. A.7, does not contain a common line B.

Affine parallelism is transitive; parallelism as defined in Definition 7 is not. Fig.

5.2.

A.8 shows

a counterexample with three digital line segments

Parallelism

S1 , S2 , and S3 . For the preimages, depicted in Parallelism is also defined in terms of domains.

Fig. A.8(a), we can find two parallel lines B1 and

For a subset S of R2 , we let πα (domx (S; τ ))

B2 such that B1 lies in the preimage of S1 , and

denote the orthogonal projection of domx (S; τ )

B2 in the preimage of S2 . Hence, S1 and S2 are

upon the α-axis. For example, Fig. A.8(b) shows

parallel. Similarly, S2 and S3 are parallel since we

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can find two parallel lines B20 and B3 (not shown

defined using domains, we start defining concur-

in the figure) that lie in the preimages of S2 and

rency in terms of preimages.

S3 , respectively. In contrast with affine geometry, however, this does not imply that S1 and S3 are parallel. Completely in accordance with this finding, πα (domx (S1 ; 1)) ∩ πα (domx (S2 ; 1)) and πα (domx (S2 ; 1)) ∩ πα (domx (S3 ; 1)) are both nonempty, but πα (domx (S1 ; 1)) ∩ πα (domx (S2 ; 1)) ∩ πα (domx (S3 ; 1)) is empty,

as illustrated in

Fig. A.8(b). Parallelism as defined in Definition 7 has, however, the following Helly type property.

Definition 8.

Let S1 , . . . , Sn be a collection of

finite subsets of R2 , and let σx (Si ) < τ . We say that the sets Si are concurrent relative to the xaxis if and only if the intersection ∩i dig−1 x (Si ; τ ) is non-empty. It is straightforward, however, to give an equivalent formulation in terms of domains. The following proposition, which relates concurrency to line transversals in the αβ-plane, follows immediately,

Theorem 9.

2

A finite collection of sets in R

and is stated without proof.

is parallel relative to the x-axis if and only if each subcollection of two sets is parallel relative to the

Proposition 8.

Let S1 , . . . , Sn be a collection

x-axis.

of finite subsets of R2 , and let τ be a maximal acceptable thickness that exceeds the thickness of all

Proof:

Apply Helly’s Theorem to the intervals

sets Si . Then these sets are concurrent relative to the x-axis if and only if their domains domx (Si ; τ )

πα (domx (Si ; τ )). Theorem 9 is not in contradiction with

have a common line transversal.

Fig. A.8, since S1 is not parallel with S3 . Hence,

In affine geometry the concurrency of two

S1 , S2 and S3 do not have to form a collection of

straight line triples that have two lines in common

parallel sets.

induces the concurrency of all four lines. To be precise, if the lines B1 , B2 , B3 are concurrent, and

5.3.

B2 , B3 , B4 are concurrent, then all four lines pass

Line concurrency

through a single point. For concurrency as defined The last geometrical concept that we adapt for the

in Definition 8, this is no longer valid. Fig. A.9(a)

digital plane is the concurrency of lines. In con-

shows, as a counterexample, a collection of four

trast with collinearity and parallelism, which were

digital sets S1 , . . . , S4 and their preimages. Since

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the sets S2 and S3 are difficult to distinguish, the

translate of B that strictly separates the two sets.

points of S2 have been encircled. Fig.

A.9(b)

Then F admits a line transversal if and only if

shows the corresponding domains. It is important

each 3-membered subcollection of F admits a line

to note that, in this example, the intersection of

transversal.

domx (S2 ; τ ) and domx (S3 ; τ ) is empty, although their closures have one point in common. (Neither of the two preimages contains a horizontal line. Note that a preimage is neither open nor closed, and that only the highlighted boundary

Klee and Gr¨ unbaum’s Theorem is not an exclusive result on line transversals in the plane. In fact, Danzer et al list several similar results of at least equal importance [6]. For our purposes, though, Theorem 10 stands out as it can be re-

segments belong to the preimages.)

lated directly to parallelism. In Fig. A.9(a), each 3-membered subcollection is concurrent. This can be seen from the preim-

Theorem 11.

ages, since the intersection of each 3-membered

of R2 such that each pair Si , Sj is not parallel

collection of preimages is non-empty, and also

relative to the x-axis (where parallelism is defined

from the domains, since each 3-membered collec-

as in Definition 7) . Then these sets are concur-

tion of domains has a line transversal. There is,

rent relative to the x-axis if and only if each triple

however, no straight line that transverses all four

of sets is concurrent relative to the x-axis.

domains. Or, equivalently, there are no four concurrent lines B1 , . . . , B4 such that each Bi lies in the preimage of Si .

Proof:

Let S1 , . . . , Sn be finite subsets

Since each pair of sets is non-parallel,

it follows immediately from Definition 7 that each pair of domains domx (Si ; τ ), domx (Sj ; τ ) can be

For concurrency also we derive a Helly type

strictly separated in the αβ-plane by a line that is

property. We employ Valentine’s formulation (

parallel to the β-axis. Thus we can apply Theo-

[27], Theorem 14) of a theorem found indepen-

rem 10, since, as explained in the Appendix, this

dently by Klee and Gr¨ unbaum [6].

theorem is also valid for a finite collection of noncompact sets. Let F be

Note that in the proof of Theorem 11 we do not

a finite collection of compact convex sets in R2 .

use the full power of Klee and Gr¨ unbaum’s The-

Assume that there exists a line B in the plane

orem, because we restrict ourselves to separating

such that for each pair of sets in F there is a

lines that are parallel to β-axis. Hence, it may

Theorem 10. (Klee, Gr¨ unbaum)

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very well be possible to find more general Helly

sals are general enough to handle the combination

type results on concurrency that are valid for a

of both schemes. The main idea is that the previ-

collection of lines in which some lines are paral-

ously defined domains domx (S; τ ) and domy (S; τ )

lel and that are meaningful from the geometrical

can be embedded in two parameter planes that lie

viewpoint.

in a more general parameter space. In this space,

Also note that neither Theorem 11, nor the

both schemes can be combined by projecting the

stronger Theorem 10, apply to the situation shown

domains corresponding to one of the schemes into

in Fig.

A.9. The four domains cannot be sepa-

the parameter plane of the other scheme. Since

rated by a family of parallel lines. The example

projections preserve transversals, we thus obtain

in Fig. A.9 is rather special, however, since the

scheme independency.

sets Si are very small. For any reasonably sized problem, the domains of non-parallel subsets are almost always sufficiently small so that they can be separated.

6.

6.1.

Combining different schemes

To combine the two digitization schemes, we introduce a more general definition for the domain

Digitization scheme independence

Up to now, all properties have been derived within a single digitization scheme, that is, properties relative to the x-axis, or properties relative to the y-

of a set. Definition 9.

Let S be a finite subset of R2 ,

and let τ be the maximal acceptable thickness. Then

axis. But in a real application both schemes must

dom(S; τ ) = {(α, β, γ) ∈ R3 :

be used simultaneously.

−τ /2 < αxi + β + γyi ≤ τ /2, (xi , yi ) ∈ S}

For example, we may

have to examine the concurrency of a number of

is called the generalized domain of S.

sets where for some sets the x-scheme is the most appropriate (e.g., digitizations of nearly horizontal

Clearly, if we identify the parameter points

lines), for some sets the y-scheme (e.g., nearly ver-

(α, β) mentioned in Definition 3 with the points

tical lines), and where for some sets the choice is

(α, β, −1)) in the γ = −1 plane in R3 , the pre-

not at all clear (e.g., diagonals). In this section, we

viously defined domain domx (S; τ ) becomes iden-

examine how the x-scheme and the y-scheme can

tified with the intersection of dom(S; τ ) and the

be used together. We shall prove that transver-

plane γ = −1. Similarly, by identifying the pa-

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rameter points (β, γ) with the points (−1, β, γ),

To obtain scheme independence, it is crucial

the y-related domy (S; τ ) is identified with the in-

that all the basic properties that we have defined

tersection of dom(S; τ ) and the α = −1 plane.

are transversal properties (where intersection is considered as a transversal by a common point) and that projections preserve transversals. As a result, it does not matter whether we look for

By projections in the αβγ parameter space we can now compare the y-related domain with the x-related domain.

transversals in the γ = −1 plane or for transversals in the α = −1 plane.

We let πγβ (domx (S; τ )) de-

note the (non-orthogonal) projection of the set domx (S; τ ) upon the α = −1 plane, with the origin as the projection center. Likewise, we let παβ (domy (S; τ )) denote the projection of the set domy (S; τ ) upon the γ = −1 plane, also through the origin. Note that παβ (domy (S; τ )) is a convex set provided domy (S; τ ) does not intersect the plane γ = 0. This intersection would be nonempty if the domain domy (S; τ ) contains parameter points (γ, β) of the form (0, β), or in other

Proposition 9.

For a given maximal accept-

able thickness τ , let S1 , . . . , Sn be a finite collection of subsets of R2 with domx (Si ; τ ) 6= ∅.

Similarly, let T1 , . . . , Tk be a finite collec-

tion of subsets of R2 with domy (Tj ; τ ) 6= ∅. Let Dx denote the collection {domx (S1 ; τ ), . . ., παβ (domy (T1 ; τ )), . . .}, and let Dy denote the collection {πγβ (domx (S1 ; τ )), . . ., domy (T1 ; τ ), . . .}. Then the following assertions hold:

words, if the preimage of S contains a vertical line. In practice, when this occurs in an image pro-

1. Dx can be transversed by a line in the αβ-

cessing application we can almost always resolve

plane if and only if Dy can be transversed by

this problem by projecting the domains upon the

a line in the γβ-plane;

α = −1 plane instead of upon the γ = −1 plane,

2. Dx can be transversed by a line in the αβ-

so that all projections remain convex. Only when

plane that is parallel to the β-axis if and only

some of the preimages contain horizontal lines and

if Dy can be transversed by a line in the γβ-

some of the preimages contain vertical lines, we

plane that is parallel to the β-axis;

must take into account the non-convexity of some

3. the sets in Dx have a non-empty intersection

of the projections, as will be briefly discussed later

if and only if the sets in Dy have a non-empty

on.

intersection.

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Proof:

Since a projection preserves all in-

cidence relations of lines and points, the col-

23

transversal parallel to the β-axis, or have a nonempty intersection.

lection Dx admits a line (or point) transversal if and only if the collection {πγβ (domx (S1 ; τ )), . . ., πγβ (παβ (domy (T1 ; τ ))), . . .} admits a line (or point) transversal. Since πγβ (παβ (domy (Tj ; τ ))) = domy (Tj ; τ ), the latter collection is the same as Dy . Hence, the first and third assertions follow immediately. By noting that παβ projects a line

Note that in the special case of digital straightness, this definition complies with the standard definition that a digital set S is called digitally straight if it is digitally straight according to at least one of the digitization schemes, that is, we must have min(σx (S), σy (S)) < 1 (see [24, 28]).

in the plane α = −1 that is parallel to the β-axis

According to Definition 10, we only have to se-

upon a line in the plane γ = −1 that is also par-

lect for each set the most appropriate domain.

allel to the β-axis, the second assertion follows in

That is, we must decide whether for a given set

a similar way.

S, we either use the domain domx (S; τ ) or the domain domy (S; τ ) to examine geometric properties. Once an appropriate domain has been selected, it

This leads us to the following scheme indepen-

makes no difference whether we look for transver-

dent definition for concurrency, parallelism and

sals and intersections in the αβ-plane or in the γβ-

collinearity.

plane. The remaining problem, however, is that according to Definition 10, an appropriate choice is any choice for which we can find a transversal, and that, for a collection of n sets we may have to

Definition 10.

Let τ be a given maximal

acceptable thickness, and let S1 , . . . , Sn be a fi2

nite collection of subsets of R whose thickness is smaller than τ . We say that the sets are ei-

verify, in theory, all 2n possible choices for the domains of the sets (2 possible schemes for each set). We now show that in almost all practical cases the choice of the domain can be made in advance.

ther concurrent, parallel, or collinear if we can partition this collection into two subcollections

6.2.

Preference of scheme

T1 , . . . , Tm and U1 , . . . , Un−m such that the domains domx (T1 ; τ ), . . . , παβ (domy (U1 ; τ )), . . . ei-

According to the following proposition, proven in

ther admit a line transversal, admit a line

[29], a finite set has almost always a clear pref-

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erence for one of the digitization schemes. More

Only those sets for which Proposition 10 does not

precisely, it suffices that a set does not resemble

immediately yield a preferred choice of domain,

the digitization of a diagonal line.

must be examined more closely.

Proposition 10.

Let S be a finite subset of R2 .

Let 0 < τ . Then the following assertions hold:

6.3.

Parameter space decomposition

There is a weaker but more general result than Proposition 10, according to which we can parti-

1. If there are two points (x1 , y1 ), (x2 , y2 ) in S such that |y1 − y2 | ≥ |x1 − x2 | + τ , then −1 dig−1 x (S; τ ) ⊆ digy (S; τ );

tion the γ = −1 parameter plane into two parts U and V such that inside each of these two parts the domain related to one of the schemes is always a

2. If there are two points (x1 , y1 ), (x2 , y2 ) in S

subset of the domain related to the other scheme.

such that |x1 − x2 | ≥ |y1 − y2 | + τ , then −1 dig−1 y (S; τ ) ⊆ digx (S; τ );

Proposition 11.

Let S be a subset of R2 , and

let τ be chosen such that both domx (S; τ ) and −1 Furthermore, if dig−1 y (S; τ ) ⊆ digx (S; τ ), then

it follows by Proposition 4 that παβ (domy (S; τ )) ⊆ domx (S; τ ) as well as domy (S; τ ) ⊆ πγβ (domx (S; τ )). This tells us that almost always we can select be-

domy (S; τ ) are non-empty. Let U denote the set of parameter points (α, β, −1) in the plane γ = −1 for which |α| ≤ 1. Similarly, let V denote the set of parameter points (α, β, −1) for which |α| > 1. Then the following assertions hold:

forehand an appropriate digitization scheme for each set. For example, if a S has a pair of points for which |y2 − y1 | + τ ≤ |x2 − x1 |, then any

1. (domx (S; τ ) ∩ V ) ⊆ παβ (domy (S; τ )), and (παβ (domy (S; τ )) ∩ U ) ⊆ domx (S; τ ).

transversal through παβ (domy (S; τ )) also trans-

2. If domx (S; τ ) ∩ παβ (domy (S; τ )) is empty,

verses domx (S; τ ), since the latter is a superset

then domx (S; τ ) ⊆ U and παβ (domy (S; τ )) ⊆

of the former. Thus, since domx (S; τ ) also covers

V.

all possible transversals through παβ (domy (S; τ )), we do not have to examine παβ (domy (S; τ )) as a

Proof:

separate case, and as a result, domx (S; τ ) is al-

verses all horizontal segments Cy ((xi , yi ); τ ) of the

ways an appropriate choice for the domain of S,

set S, also transverses all the vertical segments

regardless of the plane on which it will projected.

Cx ((xi , yi ); τ ), provided that −1 < α ≤ 1. It

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follows that any parameter point (α, β, −1) in

More precisely, we have to examine whether the

παβ (domy (S; τ )) for which −1 < α ≤ 1, also

sets domx (Si ; τ ) have a transversal parallel to the

belongs to domx (S; τ ).

In a similar way we

β-axis that lies in the set U , or whether the pro-

can prove that any parameter point (α, β, −1) in

jections παβ (domy (Si ; τ )) have a transversal par-

domx (S; τ ) that belongs to either the halfspace

allel to the β-axis that lies in the set V . The sets

α > 1, or the halfspace α < −1, also belongs to

Si are parallel if and only if at least one of these

παβ (domy (S; τ )). Second, (ii) follows immediately

transversals exists.

from (i). 6.4.

Overlapping domains

Hence, although for some sets we may not know

Proposition 11 can only be used when verifying

beforehand which of the domains domx (S; τ ) and

collinearity or parallelism. As for concurrency, a

παβ (domy (S; τ )) will give us the best chance of

line transversing the domains does not have to

finding a transversal in the parameter space, by

be parallel with the boundary between U and

considering the γ = −1 plane as the union of two

V.

disjoint subsets U and V , in each subset the choice

we cannot consider the line transversals in the

may become clear. To be precise, within U the

U and V part separately. To remedy this situa-

domain domx (S; τ ) covers all possible transver-

tion, we give another result, which handles almost

sals, while in V all transversals are covered by

all the cases for which Proposition 10 does not

παβ (domy (S; τ )). By decomposing thus the pa-

yield an exclusive choice. The basic idea is that

rameter plane into two subsets U and V , we can

when a line transverses the convex hull of two non-

solve all problems with respect to collinearity and

disjoint convex sets, it transverses at least one of

parallelism. As for collinearity, a common inter-

the sets. We therefore prove that the intersection

section point of the domains either lies in U or in

domx (S; τ ) ∩ παβ (domy (S; τ )) is non-empty if the

V , and it suffices to examine these two cases sep-

pair of enclosing lines of S is the same for both

arately. As for parallelism, the boundary between

schemes, which is always the case except for very

U and V consists of two lines that are parallel to

special sets.

Since transversals can cross the boundary,

the β-axis. Hence, any line transversal parallel to Let S be a finite subset of R2 ,

the β-axis either lies in U or in V . Hence, also in

Proposition 12.

this case both parts can be examined separately.

and let τ be chosen such that both domx (S; τ )

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If reprx (S) =

fact, it suffices to choose τ such that domx (S; τ )

repry (S), then domx (S; τ ) ∩ παβ (domy (S; τ )) is

contains only a small region around the parame-

non-empty.

ters of the line in reprx (S). Similarly, for such a

and domy (S; τ ) are non-empty.

value of τ the projected domain παβ (domy (S; τ )) If the domain domx (S; τ ) is non-empty

will contain only a small region around the param-

then it always contains the parameters of the

eters of the line in repry (S). Since the two lines

line reprx (S).

are distinct, the intersection of the two domains

Proof:

Likewise, if domy (S; τ ) is non-

empty then it contains the parameters of repry (S). Since reprx (S) = repry (S), the same parame-

will be empty. Fig.

A.10 shows an example of a set for

ters occur in both domains. Hence, domx (S; τ ) ∩

which domx (S; τ ) ∩ παβ (domy (S; τ )) is empty.

παβ (domy (S; τ )) is non-empty.

Fig. A.10(a) also shows two preimages of the set.

Now suppose we have a set S for which the intersection of domx (S; τ ) and παβ (domy (S; τ )) is non-empty.

When verifying concurrency we

can replace domx (S; τ ) and παβ (domy (S; τ )) by the convex hull of their union. It is clear that any transversal that intersects the convex hull of domx (S; τ ) ∪ παβ (domy (S; τ )) also intersects at least one of these sets, provided their intersection

The first preimage is defined relative to the x-axis and consists of all points that lie on lines that transverse the vertical segments Cx ((xi , yi ); τ ). The second preimage is defined relative to the yaxis, and is generated by the lines that transverse the horizontal segments Cy ((xi , yi ); τ ). Although the preimages have a non-empty intersection, this intersection does not contain a straight line. That this is in fact a rare example is shown by the

is non-empty.

following proposition, which gives a simple and sufficient condition to have reprx (S) = repry (S). 6.5.

Rare unsolvable cases

Recall that Mx denotes the collection of 3-point subsets P of S that satisfy σx (P ) = σx (S), and

To a certain extent, the converse of Proposition 12 is also true in the rare case that reprx (S)

that My denotes those subsets for which σy (P ) = σy (S).

and repry (S) are distinct and both consist of one line. If this happens, we can always find a suffi-

Proposition 13.

ciently small value for τ such that the intersection

R2 such that Mx ∩ My contains at least one

domx (S; τ ) ∩ παβ (domy (S; τ )) becomes empty. In

3-point subset P whose points satisfy the follow-

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Let S be a finite subset of

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ing betweenness constraint: x2 ∈]x1 , x3 [ and y2 ∈

sets occur, which is extremely rare, then we must

]y1 , y3 [. Then we have reprx (S) = repry (S).

verify 2m possibilities. Similarly, when the preimage dig−1 y (S; τ ) of a set S contains a vertical line,

Proof:

Since x2 ∈]x1 , x3 [ there is, according

then the projection παβ (domy (S; τ )) will consist

to Proposition 2, a unique pair of enclosing lines

of two disjoint convex parts. This means that

{B1 , B2 } relative to the x-axis, where B1 is the

when we have to verify the concurrency of a collec-

line passing through (x1 , y1 ), (x3 , y3 ), and B2 is

tion of lines where at the same time some preim-

the line parallel to it and passing through (x2 , y2 ).

ages contain horizontal lines and some preimages

Since we have y2 ∈]y1 , y3 [, B1 , B2 also form a

contain vertical lines, then we cannot avoid that

unique pair of enclosing lines relative to the y-axis.

some of the sets are represented by a domain that consists of two parts, no matter which of the pa-

Note that the converse is not true. Fig. A.11 shows an example of a set for which reprx (S) and repry (S) are identical and consist of a single line,

rameter planes γ = −1 or α = −1 is chosen. Hence, also in this case we must verify transversals for both parts.

but where there is no triple of points that satisfies the betweenness condition of Proposition 13. Hence, the condition is sufficient but not necessary, and the example shown in Fig. A.10 is even rarer than what can be derived from Proposition 13.

7.

Concluding remarks

The goal of this paper was to adapt three important geometric incidence properties so that they can be employed in the digital plane: collinearity, parallelism, and concurrency. To this end they

We conclude that when we have to verify con-

have been reformulated as problems on transver-

currency there remain rather special sets for which

sals: collinearity corresponds to a non-empty in-

none of the above results can be applied, and for

tersection of domains (a point transversal), paral-

which we cannot even replace the x and the y-

lelism corresponds to a transversal by a line paral-

related domains by their convex hull. As a result,

lel to the β-axis in the parameter space, and con-

for such a set S we must take into account that

currency corresponds to a line transversal without

a transversal may exist either for the x-related

restriction on the direction of the line. Helly type

domain or for the y-related domain, and that we

theorems then follow almost immediately. Dur-

must verify both possibilities. In fact, if m of these

ing this translation process, however, the classical

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definitions of affine geometry are often translated

have if the properties would be equivalence prop-

into more elaborate concepts. For instance, we re-

erties.

place the parameters of a line by a domain, a line passing through a pair of points by the preimage of the pair, and the intersection point of two lines by the intersection region of two preimages. Furthermore, the Helly type theorems are weaker than the classical ones. For example, an equivalence property such as parallelism is replaced by a property that is not transitive. Nevertheless, Helly type theorems still bring much clarity and structure to the geometry of the digital plane. They enable us to obtain additional geometrical results, for example, results that concern the shape of the preimage, which would be difficult to prove without the Helly type result of Theorem 7. Helly type theorems often point to special cases and counterexamples, such as the ones illustrated in Figures A.7, A.8, and A.9. Each of these examples shows that the derived Helly type theorems cannot be improved upon with respect to the size of the subcollections needed to verify a certain geometric property. Finally, from the computational viewpoint, Helly type theorems show that many geometrical problems can be solved in polynomial time, which may be considered to be the next best thing to the linear time algorithms that we would

In principle, the results obtained for the digital plane can be extended to digital spaces of arbitrary dimension. Danzer et al, and Valentine provide several Helly type theorems for transversals in spaces of arbitrary dimension [6, 27]. However, although the proposed reformulation of a geometric incidence property as a transversal property seems natural, it may not always be clear how we can choose the most appropriate Helly type theorem for a given transversal problem. For instance, in this paper we have chosen Klee and Gr¨ unbaum’s Theorem to adapt concurrency, but this theorem is only one of the many results on transversals in the plane. In fact, there seem to exist two distinct kinds of transversal theorems [6]: (i) transversal theorems for sets that satisfy restrictions on their relative positions, such as in Klee and Gr¨ unbaum’s Theorem; (ii) theorems for sets that satisfy restrictions with regard to their shape. It may well be possible that some of these additional theorems lead to equally meaningful geometric properties, and that the number of possible choices increases considerably in spaces of higher dimension.

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digx−1 (S; τ ), p ∈ B1− , and p ∈ B2+ , then p ∈

Appendix

digx−1 (S; τ ). Theorem 10 has been formulated for compact convex sets. However, this theorem is also true for a finite family of (not necessarily compact) convex sets. This is the way in which the theorem is used in the proof of Theorem 11. Let Si , with i ∈ I,

Proof:

Let B1 , B2 be two lines that satisfy the

be a finite family of convex sets such that each 3-

conditions of the lemma. Note that p cannot lie

membered subcollection Si , Sj , Sk , with i, j, k ∈ I,

on either of these lines. If B1 , B2 are non-parallel

and i < j < k, admits a transversal, which we

lines, then let B3 be the line passing through p and

denote as Lijk . Since this is a finite family, we

the intersection point of B1 and B3 . On the other

can always find compact subsets Sˆi ⊂ Si such

hand, if B1 and B2 are parallel , then let B3 be the

that each transversal Lijk also transverses the sets

line passing through p and parallel to B1 and B2 .

Sˆi . In fact, for each set Si we can choose points

Now let A1 and A2 be two arbitrary parallel lines

qjk ∈ Lijk ∩Si for all the lines Lijk that transverse

in the plane. By a basic property of geometry,

Si . Then the convex hull of the points qjk , with

these lines will intersect the three lines B1 , B2 and

j, k ∈ I and j < k, is a compact convex subset of

B3 either in the same order, or in inverted order.

Si , which is transversed by Lijk . Thus we find a

In particular, this is true for vertical lines of the

finite family of compact convex sets Sˆi such that

form x = c. Now consider the vertical line passing

each 3-membered subcollection has a transversal

through p. Since p ∈ B1− , and p ∈ B2+ , the in-

Lijk . Hence, according to Theorem 10, the finite

tersection of this vertical line with the line B3 lies

family of sets Sˆi admits a transversal. Since each

between its intersection with the lines B1 and B2 .

Sˆi is a subset of Si , it follows that also the finite

It follows that for any vertical line the intersection

family of sets Si admits a transversal.

with B3 lies between the intersections with B1 and B2 . Because B1 and B2 both cross all the vertical line segments Cx ((x, y); τ ), (x, y) ∈ S, also the

Lemma 2.

Let S be a finite subset of R2 , let

σx (S) < τ and let p be a point in R2 . If we

line B3 crosses all segments Cx ((x, y); τ ). Hence, −1 B3 ⊂ dig−1 x (S; τ ), and therefore p ∈ digx (S; τ ).

can find two lines B1 , B2 such that B1 , B2 ⊂

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Figure captions Figure 1: Image in which subsets have been recognized that represent straight line segments. Figure 2: Four different digitization schemes. Figure 3: Applying (a) Santalo’s and (b) Helly’s Theorem to digital point sets. Figure 4: (a) A pair of enclosing lines and the best fit of a set S; (b) Set with more than one pair of enclosing lines. Figure 5: (a) Collinearity region and preimage of a two-point set; (b) The preimage of a pair of points is the complement of two convex regions T + and T − . Figure 6: Collinearity region of a five-point set. Figure 7: Each pair of segments is collinear, but the collection itself is not collinear. Figure 8: In the digital plane, parallelism is not a transitive relation. Figure 9: Each three sets are concurrent (points of S2 are circled), but the entire collection of four sets is not. Figure 10: Example where two schemes yield disjoint domains. Figure 11: Betweenness conditions are not satisfied in this example.

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Fig. A.1.

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Fig. A.2.

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Fig. A.3.

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Fig. A.4.

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Fig. A.5.

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Fig. A.6.

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Fig. A.7.

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Fig. A.8.

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Fig. A.9.

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Fig. A.10.

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Fig. A.11.

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