2012 Vol. X No. XX, 000–000

11institutetext: National Astronomical Observatories,Chinese Academy of
Sciences,Beijing 100012, China<EMAIL_ADDRESS>

# The Deconvolution of Lunar Brightness Temperature based on Maximum Entropy
Method using Chang’E-2 Microwave Data ∗ 00footnotetext: $*$ Supported by the
National Natural Science Foundation of China.

Shu-Guo Xing  Yan Su  Jian-Qing Feng  Chun-Lai Li

###### Abstract

A passive and multi-channel microwave sounder onboard Chang’E-2 orbiter has
successfully performed microwave observation of the lunar surface and
subsurface structure. Compared with Chang’E-1 orbiter, Chang’E-2 orbiter
obtained more accurate and comprehensive microwave brightness temperature data
which is helpful for further research. Since there is a close relationship
between microwave brightness temperature data and some related properties of
the lunar regolith, such as the thickness, temperature and dielectric
constant, so precise and high resolution brightness temperature is necessary
for such research. However, through the detection mechanism of the microwave
sounder, the brightness temperature data acquired from the microwave sounder
is weighted by the antenna radiation pattern, so the data is the convolution
of the antenna radiation pattern and the lunar brightness temperature. In
order to obtain the real lunar brightness temperature, a deconvolution method
is needed. The aim of this paper is to solve the problem in performing
deconvolution of the lunar brightness temperature. In this study, we introduce
the maximum entropy method(MEM) to process the brightness temperature data and
achieve excellent results. The paper mainly includes the following aspects:
firstly, we introduce the principle of the MEM, secondly, through a series of
simulations, the MEM has been verified an efficient deconvolution method,
thirdly, the MEM is used to process the Chang’E-2 microwave data and the
results are significant.

###### keywords:

Space vehicles—instruments: microwave sounder—Moon: brightness
temperature—method:maximum entropy method

##  1 Introduction

Chang’E-2 orbiter was launched on October 1, 2010. The microwave sounder is
one of the main payloads on both Chang’E-1 and Chang’E-2 orbiter, which is
passive and has four channels(3GHz, 7.8GHz, 19.35GHz, 37GHz)(Jiang et al.
2010). During on-orbit observation, both of the two microwave sounders
obtained a lot of microwave radiation brightness temperature data. Many
scholars have begun to carry out extensive researches using these microwave
brightness temperature data(Fa & Jin 2010,Wang et al. 2010). Benefiting from
the lower orbit altitude, the spatial resolution of the microwave sounder on
Chang’E-2 orbiter is nearly twice as good as Chang’E-1. In addition, Chang’E-2
measurements covered the lunar surface more completely and got more tracks of
swath data than Chang’E-1. Therefore, more available data can be used to
research radiation characteristics of lunar surface and subsurface structure.
In this paper, we choose the 37GHz microwave data, which better reflect the
lunar surface microwave radiation. Moreover, the Laser Altimeter and the
Charge Coupled Device(CCD) camera are also the main payloads on the Chang’E-2
orbiter, thus the Digital Elevation Model (DEM) and CCD images are direct
response to lunar surface structure. So we use the DEM and CCD images to
verify the results of the deconvolution.

It is known that in the detection mechanism of the microwave sounder, antenna
brightness temperature has been weighted by antenna radiation pattern. It is
the convolution of the antenna radiation pattern and the lunar brightness
temperature. The result of this impact is shown in Figure 1. Because of the
impact of the antenna pattern, points close together on the original map (Fig.
1a) can’t be distinguished in the observation map (Fig.1c). Actually, when
using the brightness temperature data to research, we suppose the lunar
brightness temperature is approximately equal to the antenna brightness
temperature, but this will cause the loss of details and a decrease in spatial
resolution(Feng et al. 2013). The paper introduces a deconvolution method
named Maximum Entropy Method(MEM) to eliminate the influence of the antenna
pattern. In the field of image reconstruction of radio astronomy, MEM is a
valuable tool that has many advantages in dealing with both the extended
sources and the point sources(Starck et al. 2002).

Figure 1: The diagram of the impact of the antenna pattern. (a) the original
map (b) the antenna pattern (c) the observation map(the convolution of (a) and
(b)).

The paper is organized as follows. In Section 2, a brief description on the
principle of MEM is given. In Section 3, a series of simulations are carried
out to verity the validity of MEM. In this section, we introduce two indices,
Mean Squared Error(MSE) and Peak Signal to Noise Ratio(PSNR), to evaluate the
quality of the deconvolution. Through the simulations, we come to a conclusion
that the MEM is an efficient deconvolution method. In Section 4, the MEM is
used to process the Chang’E-2 brightness temperature data. A new brightness
distribution map is shown and compared with the original map, and we also
choose several local regions for discussions of the results. Finally,
conclusion of this work is stated in Section 6.

##  2 The principle of MEM

In the field of image restoration, Frieden first introduced the concept of
image entropy in 1972(Frieden 1972) and gave the expression,
$S=-\sum{x(i,j)\ln x(i,j)}$, where $x$ defined the gray value of the image at
the point $i$ and $j$. With the introduction of image entropy, a new
constraint condition has been applied in the image restoration. Based on the
different background and the limitations of the application of the MEM,
Frieden, Gull and Bryan proposed different MEMs named by the authors name
respectively(Frieden 1972;Gull & Daniell 1978;Skilling & Bryan 1984). The MEM
used in the paper is based on method proposed by N.L.Bonavito in 1993
(Bonavito et al. 1993).

Here, a model used in the paper is shown as below.

$\left\\{\begin{array}[]{l}subject~{}to~{}y(i,j)=x(i,j)\otimes PSF\\\
max~{}S=-\sum{x(i,j)\cdot\ln x(i,j)}\\\ find~{}x(i,j)\end{array}\right.$

In the above model, $x$ means the original map, $y$ means the observation map,
$PSF$ means the antenna pattern, and the model indicates that the best
solution of the deconvolution need obey the constraint of the maximum entropy.

With the lagrange multiply method, we get the maximum entropy distribution,

$x(i,j)=\exp(-\sum\limits_{m,n=1,1}^{i,j}{\lambda mn}\cdot
PSF)/Z({\lambda_{11}},\cdot\cdot\cdot,\lambda ij)$ (1)

$Z({\lambda_{11}},\cdot\cdot\cdot,\lambda
ij)=\sum\limits_{1,1}^{i,j}{\exp(-\sum\limits_{m,n=1,1}^{i,j}{\lambda mn\cdot
PSF})}$ (2)

where $Z$ means the partition function, $\lambda$ means lagrange multipliers.
Also, a parameter is defined as

$G=x(i,j)\otimes PSF-y(i,j)$ (3)

Here we use a more storage-efficient method of successive approximation. Our
recursion relation is defined as

$\lambda(New)=\lambda(old)-\ln(\frac{y}{{y-G}})$ (4)

Steps are as follows: the first step, we set all $\lambda=0$; the second step,
with the $\lambda=0$, we get an original $x$ from Equation(1); the third step,
we put $x$, $PSF$ and $y$ in the Equation(3), and we calculate $G$; the fourth
step, having the value of $G$, we get the new $\lambda$ from Equation(4). Then
based on the new $\lambda$, we repeat the second step to fourth step, until
$\lambda$ converges. This method starts with $\lambda=0$, which corresponds to
a scene that is a flat map(a map with the maximum possible entropy). The
entropy of the scene decreases with each iteration as it converges on the
scene with the maximum entropy that satisfies the constraints of the original
map and the $PSF$. For more details, please refer to the Bonavito
paper(Bonavito et al. 1993).

##  3 The simulation and verification

The simulation includes three aspects, namely the simulation based on point
source, the simulation based on an extended source and the simulation based on
the Chang’E-2 microwave brightness temperature data, respectively. And two
indices, Mean Squared Error(MSE) and Peak Signal to Noise Ratio(PSNR) are used
to evaluate the quality of the deconvolution. A lower MSE and a higher PSNR
indicate better deconvolution results. The definitions of the MSE and PSNR are
given as below.

$MSE=\frac{1}{{mn}}\sum\limits_{i=0}^{m-1}{\sum\limits_{j=0}^{n-1}{{{(Y(i,j)-X(i,j))}^{2}}}}$
(5)

$PSNR=10\cdot{\log_{10}}(\frac{{MA{X^{2}}}}{{MSE}})$ (6)

where $X$ means the original map, $Y$ means the map to be evaluated and $MAX$
means the maximum value in $X$ data.

Here, a pattern function used as the fitting the antenna pattern is shown as
bellow,

$f(\theta)=\sin c(\frac{{D*\sin(\theta)}}{\lambda})$ (7)

where $D$ is the diameter of the ground antenna, $\lambda$ is the observation
wavelength and $\theta$ is the antenna pattern angle. The function is used in
Simulation 3.1 and Simulation 3.2 .

###  3.1 Simulation based on point sources

Figure 2: The flow figure of simulation based on the point sources (a) the
original map (b)the antenna pattern (c)the blurred map (d) the deconvolution
map.

The purpose of the simulation is to verify the validity of the method when
dealing with a point source. The steps of simulation are as follows.

Firstly, in Figure 2a, the original data contains three point sources and
values of the point sources are higher than the background. The second step is
fitting an antenna pattern using Equation 7. Figure 2b shows a two-dimensional
curve of the antenna pattern. The third step is convoluting the original data
with the antenna pattern, then the blurred map is shown in Figure 2c. The
fourth step is using the MEM to process the blurred map, and the result of
deconvolution is shown in Figure 2d. In addition, the shadows on the lower
left of Figure 2c and Figure 2d represent the projection of 3dB beam width on
the original map and the deconvolution map, respectively.

Because of the impact of the beam width of antenna pattern, the two lower
point sources in Figure 2c become nearly one point source. However, after the
deconvolution of the MSE, the lower two point sources have been clearly
distinguished in Figure 2d. Compared with the blurred map, the deconvolution
map based the MEM recovers the details which can be found in the original data
map. Moreover, we calculate the MSE and PSNR. The MSE and PSNR of the
deconvolution map are 0.0113 and 58.3980 respectively, while the values are
0.6309 and 40.9298 for the blurred map. This result proves the process of
deconvolution improves PSNR and reduces MSE.

###  3.2 Simulation based on an extended source

Figure 3: The simulation based on the ground antenna,(a) the original source
(b) the antenna pattern (c) the blurred map (d) the deconvolution map. Figure
4: The residual map of the blurred map and the maximum entropy map (a) the
blurred one (b) the deconvolution one; the maps are abtained from the blurred
map and the deconvolution map minus the original map respectively.

In this simulation, first, an original extended source is shown in Figure3a.
The value in the middle area is 230k, while the value in other area is 0 k.
And two forms, the discrete sequence image and the plane image are shown in
order to describe the changes between different images clearly. Second, based
on the Equation 7, the antenna pattern is shown in Figure 3b. Then, by
convoluting the original data with the antenna pattern, the blurred image is
shown in Figure 3c. Finally, using the MEM to process the blurred image, the
result of deconvolution is shown in Figure 3d.

We calculate the MSE and PSNR. The MSE and PSNR of the blurred map are 0.6608
and 40.1589 respectively, while the values are 0.0466 and 51.6718 in the
deconvolution map. Figure 4 shows the residual map of the blurred map and the
deconvolution map. The value range of the residual map of the blurred map is
from -150 K to 100 K but the value range is about from -0.13 K to 0.17 K in
deconvolution map. So when processing the extended source data, the MEM
produced a better result.

###  3.3 Simulation based on Chang’E-2 microwave data

During the ground calibration testing, we tested antenna pattern and gain of
the 37GHz channel and then got four important parameters: (1) 3dB beam width
of approximately $10^{\circ}$, (2) the main beam width of about $26^{\circ}$,
(3) the attenuation of first side lobe is about 22dB, (4) the first side lobe
beam width of approximately $6^{\circ}$. Based on Gaussian function, we fitted
the antenna pattern function using the above four parameter(the contribution
of main lobe is about 95.9%(Wang et al. 2009), so here we only considered the
main lobe and first side lobe, which is shown in Equation 8. The antenna
pattern is shown in Figure 5.

Moreover, considering that bandwidth of 37GHz channel is about 500MHz, we
tested the antenna pattern and gain in three frequency points (36.47GHz,
37GHz, 37.53GHz) during ground calibration testing. Taking 3dB beam width for
instance, three frequency points(36.47GHz, 37GHz, 37.53GHz) corresponded to
$10.07^{\circ}$, $10.17^{\circ}$ and $10.03^{\circ}$ respectively. The
difference is less than $0.15^{\circ}$, which is about 0.25km corresponding to
lunar surface. However, the spatial resolution of 37GHz channel is 13km, far
larger than 0.25km. So in the paper we only use antenna pattern of 37GHz on
behalf of the whole bandwidth.

$f(\theta)=\left\\{\begin{array}[]{l}-30\cdot{e^{-{{(\frac{{\theta+13}}{{5.3}})}^{2}}}}-30\cdot{e^{-{{(\frac{{\theta-13}}{{5.3}})}^{2}}}}\begin{array}[]{*{20}{c}}{}\hfil&{\begin{array}[]{*{20}{c}}{}\hfil&{\begin{array}[]{*{20}{c}}{}\hfil&{}\hfil\end{array}}&{\begin{array}[]{*{20}{c}}{}\hfil&{}\hfil&{}\hfil&{}\hfil&{}\hfil\end{array}}\end{array}}&{-{{13}^{\circ}}\leq\theta\leq{{13}^{\circ}}}\end{array}\\\
-8\cdot{e^{-{{\left({\frac{{\frac{{\left|\theta\right|-13}}{{0.23}}}}{{5.3}}}\right)}^{2}}}}-8\cdot{e^{-{{\left({\frac{{\frac{{\left|\theta\right|-13}}{{0.23}}-26}}{{5.3}}}\right)}^{2}}}}{\rm{-}}22\begin{array}[]{*{20}{c}}{\begin{array}[]{*{20}{c}}{}\hfil&{}\hfil\end{array}}&{{{13}^{\circ}}\leq\left|\theta\right|\leq{{19}^{\circ}}}\end{array}\end{array}\right.$
(8)

Figure 5: The antenna pattern of 37GHz, (a) Gain(dB) antenna pattern, (b)
Normalized antenna pattern. According to Wang et al. 2009, the antenna pattern
of 37GHz is figured out and used in the following deconvolution processing
procedure. Figure 6: The result of simulation based on Chang E-2 microwave
data (a) the original brightness temperature map (b) the antenna pattern (c)
the gaussian noise (d) the blurred map. Figure 7: The deconvolution results
from differernt methods.

In this simulation, Chang’E-2 microwave data is used as the original data. The
purpose of the simulation is to verify the method used in this step can work
well when processing the Chang’E-2 microwave data with corresponding antenna
pattern. Firstly, a region about ${20^{\circ}}*{20^{\circ}}$ chosen from
Chang’E-2 microwave brightness temperature map is shown in Figure 6a. As we
choose the 37 GHz microwave data, so the corresponding beam width of the
antenna pattern is about $10^{\circ}$ (Wang et al. 2009) and the antenna
pattern map is shown in Figure 6b. Given the sensitivity of the Chang’E-2
microwave sounder is approximately 0.5K (Wang et al. 2009), a gaussian noise
map is shown in Figure 6c, whose mean and standard deviation are 0 and 0.5
respectively. Figure 6d shows the blurred map, which is obtained by
convoluting the original map and the antenna pattern and then adding the
noise.

Here, different methods such as the Richardson Lucy (R-L) deconvolution method
and the blind decon-volution method (Ayers & Dainty 1988;Biggs & Andrews 1997)
are chosen to process the same data, and we compare the processing results.
The comparison is shown in Figure 7. And we also calculate the MSE and PSNR of
the different method, which is shown is Table 1. By comparison, we find that
the MEM shows a significant advantage both on restoring the details and
restraining the noise.

Table 1: The PSNR and MSE of the different methods

| the blurred | the MEM | the blind method | the P-L method
---|---|---|---|---
PSNR | 52.8063 | 57.8611 | 50.4692 | 52.7997
MSE | 0.3404 | 0.1039 | 1.2300 | 0.3333

0.65The PSNR and MSE are calculate refer to Equation 6 and 5.

###  3.4 Error Analysis

Through the above three simulations, we found that the MEM is a good method to
solve deconvolution problem. In this section, we discuss the radiometric error
introduced by MEM.

Figure 8: Error produced in the above three simulations, (a) simulation based
on point sources (b) simulation 3.2 based on extended source (c) simulation
based on real microwave data.

Figure 8 shows the error produced in above three simulations, which is got by
the deconvolution map minus the original map respectively. We found that in
simulation 1, the range of the error is from -0.35K to 0.4K with standard
deviation $\sigma=0.0676K$; in simulation 2, the error is from -0.13K to 0.17K
with $\sigma=0.0089K$, and in the simulation 3, because of the Gaussian
noise(-1.8K to1.8K) introduced, the error is about from -2K to 1.5K with
$\sigma=0.3054K$. The errors introduced by MEM in the above three simulations
are all less than the sensitivity of the microwave sounder which is 0.5K(Wang
et al. 2009).

Through the above simulations and error analysis, we can draw the conclusion
that the MEM is an efficient method when dealing with the deconvolution
problem and it can further be used to process the microwave brightness
temperature data of the Chang’E-2 orbiter.

##  4 Data Processing Based on Chang’E-2 Microwave Data

In this section, the MEM is used to process the real microwave data. The data
is the 37 GHz nighttime microwave data acquired by Chang’E-2 orbiter, which is
normalized with the Lunar local time. Refer to the paper (Zheng et al. 2012),
we normalize the lunar hour angle to $0^{\circ}$, which implies the lunar time
is the midday of the Moon.

There is also a problem worthy of notice, which is how to keep the scale of
the microwave data consistent with the antenna pattern. Figure 9 describes the
problem clearly. In this Figure, $h$ means the observation height of the
antenna; $\theta$ means the angle of the antenna pattern; $r$ means the radius
of the moon and $\alpha$ means the central angle of moon. In order to keep the
consistent the projection of the $\theta$ and the $\alpha$ on the lunar
surface, we get approximate equation shown in Equation 9. With further
simplification, we get $\theta\approx 17.4\alpha$, which means $1^{\circ}$ on
lunar surface corresponds $17.4^{\circ}$ in the antenna angle. When the
microwave data value is taken every $0.2^{\circ}$, the corresponding antenna
pattern value is taken every $3.48^{\circ}$. Thus, the correspondence of
$0.2^{\circ}$ in microwave data and $3.48^{\circ}$ in antenna pattern is
adopted in this paper.

Figure 9: The sketch map of the antenna observation.

$\frac{\alpha}{{180}}\times\pi\times
r\approx\frac{\theta}{{180}}\times\pi\times h$ (9)

In addition, the Digital Elevation Model(DEM) data and Charge Coupled
Device(CCD) data from Chang’E-2 orbiter are used in this section. The laser
altimeter and the CCD camera are also the main payloads on the Chang’E-2
orbiter and the spatial resolution of the two payloads is higher than the
microwave sounder. So the data are used to verify the validity of the new
added details in the deconvolution map. The comparison of the original
distribution map and the deconvolution distribution map based on MEM is shown
in Figure 10.

Furthermore, three regions are also chosen to show the differences between the
original map and the deconvolution map. The range of first region is from
$14.3^{\circ}$W to $11.27^{\circ}$E and $48.2^{\circ}$S to $22.4^{\circ}$S.
The second region is about ${10^{\circ}}*{10^{\circ}}$ with centering the
Apollo 16 landing point, while the third region is centered the Luna 24
landing point. Comparisons of three regions are shown in Figure 11,12 and 13
respectively.

Figure 10: The comparison of the original map and the deconvolution map of the
whole lunar surface,(a)the original map, (b) the deconvolution map. The
original data came from the 37GHz nighttime microwave data. Figure 11: The
comparison result map of the region with the longitude range from
$14.3^{\circ}$W to $11.27^{\circ}$E, the latitude range from $48.2^{\circ}$S
to $22.4^{\circ}$S. (a) the original brightness temperature map, (b) the
deconvolution brightness temperature map, (c) the DEM map, (d) the CCD map.
Figure 12: The comparison result map of the region with the longitude range
from $57.2^{\circ}$E to $67.2^{\circ}$E, the latitude range from
$7.3^{\circ}$N to $17.3^{\circ}$N. (a) the original brightness temperature
map, (b)the deconvolution brightness temperature map, (c) the DEM map, (d) the
CCD map. Figure 13: The comparison result map of the region with the longitude
range from $10.5^{\circ}$E to $20.5^{\circ}$E, the latitude range from
$14^{\circ}$S to $4^{\circ}$S. (a) the original brightness temperature map,
(b)the deconvolution brightness temperature map, (c) the DEM map, (d) the CCD
map.

Figure 10 indicates that the deconvolution brightness temperature distribution
map has higher resolution than the original map. In order to clearly compare
the deconvolution map and the original map, we chose three regions. From
Figure 11, 12 and 13, we find that there are some new added details which have
bounded by blue lines in the deconvolution map. Furthermore, in the position
of the domains, we find similar features in both the DEM and the CCD map.

In deconvolution maps, under the conditions of only considering the influence
of the antenna pattern, the brightness temperature error introduced by MEM is
less than 0.5K.

##  5 Conclusion

In this paper, the MEM is first used to deconvolute the Chang’E-2 microwave
data and the results are satisfactory. Compared to the original brightness
temperature map, the map processed by MEM has more details and higher
resolution. Only considering the influence of the antenna pattern, the new
data obtained by MEM processing is very useful for further research and the
brightness temperature error introduced by MEM is less than 0.5K. We conclude
the following.

Given the impact of the antenna pattern, it is necessary to find a suitable
deconvolution method used to process Chang’E-2 microwave data. And by applying
the simulations above, we found on the one hand the impact of the antenna
pattern could blur the original map heavily, but on the other hand MEM was
effective in increasing PSNR and reducing MSE, for both extended sources and
point sources, eventually restoring the original distribution information.
Furthermore, in comparison to other methods MEM shows significant advantages,
both for restoring detail and reducing noise.

After comparison of the whole lunar brightness temperature distribution map
and several local regions, we found the deconvolution maps got more details
and higher resolution than the original maps. So the more precise and higher
resolution brightness temperature has been achieved.

By comparison between the brightness temperature map and the topographic map,
we found the new added details also have a correspondence in the CCD map and
DEM map. This, for one thing, verifies the validity of the new added details,
for another reveals a relativity between brightness temperature and lunar
terrain and geological formation. Thus, the precise brightness temperature
data processed by MEM is very useful to study the lunar surface and subsurface
structure for further research.

###### Acknowledgements.

The work is supported by the National Natural Science Foundation of China
(Grant Nos.11173038 and 11203046). We thank the Lunar and Deep Space
Exploration Department, National Astronomical Observatories, Chinese Academy
of Sciences(NAOC) for providing the microwave data, CCD and DEM map.

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