# Macro-scale roughness reveals the complex history of asteroids Didymos and
Dimorphos.

Jean-Baptiste Vincent  DLR Institute of Planetary Research, Berlin, Germany
Erik Asphaug  Lunar and Planetary Laboratory, University of Arizona, Tucson,
AZ 85721, USA  Olivier Barnouin  Johns Hopkins University Applied Physics
Laboratory, Laurel, MD, USA  Joel Beccarelli  INAF-OAPD Astronomical
Observatory of Padova, Italy  Paula G. Benavidez  IUFACyT - DFISTS,
Universidad de Alicante, Spain  Adriano Campo-Bagatin  IUFACyT - DFISTS,
Universidad de Alicante, Spain  Nancy L. Chabot  Johns Hopkins University
Applied Physics Laboratory, Laurel, MD, USA  Carolyn M. Ernst  Johns Hopkins
University Applied Physics Laboratory, Laurel, MD, USA  Pedro H. Hasselmann
INAF-OAR, Rome, Italy  Masatoshi Hirabayashi  Daniel Guggenheim School of
Aerospace Engineering, Georgia Institute of Technology, USA  Simone Ieva
INAF-OAR, Rome, Italy  Özgür Karatekin  Royal Observatory of Belgium,
Brussels, Belgium  Tomáš Kašpárek  Brno University of Technology, Czech
Republic  Tomáš Kohout  Department of Geosciences and Geography, University of
Helsinki, Finland  Institute of Geology of the Czech Academy of Sciences,
Czech Republic  Zhong-Yi Lin  Institute of Astronomy, National Central
University, No. 300, Zhongda Rd., Zhongli Dist., Taoyuan City 32001, Taiwan
Alice Lucchetti  INAF-OAPD Astronomical Observatory of Padova, Italy  Patrick
Michel  Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS,
Laboratoire Lagrange, Nice  The University of Tokyo, Department of Systems
Innovation, School of Engineering, Tokyo, Japan  Naomi Murdoch  Institut
Supérieur de l’Aéronautique et de l’Espace (ISAE-SUPAERO), Université de
Toulouse, Toulouse, France  Maurizio Pajola  INAF-OAPD Astronomical
Observatory of Padova, Italy  Laura M. Parro  IUFACyT - DFISTS, Universidad de
Alicante, Spain  Sabina D. Raducan  University of Bern, Switzerland  Jessica
Sunshine  Department of Astronomy and Department of Geology, College Park, MD,
USA  Gonzalo Tancredi  Facultad de Ciencias de la Universidad de la República,
Uruguay  Josep M. Trigo-Rodriguez  Institute of Space Sciences (CSIC-IEEC),
Campus UAB, c/Can Magrans s/n, Cerdanyola del vallès, Barcelona, Catalonia,
Spain  Angelo Zinzi  ASI-Space Science Data Center, Roma, Italy

(Received May 29, 2024)

###### Abstract

Morphological mapping is a fundamental step in studying the processes that
shaped an asteroid surface. Yet, it is challenging and often requires multiple
independent assessments by trained experts. Here, we present fast methods to
detect and characterize meaningful terrains from the topographic roughness:
entropy of information, and local mean surface orientation. We apply our
techniques to Didymos and Dimorphos, the target asteroids of NASA’s DART
mission: first attempt to deflect an asteroid. Our methods reliably identify
morphological units at multiple scales. The comparative study reveals various
terrain types, signatures of processes that transformed Didymos and Dimorphos.
Didymos shows the most heterogeneity and morphology that indicate recent
resurfacing events. Dimorphos is comparatively rougher than Didymos, which may
result from the formation process of the binary pair and past interaction
between the two bodies. Our methods can be readily applied to other bodies and
data sets.

asteroids, roughness, image processing, DART, Didymos, Dimorphos

††journal: PSJ

##  1 Introduction

NASA’s Double Asteroid Redirection Test (DART) is the first planetary defense
mission to attempt the deflection of an asteroid orbit. Its target was
Dimorphos, a $151\pm 5m~{}$ diameter S-type asteroid, the secondary component
of the 65803 Didymos system (primary’s diameter: $761\pm 26~{}m$) Daly et al.
(2023). The impact successfully occurred on 26 September 2022, at 23:14:24 UTC
Daly et al. (2023) and changed the orbital period of Dimorphos around Didymos
by $-33\pm 1~{}min$ Thomas et al. (2023). Shortly before the impact, DART
acquired and transmitted high-resolution images of both asteroids. The data
revealed complex surfaces with several types of terrains and morphological
features like craters and boulders similar to those previously observed on
other asteroids Daly et al. (2023); Barnouin et al. (2023).

The first step in understanding the evolutionary processes that shaped the
surfaces of Didymos and Dimorphos consists of mapping the terrains and
identifying the different units comprising the overall morphology. This
approach is particularly useful for analyzing this binary system as these
asteroids will be revisited by ESA’s Hera mission in 2027 Michel et al.
(2022). The morphological mapping and all information retrieved by DART before
and after the impact therefore establish a reference for Hera to identify
changes that may have been induced by DART in 2022, or on-going, on both
asteroids.

Establishing a morphological map is, however, challenging as the data are
limited by the resolution of imaging instruments and the viewing and
illumination conditions at the time of observation. The number of images is
also limited due to the high-speed nature of the encounter, and only a handful
of observations are suited for morphological analysis. Identifying
morphological units typically requires multiple experts to independently draw
boundaries on surface images until a consensus is reached. While this approach
has been successful for previous asteroids (e.g., 25143 Itokawa Fujiwara et
al. (2006), 21 Lutetia Thomas et al. (2012), 162173 Ryugu Sugita et al.
(2019), 101955 Bennu Barnouin et al. (2019), it is very time consuming and may
suffer from human biases that are not easy to quantify. To alleviate this
issue, automated methods can be considered to serve as initial guidance and
obtain the first assessment of the surface morphology. Examples of automated
determination of morphological units on comet 67P/Churyumov-Gerasimenko can be
found in Vincent et al. (2017) (cliffs) and Thomas et al. (2018) (smooth
units). Both studies are based on the analysis of the tri-dimensional mesh of
the comet’s shape model.

This current work focuses on the topographic roughness (a.k.a. multi-scale
macro-roughness, for features larger than one pixel). This quantity describes
the variability of surface topography as a function of spatial scale. It can
be expressed in different ways, like the mean slope angle of the surface with
respect to its local neighbourhood, or the root-mean-squared height over a
reference surface. A review of techniques commonly used with remote sensing
data can be found in Shepard et al. (2001).

It is well known from previous missions that the surface of a single small
body can present a large variability in roughness, typically seen as varying
surface texture. Prominent examples are the localised fine-grained deposits on
433 Eros Thomas et al. (2002) or the Muses Sea smooth terrain on 25143 Itokawa
Fujiwara et al. (2006); both areas contrast with their rougher surroundings.
More recent studies, using high-resolution laser altimetry, are available for
asteroids Eros Susorney et al. (2019) and Bennu Daly et al. (2020) and were
used to identify regions of different evolutionary states (coupled with crater
counting), or assess the thickness of the mobile regolith layer.

As the roughness is inherently a measure of small-scale amplitude variations
in the terrain, its estimate depends on the resolution of available
observations and the scale at which it is investigated. At centimeter to meter
scale, the roughness might reveal grain size-sorting processes that mobilise
the regolith across the surface (e.g. Jawin et al. (2022)). At larger
measuring scales, it can help to identify craters, boulders, lineaments, or
find the boundaries between ponded deposits of fine material and areas of
coarser regolith. In turn, those identifications can be associated with
gravitational, albedo, compositional maps and so on, to build a full
understanding of the surface.

We note that at pixel-scale and below, the surface roughness is usually
retrieved with dedicated photometric modelling of the surface brightness per
pixel, through advanced models such as Hapke (1984). See Marshall et al.
(2018) for roughness studies of comet 67P/Churyumov-Gerasimenko at microwave
and infrared wavelengths, or Hasselmann et al. (2021) for the optical
roughness of asteroid Bennu.

Beyond small bodies, roughness and its connection to morphology have often
been studied from topographic measurements of larger bodies, e.g. the Moon
Kreslavsky et al. (2013) or Mercury Susorney et al. (2017), and show a clear
correlation with regolith formation and modification as well as bedrock
geology. Roughness is also an important factor to be considered because it
affects the balance of radiation and the thermal inertia. The amount and size
distribution of the regolith affects the roughness. For example, asteroids
smaller than 100 km are often covered by coarse regolith grains that decrease
the roughness and thermal inertia Hanuš et al. (2018). The situation for
binary asteroids could be different as consequence of the gravitational
interactions among the bodies.
In this work, we present innovative approaches that allow for a fast
determination of the surface roughness on airless bodies. Our study focuses on
the topographic roughness, in order to identify features that span multiple
pixels in the data. We show how techniques derived from information theory can
be used to quantify the topographic roughness of asteroid surfaces at multiple
spatial scales, and automate the mapping of significant units. We apply our
methods to a subset of images of asteroids Didymos and Dimorphos, acquired by
DART. We provide a first description of the spatial distribution of roughness
on each object, and derive information about the surface evolution of the two
bodies.

Our techniques are not constrained to the objects described in this paper and
can be readily applied to any other application that need to assess the
roughness of a terrain. We are already implementing these techniques in the
data analysis pipelines for ESA’s Hera mission, that will revisit Didymos and
Dimorphos. Our methods may also be considered for autonomous navigation around
asteroids or obstacle avoidance during lunar landings.

##  2 Methods

We provide here more details on how the roughness measurement techniques work
and how they are implemented. Two methods are presented: ”Roughness from
Texture” (RFT) and ”Roughness from Shape” (RFS).

###  2.1 RFT: Roughness from texture

The RFT method we developed is inspired by techniques used in Information
Theory and defined in Shannon (1948). We measure a parameter known as the
entropy of information which can be described as the number of bits needed to
encode the variability of the signal. Mathematically, it is the log-base-2 of
the number of possible outcomes for a message:

$H(X)=-\sum_{x\in\chi}p(x)\log_{2}p(x)$ (1)

For a variable $X$ in the domain $\chi$, we sum over the variable’s possible
outcomes $x$, and their probability $p(x)$ of occurrence within the signal.

The choice of a logarithm base depends on the application. Here we chose to
use $\log_{2}$, which gives an entropy of information expressed in bits.
$\log_{2}$ also has the advantage of being often implemented directly on the
hardware (i.e. instruction FYL2X in Intel x86-assembly), and therefore
extremely fast to calculate. This is particularly relevant for the use of this
technique in embedded software, e.g. for autonomous navigation on board a
spacecraft where computing resources are limited.

When applied to imaging data, we measure this parameter in a sliding window of
$n\times n$ unordered pixels, across the image. Regions that present a high
variability in pixel value will be characterised by a high entropy, while
regions with uniform pixel values will have a low entropy.

Photometric models typically consider incident, emission, and phase angles as
well as surface albedo and several correcting factors. Typically for
asteroids, brightness variation across the surface is mostly a function of the
local topography (i.e. incidence and emission). We must however be wary of
observations at low phase angles (typically $i,e<10^{\circ}{}$) where
additional effects such as shadow-hiding and coherent backscattering play a
major role and non-topographic parameters like composition and grain size must
be accounted for. In a similar fashion, observations at high incidence angle
can also be problematic as the long shadows will obscure part of the terrain
and also create artificial high-entropy boundaries, where the border between
shadowed and illuminated terrain is seen as a large jump in pixel values. This
is, however, a well known problem with the interpretation of imaging data for
asteroids, and we preempt it by selecting data and observing conditions that
avoid extreme values of $i$ and $e$, as well as masking potentially
problematic areas in the images (shadows). With these precautions, our
technique relies on the same principles behind digital terrain model
reconstruction methods like photoclinometry (a.k.a. ”shape from shading”),
routinely used by space missions to reconstruct the local terrain.

Therefore, we use the entropy of information (pixel value variability) as a
proxy for a measure of the variability of incidence and emission, which is
controlled by the local slope of the terrain. This means that the measure of
entropy of information informs us directly about the roughness of the surface,
at a scale defined by the size of the neighbourhood in which we calculated the
entropy.

Figure 1 shows examples of entropy measurements on several asteroids and
comets, and the clear correlation between the entropy value and the actual
roughness of the surface.

Figure 1: Entropy of Information measured on images of other small bodies. The
low and high values of entropy match the distribution of rough and smooth
terrains on all objects. All images use the same color scale. Qualitatively,
67P and Itokawa display different types of terrains, while Bennu’s surface is
more uniform. All data is publicly available, see Section A

A strong advantage of this method is that it does not require prior knowledge
of the photometric response of the surface. More specifically, the results are
independent of the viewing angles (incidence, emission, phase) as long as one
avoids extreme situations: e.g. at phases below 20°, the brightness variations
must account for albedo and material properties. In most images, however, the
phase can be ignored as the entropy measure depends on the number of state
transitions (how often do pixel values change) rather than the states
themselves (absolute difference between pixel values). This means that low-
contrast and high-contrast images of the same area have the same entropy, as
long as the topography can be detected (again, ignoring opposition and
shadows). Figures 2 and 3 shows how incidence and emission angles can
essentially be ignored when measuring the roughness with our method on Didymos
and Dimorphos, as long as we avoid the more extreme angles.

Figure 2: Comparison between the Roughness-From-Texture and the illumination
angles (incidence and emission) at the time of observation. There is no
apparent correlation between these quantities, which suggests that the
measurement is robust against a wide range of illuminations. Figure 3: Entropy
of information vs incidence and emission angles for both asteroids, mean and
standard deviations are calculated in bins of $10^{\circ}{}$. The EOI does not
correlate with incidence and emission, except for extreme angles
($i,e<20^{\circ}{}$ and $i,e>80^{\circ}{}$) where photometric effects and long
shadows start to become significant.

Besides measuring roughness, our technique can also be used to identify
surface changes. By measuring the entropy of the same region at two different
epochs, one can determine precisely which areas have seen their texture
modified. The technique has been applied successfully to track changes on
comet 67P (boulder fields covered/exposed by dust deposition/removal, dunes
displacement, formation of circular depressions, etc.), see Vincent et al.
(2021).

We note that the technique is easy to implement and readily available in most
image-processing software/libraries. This paper, for instance, used the
function available in the open-source Python library scikit-image
(https://scikit-image.org). The code implementing the entropy function for
this library can be found at https://github.com/scikit-image/scikit-
image/blob/main/skimage/filters/rank/generic.py.

###  2.2 RFS: Roughness from shape

The RFS method operates on a totally different set of data. Rather than
considering pixel values in images, we measure the variability of the
topography on a reconstructed 3-dimensional model of the object of interest. A
shape model is a collection of points in 3D space, linked together in a closed
mesh of polygons (typically triangles). For each facet, one can measure its
orientation in the body-fixed frame, and compare it to other facets in a given
neighbourhood. There are several ways to achieve this, which can be found in
the literature and have been applied to Solar System bodies, or everyday
objects (e.g. Gabriel Gowman et al. (2023)). The most commonly used technique
consists in measuring the RMS distance between facet centers and a reference
baseline surface Susorney et al. (2019). Other authors M. et al. (2017) have
also considered comparing the orientation of a facet in a high-resolution
model with the same location on a lower-resolution shape within the same
reference frame.

In this paper, we use a slightly different approach which aims to minimize the
number of free parameters. Like M. et al. (2017), we are interested in the
local variation of facet orientation. However, we use the highest-resolution
shape model of our target, and do not introduce low-resolution models, to
avoid uncertainties that arise from the decimation process. Instead, for each
facet, we measure the angle between its normal vector and those of its
neighbouring facets. The mean value of this angle is our roughness parameter.
To control the scale of the measurement, one can adjust the number of
neighbours to be considered. Those can be limited to the immediate three
neighbouring triangles, or include all facets within a certain distance of the
point of interest. ”Distance” can be measured in different ways: shortest path
on the surface, latitude/longitude difference, number of neighbours from the
starting point, etc.

This technique has the advantage of not requiring any data other than the
original shape model. It is purely geometrical, making no assumptions about
the internal state of the target (gravity is irrelevant). Because our
measurement only depends on the local facet orientation within a given
neighbourhood, the results can be directly compared to what is obtained with
the RFT method, which also depends on the same parameters.

Of course, we must mention the caveat that our technique is applied to a
secondary product (shape model). Hence, the interpretation of our results must
account for any uncertainty pertaining to the method used to create that
specific model.

The computation can be quite demanding as the file formats used for 3D models
of asteroids (e.g. OBJ) do not store information about facet neighbours.
Determining if two facets are neighbours requires checking whether they share
an edge, i.e. they have two common vertices. A naive way to do this is to load
all facets and, for each of them, loop once through the whole list and
identify neighbours. One can speed this up by stopping the secondary loop as
soon as three neighbours are found, and double-allocating neighbours (if B is
a neighbour of A, then A is a neighbour of B). Still, this approach is very
time-consuming for models that have millions of facets.

For this work, we have implemented a faster approach to find all neighbours as
we load the shape, requiring only one pass through all facets. The algorithm
is as follows:

Find all neighbours

1:list of $N$ facets, defined by their vertices: $facet[f]=[i,j,k]$ where $f$
is a facet index and $i,j,k$ are indices of elements in list of vertices

2:$edge\Leftarrow\textrm{empty array}$

3:for facet index $f$ in range $[0,N]$ do

4: $v_{0}\Leftarrow facet[f][0]$

5: $v_{1}\Leftarrow facet[f][1]$

6: $v_{2}\Leftarrow facet[f][2]$

7: $edge[3\times f+0]=[min(v0,v1),max(v0,v1),f]$

8: $edge[3\times f+1]=[min(v1,v2),max(v1,v2),f]$

9: $edge[3\times f+2]=[min(v2,v0),max(v2,v0),f]$

10:end for

11:Sort $edge$ list by first column

12:For each value of first column, sort $edge$ list by second column

13:$e\Leftarrow 0$

14:while $e<length(edge)$ do

15: $f_{a}\Leftarrow edge[e][2]$

16: $f_{b}\Leftarrow edge[e+1][2]$

17: Append $f_{b}$ to $facet[f_{a}].neighbours$

18: Append $f_{a}$ to $facet[f_{b}].neighbours$

19: $i\Leftarrow e+2$

20:end while

The algorithm guarantees that all neighbours are found uniquely, and reading
the $edge$ list two rows at a time gives a list of pairs of neighbours.

After the neighbours of each facet have been determined, it is trivial to
calculate the roughness as a mean orientation angle using simple geometry. For
a facet defined by three vertices $[v_{0},v_{1},v_{2}]$, the normal vector is
given by the cross product between the vectors positioning any pair of
clockwise consecutive vertices in the body-fixed frame (e.g.
$\overrightarrow{v_{0}}\times\overrightarrow{v_{1}}$). The angle $\theta$
between two facets is derived from the dot product between two normal vectors:

$cos(\theta)=\frac{\overrightarrow{n_{1}}.\overrightarrow{n_{2}}}{||\overrightarrow{n_{1}}||\times||\overrightarrow{n_{2}}||}$

We implemented this approach in the data analysis tool shapeViewer Vincent
(2018), a scientific software freely available at https://www.comet-
toolbox.com. Our code (C language) can process a shape model with 1 million
facets in a couple of seconds on a standard laptop (the slowest machine tested
has 2GHz CPU, and 4GB RAM). While this is sufficient for our needs, we note
that the algorithm could be sped up by parallelizing the construction of the
$edge$ list, and selecting a different sorting method (we used $quicksort$).

Figure 4 shows examples of roughness from shape measurements on several
asteroids and comets; smooth regions are clearly identified and match those
found in Figure 1 with the RFT method.

Figure 4: Roughness from shape method applied to the reconstructed tri-
dimensional models of comet 67P/Churyumov-Gerasimenko and asteroid (25143)
Itokawa. Models taken from public archives, processing and rendering done with
shapeViewer Vincent (2018). We use the same reference images as in Figure 1.
The smooth/rough terrains on 67P and Itokawa are well identified, within the
limitations of the shape model resolution. Bennu does not display much
roughness variation.

##  3 Results

Because the DART spacecraft flew by Didymos and impacted Dimorphos at
$6.14~{}km.s^{-1}$ Daly et al. (2023) on an almost straight-on trajectory
(only 16.7 degrees from vertical), all images obtained by the Didymos
Reconnaissance and Asteroid Camera for OpNav (DRACO, Fletcher et al. (2022))
present essentially the same viewing geometry and illumination conditions,
with only a change in resolution. In this work, we focus on two specific
observations: the highest resolution image of the full sunlit Didymos system
(which also contains the highest resolution image of the full sunlit primary),
and the highest resolution image of the full sunlit secondary, Dimorphos. Both
observations are shown in Figure 5. We use the calibrated data publicly
available in NASA’s Planetary Data System archive Ernst et al. (2023) with no
additional processing.

Figure 5: DRACO images of Didymos and Dimorphos acquired respectively 2m 43s
(a) and 11s (b) before impact. The yellow arrow indicates the direction of the
North pole of both asteroids (Didymos’s positive spin axis).

We first characterise the surface roughness of both asteroids by measuring the
distribution of entropy of information (EOI) in these two images. In short,
the EOI describes the average amount of information contained in a signal
Shannon (1948). A rapidly changing signal (either spatially or in time) is
considered rich in information (high entropy), while a monotonous signal with
little variability is poor in information (low entropy). When applied to
imaging data, the signal considered is the distribution of pixel values across
the image. The entropy is measured at different scales by considering sub-
regions of different sizes in the images, like usually done with spatial
filtering Gonzalez R.. & Woods E. (2008).

This technique has been used for many decades to classify signals or retrieve
information in noisy backgrounds, and is readily available in most signal
processing or digital image processing software packages. Despite its numerous
benefits, we found that the technique has seldom been used in planetary
science. To the best of our knowledge, the method has previously only been
applied to characterize the roughness of topographic maps derived from laser
altimetry (e.g. Li et al. (2015)). Our work is the first application of the
method to study primary data products such as camera images.

A strong advantage of the method is that it is mostly unaffected by the
photometric properties of the surface, as long as we avoid areas with extreme
observing and illumination angles. For instance: the opposition effect at low
phase angle, or long shadows at high incidence angles. Under typical observing
conditions encountered by space missions (i.e. $20^{\circ}<$ phase angle
$<80^{\circ}$) and for uniform albedo surfaces (most asteroids), the entropy
is directly a function of the surface topography at the scale of interest.

The method is presented in detail in 2.1, with additional examples for other
asteroids and comets. As it is based on the analysis of pixel values in the
images, we hereafter refer to the method as roughness-from-texture (RFT), in
opposition to alternative methods that are typically based on a shape model.

Figure 6: Entropy of information (EOI) measured on Didymos [a] and Dimorphos
[b] at multiple scales. Image resolution is 5m/px, the measurement scale is
given in pixels and also represented visually by a white square in the top-
left corner of each panel.

###  3.1 General observations and comparative analysis at 5 m/px

Figure 6 displays a comparative view of the entropy of information on both
asteroids at a resolution of 5 m/px. In these images, we measured the EOI at
different spatial scales, represented by sliding boxes of $n\times n$ pixels
across the images. Here we used scales of 3, 5, and 7 pixels, which correspond
to distances of 15, 25, and 35 meters on the surface. All observing conditions
(resolution, illumination, viewing angle, etc.) being equal, this allows a
direct comparison of the EOI measurement between the two bodies. Of course, at
this resolution, Dimorphos is only 20 pixels wide, while Didymos is five times
larger. Still, as long as the entropy measurement scale is smaller than the
asteroid, we can get a reliable description of the roughness across the
surface.

There are a couple of artefacts to be aware of when analysing results such as
shown in Figures 6, 7, and 9. First, we have masked all shadowed areas and
space beyond the asteroid. This is to avoid that low-level noise interpreted
as signal. It is possible for very sensitive cameras to actually ”see in the
dark” when imaging shadowed terrains faintly illuminated by scattered light
from nearby lit surfaces. That is however not the case for this dataset.
Second, the border between a shadowed and a lit region shows up as a high-
entropy edge in our analysis, because pixel values vary by a large amount over
a short distance. This leads to an increase of entropy close to the darkest
shadows, and around the asteroid limb, and must be taken into account when
interpreting the results.

As explained in the introduction, one of the goals of our method is to assist
in the definition and characterisation of morphological units. This is
achieved by defining thresholds and sorting the surface entropy values in a
limited number of bins that correspond to specific morphology. As the entropy
values are expressed in bits and do not have a direct physical meaning, we
first consider the typical entropy values of areas that would be unambiguously
characterised as smooth/rough by trained morphology experts, in the data being
investigated. We then segment the images according to these thresholds. For
bodies whose surfaces shows starker differences in roughness level, this
process could be fully automated by identifying peaks in the histogram of
entropy values.

Our results are shown in Figures 7 and 9, and discussed hereafter:

  * •

Both asteroids display significant variations in entropy of information,
unrelated to the observing conditions (no correlation between roughness values
and incident, emission, or phase angles, see 2.2 for additional information).
This is indicative of varied roughness levels at the scales considered.

  * •

Both asteroids show a non-random distribution of smooth and rough terrains. We
detect patterns that persist at multiple scales and may indicate boundaries
between morphological terrains, or sub-surface structures (see 3.2, 3.3).

  * •

Didymos displays a latitudinal trend in roughness values, with the equatorial
regions appearing smoother than the mid and high latitudes. Such a latitudinal
trend is not detected on Dimorphos.

  * •

The equatorial regions of Didymos are the smoothest terrains found in the
system.

  * •

Across the scales considered, Dimorphos’ surface displays, on average, the
same level of roughness as the rougher terrains on Didymos.

We now look at each object in more detail, and discuss the interpretation of
entropy of information at the highest available resolution.

###  3.2 Didymos

Didymos shows unambiguous patterns in its distribution of topographic
roughness, with its smoothest regions being located close to the equator (Fig.
7). This is particularly visible in the thresholded entropy image that shows a
factor two difference between the entropy level at higher latitudes and the
equatorial measurements. We find that the smooth areas do sometimes expand to
higher latitudes, particularly in the southern hemisphere, into terrains that
resemble avalanche paths. Those lobate features end with a rough ”tongue”-like
feature that overlaps the equatorial smooth areas. When thresholding the
image, we found it useful to introduce three levels of roughness which
accurately map the smooth terrains, rough terrains, and intermediate zones
that would correspond to the edges of the landslides mentioned previously. Our
determination of morphological units matches the geological investigation
Barnouin et al. (2023). This is illustrated in Figure 8 for a couple of
morphological units: smooth regions and putative mass wastings.

This mapping is consistent with our understanding of the current state of
Didymos. With a rotation period of 2.26 hours Naidu et al. (2020), and an
estimated mass of $5.6\times 10^{11}~{}kg$ Barnouin et al. (2023), the
centrifugal force at the equator of Didymos is comparable in magnitude to the
gravitational acceleration ($\simeq 2.2\times 10^{-4}~{}m.s^{-1})$. This means
that the surface at low and middle latitudes may be only held together by non-
zero mechanical strength and is likely to be easily mobilised, even by the
smallest perturbations. Additionally, several authors (e.g., Walsh et al.
(2012); Zhang et al. (2021)) have proposed that top-shaped asteroids spinning
at a short spin period may structurally fail on their surface if there is a
mechanically strong interior. If there is no such strong interior, then the
interior fails first, the deformation mode triggering surface failure, causing
catastrophic disruption Hirabayashi et al. (2022). However, recent work also
shows that the spin period also changes the failure mode Hirabayashi &
Scheeres (2019). Surface layers may be more sensitive at a longer spin period,
given almost zero cohesion; on the other hand, interiors may be more sensitive
at a shorter spin period, needing higher mechanical strength Hirabayashi et
al. (2020). In summary, these findings suggest that while Didymos’s highly
oblate shape likely came from internal failure Barnouin et al. (2023), its
current surface can also be sensitive to mass wasting without internal
failure, given the existence of internal mechanical strength to support the
current shape.

The equatorial regions are, therefore, areas where we expect to find
landslides, but also where dust might be able to stick to the surface better
than boulders (cohesion of granular matter increases with decreasing particle
size Heim et al. (1999); Scheeres et al. (2010); Sánchez & Scheeres (2014).
The spatial distribution of roughness level, as shown in Fig. 7 indicates that
multiple regions overlap each other. They display the characteristic features
of mass wasting: a sharp upper edge where the slope initially failed, a smooth
flow area with few boulders that did fully reach the end of the run, and a
rough talus at the bottom of the slope as an accumulation of the mass wasted
material. These areas appear to be preferentially oriented from higher
latitudes toward the equator, which is consistent with the model predictions.
We present an few of the afore mentioned morphological features in Figure 8,
overlayed on an image of Didymos and on the roughness map. A complete
description of the various geological units is available in Barnouin et al.
(2023), and a detailed study of boulders rolling down putative avalanche runs
can be found in Bigot et al. (2023). Here we emphasise that the roughness
analysis provides additional insight which can help refine the geological
interpretation.

Mobilized material close to the equator may leave the surface and be put in
orbit, depending on the actual asteroid mass and equatorial extents Trógolo et
al. (2023). The overlap between these mass-wasted areas indicates that several
events took place sequentially, and it is possible that regolith mobilization
will occur again on Didymos in the near future, perhaps to be detected by the
upcoming Hera mission. The ejecta mass has been measured to be at least
$6\times 10^{6}~{}kg$ Moreno et al. (2023), potentially up to $2.2\times
10^{7}~{}kg$ Graykowski et al. (2023). Some of this material may have impacted
Didymos, triggering additional mass wasting.

Figure 7: Entropy of information (EOI) measured on Didymos at 5 m/px. The
thresholds in EOI values have been chosen to enhance the location of smooth,
intermediate, and rough terrains on the asteroid. Figure 8: Example of
geological units detected on Didymos, overlayed on the original image and the
map of EOI. The dotted area shows one the contiguous smooth equatorial region.
The terrains marked with a dotted line are putative mass wasting break lines,
from which material slumped downhill, creating smooth areas bordered at their
bottom end by a pile of debris. Whether each mass wasting intercepts another
provide information on their time sequence. A complete description of the
geological units is available in Barnouin et al. (2023).

###  3.3 Dimorphos

By contrast with its larger companion, Dimorphos presents a much more
homogeneous surface, on average rougher than Didymos’ at all scales (Figs. 6)
and 9). We do not find any smooth region on the surface, which is dominated by
high-entropy terrains. However, the EOI is not randomly distributed and
sorting the values in two bins leads to a clear separation between boulders
and regolith at the chosen scale. Figure 9 shows an example of this
categorisation, where meter-size boulders are clearly identified by the
entropy measurement. This detection is consistent with the distribution
reported in Pajola et al. (2023), where the authors manually mapped 4734
boulders on the surface.

The efficient detection of meter-size objects is due to two supporting factors
(1) we selected an entropy window size smaller than the objects, and therefore
the filter acts as an edge detector; (2) there is an intrinsic difference in
roughness between boulders and regolith substrate, at the resolution
considered. This indicates that boulders are rather monolithic entities,
rather than agglomerates of pebbles with a same size distribution as the
surrounding pebbles.

We stress that for all objects, the entropy of information provides a reliable
qualitative estimate of the distribution of smooth and rough terrain, as well
as morphological features of interest within those terrains. One must however
be cautious about quantitative comparisons, which can only be performed when
all measurement parameters are kept constant. We particularly emphasise that
it is critical to use objects or areas images at similar resolution, and the
entropy is measured at the same spatial scale. In general, we recommend to
prioritise the use of high resolution images, even when measuring roughness at
large spatial scales. The low resolution may hide topographic features.

Figure 9: Entropy of information (EOI) measured on Dimorphos at 35 cm/px. The
thresholds in EOI values have been chosen to clearly separate boulders from
pebbles.

At 5 m/px, we do not find any terrain on Dimorphos that would be as smooth as
the equatorial regions of Didymos, possibly indicating the absence of small
particles, although this cannot be proven from the low-resolution images only.
Higher-resolution images of the impact site obtained in the last seconds
confirm that the surface of Dimorphos does not have expansive smooth deposits
where there were regolith particles that were unresolved at 5.5 cm/px Daly et
al. (2023).

We acknowledge that DART observed mostly the leading side of Dimorphos (with
respect to its orbit around Didymos). Other regions, in particular the areas
facing toward or away from Didymos may display a different morphology or
boulder distribution, possibly affected by tides induced by the primary. These
terrains will be observed by the European Space Agency’s Hera mission,
scheduled to launch in October 2024.

###  3.4 Roughness from shape

As we measure the entropy on the calibrated images, with no further
processing, one may have to consider the potential distortions induced by the
asteroid shape itself: a pixel of an area with high emission angle covers, for
instance, a much larger surface than a pixel at nadir. This effect must be
taken into account when discussing roughness per area, and can be corrected
using ortho-rectified images (i.e. images projected onto a common reference
frame). This is particularly important if one wants to study the evolution of
roughness over time, with multiple images of the same region that may have
been acquired from different viewpoints Vincent et al. (2021). Here, we are
less concerned about the effect of the shape, as we work on single images and
do not compare several viewing observations. We also show in Sec. 2.2, Fig. 2
that, for this data set, the viewing and illumination angles of singular
pixels do not affect the distribution of entropy values.

Still, in order to validate our approach, we also measured the roughness using
a more traditional method of directly comparing the topography with a
reference baseline, as done in other studies. However, see section 2.2, we do
approach this measurement in a new way, in which the baseline is not
constrained by free parameters, but is rather directly calculated on the shape
model itself. Our roughness from shape (RFS) measures the mean angle between
the orientation of a facet on the shape model and its neighbours. The size of
the neighbourhood determines the spatial scale at which the roughness is
measured.

We based our measurements on the highest resolution shape models obtained by
the DART team (Didymos: Palmer et al. (2022), Dimorphos: Daly et al. (2023)),
and publicly available (see A). The models have a ground sample distance
(separation between vertices) of 25 cm for Dimorphos, and 1.2 m for Didymos,
and were resampled to a resolution comparable to the smallest scale at which
we measured the entropy of information in the images (25 m for Didymos and
3.85 m for Dimorphos), using the open-source software Meshlab
(https://www.meshlab.net/) and its implementation of the resampling algorithm
provided in Garland & Heckbert (1997), which preserves the original
orientation of the surface.

We find that the roughness calculated on the shape model (Figure 10), matches
very well what we obtained with the RFT. In particular for Dimorphos, the
dichotomy between detected in the entropy (boulders vs. regolith) also appears
clearly with the RFS method: boulders are detected as areas with rapidly
changing local angles on the digital terrain model, which translates into
high-entropy regions in the images.

The match is not as good on Didymos. This is mostly due to the fact that the
tri-dimensional shape reconstruction of this object is far more challenging,
because of the limited data available. Essentially, DART returned only a
couple of images of Didymos at a resolution $<5m/px$, with very little
variation in viewing geometry. This limits the scale at which surface features
can be reliably reconstructed. On a large scale, though, the analysis of the
digital shape model confirms the presence of smoother areas in the equatorial
regions, and rougher terrains at higher latitudes.

Figure 10: Comparison between Roughness-From-Texture and Roughness-From-Shape.
Colors indicate qualitatively the distribution of smooth and rough terrains.
Both methods lead to a similar description of the surfaces of Didymos and
Dimorphos.

Overall, we find that both methods converge qualitatively toward a similar
description of the asteroid surfaces. In general, for a quantitative
assessment, all roughness analysis should be performed on high resolution 3D
shape models. Obviously, that is not always possible. Reconstructing
accurately the shape of an asteroid from space images usually requires weeks
of effort, and large computer resources. Here we have shown that a technique
like the EOI provides a reliable first description of the surface and
identification of major terrains and landforms of interest like boulders for a
much lower cost (fraction of a second). This has implications for on-board
autonomous navigation systems, for instance, which do not have the processing
resources to work with 3D models.

##  4 Discussion

In summary, we have analysed images returned by NASA’s DART mission via novel
approaches that aim to provide a fast identification of different terrain
types. We use our method to derive a first characterization of the topographic
roughness of asteroids Didymos and Dimorphos.

We show that Didymos and Dimorphos exhibit complex geomorphological features.
Didymos’ morphology presents large-scale patterns in its roughness
distribution, which clearly identify several distinct morphological units:
smooth equatorial terrains, rough highlands at high latitudes, and possible
landslides flowing from mid-latitudes toward the equator. This pattern is
consistent with our current understanding of evolutionary processes that may
occur on the surface of rapidly spinning asteroids. The layering of multiple
landslides indicates that such events took place sequentially at different
epochs and may be related to the spin-up process that led to the current
state. This interpretation is compatible with the geological study presented
in Barnouin et al. (2019), which concludes that Didymos is structurally weak
object with a shear strength in the range 1-10 Pa from surface to interior.
This implies that the surface would be extremely prone to failure from the
smallest solicitation, and mass wasting is likely to have taken place, and
perhaps still occur at the present day.

Dimorphos, on the other hand, presents a much more homogeneous surface, on
average rougher than Didymos. We note a clear bi-modal trend in the roughness
distribution, which separates decimeter-scale and larger boulders from the
finer regolith substrate. The roughness analysis finds no concentrated regions
of dust on the surface of Dimorphos, which is confirmed by high-resolution
images Daly et al. (2023); Barnouin et al. (2023). Note that this does not
mean that Dimorphos does not host dust at all, but that small particles are
not present on the surface in large deposits. Telescopic observations, which
are usually sensitive to $\textmu m$-level particles, captured the ejecta
plume consisting of such small particles. It is hard to consider that the DART
impact generated all such dust particles because highly effective
fragmentation only occurs at the impact site, meaning that the large amount of
dust observed by telescopic campaigns may have existed before the impact,
possibly stored in subsurface regions.

The lack of dusty regions on the surface of Dimorphos, though, suggests that
the binary pair was formed relatively recently, within a time scale shorter
than what is needed to create new dust from boulder erosion by meteoroid
impact or thermal fatigue (0.1 MYrs, A. Lucchetti et al. (2023)). That is
compatible with cratering studies, which indicate that the surface age of
Dimorphos is significantly younger than Didymos’, in the range 0.09-11 MYrs
Barnouin et al. (2023). We also note that a detailed analysis of the
morphology of Dimorphos boulders shows that they are primarily fragments of a
catastrophic disruption and do not display evidence for significant subsequent
erosion Colas Q. Robin et al. (2023). Whether this disruption resulted from
YORP spin-up or from collisions cannot be determined by our analysis alone.
However, dynamical studies Bagatin et al. (2023) strongly suggest that
collisions have been the dominant factor driving surface evolution.

As Didymos’s equatorial regions appear much smoother than Dimorphos, we
suggest that these areas are likely to be covered in material finer than what
we detected on Dimorphos at the highest resolution (a few centimetres).

In conclusion, several lines of evidence combine to build a picture of how the
Didymos binary system came to be: the rough surface of Dimorphos, the smooth
areas lacking boulders on Didymos’ equatorial regions, and the relative age
difference between the two bodies, may be indicative of a formation process by
which Dimorphos is preferentially formed from material that was shed from the
primary asteroid.

The methods presented in this paper provide a novel way to assess the surface
roughness of an airless body, from texture alone, or from a shape model. In
both cases, our approach strive to obtain a rapid evaluation of the surface
properties while minimising the number of free parameters. We find that the
roughness from texture, based on measuring the entropy of information in
images, is particularly robust and low in resources. This make it a good
candidate for embedded software that could support autonomous navigation, for
instance with future asteroid missions or a lunar lander. We are currently
implementing the technique in the data analysis pipeline for ESA’s Hera
mission, which is scheduled to launch in 2024 and will fully characterise
asteroids Didymos and Dimorphos at high resolution in 2026.

##  5 acknowledgements

This work was supported by the DART mission, NASA Contract No. 80MSFC20D0004.
It has also received funding from the European Union’s Horizon 2020 research
and innovation programme under grant agreement No 870377 (project NEO-MAPP)
and the Centre National d’Etudes Spatiales (CNES). AL and MP acknowledge
financial support from Agenzia Spaziale Italiana (ASI-INAF contract No.
2019-31-HH.0 and No. 2022-8-HH.0). ACB acknowledges funding by EC
H2020-SPACE-718 2018-2020 / H2020-SPACE-2019, and by MICINN (Spain) PGC2021,
PID2021-125883NB-C21. JMT-R acknowledges financial support from the project
PID2021-128062NB-I00453 funded by MCIN/AEI/10.13039/501100011033. TK is
supported by Academy of Finland project 335595 and by institutional support
RVO 67985831 of the Institute of Geology of the Czech Academy of Sciences.

##  Appendix A Data Availability

All data used in this article is publicly available on NASA’s Planetary Data
System archive (https://pds-smallbodies.astro.umd.edu/), including images,
associated documentation and advanced products like the asteroid shape models.
Reference images used in this study: 67P: N20140903T034422640ID20F22, Itokawa:
st_2474846738_v, Bennu: 20181113T045713S212_pol_L1pan, Didymos:
dart_0401929888_05627_01_iof, Dimorphos: dart_0401930040_12262_01_iof

##  Appendix B Code Availability

The image analysis performed in this paper is based on publicly available
python libraries, all links have provided in the text. The shape model
analysis is based on custom C code added to the publicly available tool
shapeViewer, (Vincent, 2018). Because the code relies on other parts of the
software to load the shape model and display the results on the screen, we
opted for providing the detailed algorithm in section 2.2 rather than the
actual implementation. The compiled software is freely available to verify our
results. Code can be made available on request.

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