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Identi cation of Martian Polygonal Patterns Using the Dynamics of Watershed Contours Pedro Pina1, Jos
e Saraiva1, Lourenco Bandeira1 and Teresa Barata2 1

Centro de Geo-Sistemas, Instituto Superior Tecnico Av. Rovisco Pais, 1049{001 Lisboa, Portugal [email protected]

Centro de Geofisica, Universidade de Coimbra Av. Dias da Silva, 3000{134 Coimbra, Portugal

2

Abstract. This paper presents a methodology to automatically identify

polygonal patterns on the surface of Mars. These structures, which are typical of periglacial regions, result from climate oscillations and present a wide variation in size, shape and topology and occur in di erent types of terrains with rather di erent constituents and spectral reectances. The proposed approach is mainly based on the analysis of the dynamics of watershed contours and is successfully applied to a set of di erent types of patterned terrains of Mars shown by MGS/MOC images.

1 Introduction Polygonal patterns occurring in periglacial regions of the surface of the Earth are the signature of climatic oscillations. Similar structures appear on the surface of Mars and their characterization could provide a better understanding of the past martian climate 11]. The occurrence on Mars of similar polygons to those found in periglacial regions of the Earth has been a matter of discussion for the last three decades. Large polygonal patterns were rst identied in the Viking images (middle 1970's), possibly originated by periglacial processes, but with a more probable origin on tectonic stresses. Later, in the late 1990's, the higher spatial resolution images of the Mars Orbiter Camera (MOC) of the Mars Global Surveyor probe (MGS) have shown small-scale polygons much more similar in size to terrestrial patterned ground 9, 10]. If these polygons are caused by periglacial processes, their study can yield information about recent modications of ice distributions and climate on Mars. The formation of patterned terrains on Earth occurs in periglacial or alpine regions that are subject to permanent or temporary freezing temperatures. The climatic eects that control their formation are the freeze-thaw (formation of sorted polygons) and the thermal contraction cycles (formation of crack networks due to the decrease in ground ice during summer). It must be pointed out that in both of these processes the presence of ice in the ground plays a fundamental role 11]. For illustrating purposes we present in gure 1 a small set of images of these martian patterns acquired by MOC/MGS on several locations of the southern hemisphere. The dierences on the size, shape and topology are quite evident, on a great diversity of terrains.

(a) M1900047

(b) R1001555

(c) R1001796

(d) S0701948

(e) R0801628

(f) M0304331

(g) E0900029

(h) E1202319

(i) E0300299

Fig. 1. Examples of some martian polygonal patterns extracted from MOC/MGS images (NASA/JPL/MSSS).

Moreover, the cracks or contours of the polygons appear sometimes darker and other times brighter than the terrain itself. In what concerns related works, there exist several studies presenting detailed characterizations of these polygons on the surface of Mars. Most of them present qualitative descriptions 15, 4, 11, 8] and the ones presenting quantitative approaches 5, 18] use manual processes to extract the relevant features. For martian terrains none of these approaches uses an automated method to extract and characterize those patterns. The only published study proposing an automated approach focus on Venus and in the detection of polygonal fractures on SAR images obtained by the Magellan probe 12]. Although the complexity of venusian radar images is high, the textural variations from location to location are not very marked and allowed a good recognition. Consequently, we developed an automated approach to identify polygonal patterns on the surface of Mars. It is constituted by two main phases and it is intended to be applied to a polygonal ground independently of the type of terrain where these patterns occur. The rst phase consists of a pre-processing stage, where noise is ltered out and contours are enhanced. This goal is achieved by applying opening and closing by reconstruction lters. In the second phase, the contours of the images are identied through the watershed transform. This is followed by their analysis according to their relevance relatively to the minima of the adjacent basins which permits to construct an image containing information about their dynamics. The adequate thresholding of this image leads to obtaining the most relevant contours that correspond in our images to the cracks separating the polygons.

2 Filtering The patterns themselves divide the terrain into a set of polygonal structures, i:e:, the soil or ice is divided by cracks constructing normally a connected network. In other situations, the cracks are branches with terminations, i:e:, branches not connected in one of its extremities. These cracks always exhibit a certain contrast with the background, sometimes darker than the terrain itself (normally when the terrain is constituted by ice), sometimes brighter than the terrain (when the terrain is mainly soil). In order to segment these images we applied the watershed transform directly to the images. However, a pre-processing stage is needed before segmentation in order to lter out some local noise and to enhance the borders of the polygons. Normally, some of the edges of the polygons in the initial images acquired by the MOC instrument are not well dened in some locations due to insucient spatial resolution or due to some unnished physical process. We admit that some of these "faulty" edges, whose distance is inferior to a certain distance
, can be recovered. In addition, the presence of some local intensity variations on the images should also be ltered out. A good approach to enhance the edges and to lter some noise is to apply a morphological lter by reconstruction which has the property of simplifying the

images while preserving contours 14]. The most common of these lters are the opening and closing by reconstruction. The opening by reconstruction removes pixels of the foreground from an image by any given criteria and reconstructs all connected components of the image that had not been totally removed. The opening  by reconstruction R of size
of an image f (R ) is dened as the reconstruction by dilation  of f from the erosion " of size
of f :

R (f ) = Rf
" (f )]: (1) The closing  by reconstruction R (R ) is the dual transform of the opening

by reconstruction and is dened as:

R (f ) = Rf"  (f )]:

(2) The application of these lters is illustrated with an image selected for the purpose, which is the one presented on gure 1c. Firstly, we apply a closing of size
= 2 (gure 2a), which has the eect of reinforcing the brighter structures (including the contours of the polygons), and to ll or group some local minima. Secondly, we apply a closing by reconstruction of the same dimension (gure 2b), where the edges are maintained and reinforced. Since not all the minima are ltered, we have applied, after the closing by reconstruction, an opening by reconstruction to lter out those structures (gure 2c). The evaluation of the results of this ltering phase will be done in the next section with the corresponding watershed results.

(a)

(b)

(c)

Fig. 2. Morphological ltering with dimension = 2: (a) Closing (b) Closing by reconstruction (c) Closing and opening by reconstruction.

3 Segmentation and dynamics The segmentation of the polygons is performed through the watershed transform this is followed by the analysis of the dynamics of its contours in order to retain only the relevant edges. Currently we force the polygon cracks in the images

to be lighter than the patterns by computing, when necessary, the negative image in order to apply correctly the watershed transform. In the future we will automatically verify this situation and proceed with the required adjustments.

3.1 Segmentation by watershed Using a ooding analogy from the minima of a topographic surface, the catchment basins CB of an image f associated with a minimum MIN are obtained through the application of the watershed transform WS 1]. These basins can be considered as the inuence zones of the minima MIN of the image, being the watershed lines given by the skeleton by inuence zones SKIZ of these minima 17]. Thus, the watershed WS of an initial input image f is then given by:

WS (f ) = SKIZf MIN (f )]:

(3) Complete overviews of the watershed transform can be found in 2, 3]. One of the most common and fastest approaches to compute the watershed transform is based on immersion simulations 19]. We then compute the watershed lines of the images resulting from the ltering phase (gure 2). The minima of the image ltered by closing are almost as abundant as in initial one, and lead naturally to an overssegmented image (gure 3a). In the image ltered with a closing by reconstruction, some of the minima are not reconstructed and the corresponding watershed is a less segmented image with a higher approximation to the real polygonal structures (gure 3b). Anyhow, most of these contours are locally very irregular, a fact which reects the irregularity of the minima. The application of an opening by reconstruction afterwards, permits to smooth these contours and to ll some of their small holes. Consequently, the watershed lines are locally more regular (gure 3c). Although the number of basins was considerably reduced with the ltering phase, we still have a large number of contours that do not correspond to cracks on the terrain (false positives). In general, the contours corresponding to the cracks are lighter than the other contours, but due to a large variation on some parts of the contours, this cannot be adopted as a xed rule. A pattern can indeed be found in the local contrast between each contour and the two minima of its adjacent basins. The exploitation of this dierence is the objective of the next section.

3.2 Dynamics of watershed contours The contours detected by the watershed transform should be analysed in order to decide which ones are relevant and those that are not. We follow the approach presented by Najman and Schmitt 13] that established a dynamics of the contours through a contrast criterion that measures the grey-level dierence between a peak and its surrounding minima.

(a)

(b)

(c)

Fig. 3. Segmentation with minima (rst row) and watershed lines (second row) for the images of gure 2: (a) Closing (b) Closing by reconstruction (c) Closing and opening by reconstruction. Let the set of catchment basins be designated by fBi g and an arc of the watershed by C . Among the set of x points that constitute each arc, we designate the one with the lowest level by s, being its value given by I (s) = minx2C I (x)]. The point s is called a saddle point. Let Bas(C ) be the set of points of the catchment basins that can be reached from s by following a path with all values lower than I (s) and mi its minimum, mi = minx2B \C I (x)]. The dynamics of the contour dyn(C ) can in this way be dened as in 13] through the expression: i

dyn(C ) = minI (S ) ; mi ]:

(4) The dyn(C ) has a range of values between 0 and +1. To measure the dynamics of each contour it is necessary to individualize each contour or edge. The removal of the vertices or multiple points of the watershed lines permits to obtain all the edges. Each contour has two neighbouring basins each one with a local minimum. The computation of the dynamics of each contour is easy to perform (gure 4b). We can visually compare this result with an image of the values of the watershed transform on the initial image (gure 4a), and conclude that this step is a major improvement enhancing the dierent importance of the contours. The thresholding of this dynamics image at adequate levels permits to obtain the image of the relevant contours, i:e:, the contours that correspond to the

real cracks (gure 5a). The corresponding ground-truth image is presented in gure 5b.

(a)

(b)

Fig. 4. Values of the watershed contours on the: (a) initial image (b) dynamics image.

(a)

(b)

Fig. 5. (a) Thresholding of contour dynamics image (b) Ground-truth contours.

4 Results We have applied the same sequence (ltering by closing and opening by reconstruction of size
= 2 followed by the analysis of watershed contour dynamics), to a subset of the images previously presented (gure 1). The evaluation of our approach was performed by measuring the length of all contours and comparing them with the ground-truth images, whose contours were manually identied. The results obtained are presented in table 1. The results obtained are highly satisfactory. The recognition rate is in all cases above 80% and, in one particular image (R801628), the success was almost complete. We are aware, that this image shows a particular type of terrain

were the cracks are almost always very well dened against a spectrally uniform background terrain. Another important conclusion to retain is the low level of detection of false positive contours. The observation of these "fake" contours leads to the conclusion that they normally have small lengths and result from strong local minima that were not suppressed during the ltering phase.

Table 1. Recognition rates of polygon contours. Image Ground-truth Automatic Automatic False Positives (pixels) (pixels) (%) (pixels) M1900047 12342 9973 80.80 18 R1001796 7918 6695 84.55 440 R0801628 11323 9294 82.08 201 M0304331 11201 9326 83.26 272 E0900029 12593 12572 99.83 21

5 Conclusions The main objective of the development of this methodology was achieved since it can be considered capable of general application to the identication of polygonal patterns in a wide variety of Martian terrains with very few algorithmic parameters to tune. Although the results obtained demonstrate the robustness of the proposed approach, we believe that there is still room for improvement. In order to increase the recognition rates, we think that further enhancement of the crack contours during the ltering phase is still possible. The introduction of some directional operators to reinforce these structures is a possible solution. In addition, the automated selection of the thresholding levels on the contour dynamics image is a feature that we intend to incorporate in the future. Finally, we intend to apply our methodology to images of larger dimension and to link this identication phase to a characterization phase where geometric and topological features will be computed.

6 Acknowledgements This paper results from the research developed in the frame of the research project PDCTE/CTA/49724/03.

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