Superposition of Qualitative Rectangles using a Quantitative Model

Takeaki Kato

∗

, Sosuke Moriguchi and Kazuko Takahashi

School of Science & Technology, Kwansei Gakuin University, 2-1, Gakuen, Sanda, 669-1337, Japan

Keywords:

Qualitative Spatial Reasoning, Superposition, Relative Direction, Knowledge Representation and Reasoning.

Abstract:

This paper describes an approach to qualitative problem-solving using the quantitative method in spatial rea-

soning. We consider the superposition of two objects, such that pre-speciﬁed parts of the objects are visible.

First, we qualify an object to create a model. It is expressed as a matrix of tiles, which are either black or

white depending on the visibility requirement. We use this to determine the location of two objects. This

process involved quantitative treatment. We describe a sound and complete algorithm that provides quantita-

tive solutions and implemented it as a system with a graphical user interface. Then, we extend this algorithm

so that we may search for a better solution considering a qualitatively equivalent model of the objects; that

is, the topological relationships between the black and white regions are identical. This approach is useful

for analyzing or designing a projection of three-dimensional objects onto a two-dimensional plane, because

it not only reduces the computational expense but also provides a better ﬁt with common sense and human

reasoning.

1 INTRODUCTION

Many problems involve locating multiple objects in a

ﬁnite area with superposition or folding, where the

geometry may be more complicated than simply a

two-dimensional plane. For example, we may wish

to allocate multiple windows on a screen of limited

area (Konishi and Takahashi, 2012), determine the

location of gates or codes in designing logical cir-

cuits (Lapaugh, 1996), place labels of building names

in maps (Freeman, 1991; Li et al., 1998), or arrange

an attractive display for goods. These problems are

related to so-called “packing problems,” which repre-

sent a class of optimization problems of pack objects

into containers. Many efﬁcient algorithms have been

developed for packing problems (Lodi et al., 2002;

Birgin et al., 2010), the aim of which is usually ei-

ther to pack a single container as densely as possible,

or to pack all objects using as few containers as pos-

sible. However, in the above kinds of problems, the

aspect of visibility is accompanied; that is, an object

may have a part that should be visible and another

part that may not be; and the optimal solution is deter-

mined depending not only on the density or the num-

ber of containers but also on where speciﬁc parts are

located. This can be reduced to a problem of super-

∗

Currently, NEC Networks & System Integration Cor-

poration

position, in which the former part is visible and the

latter part is hidden. With such problems, it is easier

to understandhow to determine approximatelythe po-

sitions of the connected parts or superposed parts of

objects at an early stage of the design process than to

compute this using precise data. More efﬁcient meth-

ods for handling these kinds of problem are required

with the increase in the number of occasions on which

three-dimensional data are analyzed or designed.

Previously, we reported a qualitative method that

handles objects consisting of should-be-visible parts

and may-be-hidden parts (Konishi and Takahashi,

2012; Ghourabi and Takahashi, 2015a; Ghourabi and

Takahashi, 2015b). The method looks for a super-

position that satisﬁes these constraints. Here the

term “qualitative” means representation and reason-

ing without using precise data. In these studies, we

use a rectangle in which the size and ratio of the edges

are variable as a target object, and consider objects as

rectangles consisting of several tiles. This is repre-

sented in matrix form, where each element requires

the attribute of visibility. We deﬁne the relative direc-

tion and distance of two tiles as a basic relationship,

and perform reasoning on this representation.

Consider the superposition of the object shown in

Figure 1(b) onto the object shown in Figure 1(a) with-

out rotation, so that the white parts are visible. We

obtain ﬁve solutions, as shown in Figures 1(c)–(g),

Kato T., Moriguchi S. and Takahashi K.

Superposition of Qualitative Rectangles using a Quantitative Model.

DOI: 10.5220/0006123404230430

In Proceedings of the 9th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2017), pages 423-430

ISBN: 978-989-758-220-2

423

which are qualitatively different. For example, (c) is

obtained by placing the black region around the bot-

tom left corner of (b) on the black region around the

top right corner in (a).

(a) (b)

(c)

(d)

(f)

(g)

(e)

Figure 1: Qualitatively different solutions of a superposi-

tion.

We formalized an algorithm for this super-

position method using the proof assistant Is-

abelle/HOL (Ghourabi and Takahashi, 2015a). This

algorithm maps the relative direction and distance of

a pair of tiles from the lower rectangle to those in the

upper rectangle. This algorithm is sound but incom-

plete; that is, solutions may exist that cannot be found

in some cases. In this paper, the algorithm is revised

to preserve completeness and is implemented as a sys-

tem with a visual interface.

Moreover, we extend this algorithm so that it can

generate a “better” solution by regardingobjects qual-

itatively. See Figure 2. If we superpose pattern (b)

onto pattern (a) without rotation, so that white parts

are visible, we obtain (c) as one solution, by placing

(b) in the position indicated by the red frame; whereas

if we superpose pattern (d) onto pattern (a), we obtain

(e) as a solution. Here, the patterns shown in (b) and

(d) are qualitatively equivalent, because their conﬁg-

urations are the same; that is, the topological relation-

ships between the black and white regions are identi-

cal. In this case, (e) may be a better solution than (c),

as more of the black parts of (a) are hidden.

(a) (b)

(d)

(c)

(e)

Figure 2: Superposition based on regarding objects qualita-

tively.

The principle idea of our approach is that we ﬁrst

determine the existence of a solution for a pair of rect-

angles by considering their conﬁgurations. This is

solved quantitativelyby using a matrix representation.

If a solution exists, then a better solution is sought us-

ing another rectangle with the same conﬁguration.

The remainder of this paper is organized as fol-

lows. In Section 2, we describe our qualitative rea-

soning and tiling approach to create qualitative rect-

angles, and also deﬁne a matrix representation along

with the associated computations. In Section 3, we

describe a sound and complete algorithm for the su-

perposition. In Section 4, the method is expanded

to ﬁnd a better solution by regarding objects quali-

tatively. Section 5 concludes the paper.

2 PRELIMINARIES

2.1 Qualitative Spatial Treatment

Numerical data such as coordinates or sizes are typ-

ically used when handling spatial data such as ﬁg-

ures, images, and animations. This leads to a signif-

icant burden in terms of data storage and processing.

Considering the current growth in handling big spa-

tial data, an efﬁcient approach is required. Qualitative

spatial reasoning (QSR) is one such approach to rep-

resenting spatial data without numerical data. QSR

enables reasoning via a symbolic representation; thus,

it not only reduces computationalexpense, it also pro-

vides a better ﬁt with common sense and human rea-

soning.

There havebeen a number of studies of QSR in the

ﬁeld of artiﬁcial intelligence (Stock, 1997; Cohn and

Hazarika, 2001; Cohn and Renz, 2007; Chen et al.,

2013). There are various types of QSR depending on

which aspect one is interested in; for example, direc-

tion, size, and mereology (i.e., part-whole relation-

ships). Spatial occlusion is a concept closely related

to superposition, but studies of spatial occlusion do

not consider the location of the occluded part of an

object (e.g., (Randell et al., 2001)). In our approach

to the superposition of qualitative rectangles, we fo-

cus on the relative direction between the tiles used to

construct an object. There are several existing QSR

systems that focus on the relative directions of two

objects (Goyal and Egenhofer, 2001; Renz and Mitra,

2004; Skiadopoulos and Koubarakis, 2005). Of these,

the tiling approach considers the shapes of objects

and divides the plane into rectangular tiles (Goyal and

Egenhofer, 2001; Li and Liu, 2015). We use a vari-

ation of the tiling approach that expresses directional

relations between objects with better precision.

2.2 Tiling Approach

The target object is a closed ﬁnite connected region,

which may include holes, and where the should-be-

ICAART 2017 - 9th International Conference on Agents and Artiﬁcial Intelligence

424

visible parts and may-be-hidden parts are indicated in

advance

We assume that objects are rectangles, which have

should-be-visible parts and may-be-hidden parts;

each edge of these parts is parallel either with the x-

or y-axes. We extend these edges to divide the entire

rectangle into r ∗ c tiles (1 ≤ r, c), as shown in Fig-

ure 3, where the should-be-visible part is shown in

white and the may-be-hidden part in black.

Figure 3: Tiling for an object: rectangle.

If an object cannot be described in such a simple

form, we take a minimum bounding rectangle, which

is an expression of the maximum extents of an ob-

ject within a Cartesian coordinate system. As shown

in Figure 4(a), where the green area shows a should-

be-visible part, we approximate the border lines of

objects as well as the border lines of the should-be-

visible part so that each of them is parallel with either

the x- or y-axes (Figure 4(b)). We now obtain a rectan-

gle consisting of should-be-visible parts and may-be-

hidden parts, where each of the edges of these parts

are parallel with either the x- or y-axes. Any part not

occupied by an object is regarded as a may-be-hidden

part (Figure 4(c)). We then perform a similar process,

as shown in Figure 3.

minimal bounding

rectangle

(a) (c)

(b)

Figure 4: Preprocessing for tiling an object of any shape.

For any object, we can obtain a rectangle consist-

ing of r∗c tiles (1≤ r, c) that includes no identical pat-

terns of black and white in each adjacent row or col-

umn. This rectangle is called a unit and has the con-

ﬁguration of the target ﬁgure in the least reﬁned form.

If additional lines are drawn between the adjacent red

parallel dotted lines in Figure 3, a rectangle consisting

of more tiles is got, which is a more reﬁned model of

In previous reports, we assumed that a given object is a

rectangle; however, in principle we may consider arbitrary

geometries.

the object. For a pair of models, if their topological

conﬁguration of black and white regions are identical,

then they are said to be qualitatively equivalent. All

models of an object are qualitatively equivalent.

For example, all ﬁgures in Figure 5 are qual-

itatively equivalent, but they are not quantitatively

equivalent. In this case, Figure 5(a) is taken as a unit.

For any object, there exists a unique unit. If a tile is

occupied by a should-be-visible part of an object, then

it is called a white tile; otherwise it is called a black

tile.

(a) (b) (c)

Figure 5: Qualitatively equivalent models.

2.3 Expression for a Unit Conﬁguration

For a unit, each tile is referred as t

, where the top

left of the unit is t

and the value of i increases in the

top down direction and the value of j increases in the

left-to-right direction.

Deﬁnition 2.1 (granularity). For a unit in which a row

is r and a column is c, r ∗ c is called the granularity

of the unit

. Let U

be a unit with granularity r

∗ c

and let U

be a unit with granularity r

∗ c

. If either

≤ r

, c

< c

or r

< r

, c

≤ c

, then it is said that

is more granular than U

The tiles of a unit are classiﬁed into black and

white tiles. Let T be a set of tiles of a unit, and

let B(T) and W (T) be sets of the black and white

tiles of T, respectively. B (T) ∪ W (T) = T and

B(T) ∩ W (T) =

0 hold.

Figure 6 shows a unit with granularity 2∗ 3. Let T

be a set of tiles of this unit. B (T) = {t

}, and

W (T) = {t

Figure 6: Unit expression.

Let t

and t

be tiles of a unit with granularity

r ∗ c (0 ≤ i, p < r, 0 ≤ j, q < c). We deﬁne function

dir that computes the direction of target tile t

wrt

reference tile t

as follows, where k = |i− p|, h = | j−

q|:

Note that the meaning of “granularity” here differs

from that used in (Ghourabi and Takahashi, 2015b).

Superposition of Qualitative Rectangles using a Quantitative Model

425

dir(t

) =











same if p = i, q = j

if p < i, q = j

down

if p > i, q = j

le ft

if p = i, q < j

right

if p = i, q > j

◦ left

if p < i, q < j

◦ right

if p < i, q > j

down

◦ left

if p > i, q < j

down

◦ right

if p > i, q > j

For example, in Figure 6, dir(t

) = down

and dir(t

) = down

◦ left

For a speciﬁc reference tile, we deﬁne a set of rel-

ative directions in which black/white tiles are located.

Deﬁnition 2.2 (Direction of B/W-tiles). For a spe-

ciﬁc tile t, D

(t) = ∪

′

∈B(T)

{dir(t, t

′

)} is called

the set of directions of B-tile wrt t, and D

(t) =

∪

′

∈W (T)

{dir(t, t

′

)} is called the set of directions of

W-tile wrt t.

For example, in Figure 6, D

) =

{same, left

, down

◦ left

} and D

) =

{left

, down

◦ left

, down

3 ALGORITHM FOR

SUPERPOSITION

3.1 Superposition of a Unit

Here superposition refers to placing a foreground unit

denoted byU

onto a background unit denoted byU

in such a way that a tile ofU

is placed on a tile ofU

We deﬁne the success of superposition using di-

rections.

Deﬁnition 3.1 (Success of Superposition.). Let T

and T

be sets of tiles of a foreground unit U

and a

background unit U

, respectively. For t

∈ T

, t

∈ T

if D

) ∩ (D

) ∪ D

)) =

0, then it is said

that superposition of t

on t

succeeds. If there exists

such a pair t

and t

, then it is said that superposition

of U

on U

succeeds.

When a superposition of t

on t

succeeds, for all

d ∈ D

), there does not exist t

′

∈ T

such that

dir(t

′

) = d. Intuitively, there is no tile in direction

d wrt t

on U

, where d is the direction of a white tile

wrt t

on U

3.2 Computing Superposition

For a speciﬁc black tile t

of T

, we generate D

a set of directions of all white tiles wrt to t

. For each

element d of the set, we deﬁne a function atiles that

generates a set of tiles of T

an element of which can

be put on t

without hiding a white tile in the direction

of d

. That is, atiled(d) is a set of tiles that does not

have any tiles in the direction of d.

Let r∗ c be a granularity of U

atiles(same) =

atiles(up

) =

|0 ≤ i < min(k, r), 0 ≤ j < c}

atiles(down

) =

|max(0, r− k) ≤ i < r, 0 ≤ j < c}

atiles(left

) =

|0 ≤ i < r, 0 ≤ j < min(k, c)}

atiles(right

) =

|0 ≤ i < r, max(0, c − k) ≤ j < c}

atiles(dir ◦ dir

′

) = atiles(dir) ∪ atiles(dir

′

)

We then take the intersection of the obtained sets

of tiles for each d:

puton(t

) =

d∈D

)

atiles(d) ⊆ T

If puton(t

) 6=

0, the superposition succeeds and

we obtain the set of solutions:

sol(T

, T

) = {(t

) | t

∈ puton(t

)}.

UfUb

superposition

t’

Figure 7: Superposition.

For example, in Figure 7, let

= {t

′

} and T

}. B(T

) = {t

Let t

be a tile that is to be superposed. Then

) = {left

, down

◦ left

, down

}. In this

case, atiles(left

) is the set of tiles in U

to put

on t

without hiding t

, and we obtain {t

′

Similarly, atiles(down

◦ left

) = {t

′

}

and atiles(down

) = {t

′

}. Therefore,

puton(t

) = atiles(left

) ∩ atiles(down

◦ le ft

) ∩

atiles(down

) = {t

′

} 6=

0, and hence the superposi-

tion of t

′

on t

succeeds.

The ﬁgures obtained by these superpositions may

be identical, even if the solutions in sol(T

, T

) are

distinct. For example, in Figure 8, solutions (t

′

)

and (t

′

) both generate the same ﬁgure.

The name atiles indicates the tiles that are “allowed” to

be placed.

ICAART 2017 - 9th International Conference on Agents and Artiﬁcial Intelligence

426

UfUb

superposition

t’

Figure 8: Different solutions give the same result.

3.3 Soundness and Completeness

Here we prove the soundness and completeness of the

algorithm.

3.3.1 Soundness

The following lemma indicates that, if there exists a

tile that is allowed to be placed, then no tile exists that

hides a white tile.

Lemma 3.1. Let T be a set of tiles of a unit with

granularity r ∗ c. For any direction d, if there exists

∈ T such that t

∈ atiles(d), then there does not

exist t

∈ T such that dir(t

) = d.

Proof. Assume that there exists t

∈ T such that

dir(t

) = d (0 ≤ p < r, 0 ≤ q < c). Here, we

show a proof for the case whereby d = up

. In this

case, k = i − p from the deﬁnition of dir. If k > r,

then 0 ≤ i < min(k, r) = r, since t

∈ atiles(up

Therefore, p = i − k < i − r < 0 holds, which con-

tradicts 0 ≤ p. If k ≤ r, then 0 ≤ i < min(k, r) = k,

since t

∈ atiles(up

). Therefore, p = i− k < 0 holds,

which contradicts 0 ≤ p. For other directions, the

proof proceeds similarly.

Theorem3.1. Let t

∈ T

andt

∈ T

. For all (t

) ∈

sol(T

, T

), superposition t

on t

succeeds.

Proof. Assume that there exists (t

) ∈ sol(T

, T

)

such that superposition t

on t

does not succeed.

Then there exists d such that d ∈ D

)∩(D

)∪

)); that is, d ∈ D

) ∪ D

). Therefore,

there exists t

′

∈ T

such that dir(t

′

) = d. On

the other hand, t

∈ atiles(d) for all d ∈ D

from t

∈ puton(t

). There does not exist t

′

that

satisﬁes dir(t

′

) = d for any d ∈ D

), from

Lemma 3.1. It is a contradiction. Therefore, for all

) ∈ sol(T

, T

), the superposition of t

on t

suc-

ceeds.

3.3.2 Completeness

The following lemma indicates that if there exists a

tile that cannot be placed, there must exist a tile that

hides a white tile.

Lemma 3.2. Let T be a set of tiles of a unit with gran-

ularity r∗ c. For any direction d, if there exists t

∈ T

such that t

6∈ atiles(d), then there exists t

∈ T such

that dir(t

) = d.

Proof. Here, we show a proof for the case whereby

d = up

. If we assume k > r, then for all t

∈ T,

is included in atiles(d), which contradicts the con-

dition; therefore, k ≤ r. k ≤ i < r holds, since t

6∈

atiles(up

). It follows 0 ≤ i− k < r − k < r. There-

fore, if we take p = i − k, q = j, then t

∈ T and

dir(t

) = up

hold. For other directions, the proof

proceeds similarly.

Theorem 3.2. Let t

∈ T

and t

∈ T

. If superposi-

tion t

on t

succeeds, then (t

) ∈ sol(T

, T

Proof. Assume that (t

) 6∈ sol(T

, T

). Then, there

exists d ∈ D

) such that t

6∈ atiles(d) holds,

and there exists t

′

such that dir(t

′

) = d from

Lemme 3.2. dir(t

′

) ∈ D

) ∪ D

), and thus

d ∈ D

) ∪ D

). Hence, D

) ∩ (D

) ∪

)) 6=

0, and it follows that the superposition

of t

on t

fails. It is a contradiction. Therefore,

) ∈ sol(T

, T

3.4 Effectiveness

There may be multiple solutions of the superposition

problem. Here, we take the number of black tiles hid-

den by the superposition as a measure of the evalua-

tion, because effective usage of a ﬁnite space is often

considered as a requirement.

Hereafter, we refer to a solution for the superpo-

sition described so far as a “quantitative solution” to

distinguish it from a “qualitative solution,” which will

be introduced later.

Deﬁnition 3.2 (Quantitative Solution). Let T

and T

be tiles of U

and U

, respectively. An element of

sol(T

, T

) is called a quantitative solution for the su-

perposition of U

on U

Deﬁnition 3.3 (Covered Tile). Let (t

) ∈

sol(T

, T

). We call the black tiles hidden by

the superposition covered tiles. The number of the

covered tiles is deﬁned as follows:

cov

) = |D

) ∩

[

′

∈T

dir(t

′

It is clear that N

cov

) is less than or equal to

the smaller value of |B(T

)| and |T

Deﬁnition 3.4 (Quantitative Optimal Solution). Let

and T

be sets of tiles of U

and U

, respec-

tively. For (s, s

′

), (t, t

′

) ∈ sol(T

, T

), if N

cov

(s, s

′

) <

cov

(t, t

′

) it is said that (t,t

′

) is a better solution than

Superposition of Qualitative Rectangles using a Quantitative Model

427

(s, s

′

). If there exists no (t, t

′

) ∈ sol(T

, T

) such that

cov

(s, s

′

) < N

cov

(t, t

′

) holds, then (s, s

′

) is called a

quantitative optimal solution for the superposition of

on U

For example, compare Figures 7 and 8, which

show the superpositions of the same pair of U

and U

. N

cov

′

) = 1 in Figure 7, whereas

cov

′

) = 2 in Figure 8. Therefore, the latter is a

better solution.

3.5 Implementation

We implemented a system based on this method in

Java (Figure 9). The system initially shows a ﬁeld

consisting of 5∗ 5 cells for each rectangle by default.

A user inputs patterns of pairs of black and white rect-

angular tiles, and then the system makes a unit and

solves the problem. All quantitative optimal solutions

that are found are displayed, and if a user selects one

of them, the resizing and moving processes of the su-

perposition are shown as an animation, which pro-

vides a visual aid to help the user’s understanding.

Figure 9: A screen shot of the system.

4 SYSTEM EXTENSION

4.1 Maximal Black Rectangle

A unit produced by a tiling approach represents a con-

ﬁguration; that is, a set of topological relationships

between black and white regions. A superposition is

deﬁned as the manner in which we place a tile onto

another tile. Each tile corresponds to a subregion of

an original target object, and the exact size is ignored

when creating a unit. This implies that there may be

another solution whereby a greater number of tiles are

covered if we admit superposing one tile onto more

than one other tile.

For example, in Figure 10, when superposing U

onto U

, (a) is the quantitative optimal solution when

superposing one tile onto a single tile, whereas (b)

and (c) show the solutions if allowed to place a tile

onto multiple tiles. In these cases, we have a larger

number of covered tiles. Such a phenomenon occurs

when the granularity of two units differ, and indicates

that we must take into account such qualitative factors

in merging two representations.

Ub Uf

(a) (b) (c)

Figure 10: Superpositions of two units with different gran-

ularity.

To this end, we extend the representation of a unit

while preserving the condition whereby we place a

single tile onto another single tile. We ﬁx a back-

ground unit and extend only a foreground unit, since

the number of the black tiles is used as an evaluation

measure. A maximal black rectangle, corresponding

to the limit of the extension of the background unit, is

determined. Intuitively, a maximal black rectangle is

the connected set of black tiles of which the shape is

a rectangle.

Deﬁnition 4.1 (Maximal Black Rectangle). Let T be

a set of tiles of a unit with granularity r ∗ c. A black

rectangle R(x, y, r

′

, c

′

) included in the unit where 0 ≤

x < r− r

′

, 0 ≤ y < c− c

′

is deﬁned as follows:

R(x, y, r

′

, c

′

) =

| t

∈ B (T), x ≤ i < x+ r

′

, y ≤ j < y+ c

′

R(x, y, r

′

, c

′

) is called a maximal black rectangle of

the unit, if none of the following conditions are satis-

ﬁed:

1. y 6= 0 and ∀t

; x≤ i < x+r

′

, j = y− 1, t

∈ B (T).

2. y 6= c− c

′

and ∀t

; x ≤ i < x+ r

′

, j = y+ c

′

, t

∈

B(T).

3. x 6= 0 and ∀t

; i = x−1, y ≤ j < y+ c

′

, t

∈ B (T).

4. x 6= r − r

′

and ∀t

; i = x + r

′

, y ≤ j < y+ c

′

, t

∈

B(T).

′

∗ c

′

is called its granularity.

For example, for a unit in Figure 11, one of

the maximal black rectangles is highlighted with the

green frame.

Figure 11: A maximal black rectangle.

ICAART 2017 - 9th International Conference on Agents and Artiﬁcial Intelligence

428

4.2 Extension of a Unit

Assume that U

is more granular than U

. If there ex-

ists a maximal black rectangle of U

in which the cov-

ered tiles in a quantitative optimal solution are prop-

erly included, then an extension of U

is performed

to the extent of the maximal black rectangle. The ex-

tended unit is qualitatively equivalent to the original

unit.

Let U

be a unit with granularity r ∗ c, and let

R(x, y, r

′

, c

′

) be a maximal black rectangle of U

. An

extension of U

, denoted by U

′

, is a unit with granu-

larity r

′

∗c

′

, and attributes (black/white) of its tiles are

deﬁned as follows:

Let T and T

′

be sets of tiles of U

and U

′

, respec-

tively.

• t

′

∈ B(T

′

) iff











∈ B(T) (0 ≤ i < r, 0 ≤ j < c)

i c−1

∈ B (T) (0 ≤ i < r, c ≤ j < c

′

)

r−1 j

∈ B(T) (r ≤ i < r

′

, 0 ≤ j < c)

r−1 c−1

∈ B(T) (r ≤ i < r

′

, c ≤ j < c

′

)

• t

′

∈ W (T

′

) otherwise.

It is the number of tiles and not the attribute of

tiles of U

that affects the number of covered tiles.

Therefore, we are not concerned about the attributes

of added tiles but hold thatU

andU

′

are qualitatively

equivalent. Here, with vertical extension, the tiles in

new rows have the same attributes as those in the last

row of the original matrix, and those with horizontal

extension are deﬁned similarly.

Figure 12(a) shows an example of an extension

from unit U

, with granularity 2 ∗ 2 (Figure 12(b)) to

′

and with granularity 4∗ 4 (Figure 12(c)). Note that

and U

′

are qualitatively equivalent, but quantita-

tively different.

Deﬁnition 4.2 (Qualitative Solution). Let U

′

be an

extension of U

. A solution for superposing U

′

on U

is called a qualitative solution for the superposition of

on U

For example, consider the superposition of U

onto U

shown in Figure 13. A maximal black rect-

angle of U

is highlighted as the green frame. Fig-

ure 13(a) shows a quantitative optimal solution for a

superposition of U

on U

, and Figure 13(b) is a qual-

itative solution for a superposition of U

on U

4.3 Properties

The following properties immediately hold from the

deﬁnition of extension of a unit and the completeness

of the superposition algorithm shown in section 3.2.

r’

(b)Uf (c)Uf’

c’

r’

(a)extention of a unit

Figure 12: Deﬁnition of extension of a unit.

extend

Ub Uf

(a)superposition of

Uf on Ub

Uf’

(b)superposition of

Uf’ on Ub

Figure 13: (a) Quantitative solution and (b) qualitative so-

lution.

Proposition 4.1. For a superposition of a unit U

on U

, if all covered tiles for the quantitative opti-

mal solution are properly included in some maximal

black rectangle, then a qualitative solution in which

the number of covered tiles is the granularity of the

maximal black rectangle can be obtained.

Proposition 4.2. If no quantitative solution exists for

a superposition of U

on U

, then there is no qualita-

tive solution for a superposition of U

on U

5 CONCLUSION

We have described a qualitative treatment of two-

dimensional superposed objects consisting of should-

be-visible parts and may-be-hidden parts. We con-

sider this as the problem of superposing rectangles

while maintaining some pre-speciﬁed parts visible.

Our approach is as follows. First, we qualify an

object to make a least reﬁned model and represent it

Superposition of Qualitative Rectangles using a Quantitative Model

429

in matrix form. We ﬁnd a solution with a quantitative

computation using the matrix, then search for a better

solution considering a qualitatively equivalent model

of the object.

The main contributions of this work are as fol-

lows:

• We presented a sound and complete algorithm to

provide quantitativesolutions, and implemented it

as a system with a graphical user interface.

• We extended the algorithm so that a qualitative so-

lution can be obtained when the granularities of

units differ.

The method described is useful for analyzing or

designing a projection of three-dimensional objects.

An evaluation measure on the obtained solutions

is determined based on the user’s purpose. When su-

perposing more than two units, the black tiles that are

not hidden provide room for a third unit to be placed.

Therefore, it is not always true that fewer covered tiles

offer a better solution. The location of a black tile of

a resulting ﬁgure is also an evaluation measure candi-

date.

As part of future work, we plan to evaluate the

obtained solutions to handle superpositions of more

than two units.

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