CONSTRAINT-FREE TOPOLOGICAL MAPPING AND PATH

PLANNING BY MAXIMA DETECTION OF THE KERNEL SPATIAL

CLEARANCE DENSITY

Panagiotis Papadakis, Mario Gianni, Matia Pizzoli and Fiora Pirri

ALCOR Vision Perception and Cognitive Robotics Laboratory, Department of Computer and System Sciences

University of Rome “La Sapienza”, Rome, Italy

Keywords:

Topological mapping, Path planning, Mean shift, Distance transform.

Abstract:

Asserting the inherent topology of the environment perceived by a robot is a key prerequisite of high-level

decision making. This is achieved through the construction of a concise representation of the environment

that endows a robot with the ability to operate in a coarse-to-ﬁne strategy. In this paper, we propose a novel

topological segmentation method of generic metric maps operating concurrently as a path-planning algorithm.

First, we apply a Gaussian Distance Transform on the map that weighs points belonging to free space according

to the proximity of the surrounding free area in a noise resilient mode. We deﬁne a region as the set of all the

points that locally converge to a common point of maximum space clearance and employ a weighed mean-

shift gradient ascent onto the kernel space clearance density in order to detect the maxima that characterize the

regions. The spatial intra-connectivity of each cluster is ensured by allowing only for linearly unobstructed

mean-shifts which in parallel serves as a path-planning algorithm by concatenating the consecutive mean-shift

vectors of the convergence paths. Experiments on structured and unstructured environments demonstrate the

effectiveness and potential of the proposed approach.

1 INTRODUCTION

In real-world robotic applications where environmen-

tal constraints are minimal, formulating the perceived

knowledge acquired from incoming sensory data into

meaningful, higher-level representations of the envi-

ronment, is a prerequisite of high-level decision mak-

ing (Murphy, 2004; Maxwell et al., 2004). Among the

various robotic sensors, mobile robots are commonly

equipped with range scanners that allow for mapping

of the environment and navigating within the created

map. By modelling a map as an occupancy grid map

(either 2D or 3D), a robot can be initially endowed

with a set of basic functionalities ranging from obsta-

cle avoidance and low-level path planning to frontier-

based exploration.

An important prerequisite for the development of

higher-level functionalities is a representation of the

environment that captures the topological structure of

the free area. This is the result of a topological seg-

mentation of the metric map, a process guided by the

morphological features that determine the distinction

between regions. By building a graph structure of

the environment where nodes correspond to topologi-

cal regions and edges determine the traversability be-

tween adjacent regions, a robot is able to operate in

a coarse-to-ﬁne strategy (Kuipers et al., 2004), (Epp-

stein et al., 2006),(Gianni et al., 2011) using a hierar-

chical arrangement of the perceived knowledge about

the environment.

In this paper, we propose a novel method for topo-

logically segmenting metric maps acquired by mo-

bile robots in order to concisely capture the spatial

characteristics of the environment that are involved in

higher-level decision making. The input to the pro-

posed method is an occupancy grid wherein each grid

cell corresponds to space that is designated as free,

occupied or unknown. Merging the latter two cases,

we obtain a boolean occupancy grid that is provided

as input to a Gaussian Euclidean Distance Transfor-

mation (GEDT). By setting a threshold on the GEDT

of the map we classify the free areas into those that are

safely accessible by a robot in terms of distance from

surrounding obstacles. In the sequel, we cluster these

areas by detecting the local maxima of the GEDT via

a weighed mean-shift (Comaniciu and Meer, 2002;

Fukunaga and Hostetler, 1975) gradient ascent, where

cells are weighed by the corresponding GEDT value.

Papadakis P., Gianni M., Pizzoli M. and Pirri F. (2012).

CONSTRAINT-FREE TOPOLOGICAL MAPPING AND PATH PLANNING BY MAXIMA DETECTION OF THE KERNEL SPATIAL CLEARANCE

DENSITY.

In Proceedings of the 1st International Conference on Pattern Recognition Applications and Methods, pages 71-79

DOI: 10.5220/0003735300710079

 SciTePress

Each cluster is formed by all the points that locally

converge to the same point of maximum GEDT value,

i.e. maximum space clearance. To prevent from clus-

tering together cells that are spatially disjoint, we con-

strain the mean-shift vectors by allowing only for lin-

early unobstructed line segments. This ensures that

cells in the same cluster are actually reachable by the

robot by using the convergence paths of the mean-

shift clustering process that is in parallel functioning

as a path planning algorithm.

The proposed formulation resembles the intuitive

process for topological segmenting the free space in

the absence of prior knowledge about the environ-

ment, i.e. a topological region comprises of all the

points that locally converge to a common point of

maximum clearance-openness of free space where the

convergence path is composed of steps only within

free space.

The contributions of the present work are summa-

rized as follows:

• We propose a novel topological mapping method

concurrently operating as a path planning algo-

rithm.

• The proposed topological mapping method is ro-

bust to small topological changes, resilient to

noise and is applicable to structured as well as

unstructured-outdoor environments.

• The proposed path planning method, implicitly

derived from the proposed topological mapping

framework, yields the optimal paths for a robot

in terms of safety distance from surrounding ob-

stacles and maximum peripheral visibility.

The remainder of this paper is organized as fol-

lows: In Section 2, we brieﬂy recall the state-of-the-

art in topological mapping methods identifying the

respective prominent streams of research. In Sec-

tion 3, we describe in detail the framework in which

we compute the GEDT of a map and employ an

adapted mean-shift gradient ascent for clustering the

free space and topologically segmenting the metri-

cal map. In Section 4, we evaluate the proposed

method through experiments in structured and semi-

structured environments that demonstrate its feasibil-

ity and effectiveness and ﬁnally, in Section 5, we sum-

marize the contributions of this paper.

2 RELATED WORK

Topological maps capture the structure of the per-

ceived environment in a concise representation that

allows high-level reasoning and planning together

with an easier human-robot interaction. In compar-

ison to metric maps, topological maps are more com-

pact and less sensitive to errors that are accumulated

due to noisy sensor readings.

Depending on the sensors that are used for per-

ceiving the environment, we could distinguish topo-

logical mapping methods into those that use visual

sensory data from cameras (appearance-based) and

those that use geometric sensory data from range

scanners (geometry-based).

Appearance-based methods employ scene under-

standing and object detection in order to derive the

semantics of regions within the explored environment

(Goedeme et al., 2007),(Goedeme et al., 2008),(Liu

et al., 2009),(Montijano and Sagues, 2011),(Murillo

et al., 2007),(Valgren and Lilienthal, 2008),(Murillo

et al., 2009),(Paul and Newman, 2010) and build a

topological graph wherein nodes are assigned to de-

tected landmarks. Creating a topological graph of

the environment is imperative in order to efﬁciently

process and maintain the huge amount of information

that is acquired from the cameras.

In this paper, we focus on the second category of

topological mapping methods that are based on the

morphological features of the shape of the environ-

ment as perceived by range sensors such as lasers.

In structured-indoor environments, the most common

feature that is used regards the distinction of free

space into large open spaces that are interconnected

through narrow passages or gateways. These pas-

sages determine the borders among distinct regions

that comprise the topological map.

Within this spirit, a number of relevant works

can be identiﬁed. In the pioneering work of Thrun

(Thrun, 1998) the borders of the topological regions

were determined by ﬁnding the critical points within

the Voronoi diagram of the free space that denote the

points where space clearance is locally minimized, a

concept that was also initially adopted by Beeson et

al. in (Beeson et al., 2003). In their following work

(Beeson et al., 2005), they proposed the Extended

Voronoi Graph deﬁned as the subset of the General-

ized Voronoi Graph (GVG) that is closer than a ﬁxed

threshold from any obstacle, added to the set of all

equidistant points from the closest obstacle. Myung

et al. (Myung et al., 2009) extract the GVG of an

eroded metric map in order to detect the position of

the narrowest passages while Joo et al. (Joo et al.,

2010) compute the curvature of the map border to de-

tect candidate cutting points. In the work of Fabrizi et

al. (Fabrizi and Safﬁotti, 2000; Fabrizi and Safﬁotti,

2002), concepts of digital topology are used as the ba-

sis of the closure and opening mathematical morpho-

logical operators that are applied in fuzzy gridmaps

ICPRAM 2012 - International Conference on Pattern Recognition Applications and Methods

producing constrained regions along with geometric

information using moments.

Spectral clustering has been widely used for topo-

logical segmentation of metric maps as an approxi-

mate solution to the normalized cut of a graph (Shi

and Malik, 2000). In the work of Brunskill et al.

(Brunskill et al., 2007), the initial set of graph nodes is

constructed by random sampling of the free space and

edges are created depending on the visibility between

the nodes while the method proposed by Choi et al.

(Choi et al., 2009) is initially driven by a quadtree cell

decomposition of free space. To support incremental

construction of the topological graph during explo-

ration the former detect discontinuities between the

eigenvalues of the similarity matrix, while the latter

measure the convexity of the regions. Zivkovic et al.

(Zivkovic et al., 2006) employed spectral clustering

at the ﬁnest level of resolution of the occupancy grid

and determine the optimal number of clusters by min-

imizing a cost function that relies on the structure of

the eigenvectors of the Laplacian. In recent work, Liu

et al. (Liu et al., 2011) highlighted limitations in spec-

tral clustering based topological segmentation related

to its instability under varying settings of parameters

and map examples. In their approach they compute

the mutual information graph of the occupancy grid

and segment it using Chow-Liu tree factorization.

While a study of the current approaches provides

a thorough understanding of the topological mapping

problem, it appears that the majority of methods is

formulated and evaluated in structured-indoor envi-

ronments where the segmentation process is driven

by the presence of rooms, corridors, doors, etc. On

the contrary, in applications of minimal environmen-

tal constraints, such as Urban Search and Rescue, an

explicit underlying structure of the explored environ-

ment is unlikely to exist. In addition, the noise in-

duced from the sensors becomes more pronounced as

a result of operating under a broader range of environ-

mental conditions such as the presence non-reﬂecting

surfaces, smoke or ﬁre, etc. In the next section we

unfold the proposed topological segmentation method

that is accordingly formalized to be robust to noise

and relatively small changes of the environment.

3 TOPOLOGICAL MAPPING

In this Section we describe the method that we pro-

pose to segment a generic metrical map given in the

form of a boolean occupancy grid, into a set of topo-

logical regions. The proposed segmentation method

adheres to the following criteria:

i. A topological region is identiﬁed by a point of

maximum space clearance, i.e. whose distance

from surrounding obstacles is locally maximized.

ii. All points that locally converge to the same point

of maximum space clearance belong to the same

region.

iii. The convergence path of a point to the point of lo-

cally maximum clearance comprises only of steps

within free space.

3.1 Distance-based Classiﬁcation of

Free Space

The ﬁrst criterion formulates a space segmentation

approach that is beneﬁcial for a mobile robot in terms

of safety and visibility. With respect to safety, a point

in space that has locally the furthest distance from sur-

rounding obstacles corresponds to a position that the

robot can efﬁciently reach by following a path with

reduced obstacle interference. In terms of visibility, a

point of maximum space clearance corresponds to the

position where occlusions are minimized and a robot

can have a broad overview of the surrounding area.

Furthermore, this criterion resembles the intuitive hu-

man process of space segmentation into rooms and

areas in general. Whenever we stand in a position

very close to occupied space where visibility is rela-

tively constrained we do not identify the center of a

region, rather, when we stand in the center of a region

we commonly have a relatively large distance from

obstacles and a broad, clear view of the surrounding

area.

To obtain a space classiﬁcation of the metric map

according to the distance from surrounding obstacles,

the proposed approach adheres to the Distance Trans-

form.

More formally, we denote a boolean occupancy

grid of cells as a function C : p → {0, 1} where p =

, y

)|i = 1, 2, ..., n} is the set of cells and n is the

cardinality of the set. Letting F = {p|C(p) = 0} de-

note the set of free cells and O = {p|C(p) = 1} the set

of occupied cells, the Euclidean Distance Transform

(p) of the free area is given by:

(p) = min

∈O

kp − p

k (1)

where k.k denotes the Euclidean distance.

Ideally, the distance transform would be sufﬁcient

to obtain a classiﬁcation of the points in terms of

space clearance as shown in Figure. 1 (b) for the sim-

ulated environment shown in Figure. 1 (a). In practice

however, metric maps obtained from robotic sensors

are highly susceptible to noise as shown for a typical

example in Figure. 1 (c) together with the correspond-

ing EDT image in Figure. 1 (d). Due to noisy mea-

surements, individual grid cells that are designated as

CONSTRAINT-FREE TOPOLOGICAL MAPPING AND PATH PLANNING BY MAXIMA DETECTION OF THE

KERNEL SPATIAL CLEARANCE DENSITY

occupied may affect the EDT values of all the neigh-

bouring cells. This results in an undesirable reduction

of the estimated space clearance in relatively large

open areas.

(a) (b)

Figure 1: Example occupancy grids (a),(c) and the corre-

sponding Euclidean Distance Transforms (b),(d).

To decrease the sensitivity to noise and to rela-

tively small obstacles, we compute the discrete con-

volution of C with the Gaussian kernel G(p) =

2πσ

exp(−

kpk

2σ

) given by:

∼

C(p) =

∑

i=1

C(p

)G(p − p

) (2)

According to this formulation we derive that

∼

C(p) ∈

[0, 1) hence for free cells lying close to occupied

space

∼

C(p) → 1 while for free cells lying away from

occupied space

∼

C(p) → 0. For the purpose of the im-

plementation that is followed in this paper (as detailed

in Section 3.2), we encode this information by setting

∼

D(p) = 1 −

∼

C(p) so that cells lying away from obsta-

cles will be characterized by a higher value than the

cells lying close to obstacles. The inversion is pur-

posely adopted after computing the convolution since

the computation of the latter is more efﬁcient when

sparse matrices are used (in practise we compute the

convolution using a gaussian kernel of ﬁxed size con-

taining 99% of the energy). In Figure 2 (a) we show

the initial grid map and in Figure 2 (b) the correspond-

ing

∼

In comparison to the corresponding EDT shown

in Figure 1 (d) the convolved occupancy grid is more

robust to noise and relatively small obstacles which

results in an amended classiﬁcation of free space with

respect to the distance from surrounding obstacles. In

the following, we apply a threshold t

∼

D to signify

the areas that are considered as safely accessible by a

mobile robot, in terms of distance from surrounding

(a) (b)

Figure 2: Example noisy occupancy grid (a) and the corre-

sponding Gaussian Euclidean Distance Transform (b).

obstacles. The value of the threshold depends on the

size of the footprint-radius of the mobile robot as well

as the conﬁdence to its estimated localization within

the map. In Figure 3 we show the remaining free area

as a result of setting a threshold to

∼

The magnitude of the σ of the kernel determines

the importance that is given to surrounding obstacles

in terms of their distance from a particular free cell.

By decreasing σ we give more conﬁdence to the im-

mediate close neighbourhood of the free cell while by

increasing σ we can cover bigger areas around the cell

and consider obstacles that are more distant. The de-

termination of a ﬁxed value is a balance between these

aspects and is related to the resolution of the grid map.

(a) (b)

Figure 3: Determination of safely accessible free area

(shown in green) (b) by setting a threshold on the distance

from surrounding obstacles (a).

3.2 Weighed Mean-shift Clustering of

Free Space

The next step is to detect the local maxima within

∼

D, i.e. the points of maximum space clearance that

will represent the centres of the topological regions,

together with all the points that belong to the same

region. Towards this goal, we employ a mean-shift

(Comaniciu and Meer, 2002) gradient ascent on the

∼

image of the occupancy grid, adapted to conform to

the criteria (as described in Section 3) on which the

proposed segmentation process is based . In practise,

depending on the size of the occupancy grid, a down-

sampled version of

∼

D may be preferred in order to

reduce the overall computational cost.

In the following, we give the preliminary techni-

cal details of mean-shift clustering (for the detailed

formulation see (Comaniciu and Meer, 2002)) needed

ICPRAM 2012 - International Conference on Pattern Recognition Applications and Methods

to make the paper self contained in accordance to the

notations already used.

Let u = {i|p

∈ F} then given a cell p that lies in

free space, its kernel density estimate f

is given by:

(p) =

∑



kp − p



(3)

where b is the bandwidth of a radially symmetric ker-

nel and k the corresponding proﬁle function.

By setting g(x) = −k

(x) and taking the gradient

∇ of eq. (3) we obtain:

(p) =

∑



kp−p



∑



kp−p



− p (4)

that denotes the mean-shift vector which points to-

wards the direction of maximum increase of the ker-

nel density. By repeatedly applying the mean-shift

operator, we can identify the points of locally maxi-

mum space clearance as the points where the gradient

of eq. (3) is minimized, according to a convergence

threshold ε.

For the computation of the kernel density we em-

ploy the Epanechnikov kernel which minimizes the

square error between the density and its estimate (Co-

maniciu and Meer, 2002). The proﬁle k

of the

Epanechnikov kernel is given by:

(x) =



1 − x, if 0 ≤ x ≤ 1

0, if x > 1

(5)

We adapt the former formulation of mean-shift to

consider not only the spatial position of a data point in

the estimation of the kernel density, but also its con-

tribution as speciﬁed by a weight factor. This factor

is the corresponding

∼

D(p) value that determines the

space clearance of the point. Hence, we compute the

modiﬁed shift vector as:

(p) =

∑



∼

D(p)·p−

∼

D(p

)·p



∑



∼

D(p)·p−

∼

D(p

)·p



− p (6)

The convergence to the modes of the density pro-

duces slightly different locations due to the threshold

ε that determines the termination of the mean-shift it-

erations, as well as because the value of the gradient

is very close to zero within regions where the density

estimate is constant. These artifacts are eliminated by

fusing mode candidates at a distance less than the ker-

nel bandwidth b by choosing the one corresponding to

the highest density.

The magnitude of the bandwidth b of the kernel

sets the size of the window that is considered for the

computation of the mean-shift and therefore deter-

mines the surface area of the topological regions. The

bigger bandwidth we select, the bigger areas are de-

tected and vice versa.

3.3 Constraining of Mean-shifts to

Linearly Unobstructed Vectors

According to the former formulation of mean-shift,

there are no constrains in the computation of the m

vector. This may result in computing mean-shift vec-

tors that point to positions that do not belong to free

space or that comprise of cells lying within occupied

space. Hence, a cell may converge to a point of lo-

cally maximum space clearance although the two po-

sitions are spatially disjoint. In Figure 4 (a) we show

a typical example of this case.

In the case of topological map segmentation, we

require that every point that belongs to a topological

region is spatially accessible by a robot through a path

whenever the robot is located within this region. To

avoid the costly computation of paths within the met-

ric map, we introduce this condition into the conver-

gence path of the mean-shift clustering procedure. In

detail, for each cell we condition the mean-shift vec-

tors of the corresponding convergence path to allow

only linearly unobstructed vectors. In this way, we

prevent the mean-shift vector from jumping into or

crossing through occupied space and the convergence

path simulates a path that could be followed by the

robot in real.

In order to ensure the spatial intra-connectivity of

cells, we use Bresenham’s line algorithm (Theoharis

et al., 2008) through which we can recover the cells

that lie along each mean-shift vector. If for a par-

ticular mean-shift vector all the corresponding cells

belong to free space, then mean-shift proceeds nor-

mally to the next iteration. Otherwise, we decrease

the bandwidth to b

= b − 1 and recompute the mean-

shift vector. This is repeated until all cells along the

mean-shift vector lie on free space wherein conver-

gence advances or until b

= 1 where we identify an

isolated free cell that is discarded. The consistent

clustering of the example shown in Figure 4 (a) is

given in Figure 4 (b).

The advantage of the former formulation of mean-

shift relies on the usability of the convergence paths

of all cells to the corresponding centres of the topo-

logical regions for the purpose of low-level path plan-

ning. In particular, by concatenation of the consec-

utive mean-shift vectors for each cell we obtain the

set of convergence paths that a robot can readily fol-

low in order to navigate to the center of its topologi-

cal region. By repeated concatenation of the conver-

CONSTRAINT-FREE TOPOLOGICAL MAPPING AND PATH PLANNING BY MAXIMA DETECTION OF THE

KERNEL SPATIAL CLEARANCE DENSITY

(a) (b)

Figure 4: (a) Example of clustering together spatially disconnected areas. Topological regions are identiﬁed by the index of

their cluster and their corresponding color. The cells belonging to the areas highlighted in red are erroneously incorporated in

region #1 or #2 although they are spatially disjoint; (b) Example of consistent clustering by constraining mean-shifts to allow

only for linearly unobstructed vectors by using Bresenham’s line algorithm in order to ensure the intra-connectivity of cells.

gence paths of adjacent cells that belong to different

regions, longer paths can be constructed that endow

a robot to navigate to any position within the map by

following a path that passes through the centres of the

intermediate topological regions. These paths are op-

timal with respect to the safety of the robot in terms

of overall distance from surrounding obstacles as well

as visibility in terms of space clearance. In Figure 5

we show the convergence paths of cells to the points

of locally maximum space clearance.

Figure 5: Convergence paths of a subset of grid cells (shown

in red) to the corresponding points of locally maximum

space clearance (shown in black).

4 EXPERIMENTS

In this Section we present the experiments that

demonstrate the feasibility and effectiveness of the

proposed topological segmentation method.

We divide our experiments in two parts. First we anal-

yse the role of the parameters involved in the pro-

posed methodology and second we provide a quali-

tative evaluation of the proposed topological segmen-

tation in a variety of metric maps including structured,

semi-structured and totally unstructured maps.

The parameters of the proposed segmentation

method are set by taking into account the characteris-

tics of the robot and its sensors. The values that need

to be set concern the variance σ of the Gaussian func-

tion used for the convolution (eq. 2), the threshold

for the determination of the safely accessible areas

and the bandwidth parameter b for the estimation of

the kernel density (eq. 3, 4).

The σ parameter determines the importance of

surrounding obstacles according to their distance

from a particular free cell. By decreasing σ, i.e. the

uncertainty, we increase the conﬁdence to the imme-

diate close neighbourhood of the free cell. This in-

creases the ratio of cells that are designated as hav-

ing maximum space clearance to all the other cells

that lie closer to occupied space. In other words, rela-

tively small free areas are characterized by high space

clearance. By increasing σ bigger areas around a cell

are covered by considering space that is more distant

which decreases the aforementioned ratio. Hence,

only wide open areas are characterized by having

high space clearance. The effect of various σ values

(σ = 15, 30, 45) in the estimation of space clearance

is demonstrated in Figure 6.

The threshold t

is set in proportion to the size

of the robot, since the bigger the robot the further

it should reside from occupied space and vice versa.

By increasing t

, the free area with sufﬁciently high

space clearance is decreased hence also decreasing

the area to be clustered. This may isolate portions

of free space if the access is through relatively nar-

row passages and hence not traversable by the robot.

In Figure 7 we show the effect of different thresholds

= 0.8, 0.9, 0.99) in estimating the safely accessible

free space by a robot.

ICPRAM 2012 - International Conference on Pattern Recognition Applications and Methods

(a) (b)

Figure 6: (a) Example metric map (499 × 319) and (b)-(d)

the estimated space clearance for varying σ.

(a) (b)

Figure 7: (a) Space clearance estimation of metric map and

(b)-(d) determination of safely accessible areas for a robot

(in green) using different thresholds.

The bandwidth parameter b related to the estima-

tion of the kernel space clearance density determines

the surface area of the segmented regions. It deﬁnes

the radius of the kernel window, i.e. the neighbour-

hood around a cell in the estimation of the mean-shift

vector. By increasing the radius we allow for longer

mean-shift vectors and thus clustering together cells

that are more distant from one another which typi-

cally results in less number of clusters. Setting a par-

ticular value for the bandwidth depends on the desired

granularity of the segmentation process that can vary

in order to construct either coarse or ﬁne topologi-

cal maps. In Figure 8 we show the effect of different

bandwidth values (b = 3, 5, 7) in the ﬁnal clustering.

In Figure 9 we demonstrate the stability of the pro-

posed segmentation method in the presence of noise

or relatively small obstacles within a given map. It is

fairly evident that the effect of noise or very small ob-

stacles in the clustering result is minimal as relatively

small volumes do not naturally alter the topology of

space. However, when small volumes approach one

another or reside close to bigger volumes they have a

combined effect and space topology is altered (e.g. in

(a) (b)

Figure 8: (a) Safely accessible area of metric map (in green)

and (b)-(d) topological segmentation for different values of

the bandwidth b in the kernel space clearance estimation.

(a) (b) (c)

(d) (e) (f)

Figure 9: Topological clustering of free area in the presence

of noise or relatively small obstacles; (a) initial clustering,

(b)-(f) clustering under various arrangements of small ob-

stacles or noise.

Figure 9 (d),(f) wherein the upper right region is con-

ﬁned due to the presence of an obstacle close to the

wall).

A particular setting of the t

, σ and b parame-

ters depends on the size characteristics of the robot,

the desired granularity of the segmentation process as

well as the desired level of space clearance.

In Figure 10, we show the results of segmenta-

tion under a ﬁxed setting of parameters (σ = 15,t

0.9, b = 5) in a variety of metric maps ranging from

structured to totally unstructured.

In the case of structured environments, the areas

corresponding to medium sized rooms are identiﬁed

as individual regions while larger constrained areas

such as corridors or halls, are segmented to more than

CONSTRAINT-FREE TOPOLOGICAL MAPPING AND PATH PLANNING BY MAXIMA DETECTION OF THE

KERNEL SPATIAL CLEARANCE DENSITY

(a) (b) (c)

(d) (e) (f)

Figure 10: Topological segmentation of generic metric maps; (a) parking lot, (b) tunnel car accident, (c) warehouse, (d) ofﬁce

environment, (e) simulated indoor environment, (f) hallway.

one region. Hence, by adopting this approach elon-

gated areas are segmented. Alternatively, we could

decrease the granularity of segmentation by increas-

ing the b parameter as shown in Figure 8. Another

possibility would be to fuse distinct areas that belong

to an elongated area by comparing the convexity of

the fused regions against that of the resulting region.

In the case of unstructured maps where there is no

notion of rooms or corridors, the only meaningful seg-

mentation criterion corresponds to the identiﬁcation

of regions where space clearance is locally high as

well as regions that are spatially disjoint-isolated and

hence unreachable by the robot. As can be seen from

the respective examples in Figure 8, these areas are

well deﬁned by the proposed segmentation method

The connectivity among the regions, i.e. the

edges of the corresponding topological graph can

be straightforwardly obtained by checking the 6

-8

neighbourhood of each cell.

The average segmentation time for the maps

shown in Figure 10 using a Linux-based system

equipped with an Intel Core i7 CPU 860 was 0.41

secs. The overall computational cost of the proposed

algorithm is dominated by the mean-shift clustering

stage as the preceding stage involving the computa-

tion of the space clearance can be efﬁciently com-

puted by exploiting the separability property of the

gaussian kernel. Furthermore, if necessary, the efﬁ-

ciency of the mapping process could be improved by

clustering different cells in parallel.

5 CONCLUSIONS AND FUTURE

WORK

We have presented a new topological mapping

method that is robust to noisy sensory data and does

not presume a particular structure of the environment.

By an initial parameterization that is imposed by the

characteristics of the robot, it can be readily applied to

generic metric maps of minimal environmental con-

straints, as demonstrated in a variety of real-world ex-

amples. As a result of its formulation, the proposed

approach can in parallel serve as a low level path-

planning algorithm by using the consecutive mean-

shift vectors of the convergence paths. This feature

together with the real-time performance of the topo-

logical segmentation increases the overall efﬁciency

of the robot by allowing for allocation of its compu-

tational resources to tasks other than path-planning.

As part of our future work, we are considering

the application of the proposed method for 3D topo-

logical mapping as well as segmentation of generic

3D volumes, since the notion of 2D space clearance

can be readily extended to 3D. In addition, it appears

that the proposed algorithm could be improved by in-

troducing appropriate criteria for fusing topological

regions that are part of constrained elongated areas,

whenever this is required.

ICPRAM 2012 - International Conference on Pattern Recognition Applications and Methods

ACKNOWLEDGEMENTS

This paper describes research done under the EU-FP7

ICT 247870 NIFTI project.

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CONSTRAINT-FREE TOPOLOGICAL MAPPING AND PATH PLANNING BY MAXIMA DETECTION OF THE

KERNEL SPATIAL CLEARANCE DENSITY