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PSS (preprint)

Polygonal terrains on Mars:

A contribution to their geometric and topological characterization

Pedro Pina * , José Saraiva, Lourenço Bandeira and Joana Antunes

Instituto Superior Técnico

**CERENA** – Centro de Recursos Naturais e Ambiente

Av. Rovisco Pais, 1049-001 Lisboa, PORTUGAL

ppina@ist.utl.pt, jose.saraiva@ist.utl.pt, lpcbandeira@ist.utl.pt, jbantunes@gmail.com

Tel: +351 218417425, Fax: +351 218417389

Abstract: Detailed examination of large extensions of polygonal terrains on the surface of

Mars and extraction of some characteristic geometric and topological parameters is made

possible by the application of image analysis methods to scenes of the Martian surface

acquired from orbit. This is illustrated by the analysis of a set of diverse Martian networks,

clearly visible in MOC/MGS images with high spatial resolution. It is shown that these

networks present, in average, a hexagonal habit, and that they verify two classic laws relative

to 2D random networks (those of Lewis and Aboav-Weaire).

Keywords: Mars; Polygonal terrains; Geometry; Topology

* Corresponding author

PSS (preprint)

1. Introduction

The occurrence of small-scale polygonal terrains on the Martian surface has been known since

Viking days. It raises the possibility that freeze-thaw processes take place in some areas of the

planet, an idea based on the similarities between many of the Martian polygons and terrestrial

periglacial analogues. Still, the analysis of small-scale polygons on Mars had to wait until the

last decade, when high (spatial) resolution images of the surface became widely available;

relevant works since published were concerned with different aspects of these features, such

as areas of occurrence [Seibert and Kargel, 2001; Kuzmin and Zabalueva, 2003], correlation

with ground ice [Mangold et al., 2004], dimensions and classification based on morphological

characteristics [Mangold, 2005], evolution along the seasonal cycle [van Gasselt et al., 2005],

and modelling according to the effects of thermal contraction [Mellon, 1997]. The elucidation

of the true nature and origin of the features was dealt with in most of these works, but no

definite answers were put forward. One major drawback in most of these studies is that they

seem to encompass relatively small datasets; this puts into question the statistical significance

of any measurements made on the polygons. The use of methods from the field of image

analysis, however, can easily permit the automated identification of the polygon contours,

thus delineating a network whatever its dimensions; furthermore, it can lead to its

characterization, thanks to the swift and comprehensive measurement of a broad set of

properties of the polygons.

2. Image survey

In order to obtain examples of the widely diverse settings in which polygonal patterns are

present on the surface of Mars, we conducted a survey of images acquired between 1998 and

2006 by the MOC/MGS camera in its narrow-angle mode, subject to two conditions: the

latitude of the centre of the image (higher than 50º, North and South) and its spatial resolution

PSS (preprint)

(better than 6 meters per pixel). This was decided according to previous works about the

distribution of these features and its relation with ice content in the soil. Thus, a total of 15855

images were visually inspected to assess the presence of polygonal terrains, a fact confirmed

in 1184 cases. However, only 188 of these showed adequate extensions of clearly discernible

polygonal networks; 35 of these images (15 from the northern and 20 from the southern

hemisphere), covering every type of visually distinct network, were then selected for this

work.

3. Polygon identification

The 35 images selected for this study were individually processed by an automated procedure

[Pina et al., 2006] that segmented the image and produced a new binary image (with

polygonal cells in black and pixel-thick contours in white). This was confronted with the

correspondent ground-truth (as detected by visual inspection of the original images), and

subject to the adequate corrections. Thus, we ended with digital models of the real networks,

from which the collection of quantified parameters could proceed with ease.

4. Network analysis

For measurement purposes, the networks contain only the polygons that are fully included in

the images, and whose neighbors respect that condition too. Squared samples (with a side of

about 500 m) of the typical visual aspect of each of the selected networks are presented in

Figure 1. On one image (E19-00409) two distinct networks were found, and subject to

individual analysis. It must be stressed that, at this point, no genetic hypothesis is taken into

account: if a polygonal pattern was present in an image, it was processed and analyzed.

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The extension of the studied networks is quite variable, covering areas (AN) from a few to

some thousand km 2 ; the number of polygons (N) that constitute them also presents a wide

range, from tens to thousands of cells. A density d (in polygons per km 2 ) was computed,

allowing for a first comparison between the diverse networks. These values are presented in

Table 1, along with identification of the image, location, and other parameters mentioned

below.

Another parameter that was computed for each network was AL, the percentage of the global

area of a network that is occupied by the contour lines. This was plotted against the mean

major axis (average of the major axis of the polygons in each network); the result can be seen

in Figure 2. There is a marked decrease of AL with increase of Lmaj, but this soon levels out,

and AL remains approximately constant irrespective of increasing mean major axis.

5. Polygon parameters and characterization of networks

The measurements individually performed on each polygon of a network included some

geometric parameters such as area, length of major and minor axes, shape factor, compactness

index and orientation. The average values (for each network) of the most relevant of these

parameters can be seen on Table 1.

It can be appreciated that there seems to be no significant gap in the distribution of sizes (as

given by the mean major axis) of the polygons, except between 250 and 350 m; this

somewhat contradicts other reports that separate polygons into well-defined size classes, and

may be explained by a non-exhaustive search for all types of networks, or by the inclusion, in

our case, of networks that may not be of periglacial origin.

Topological parameters have been frequently used on many scientific areas to characterize

and model the patterns that arise from the partitioning of space, from ceramics and

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metallurgical alloys [Aboav, 1970, 1980; Weaire, 1974], to biological tissues [Lewis, 1928,

1931; Pina and Fortes, 1996], foams and geologic materials [Weaire and Rivier, 1984] and

even solar granulations [Noever, 1994]. However, when it comes to the topological properties

of polygonal terrains there is only one study in which the number of sides of polygons in a

small number of Martian and terrestrial networks was counted [Yoshikawa, 2003].

Topological characterization relies, in fact, on the quantification of neighborhood relations of

the polygons. A link can be established between the number of neighbors of a polygon and its

dimensions (Lewis law), and the number of neighbors of its first-order neighbors (Aboav-

Weaire law). The experimental verifications of these laws yield parameters that may be used

for the characterization of the networks.

To collect this topological data we employed an improved algorithm based on mathematical

morphology, developed recently [Bandeira et al., 2008]. The average number of neighbors

per polygon () for the 35 networks was close to 6 (one exception is 4.89, a value obtained

for a network with only 9 polygons; in fact, this number becomes closer to 6 with increasing

number of polygons in a network). This was expected, since that is the value determined for

an indefinite trivalent network (with three edges at each vertex) [Weaire, 1974], and our

networks are mostly of that nature (though there is a number of tetravalent vertices in

particular regions of some networks). Furthermore, the minimum number of neighbors (imin)

detected was 3 for a vast majority of the networks (only one 2-sided polygon and three 4-

sided polygons were detected as minimum values, all in networks with less than 10 cells),

whereas the maximum number of neighbors (imax) ranges from 7 to 13. In accordance with

these low dispersions, the corresponding variances (μ2) are distributed along a narrow range,

between 1.20 and 3.46.

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The Lewis law, derived from the study of biological tissues [Lewis, 1928, 1931], combines

dimensional and topological parameters and describes a linear relation between the average

area (i) of polygons with i neighbors and i:

< A> =< A> [1 + λ(

i−

6)]

(1)

i

where is the average area of all polygons in a network and λ is a constant. This relation

reflects topological constraints on the polygonal structure imposed by the need to completely

fill the space; the slope of the line expressing this law is related to the topological aging or

coarsening in a polygonal network [Rivier, 1985].

Plotting the computed average area of polygons with i neighbors as a function of i (example

in Figure 3), we verified there was an approximately linear relation for all networks (the worst

least-square fitting yielded R 2 =0.7015, but 28 out of 35 networks have a value above 0.85).

Hence, the Lewis law is verified, and λ can be computed from equation (1) for the 35

networks (the values obtained are presented in Table 1). Its range is limited (from 0.26 to

0.51), and this parameter shows small promise for the distinction between networks. Note

however that the highest values of λ correspond to networks with worse linear fitting of the

Lewis law (R 2 lower than 0.85).

The Aboav-Weaire law [Aboav, 1970, 1980; Weaire, 1974], describes the correlation between

i, the number of neighbors of a polygon, and mi, the average number of neighbors of its

adjacent polygons:

m = 6 − a+ (6 a+ μ )/ i

(2)

i

2

where a is a constant and μ2 is the variance of i. This is a structural equation of state for

polygonal networks, describing a statistical equilibrium and assuming that there is no

correlation in polygon neighborhood beyond the nearest neighbors [Weaire and Rivier, 1984].

Again, if the relation between i.mi and i is approximately linear we can compute a, and use it

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as a parameter to globally characterize a network. After plotting the corresponding values for

the 35 networks, we confirmed a remarkable linear least-square fitting of our data to the

Aboav-Weaire law (the worst R 2 value is around 0.95, but 28 out of 35 are better than 0.99).

An example is presented in Figure 4.

6. Conclusions and future work

The need for an automated tool capable of collecting parameters that can lead to a quantified

and objective characterization of polygonal networks on Mars is clearly felt. This stems from

the presence of this type of features in many different areas of the surface of Mars, including

low latitudes [Page, 2007], and that occurrences at ever smaller scales are currently being

uncovered (see the HiRISE images of the Phoenix probe landing area) [Levy et al., 2008].

The use of image analysis in the acquisition of geometric and topological parameters from

Martian polygonal networks turns this task into a simple, fast and reproducible procedure,

allowing for the complete analysis of extensive areas, such as some of the examples shown in

this work; the results, while mostly confirming earlier studies, present an improved statistical

significance, since they are based on the analysis of large numbers of polygons. Some

topological results are reported for the first time for Martian polygonal terrains, and the

measured parameters confirm the general applicability of laws applying to 2D random

networks: the average global hexagonal pattern is confirmed for every network (the average

number of neighbors per polygon is close to 6), and both Lewis and Aboav-Weaire laws are

applicable.

The next phase in this work will include multivariate analysis of the whole set of parameters

extracted, and its combination with ancillary data related to topography, ground ice content,

and regional geology. The aims are a full characterization of the networks, an exploration of

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possible classification schemes, and taking steps in understanding origin and dynamics of the

diverse types of polygonal networks on Mars. To achieve this, we plan to analyze an enlarged

set of networks, to extend our research into smaller scales (using HiRISE images) and to

probe terrestrial analogues of this type of features.

Acknowledgements: This work was developed within project TERPOLI (PTDC/CTE-

SPA/65092/2006), funded by FCT, Portugal, which also financially supported JS

(SFRH/BD/37735/2007) and LB (SFRH/BD/40395/2007).

References

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List of figure captions

Figure 1. Samples from the polygonal networks studied, extracted from MOC/MGS images

(in some of the images the contrast was enhanced for visualization purposes) [image credits:

NASA/JPL/MSSS].

Figure 2. Plot of AL against Lmaj.

Figure 3. Verification of Lewis law (image M02-04505).

Figure 4. Verification of Aboav-Weaire law (image M02-04505).

List of tables

Table 1 – Geometrical and topological features measured on polygonal networks.