Computation of the φ-Descriptor in the Case of 2D Vector Objects

Jason Kemp, Tyler Laforet and Pascal Matsakis

School of Computer Science, University of Guelph, Stone Rd E, Guelph, Canada

Keywords:

Image Descriptors, Relative Position Descriptors, φ-Descriptor, Spatial Relationships, Vector Data.

Abstract:

The spatial relations between objects, a part of everyday speech, are capable of being described within an

image via a Relative Position Descriptor (RPD). The φ-descriptor, a recently introduced RPD, encapsulates

more spatial information than other popular descriptors. However, only algorithms for determining the φ-

descriptor of raster objects exist currently. In this paper, the ﬁrst algorithm for the computation of the φ-

descriptor in the case of 2D vector objects is introduced. The approach used is based on the concept of Points

of Interest (which are points on the boundaries of the objects where elementary spatial relations change)

and dividing the objects into regions according to their corresponding relationships. The capabilities of the

algorithm have been tested and veriﬁed against an existing φ-descriptor algorithm for raster objects. The new

algorithm is intended to show the versatility of the φ-descriptor.

1 INTRODUCTION

A Relative Position Descriptor (RPD) provides a

quantitative description of the spatial position of two

image objects with respect to each other (Matsakis

et al., 2015). It can be used, for example, to give a

quantitative answer to the question: “Is object A to

the left of object B?”. The descriptor may indicate

whether A is exactly, not at all, or to some degree

to the left of B. There are many RPD algorithms de-

signed for raster objects, i.e., sets of pixels (2D case)

or voxels (3D case). There are signiﬁcantly fewer

RPD algorithms designed for vector objects, i.e., sets

of vertices deﬁning polygons (2D case) or polyhedra

(3D case).

The force histogram (Matsakis and Wendling,

1999) might be the best known RPD and has found

many applications. Algorithms exist to compute the

force histogram in the case of 2D raster objects, 2D

vector objects, 3D raster objects and 3D vector ob-

jects. In this paper, we focus on another, more re-

cent RPD: the φ-descriptor (Matsakis et al., 2015)

which encapsulates much more spatial relationship

information than the force histogram, and appears to

be much more powerful (Francis et al., 2018). Cur-

rently, there only exists an algorithm for calculating

the φ-descriptor of 2D raster objects, and there is no

algorithm for calculating the φ-descriptor of 2D vec-

tor objects.

Here, we introduce the ﬁrst algorithm to calculate

the φ-descriptor of 2D vector objects. Background

information is provided in Section 2. The new algo-

rithm is presented in Section 3 and tested in Section

4. Conclusions and future work are in Section 5.

2 BACKGROUND

Spatial relations can be categorized as directional,

topological, and distance. Freeman (1975) was the

ﬁrst to suggest that spatial relations could be used

to model relative position. He also suggested that a

fuzzy system be used to allow for these relations to be

given a degree of truth. Miyajima and Ralescu (1994)

proposed that an RPD could be created to encapsu-

late spatial relationship information. An RPD pro-

vides a quantitative description of the spatial relations

between two image objects. RPDs are visual descrip-

tors like color, texture and shape descriptors. Symbol

recognition (Santosh et al., 2012), robot navigation,

geographic information systems, linguistic scene de-

scription, and medical imaging are some areas that

beneﬁt from RPDs (Naeem and Matsakis, 2015).

Existing work in RPDs is explored in Sections 2.1,

2.2, and 2.3, and methods to compare how similar two

φ-descriptors are is explored in Section 2.4.

Kemp, J., Laforet, T. and Matsakis, P.

Computation of the -Descriptor in the Case of 2D Vector Objects.

DOI: 10.5220/0008984500600068

In Proceedings of the 9th International Conference on Pattern Recognition Applications and Methods (ICPRAM 2020), pages 60-68

ISBN: 978-989-758-397-1; ISSN: 2184-4313

Figure 1: On the left, the original image with two intersecting objects. Areas where the objects overlap are shown in dark grey.

On the right, that image divided into φ-regions in the horizontal left-to-right direction θ. Regions without a corresponding

φ-group are left blank.

Figure 2: On the left, an example pair of objects. Areas where the objects overlap are shown in dark grey. On the right,

a histogram of the area corresponding to the overlaps relation between the two objects in all directions. Area has been

normalized to a percentage of the Region of Interaction, as explained in Sections 2.3.2 and 4.

2.1 Existing RPDs

Many RPDs exist, and have built upon each other over

time. The angle histogram might be the ﬁrst true

RPD. It is able to model directional relations such

as “to the right of”, “above” etc, (Keller and Wang,

1995) (Keller and Wang, 1996) (Krishnapuram et al.,

1993) (Miyajima and Ralescu, 1994). The force his-

togram generalizes the angle histogram, and also im-

proves upon it (Matsakis and Wendling, 1999). While

the angle and force histograms can model directional

relations, there are others such as the R Histogram

and the Allen Histogram that can model distance or

topological relations (Naeem and Matsakis, 2015).

The recently introduced φ-descriptor (Matsakis et al.,

2015) is the ﬁrst RPD able to model a variety of di-

rectional, topological, and distance relations (Francis

et al., 2018). In other words, it can describe a larger

set of relations than any other existing RPD. A more

in-depth review of RPDs is available in (Naeem and

Matsakis, 2015).

2.2 The Force Histogram

The force histogram sees objects as physical plates

and determines the sums of forces (like gravitational

forces) exerted by the particles of one object on the

particles of another object. A sum is calculated for

each direction in space. There are algorithms to cal-

culate the force histogram in the case of 2D vector

objects (Recoskie et al., 2012), as well as 3D vector

objects (Reed et al., 2014). In the 2D case, horizon-

tal lines partition the objects into trapezoidal pieces.

The areas of these pieces are determined and used to

calculate the value of the descriptor in the horizontal

direction. The objects are then rotated to the next di-

rection in a direction set, and the process is repeated

for each direction. In the 3D case, planes are used to

intersect the 3D objects into 2D objects, and the 2D

algorithm is applied to the resulting objects.

Computation of the -Descriptor in the Case of 2D Vector Objects

2.3 The Phi-descriptor

For each direction in a direction set, the φ-descriptor

algorithm divides the objects into subregions (φ-

regions). Each φ-region corresponds to an elemen-

tary spatial relationship in that direction. The area and

height of each φ-region are determined, and the val-

ues that correspond to the same group of relationships

(φ-group) are summed. The process is repeated in all

directions to form the complete φ-descriptor (see Fig-

ures 1 and 2). The descriptor can be conceptualized

as a set of histograms, one for each φ-group, with the

directions as bins. The value in each bin is an area or

height.

2.3.1 Phi-groups

An important aspect of this algorithm we refer to as

φ-groups. These are groups of elementary spatial re-

lations, each group sharing some characteristic de-

scribed by the verb naming the group. For example,

the overlaps group contains all relations where two

objects overlap one another. The trails group con-

tains all relations where there is a gap between the two

objects, i.e. one object trails behind the other. Each

group, along with the elementary spatial relations that

they consist of, can be seen in Figure 3. One of the

core elements of the algorithm is to ﬁnd regions, φ-

regions, that correspond to a single φ-group.

Figure 3: Elementary spatial relationships in a given direc-

tion and their corresponding φ-groups. Each spatial rela-

tionship is represented by horizontal slices of two objects A

and B. Slices for different objects have been split apart for

clarity; areas where object A is shown stacked on object B

represent areas where two objects overlap.

2.3.2 φ-Descriptor

The descriptor can be seen as a matrix of numbers, a

column for each direction and a pair of rows for each

φ-group, as seen in Table 1. The pair of rows contain

areas and heights. Each element of the matrix is the

sum of the areas or heights of all φ-regions sharing a

φ-group in a single direction. For example, if there

are two regions with the trails φ-group in direction θ

then the cells for trails in direction θ would contain

one for the sum of the areas, and one for the sum of

the heights. In addition to the φ-groups, the ﬁrst two

rows of the descriptor show the area and height of the

Region of Interaction in each direction (see Section

3.1). These two values are the sum of the areas and

heights, respectively, of all φ-regions in the direction.

Table 1: An example of part of a φ-descriptor. 72 directions

were tested for this example. Each row represents an ele-

mentary spatial relation (or the Region of Interaction) and

the corresponding sums of areas and heights for each tested

direction. For example, the sum of the areas of the φ-regions

corresponding to the trails φ-group in direction 5

◦

is 10461

(shown below in bolded font).

φ-Group

◦

...

RoI

area 310009 311138 ...

height 516.681 533.891 ...

Void

area 404 234 ...

height 202 36.137 ...

Arg

area 7769 7819 ...

height 75.062 70.710 ...

Ref

area 7863 9200 ...

height 129.966 120.720 ...

Trails

area 7481 10461 ...

height 124.683 129.641 ...

... ... ... ... ...

2.4 Similarity Measures

Two similarity measures, MinMax and SubAdd, are

used to compare the similarity of φ-descriptors. These

measures have previously been used and detailed in

(Matsakis, 2016) to compare φ-descriptors of pairs of

raster objects. They each require that the descriptors

use the same direction set, and that all values are non-

negative. The values from one descriptor are com-

pared against their counterpart in the other descriptor;

the pair of values d1(θ) and d2(θ) would refer to the

same element in each descriptor (ex. trails area in di-

rection θ).

2.4.1 MinMax Similarity

The MinMax of the two descriptors is essentially the

ratio between the sum of the minima and the sum of

the maxima the pairs of values:

∑

min(d1(θ), d2(θ))

∑

max(d1(θ), d2(θ))

(1)

ICPRAM 2020 - 9th International Conference on Pattern Recognition Applications and Methods

The MinMax similarity is 1.0 if both descriptors

are identical, and lower values indicate less similarity.

2.4.2 SubAdd Similarity

The SubAdd of the two descriptors is essentially the

ratio between the sum of the absolute differences and

the sum of the sums of the pairs of values:

1 −

∑

|d1(θ) −d2(θ)|

∑

|d1(θ) +d2(θ)|

(2)

The SubAdd similarity is 1.0 if both descriptors

are identical, and lower values indicate less similarity.

3 ALGORITHM

The φ-descriptor is built upon the principle that pairs

of objects can be divided into regions (φ-regions), for

each direction in a direction set, and these regions

correspond to speciﬁc sets of elementary spatial re-

lations (φ-groups). The primary goal then of any φ-

descriptor-based algorithm is to ﬁnd an appropriate

way to create these regions and identify the spatial

relations they correspond to. Using the areas and

heights of these φ-regions, the descriptor can be built

in one direction; to complete the φ-descriptor, the pro-

cess is repeated in all directions. The goal of this new

Figure 4: This image shows the Region of Interaction, as a

dotted box, in direction θ. The PoI are shown: Max0 (black

square), Min0 (grey square), Intersections (white circle with

black border), Local Minima (black circle), and Local Max-

ima (grey circle).

algorithm is to calculate the φ-descriptor for a pair of

vector objects; this process is summarized in Algo-

rithm 1.

Algorithm 1: Vector φ-Descriptor Computation.

1: for all θ ∈ direction set do

2: Rotate original objects by −θ.

3: Find vertices of the inputted objects where ele-

mentary spatial relations change, referred to as

Points of Interest (PoI).

4: for all PoI do {Create Rays}

5: Cast horizontal “rays” through the PoI.

6: Determine the points where the ray inter-

sects with the objects (endpoints).

7: Determine PoI φ-groups for directions ”left-

to-right” and ”right-to-left”.

8: Add edges between endpoints closest to PoI.

9: end for

10: Divide the objects into φ-regions based on the

boundaries of the original objects that intersect

the rays.

11: Calculate areas and heights of φ-regions.

12: Save them to descriptor.

13: end for

3.1 Points of Interest

We focus here on the ”left-to-right” direction (or di-

rection θ=0

◦

). For any other direction θ, rotate the

objects by −θ beforehand. To ﬁnd the φ-regions we

must look at what would make one portion of the ob-

jects change to a different elementary spatial relation-

ship. The points in which these changes occur we

call Points of Interest (PoI) (see Figure 4) and are

conﬁned to the region of interaction, the space which

both objects share in direction θ=0

◦

. The top-most

and bottom-most points of the region of interaction

are both PoI, as well as any local extrema within that

region, and any point where the boundaries of the ob-

jects intersect.

The Region of Interaction is deﬁned by two ver-

tices, Max0 and Min0 (the top-most and bottom-most

points). The extrema are vertices with both neigh-

bours either above (minima), or below (maxima) the

vertex. We consider that a pair of vertices can form

an extrema pair if both vertices share a Y value, they

are neighbours of each other, and their remaining two

neighbours are both either above (minima), or below

(maxima) the pair. After a rotation the set of PoI

which correspond to these two types may change (i.e.

a minima may no longer be a minima after a rotation),

and thus they must be recalculated for each direction.

The intersections are points where an edge from

one object intersects an edge from the other. While

Computation of the -Descriptor in the Case of 2D Vector Objects

the other types of PoI may change after each rotation,

the intersections remain the same throughout the algo-

rithm. For this reason, they are calculated only once

in a pre-processing step.

3.2 Rays

At each PoI we pass a horizontal ”ray” and determine

the locations in which it intersects the edges of the two

objects. We call these intersection points endpoints.

We sort the endpoints left to right, and by counting

how many endpoints are on either side of an endpoint

we can determine whether it lies on object A, B, both

or neither; this is the endpoint’s source.

Edges are added between the endpoints along the

ray. These edges serve to divide the objects into φ-

regions based on the changing elementary spatial rela-

tionships (as deﬁned by the PoI). The endpoints along

the top-most and bottom-most rays are all connected

by edges, while intersection and extrema rays only

have edges added directly adjacent to PoI. An exam-

ple of the process to create rays is shown in Figure 5

by creating two rays from local minima.

(a) (b)

Figure 5: An example creating two rays from local minima.

a) Local minima A1 and B5 along with the rays passing

through. b) All ray intersections located. c) The edges and

endpoints formed by the ray are determined. d) The new

edges (dotted-lines) and endpoints added to the original ob-

jects.

3.3 φ-Regions

While creating the rays, the sources of the endpoints

were determined, and edges added along the ray.

By looking at the sources of an endpoint and its

two neighbouring endpoints, we can determine the φ-

group of the center endpoint. For example, if a point

has the source A, and the neighbouring sources belong

to neither object, we can know (Figure 3) that the end-

point corresponds to the φ-group Argument. Once we

have identiﬁed the φ-groups, and all rays have been

created, we can begin forming the φ-regions. To ac-

complish this we traverse the edges above each point

on a ray in a counter-clockwise direction; the φ-region

is the smallest cycle above and to the right that con-

tains the starting endpoint on the ray. As we traverse

the region, we remove the directed edges that form it;

this reduces computation costs for traversing subse-

quent φ-regions.

3.4 φ-Region Areas and Heights

The φ-descriptor requires the size of each φ-region.

The area of the φ-region is used for the size, as well

as the height of the φ-region. Algorithm 2 shows the

process to ﬁnd the area of a polygon (φ-region) (Page,

2011). To calculate the height of the φ-region, take

the difference of the maximum and minimum Y val-

ues of all points in the φ-region. After the sizes of all

φ-regions are determined for a direction θ, we look

at each φ-region and add the appropriate areas and

heights to the θ and θ+180

◦

columns at the two rows

associated with the φ-groups of the region.

Algorithm 2: Find φ-Region Area.

1: numPoints = number of points in polygon[].

2: area = 0

3: j = numPoints -1

4: for i = 0 to i = numPoints do

5: area+ = (polygon[ j].x + polygon[i].x ) ∗

(polygon[ j].y − polygon[i].y)

6: j = i

7: end for

8: area = area / 2

The φ-descriptor contains the areas and heights of

φ-regions in multiple directions. The number of direc-

tions k must be an even number greater than zero, i.e.,

k = 2 j where j ∈ Z, j ≥ 1. This is because directions

are processed in pairs; when a φ-group in direction

θ is determined the φ-group for direction θ+180

◦

easily determined as well.

The objects are rotated about the centroid of the

two objects. We require the centroid of the two ob-

ICPRAM 2020 - 9th International Conference on Pattern Recognition Applications and Methods

jects such that they can rotate around the same origin;

this will preserve their relative position. This centroid

is calculated once at the start of the algorithm and

used for each rotation. The x-coordinate of the cen-

troid, for example, is the average of the x-coordinates

of all vertices from both objects.

In addition to rotating, we remove all remaining

points and edges we added when creating the rays.

We want to return the objects to their original state,

i.e., with only the initial vertices, object intersections,

and edges on the boundaries of the objects between

these vertices and intersections.

Algorithm 3: Rotate Object.

1: centroid = (mean(X values), mean(Y values)

2: for all Point p in object do

3: ∆x = p.x − centroid.x

4: ∆y = p.y − centroid.y

5: p.x = centroid.x +∆x ∗ cosθ − ∆y ∗ sinθ

6: p.y = centroid.y + ∆x ∗ sinθ + ∆y ∗ cosθ

7: end for

4 EXPERIMENTS

The testing plan to validate the new algorithm pre-

sented in this paper can be broken into ﬁve steps:

1. Generate random vector object pairs.

2. Calculate φ-descriptors of vector object pairs us-

ing the new algorithm.

3. Convert vector object pairs to raster object pairs.

4. Calculate φ-descriptor of raster object pairs using

the existing algorithm.

5. Compare descriptors and processing time.

It is expected that a vector object pair and the

corresponding rasterized object pair should produce

near identical descriptors, and the higher the resolu-

tion of the rasterized objects the higher the similar-

ity between the two descriptors. Note, however, that

the descriptors must ﬁrst be “normalized” using the

size of the Region of Interaction mentioned in Section

2.3.2. For each direction, each area in the descriptor

is divided by the area of the Region of Interaction,

and each height is divided by the height of the Region

of Interaction. In the end, the normalized descriptors

contain percentages, rather than actual size data; for

example, “trails area in direction θ accounts for 2.1%

of the Region of Interaction area”. When comparing

the descriptors we ask the following question, ”Does

this region account for the same percentage in both

descriptors?”

The random generation of vector object pairs is

brieﬂy discussed in Section 4.1, the results of the ex-

periments are presented in Section 4.2, and the limi-

tations with the algorithm are stated in Section 4.3.

4.1 Test Data

Random vector objects pairs were generated using

Ounsworth’s algorithm (Ounsworth, 2012). A num-

ber of random directions is selected. A point is placed

at a random distance in each direction, maintaining a

clockwise progression. The algorithm contains some

variables to have some control over the output, such

as the number of directions, the center of the poly-

gon, the “average” distance from the center for the

points, the “spikiness” (which determines how much

variance there is from the average distance) and the

“irregularity” (which controls how even the direction

set is). Two examples of randomly generated object

pairs are shown in Figure 6.

Figure 6: Two example object pairs created with the polygon generating algorithm.

Computation of the -Descriptor in the Case of 2D Vector Objects

●

●●

●

0.25

0.50

0.75

1.00

100 300 500 700 900 1100 1300 1500 1700 1900

Image Width (Pixels)

MinMax Similarity

Function

Height

Area

Figure 7: Results of the MinMax similarity tests over vari-

ous resolutions. 300 object pairs were tested at each resolu-

tion. As the resolution increases, the similarity improves.

●

●●

●

0.4

0.6

0.8

1.0

100 300 500 700 900 1100 1300 1500 1700 1900

Image Width (Pixels)

SubAdd Similarity

Function

Height

Area

Figure 8: Results of the SubAdd similarity tests over various

resolutions. 300 object pairs were tested at each resolution.

As the resolution increases, the similarity improves.

4.2 Results

Three hundred vector object pairs were created, con-

verted to raster pairs with various resolutions, and

tested against both the raster and vector φ-descriptor

algorithms. Figures 7 and 8 show the MinMax and

SubAdd similarity results. The similarities tend to im-

prove as the resolution of the raster objects increases.

The graphs of the results show some outliers for

pairs of descriptors that are particularly dissimilar.

These results are from images that have very small

regions, as shown in Figures 9 and 10. When rasteriz-

ing φ-regions that are very thin, portions of the objects

may ﬁt between rows or columns of pixels; the result-

ing raster image may not show any pixels for these

Figure 9: These are two raster images of object pair 73. The

image on the left has a width of 100 pixels, and the image

on the right 1,100. The image on the left shows gaps in the

thin object, which was too thin to be correctly rasterized. It

therefore produced a very low similarity score.

Figure 10: These are two raster images of object pair 43.

The image on the left has a width of 100 pixels, and the

right 1,100. There are multiple locations in the left image

where a pixel from object A is directly adjacent to a pixel

from object B. However, this never actually occurs in the

vector image, as there is either an overlap or a space be-

tween the object. The left image therefore produced a very

low similarity score.

regions at all, as shown in Figure 9. When objects

have edges that are very close, the raster image may

have pixels from the two objects directly adjacent,

while the vector image shows the objects to overlap,

or have a gap, as shown in Figure 10. It is also im-

portant to note that we are not comparing the original

descriptors, but rather the normalized descriptors, as

explained in Section 4. If we look at Figure 9 it is

apparent that the region of interaction will change de-

pending on what pixels are detected for each object.

While these outliers show poor similarity, this

does not indicate that either algorithm is producing

incorrect results. Both algorithms are giving the cor-

rect results for the input data provided.

When analysing the time required to calculate the

φ-descriptor we must consider the number of vertices

in the vector object. Figure 11 shows the processing

time for 100 randomly generated object pairs, with

random numbers of vertices (the total number of ver-

tices of the pairs ranges from 12 − 181). We can see

that the number of vertices has a large impact on the

processing time; Figure 12 shows the object pair that

required the most processing time, and had the most

vertices.

ICPRAM 2020 - 9th International Conference on Pattern Recognition Applications and Methods

Figure 11: The processing times of 100 vector object pairs.

The number of vertices refers to the sum of number of ver-

tices of both objects in each pair. It is clear that a larger

number of vertices increases the processing time of the φ-

descriptor computation for vector objects.

Figure 12: The vector object pair that required the most

processing time. There are a total of 181 vertices between

the two objects.

Figure 13: A comparison of the processing times of 100

pairs of objects. We consider vector objects, and raster ob-

jects with a pixel width of 100, 500, 1100, and 1900.

Figure 13 compares the processing time of

these same 100 vector object pairs to their raster

counterparts. We examine the raster objects at

100, 500, 1100, and 1900 pixels in width. We see

the computation time for the φ-descriptor of raster

objects is very consistent for a given object size, and

as expected the more pixels in the raster object, the

more processing time is required. When the raster

objects are between 500 and 1100 pixels in width,

and the vector object pair has less than 100 vertices,

the φ-descriptors are often processed in roughly the

same time range.

4.3 Current Limitations

There are some limitations with the algorithm in re-

gard to the current implementation that should be

noted. In particular, the handling of objects with holes

or made of multiple connected components has not

been implemented. It should be noted that these limi-

tations are not a fault of the algorithm itself, but rather

is a fault of how it was implemented. Work is cur-

rently being done to eliminate these limitations and to

test robustness of the algorithm against these cases.

5 CONCLUSION

We have introduced the ﬁrst algorithm to calculate the

φ-descriptor in the case of 2D vector objects. Our mo-

tivation was to show that the φ-descriptor is a capable

successor to a well-known Relative Position Descrip-

tor (the force histogram), and is equally versatile. The

algorithm was tested against 300 randomly generated

pairs of 2D vector objects. Each pair had the 2D vec-

tor φ-descriptor computed using the new algorithm,

the corresponding 2D raster φ-descriptor was com-

puted as well, and the two descriptors were compared.

Our results have shown that in most cases the vector

and raster descriptors were very similar as expected,

and the higher the resolution of the rasterized objects

the higher the similarity. Cases with poor similarity

were found to have very thin regions that could not be

properly rasterized, i.e., the low similarity scores were

due to data loss in the conversion, not a fault of either

algorithm. Potential applications of this work are in

areas where a powerful RPD is required to encapsu-

late a great amount of spatial relationship information

between 2D vector objects (e.g., human-robot com-

munication and geographical information systems).

Work is currently being done to eliminate the lim-

itations discussed in Section 4.3. Future versions of

this algorithm are planned to be capable of handling

objects with holes or made of multiple connected

Computation of the -Descriptor in the Case of 2D Vector Objects

components. This algorithm also has the potential to

be expanded to the case of 3D vector objects.

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ICPRAM 2020 - 9th International Conference on Pattern Recognition Applications and Methods