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Global Mapping of the Surface Composition on an Exo-Earth Using Color Variability
Hajime Kawahara
1,2
1
Department of Earth and Planetary Science, The University of Tokyo, 7-3-1, Hongo, Tokyo, Japan; kawahara@eps.s.u-tokyo.ac.jp
2
Research Center for the Early Universe, School of Science, The University of Tokyo, Tokyo 113-0033, Japan
Received 2020 March 1; revised 2020 April 3; accepted 2020 April 6; published 2020 May 6
Abstract
Photometric variation of a directly imaged planet contains information on both the geography and spectra of the
planetary surface. We propose a novel technique that disentangles the spatial and spectral information from the
multiband reflected light curve. This will enable us to compose a two-dimensional map of the surface composition
of a planet with no prior assumption on the individual spectra, except for the number of independent surface
components. We solve the unified inverse problem of the spin–orbit tomography and spectral unmixing by
generalizing the nonnegative matrix factorization using a simplex volume minimization method. We evaluated our
method on a toy cloudless Earth and observed that the new method could accurately retrieve the geography and
unmix spectral components. Furthermore, our method is also applied to the real-color variability of the Earth as
observed by Deep Space Climate Observatory. The retrieved map explicitly depicts the actual geography of the
Earth, and unmixed spectra capture features of the ocean, continents, and clouds. It should be noted that the two
unmixed spectra consisting of the reproduced continents resemble those of soil and vegetation.
Unified Astronomy Thesaurus concepts: Exoplanet surface characteristics (496);Exoplanet surfaces (2118);
Habitable planets (695);Direct imaging (387);Astronomical methods (1043);Exoplanet surface composition
(2022);Exoplanet surface variability (2023)
1. Introduction
Direct imaging of terrestrial planets around a nearby solar-
type star is an important goal for future astronomy. In the 2020
decadal surveys, both HabEx and LUVOIR have shown the
capability to search for these planets, even in the habitable
zone. Direct imaging with spectroscopy provides information
regarding the molecules in the atmosphere of the planet, which
enables us to search for biosignatures such as oxygen, carbon
dioxide, and water. Moreover, surface inhomogeneity can be
explored with photometric monitoring of the reflected light as
proposed by Ford et al. (2001). The color variability of the
reflected light has been studied as a probe of surface
composition (e.g., Cowan et al. 2009; Fujii et al. 2011). The
spatial distribution of the planet surface can also be inferred
from the photometric variation. Diurnal variation, due to the
rotation of the planet, provides the spin rotation period and a
one-dimensional distribution of the surface (Pallé et al. 2008;
Cowan et al. 2009; Oakley & Cash 2009; Fujii et al.
2010,2011; Lustig-Yaeger et al. 2018). Furthermore, the axial
tilt can be obtained from the analysis of the frequency
modulation of the periodicity (Kawahara 2016; Nakagawa
et al. 2020). The analytic expression of the reflected light
curves has been studied (Cowan et al. 2013; Haggard &
Cowan 2018).
A full two-dimensional inversion technique called “spin–
orbit tomography”(analogous to computer tomography)was
proposed by Kawahara & Fujii (2010)and has been studied in
terms of the inverse problem (Kawahara & Fujii 2011; Fujii &
Kawahara 2012; Farr et al. 2018; Berdyugina & Kuhn 2019;
Aizawa et al. 2020)and obliquity measurement (Schwartz et al.
2016; Farr et al. 2018). Recently, Luger et al. (2019)analyzed a
single-band light curve of the Earth with Transiting Exoplanet
Survey Satellite data and inferred a rough two-dimensional
cloud distribution. Fan et al. (2019)successfully retrieved
a global map that was analogous to the distribution of a
continent, from data that was obtained by DSCOVR by
monitoring the Earth for two years (Jiang et al. 2018). They
used the second principle component (PC2)of a multicolor
light curve. Aizawa et al. (2020)improved the retrieved map
from DSCOVR using sparse modeling. These examples
showed that a global map could be retrieved from a time
series of a single band or PC. However, there exists a level of
ambiguity when interpreting the derived maps when we do not
have prior knowledge on the surface compositions.
Moreover, a blind retrieval of the reflectance spectra of the
surface components from the integrated light is known as
“spectral unmixing”in remote sensing. Cowan & Strait (2013)
formulated the spectral unmixing as a disentanglement of
geography by spin rotation. However, the longitudinal map
inferred from the EPOXI data did not match the real
geographies because of the degeneracy of the inferred
geometric distribution and spectral components (Fujii et al.
2017). The ambiguity of spectral unmixing originates from the
matrix factorization not being unique, which has been
extensively studied in the field of remote sensing. These
studies found that additional constraints such as the simplex
volume minimization of spectral components guarantee a
unique solution to the unmixed spectra (Craig 1994; Fu et al.
2015,2019; Lin et al. 2015; Ang & Gillis 2019). In practice,
nonnegative matrix factorization (NMF)with regularization
terms easily retrieves the surface components in hyperspectral
unmixing (Ang & Gillis 2019). These techniques in remote
sensing are worth considering in their application to multicolor
light curves of directly imaged exoplanets.
This paper aims to formulate a single inverse problem that
unifies the spin–orbit tomography and spectral unmixing using
a novel technique via remote sensing. To achieve this, we unify
the NMF-based spectral unmixing technique and spin–orbit
tomography to retrieve both the spectra and geographies of a
disk-integrated light curve from an exoplanet. We demonstrate
its capabilities using the simulated data and real data from Deep
The Astrophysical Journal, 894:58 (14pp), 2020 May 1 https://doi.org/10.3847/1538-4357/ab87a1
© 2020. The American Astronomical Society. All rights reserved.
1
Space Climate Observatory (DSCOVR). The rest of the paper
is organized as follows. In Section 2,wefirst review the spin–
orbit tomography and spectral unmixing. Next, we construct a
unified retrieval model using NMF; the optimization scheme is
also provided. In Section 3, we test the technique by applying it
to a cloudless toy model. In Section 4, we demonstrate this new
technique by applying it to real observational data of the Earth
recorded by DSCOVR. Finally, in Section 5, we summarize
our results.
2. Formulation of Spin–Orbit Tomography with Spectral
Unmixing
2.1. Spin–Orbit Tomography
Space direct imaging in optical and near-infrared bands aims
to detect the reflected light (or scattered light)from a planet.
The reflected light is a summation of photons from the
illuminated and visible side of a planet. This integrated-
reflected light is expressed as
ò
pJJj J J=W
ffR
adR ,, coscos, 1
p
ps
2
2IV 101 0 1
() ()
where
fis the stellar flux, R
p
is the radius of the planet, ais the
star–planet distance, IV is the illuminated and visible area, and
W1is the solid angle of the planet’s sphere. JJj
R
,,
s01
()
represents the bidirectional reflectance distribution function
(BRDF)of the surface element s.
J
0is the solar zenith angle,
J
1
is the zenith angle between the direction toward an observer
and the normal vector of the surface, and jis the relative
azimuth angle between the line of sight and stellar direction.
The derivation of Equation (1)is given in Appendix A.An
isotropic approximation of the surface reflectance (the Lambert
approximation),JJj=
R
R,,
ss
01
()
, significantly reduces the
complexity of the problem. We also define the spherical
coordinate fixed on the surface by (θ,f)and express the surface
component, s, in spherical coordinates qf=
R
m,
s()
as the
time-independent quantity (static surface approximation).
Then, we obtain
ò
pqf J J=W
ffR
adm,coscos. 2
p
p
2
2IV 101
() ()
We note that the IV area, Jcos
0
, and Jcos 1are time-dependent.
The terms of Jcos
0
and Jcos 1also depend on the position of
the planet surface, (θ,f), and the axial tilt parameters,
z=Qg,eq
(
)
, where ζis the planet obliquity and
Q
e
q
is the
orbital phase at the equinox. We define the geometric kernel
introduced by Kawahara & Fujii (2010)as
qf JJ J J
=>
p
Wt,, cos cos for cos , cos 0
0otherwise,
3
g
fR
a01 0 1
p
2
2
() ()
⎪
⎪
⎧
⎨
⎩
where the positive condition of Jcos
0
and Jcos 1restricts the
surface integral to pixels on the IV area. Assuming thatgis
fixed, we obtain the Fredholm integral equation of the first kind
òqf qf=Wft d Wt m,, , . 4
g
p() ( ) ( ) ( )
Discretization of the time tt
i
and planetary surface
qf q f,,
jj
(
)(
)
reduces Equation (4)to the linear inverse
problem
å
=dWm,5
i
j
ij j ()
or, using the vector form, we can express it as
=dmW,6()
where =
d
ft
ipi
()
for =¼-iN0, 1, , 1
iand qf=mm,
jj
j
()
for =¼-jN0, 1, , 1
j. The explicit expression of the
geometric kernel qf=
W
Wt,,
gij i j j
()
in the spherical coordinate
is given in Appendix A.
Because the inverse problem of Equation (6)is ill-posed, an
additional constraint or regularization is needed to solve the
problem. Various types of regularizations have been attempted
so far. Kawahara & Fujii (2010)used nonnegative regulariza-
tion and the requirement of an upper limit of albedo as
regularization using the bounded variable least-squares solver
(Lawson & Hanson 1995). Kawahara & Fujii (2011)used the
Tikhonov regularization, which minimizes the cost function
l
=- +dm mQWminimize 1
22
,7
A
2
2
2
2
∣∣ ∣∣ ∣∣ ∣∣ ( )
where λ
A
is the spatial regularization parameter, and 2
2
∣
∣·∣∣ is the
squared L2 norm. To construct the model on the Bayesian
framework, Farr et al. (2018)used a Gaussian process to
regularize the map while Berdyugina & Kuhn (2019)used an
Occamian approach algorithm. Recently, Aizawa et al. (2020)
demonstrated that the L1 +total square variation (TSV)provided
better results than a simple L2 (Tikhonov)regularization.
The value of
d
depends on which features we want to extract
from the multicolor light curve. Kawahara & Fujii (2011)used
a single-band light curve to retrieve a cloud map of the
simulated Earth. They also demonstrated that a rough two-
dimensional distribution of the continent or ocean can be
retrieved from a color difference between 0.85 μm and 0.45 μm
or 0.85 μm and 0.65 μm, owing to the near flatness of the cloud
spectrum. Cowan et al. (2009)utilized principle component
analysis (PCA)for their longitudinal mapping of EPOXI data.
Fan et al. (2019)used the second component of PCA of the
multicolor light curve of DSCOVR. Comparing with the
ground truth, they found that the resultant map was similar to
the global continent/ocean map of the Earth. However, these
two examples required prior knowledge of the surface
composition or the ground truth of the geography. The
ambiguity in the interpretation of the map is a limitation of
the spin–orbit tomography.
2.2. Spectral Unmixing
Spectral unmixing is a procedure that disentangles mixed
spectra by finding the end members. The mixing model of the
spectra of multiple surface compositions is required to unmix
the spectra. The simplest model is the linear mixing model,
expressed as
å
l==dt D A X,,8
il il
k
ik kl
(˜)()
or simply
=DAX,9()
where =
A
at
ik k i
(
)
is the contribution of the k-th component at
time =tt
i
to the intensity of light, and l=Xx
kl k l
(˜)for
2
The Astrophysical Journal, 894:58 (14pp), 2020 May 1 Kawahara
=¼-lN0, 1, ,
1
lis the reflection spectra of the k-th
component at wavelength
l
l
˜.
3
We need to solve the matrix
factorization of Aand X. Generally, the matrix factorization is
formulated as the minimization of the cost function, where the
cost function can either be the squared Euclidean distance or
the Kullback–Leibler distance. In this paper, we use the
squared Euclidean distance
=- +QDAXRAX
1
2,10
F
2
∣∣ ∣∣ ( ) ( )
where F
2
∣
∣·∣∣ is the squared Frobenius norm defined by
åå
ºYY,11
F
ji
ij
22
∣∣ ∣∣ ( )
and
R
AX,()
is the regularization term.
2.2.1. Principle Component Analysis
PCA is a traditional technique used to disentangle the
spectral components of multicolor light curves, as observed in
Cowan et al. (2009). It was also used in a global map
reconstruction of the Earth by Fan et al. (2019)and Aizawa
et al. (2020). PCA can also be formulated as a minimization of
the cost function, from the perspective of optimization,
=-QDAXminimize 1
212
F
2
∣∣ ∣∣ ( )
s==SAAsubject to diag , 13
TAA
() ()
s==SXX diag , 14
TXX
() ()
where sdiag(
)
is a diagonal matrix whose elements are
si
. The
drawback of the PCA as a matrix factorization method is the
strong assumption of orthogonality for Aand X. However, its
orthogonality is useful for visualizing the simplex by reducing
its dimensionality Cowan & Strait (2013). In this paper, we
denote the orthogonal PCA basis by =S
-
U
X
XXT
12
()
, i.e.,
=
U
UI
X
TX(Iis an identity matrix). An arbitrary matrix Mcan
be decomposed by row vectors of U
X
as
å
=puM,15
k
kk
T()
where
u
kis the k-th row of U
X
. The projection of Monto PCkis
computed by
=puM.16
kk()
2.2.2. Nonnegative Matrix Factorization
In the field of remote sensing, a wide variety of spectral
unmixing has been studied. Among these techniques, NMF
decomposes a single matrix Dto two matrices Aand Xwhose
elements are nonnegative; that is, =
D
AX (Paatero &
Tapper 1994; Lee & Seung 1999). NMF can be defined by
the minimization of the cost function. For instance, using the
squared Euclidean distance, NMF is formulated as
=- +QDAXRAXminimize 1
2,17
F
2
∣∣ ∣∣ ( ) ( )
AXsubject to 0, 0. 18
ik kl ()
NMF is known to be NP-hard (Vavasis 2009); therefore, the
optimization of NMF is difficult to achieve. Nevertheless,
various efficient optimization methods have been proposed
(Lee & Seung 1999; Cichocki et al. 2009, references therein).
In particular, NMF combined with the simplex volume
minimization technique can accurately reproduce the high-
resolution spectrum components from remote-sensing satellite
data (Craig 1994; Fu et al. 2015,2019; Lin et al. 2015; Ang &
Gillis 2019). The concept of the simplex volume minimization
can be summarized as follows: if the data are sufficiently
spread in the convex hull defined by the end members, the data-
enclosing simplex whose volume is minimized identifies the
true end members.
In Figure 1, we plot three simplexes that enclose all of the
data points. Each simplex provides its vertices as a solution of
NMF. The simplex volume minimization choose the vertices of
the volume-minimum simplex (dashed triangle)as the end
members of NMF. When there is at least one pure pixel of each
end member in the data, the volume-minimum simplex
obviously identifies the true end members. Even in the case
without pure pixels, Lin et al. (2015)showed that the true end
members could be identified by the volume-minimum simplex
under the condition of the pixel purity level that applies
uniformly to all of the end members. In Figure 1, if the data
points on the red dashed lines have high purity levels, that is,
they are on the boundaries of the simplex defined by the true
end members, then the volume-minimum simplex identifies the
true end members.
Figure 1. Schematic picture of the simplex volume minimization, which is
based on Figure 1 in Lin et al. (2015; see also Fujii et al. 2017). The black dots
represent the observed data, and the three triangles indicate a simplex that
encloses all of the data points. The dashed triangle is the simplex whose
volume is minimized. The end members are defined by three vertices of the
dashed triangle.
3
We note that the spectral unmixing in remote sensing is often expressed in
the form of ¢=¢¢
D
XA
T
() instead of that appearing in Equation (9), i.e.,
“spectral component first”, where ¢=¢=¢=
D
DX XA A,,
TT. We select the
form of Equation (9)because of the connectivity between the unmixing and
spin–orbit tomography as seen in Section 2.3.
3
The Astrophysical Journal, 894:58 (14pp), 2020 May 1 Kawahara
As the regularization term for the simplex volume minimiza-
tion, the Gram determinant of spectral components (VRDet)
l
=RA X XX,2det 19
XT
() () ()
å
l
=¢¢
XX
2det , 20
Xkk
l
kl k l,() ()
⎡
⎣
⎢⎤
⎦
⎥
was used (e.g., Schachtner et al. 2009; Zhou et al. 2011; Xiang
et al. 2015; Ang & Gillis 2019; Fu et al. 2019), where λ
X
is the
spectral regularization parameter. The Gram determinant,
Equation (19), is a surrogate of the volume of a convex hull
of spectral vectors, -
xx x, ,.... N01 1
k
(
), where l=xx
kkl
{(
˜)for
=¼-lN0, 1, , 1
;
l}such a convex hull is identifiable for its
well-spread data (Lin et al. 2015).
4
The minimization of the
convex hull of spectral vectors can be achieved by minimizing
Equation (19).
2.3. Unified Retrieval Method of Mapping and Spectra
Our task is to unify the spectral unmixing of Equation (9)
and spin–orbit tomography of Equation (6). To achieve this, we
assume a pixel-wise spectral unmixing
å
qfl==mmAX,, , 21
jjljl
k
jk kl
(˜)()
where X
kl
is the reflectivity of the k-th component at
wavelength
l
l
˜, and qf=
A
a,
jk k j j
()
is the surface distribution
of the k-th surface component at the j-th pixel instead of time in
Equation (9). Combining Equation (21)with the multicolor
version of Equation (6), we obtain
åå
==DWmWAX 22
il
j
ij jl
jk
ij jk kl ()
or simply,
=DWAX.23()
Equation (23)provides the general form of the spin–orbit
tomography with spectral unmixing.
The two-dimensional mapping thus far estimated the spectra
using PCA or a color difference prior to retrieving the
geographic distribution. This “unmixing first”strategy could
not feed back information on the fitting accuracy of the
geographic retrieval to spectral unmixing. The improvement of
Equation (23)over the spin–orbit tomography is that we fit
both the spectral components and geography to data in a
consistent manner.
Cowan & Strait (2013)solved an equation similar to
Equation (23)for Aand Xusing the multicolor light curve
provided by the EPOXI satellite as D. They retrieved a
longitudinal map of the surface components from the diurnal
rotation of the light curve. This procedure is referred to as
“rotational unmixing.”They used the Markov Chain Monte
Carlo method to determine the best parameters of Xand D.
Their optimization corresponds to the minimization of
=-QDWAX
1
224
F
2
∣∣ ∣∣ ( )
å
AXsubject to 1 0, 1 0, 25
k
jk kl ()
where
W
is the latitudinal average of the kernel.
5
The
minimization of Equation (24)under the constraint of
Equation (25)is formally identical to the weighted NMF (we
explain this in Section 2.4)with no regularization +the upper
limits of Aand X.
In general, matrix factorization has a degeneracy of
solutions. The transformation of ¬-
A
AG 1and ¬XGX
for
a regular matrix Gunder the constraint of Equation (25)does
not change the value of the cost function, Equation (24). This
means that if we change the spectral basis by a rotation of G,
then the inferred map should change too. This degeneracy is
known in the field of blind signal separation (see, e.g., Cichocki
et al. 2009). The nonuniqueness of the blind signal separation
might explain the mismatch between the inferred longitudinal
map by Cowan & Strait (2013)and the actual geography
suggested by Fujii et al. (2017).
The nonuniqueness feature of NMF can be avoided by
adding regularization when neglecting unavoidable scaling and
permutation ambiguities (Cichocki et al. 2009; Lin et al. 2015).
Our task is to find a unique (identifiable)solution to the spectral
unmixing and spin–orbit tomography. To achieve this, we
consider the cost function with regularization for both Xand A.
In this paper, we used the squared Euclidean distance as the
cost function for Equation (23),
=- +QDWAXRAX
1
2,26
F
2
∣∣ ∣∣ ( ) ( )
where
R
AX,()
is the regularization term. Similar to the rotation
unmixing for longitudinal mapping (Cowan & Strait 2013),we
call the two-dimensional mapping+unmixing+regularization
of Equation (26)“spin–orbit unmixing”in this paper.
2.4. Weighted Nonnegative Matrix Factorization
The nonnegative condition of Equation (26)yields an NMF
version of the unified retrieval model
=- +QDWAXRAXminimize 1
2,27
F
2
∣∣ ∣∣ ( ) ( )
AXsubject to 0, 0. 28
jk kl ()
As this formulation differs from a standard NMF,
Equation (17), for a weight W, we require an extension for
the optimization of a standard NMF to the weighted NMF.
2.4.1. Regularization
The regularization term suppresses the instability of the
retrieved map due to overfitting, otherwise known as over-
training in machine learning. A Tikhonov regularization (or an
L2 regularization)used in the original spin–orbit tomography
(Kawahara & Fujii 2011)can be extended to the regularization
term using the Frobenius norm, µ
R
AX A,F
2
( ) ∣∣ ∣∣ . Hence, a
4
The Gram determinant can be rewritten by the wedge product of the spectral
vectors as =
-
xx xXXdet ...
Tk01 1
2
( ) ∣∣ ∣∣ . Therefore, XXdet T
(
)
can be
regarded as the squared volume of the spectral vectors.
5
Besides the additional constraints on Xand A, the difference between the
equation in Cowan & Strait (2013)and Equation (23)is W. As rotational
unmixing performs the longitudinal mapping according to spin rotation, the
geometric kernel should be integrated unto the latitudinal direction,
f=WW(
)
. In the frame of the spin–orbit tomography, we need to use a
two-dimensional discretization of a sphere, qfW,(
)
.
4
The Astrophysical Journal, 894:58 (14pp), 2020 May 1 Kawahara
simple extension of the spin–orbit tomography is expressed as
l
=-RA X A,2L2 Unconstrained . 29
A
F
2
( ) ∣∣ ∣∣ ( ) ( )
However, in the case of “L2,”we do not have any
regularization for X(“L2-Unconstrained”). In spectral unmix-
ing, an assumption made on the convex hull of spectral
components can be expressed as a regularization term (a
function of X), as explained in Section 2.2.2. We consider a
combination of the Tikhonov regularization for mapping, and
the Gram determinant-type volume regularization for spectral
components, expressed as
ll
=+ -RA X A XX,22
det L2 VRDet ,
30
A
F
XT
2
( ) ∣∣ ∣∣ ( ) ( )
()
where λ
A
and λ
X
are the regularization parameters for Aand X,
respectively. We call this model “L2-VRDet”.
2.4.2. Optimization
The minimization of the cost function is performed by a
block coordinate descent, which consists of two separate
optimizations for Aand X(see Kim et al. 2014, as a review
paper). These optimizations are solved by minimizing the
quadratic forms
=-aabaq1
231
Ak
TAk A
Tk()
=-xxbxq1
2.32
Xk
TXk X
Tk()
of
a
k(the k-th column vector of A)and xk(the k-th row vector
of X)for k=0to -
N
1
k, iteratively (Zhou et al. 2011), where
A,
X,
b
A, and
b
Xare placeholders that depend on the cost
function.
As the L2 regularization of A, we minimize
=+-aalaq1
2L2 33
Ak
TAAk
A
Tk
() () ()
=xxWW 34
Ak
TkT()
=DlxW35
ATk()
l=I36
AAJ ()
where
D
=-
åå
¹
DWAX
il il sk j ij js sl,δ
I
JNN
jj
is an identity
matrix . The Xcomponent of the quadratic problem for the
VRDet model is calculated by
=+-xxlxq1
2VRDet , 37
Xk
TXXk
X
Tk
() ()()
=aWI 38
XkL
2
2
∣∣ ∣∣ ( )
=DlaW39
XTk()
l=-
-
XX I X XX Xdet , 40
XX k
k
T
Lk
T
kk
T
k
1
()[ ()] ()
where Xk
is a submatrix of Xwhen the k-th row of Xis
removed, and δ
I
LNN
ll
is the identity matrix. The derivations
of these terms are given in Appendix B.1.1.
Following Ang & Gillis (2019), we use the accelerated
projected gradient descent +restart (APG+restart)to optimize
the quadratic problems with nonnegative conditions. The APG
+restart is based on a projected gradient descent onto a positive
orthant with Nesterov’s acceleration and the restarting method.
The algorithm is summarized as follows.
Algorithm 1.
NMF/Block Coordinate Descent for Spin–Orbit Unmixing
Minimize -+DWAX RAX,
F
1
2
2
∣∣ ∣∣ (
)
s.t.
A
X,0
Initialize
A
X,
00() ()by random nonnegative values
while Condition do
for kin(0, N
k
−1)do
Update
x
kusing APG+restart
Update
a
kusing APG+restart
end for
end while
A more detailed description of the APG+restart algorithm is
given in Appendix B.2. Additionally, we show that the
traditional multiplicative update algorithm is extended for the
weighted NMF in Appendix C. The code for optimization is
publicly available.
6
3. Testing the Spin–Orbit Unmixing Using a Cloudless Toy
Model
We test the spin–orbit unmixing by using the volume-
regularized NMF and applying it to a toy model. The toy model
assumes three surface types on a planet: ocean, land, and
vegetation. The reflection spectra for land and vegetation were
taken from the ASTER spectral library (Baldridge et al. 2009),
and the ocean albedo is from McLinden et al. (1997),as
indicated by the gray lines in Figure 2. The input classification
of the map is based on the moderate resolution imaging
spectroradiometer classification map in 2008, as shown in the
left panel in Figure 3. We use the geometric settings of Fujii &
Kawahara (2012), an orbital inclination of 45°, an obliquity of
23°.4, and
Q
=
90
eq . We assume that the spin rotation period
is a sidereal day of Earth, 23.9344699/24.0 days, and we
assume an orbital revolution period P
orb
of 365 d. We took
N
i
=512 homogeneous samples over a year and injected a 1%
Gaussian noise into the light curve.
Figure 2. Input (gray)and unmixed spectral components (colors with markers)
for the L2-VRDet model with
l
=-
10
A1and
l
=10
X2.
6
https://github.com/HajimeKawahara/sot
5
The Astrophysical Journal, 894:58 (14pp), 2020 May 1 Kawahara
For the retrieval, we use a HEALPix map (Górski et al.
2005)and A
jk
with ==jN1, 2 .., 307
2
pix pixels. In this test,
we assume that we know the number of spectral components,
N
k
=3. Furthermore, we assume that we know the axial tilt
parameters gand set 10
5
as the number of iterations for the
optimization.
Figures 2and 4are examples of unmixed spectra and
retrieved maps for the L2-VRDet model (
l
=-
10
A1and
l
=10
X2). Because the normalization of each component is
arbitrary, we adjust the normalization of each component to the
input spectra. In this case, the input spectra and geography are
accurately reproduced by the unmixed spectra and their
retrieved distributions of components 0, 1, and 2, which
correspond to vegetation, land, and water, respectively. These
results indicate that the spin–orbit unmixing using the volume-
regularized NMF can infer the spectral components and their
geography simultaneously.
The sparsity of the retrieved maps is a notable feature of the
spin–orbit unmixing that utilizes NMF. Because of the
nonnegative constraint, large parts of the maps remain zero.
This feature was observed in the spin–orbit tomography using
BVLS in Kawahara & Fujii (2010). In contrast, the spin–orbit
tomography that uses the Tikhonov regularization does not
exhibit such sparsity (Kawahara & Fujii 2011).
We made a color composite map from the three maps in
Figure 4, as shown in the right panel of Figure 3. These results
show that the L2-VRDet model with an appropriate regulariza-
tion can infer a global composition map for the toy model.
We remind the reader that the spectra are well mixed even
for the cloudless toy model. To illustrate how the spectra are
unmixed, we project the input light curves and unmixed spectra
as end members onto the PC1–PC2 plane in Figure 5. The PCA
is computed using the input light curve. To draw this plot, we
first compute the normalized light curve via the mixing matrix
º
A
WA
˜; that is,
=DAX.41
˜()
The light curve is normalized as =å
D
DA
il il kik
˜˜. We then
derive PC1 and PC2 using
D
il
˜. The projection of
D
il
˜and Xonto
the PC1–PC2 plane, indicated by the orange crosses and red
points, was computed using Equation (16). The light curve
does not touch the boundary of the triangle, which is defined by
end members; that is, the triangle is not a convex hull of the
light curve. This is because the spectra of the light curve are
well mixed, and the purity is low. In this case, the geometric
disentanglement is essential for the spectra unmixing because
the end members are far from the trajectory of the light curves.
Figure 3. Left panel: input map of a toy model. The three colors indicate the different surface types, land, vegetation, and ocean, corresponding to white, gray, and
black, respectively. Right panel: color composite map for the same model. The color composite is based on the retrieved components in Figure 2; the components 0, 1,
and 2 correspond to green, orange, and blue, respectively.
Figure 4. Retrieved maps for different unmixed components 0, 1, and 2 from left to right. We adopt the L2-VRDet model with
l
=-
10
A1and
l
=10
X2.
Figure 5. Input light curve (orange cross), unmixed spectral components (red
points), and disentangled spectra (blue dots)on the PC1–PC2 plane. A simplex
defined by the components 0, 1, and 2 is shown by the gray triangle.
6
The Astrophysical Journal, 894:58 (14pp), 2020 May 1 Kawahara
The “disentangled spectra”of the light curve are defined by
ºXAX
˜; that is,
=DWX.42
˜()
The blue dots are the projection of the disentangled spectra
normalized by
å
A
kjk onto the PC1–PC2 plane. The disen-
tangled spectra are well spread in the triangle, and therefore,
the triangle defines a convex hull of the disentangled spectra.
The effect of the geometric disentanglement of the spectral
unmixing is visualized as the expansion from orange crosses to
blue dots in Figure 5.
3.1. Dependence on Regularization Parameters
The over-regularization of the simplex volume (i.e., large
λ
X
)induces a worse fit of the data. This is confirmed by the
mean residual of fitting the model to the data.
º-
D
DWAX
NN
mean residual 1,43
F
li
2
∣∣ ∣∣ ()
where
D
is the mean value of the data. The top panel of
Figure 6presents the mean residual as a function of λ
X
. The
mean residual gradually increases as the spectral regularization
parameter increases.
l
=-
10
X2and 10
−1
have similar mean
residuals, which indicates that the model fits the data well for
small regularization parameters. A smaller spectral regulariza-
tion provides fewer constraints on the simplex volume
minimization. Hence, there is a trade-off relation between the
models goodness of fit and volume minimization.
The second panel from the top illustrates a surrogate of the
spectral of volume of normalized spectral components
XXdet
T
(ˆˆ )
,
where =å
XX X
kl kl lkl
ˆ. This quantity decreases as the spectral
regularization parameter increases, which indicates that the
spectral volume is minimized more as λ
X
increases. The surrogate
of the normalized spectral volume roughly converges at
l
=10
X1
A direct comparison with the ground truth is useful to see
how λ
X
affects the estimate of the spectral components and
geography. To quantify the difference between the unmixed
spectra and ground truth, we define the mean removed spectral
angle (MRSA)between the two vectors xand yas
p
=--
--
-
xy xxyy
xx yy
MRSA , 1cos , 44
T
1
22
() ()()
∣∣ ∣∣ ∣∣ ∣∣ ()
⎛
⎝
⎜⎞
⎠
⎟
where xand yare the mean of xand y, respectively, and
MRSA Îxy,0,1
(
)[
]
. The two vectors perfectly match when
the MRSA =xy,0
(
). The third panel shows the mean of
MRSA over the components
å
=
xx NMRSA MRSA , , 45
k
kkk
() ()
where
xkis the ground truth (input spectrum). Moreover,
Figure 7shows the actual shapes of the unmixed spectra for
different λ
X
. The difference is observed for Component 2
(orange, land). The unmixed spectra when
l
10
X0resulted in
a worse fit to the ground truth. Interestingly, when
l
10
X3,
the fit of the unmixed spectra to the ground truth also
worsened. This is possibly due to the over-constraint on the
spectral model, which in turn restricts its ability to explain the
data accurately (large residuals); this results in an incomplete
estimate of the spectral components.
The comparison between the retrieved map and ground truth
is quantified by the Correct Pixel Rate (CPR), which is defined
by the correct answer rate of the classification map. The bottom
panel of Figure 6shows its dependence on λ
X
. Insufficient
spectral regularization resulted in not only a worse mean
MRSA but also a worse estimate of the geography. In regards
to both the mean MRSA and CPR, an optimal range for λ
X
of
10
1
–10
3
was observed.
Although we cannot compute the MRSA and CPR for
unknown geographies and surface spectra, these results suggest
that a curve of a surrogate of normalized spectral volume, as a
function of λ
X
, can be used to determine the optimal value of
λ
X
. We suggest the following procedure: (1)Plot the mean
residual and
XXdet
T
(ˆˆ )
as a function of λ
X
.(2)Observe the
change of the spectral shape as a function of λ
X
.(3)Use λ
X
at a
turning point of the spectral shape and
XXdet
T
(ˆˆ )
and avoid a
large value for the mean residual.
Figure 6. The residuals, the surrogate of the normalized spectral volume, mean
MSRA, and CPR as a function of λ
X
from top to bottom. We fix
l
=-
10
A1in
these panels.
Figure 7. Unmixed spectra for
l
=-
10 , 10
X22
, and 10
4
. The colors are the
same as in Figure 2.Wefix
l
=-
10
A1in these panels.
7
The Astrophysical Journal, 894:58 (14pp), 2020 May 1 Kawahara
Figure 8shows similar plots to those in Figure 6but for the
spatial regularization λ
A
. The over-constraint on A(i.e., large
λ
A
)contributes to a bad fit of the data and a smaller volume of
spectral components. Smaller spatial regularization parameters
resulted in a bad estimate of the geography, as indicated by the
CPR. This is because insufficient spatial regularization creates
a noisy map due to the instability of the mapping (see Figure 9
as an example). Contrastingly, a large λ
A
will slightly decreases
the CPR because it will imply that the inferred spectra are
getting worse, and the spatial resolution of the map is
decreasing in quality. This poor resolution will result in a
large mean residual. Therefore, one should check both the
residual and the surrogate of normalized spectral volume as a
function of λ
A
because these quantities have a trade-off
relation. We suggest choosing the optimal λ
A
as the smallest
value that (1)keeps the noise in the inferred map nonsignificant
and (2)avoids a large mean residual.
We have described how spatial and spectral regularization
parameters affect the results; we have also discussed a
guideline to follow when choosing the optimal spatial and
spectral parameters. To find the optimal λ
X
(or λ
A
),wefixed λ
A
(or λ
X
)in Figure 6(or Figure 8). In practice, this procedure
should be iterative so that we can find the optimal set of λ
A
and
λ
X
. We recognize that our current guideline for choosing the
optimal parameters is not quantitative. Ideally, the performance
of the prediction can be used to choose the optimal parameters,
such as cross validation. However, the large computational
time of the optimization method is too long to perform a cross
validation. Therefore, we postpone the quantitative criterion
needed to choose the parameters for further study.
3.2. Choice of the Number of Spectral Components
So far, we have assumed the number of spectral components
N
k
=3. Generally, N
k
should be one of the free parameters.
Here, we consider the cases for when N
k
=2(over-
constrained)and N
k
=4(under-constrained). The cost function
for N
k
=2(=´
Q
610
4)is much larger than that of N
k
=3
(Q=2355)and 4 (=
Q
2565). This indicates that N
k
=2is
insufficient to explain the data. For N
k
=4, we could not reach
the convergence of the cost function, Equation (26), with the
regularization term of Equation (30), even though the number
of iterations reached 10
6
, which was where we stopped.
Although information criteria such as the Akaike Information
Criterion (AIC)are used as the model selection for a different
number of free parameters, the degrees of freedom are not clear
for the inverse problem. Ignoring this fact, if we evaluate AIC by
-
+=-DWAX2 log Likelihood 2 degree of freedom F
2
( ) ( ) ∣∣ ∣∣
s
+NN2,
kj
2where σis the standard deviation of the input
noise, we obtain AIC=135158.8, 23141.4, and 28889.3 for
=
N
2, 3,
k
and 4, respectively. These results might indicate that
N
k
=3 is the optimal number for the components.
Another problem in real data is that it is often difficult to
estimate the likelihood because we do not understand the
statistical nature of the noise. In this case, cross validation is
often used as the model selection. However, cross validation is
unrealistic because of the high computational cost of the
current scheme. We postpone the criterion that will allow us to
choose the optimal number of surface components for further
study. Hence, in this paper, we require the number of surface
components as prior knowledge for mapping.
3.3. Comparison with Spectral Unmixing on Light Curves
So far, we have explained how geography is disentangled
from spectra in spin–orbit unmixing. Here, we consider spectral
unmixing on the light curve with no disentanglement of
geometry and compare it with the unified model. By
minimizing the cost function
l
=- +QDAX XX
1
22
det 46
F
XT
2
∣∣ ˜∣∣ ( ) ( )
AXsubject to 0, 0, 47
ik kl
˜()
we obtain the unmixed spectral components for different spectral
regularization as shown in Figure 10.Wefind that both components
0 and 1, which can be interpreted as surface components on
continents, are sensitive to the volume regularization. These results
Figure 8. Mean residual, a surrogate of the normalized spectral volume, mean
MSRA, and CPR as a function of λ
A
from top to bottom. We fix
l
=10
X2in
these panels.
Figure 9. Example of the color composite map for insufficient spatial
regularization (
l
=-
10
A3). We adopt
l
=10
X2in this figure.
8
The Astrophysical Journal, 894:58 (14pp), 2020 May 1 Kawahara
show that the NMF with simplex volume minimization works even
without geometric disentanglement.
Compared with the spin–orbit unmixing, the spectrum of soil
(gray dashed)is less reproduced by component 1 (orange)even
for the best case,
l
=10
X0, and the results are more sensitive to
the choice of λ
X
(see Figure 7for comparison). This is likely
because the geometric disentanglement is essential to suffi-
ciently separate the spectrum of soil from that of vegetation.
4. Application to DSCOVR Data
In this section, we demonstrate our method using real
multiband light curves of the Earth as observed by DSCOVR
(Jiang et al. 2018). DSCOVR has been continuously monitor-
ing our Earth from the L1 point since 2015. The geometry
provided by DSCOVR is not the same as the geometry
provided by direct imaging, because DSCOVR continuously
looks almost at the dayside of Earth. However, the geometric
kernel contains latitudinal information because of the axial tilt
of the Earth. This enables us to do a two-dimensional mapping
(Fan et al. 2019). We use seven optical bands (0.388, 0.443,
0.552, 0.680, 0.688, 0.764, and 0.779 μm)in DSCOVR filters
(N
l
=7). The bandwidths are very narrow (0.8–3.9 nm), and
there is strong oxygen B and A absorption in 0.688, 0.764 μm.
Owing to computational efficiency, we use one-fourth of the
two-year data (i.e., one in each four bins)used in Fan et al.
(2019), resulting in a number of N
i
=2435 time bins.
Figure 11 shows the unmixed spectra and color composite
map when we assume that N
k
=4. For regularization
parameters, we followed the procedure described in the
previous section. Figure 12 shows the mean MRSA and
surrogate of the normalized spectral volume. It was observed
that
l
=-
10
X4.
5
and
l
=-
10
A2are the optimal values, because
a significant increase of the mean residual is observed at the
range larger than these values. Component 1 accurately
reproduced the actual geography and blue spectrum of the
Figure 10. Unmixed spectral components (colors with markers)for the direct
spectral unmixing of the light curve with
l
=-
10 , 10 ,
X10
and 10
1
. The green
circles, orange squares, and blue triangles correspond to components 0, 1, and 2,
respectively. The gray lines are input spectra the same as those in Figure 2.
Figure 12. Mean residual and the surrogate of the normalized spectral volume
as functions of λ
A
(top;
l
=-
10
X4.5)and λ
X
(bottom;
l
=-
10
A2). We take
10
−2
as the optimal value of λ
A
because of a significant increase at
l
=-
10
A1.5. Also, we take -
1
04.5 as the optimal value of λ
X
because of a
significant increase in the mean residual at
l
=-
10
X
4
.
Figure 11. Normalized unmixed spectra (top panel)and color composite map
(bottom panel)for the DSCOVR data. In the top panel, both the 0.688 and
0.764 μm bands are strongly affected by oxygen absorption (shaded by blue
vertical lines). The bottom panel shows a color composite map. We use white,
blue, green, and brown colors for components 0, 1, 2, and 3, respectively.
9
The Astrophysical Journal, 894:58 (14pp), 2020 May 1 Kawahara
ocean; components 2 and 3 reproduced the continent distribu-
tion of Earth. Component 1 is less sensitive to the choice of λ
X
compared with components 2 and 3; therefore, component 1 is
a relatively robust estimate of a surface component. From the
unmixed spectrum, component 2 resembled the spectrum of
vegetation because of the increase larger than 0.688 μm;
although, the strong oxygen absorption at 0.688 and 0.764 μm
suppressed this increase to some extent. Component 3
corresponds to the spectrum of soil or sands. In fact, the
continent of Australia (less vegetation)was painted by
component 3. The southern part of Africa and the Amazon
(large forest areas)are roughly painted by component 2. We do
not have enough spatial resolution around North Africa,
Eurasia, and Europe. Although we did not consider our scheme
being able to clearly distinguish between soil and vegetation,
we believe that the differences between components 2 and 3
reflect the variety of spectra of land continents on Earth.
Component 0 exhibited a flat spectrum except for strong
oxygen absorption bands (0.688, 0.764 μm), reproducing the
cloud or ice spectrum. Component 0 as well as component 1
are less sensitive to the choice of λ
X
compared with
components 2 and 3. However, the distribution of component
0 is patchy, except for the localization at the North Pole. The
patchy distribution probably reflects a temporal cloud distribu-
tion because real clouds do not have a static distribution, some
of which might be from the ice near the pole.
These patchy pixels have values that are roughly α=5
times larger compared to those of other components. Also, the
unmixed spectrum of component 0 is β=50 times higher on
wavelength average than the total value of those of other
components. The fraction of the patchy pixels is about
g
~1 100. Multiplying α,β, and γ,wefind that the power
of component 0 in the patchy pixels is roughly several times
higher than the total power of other components. This value is
consistent with the contribution of clouds on reflected light on
Earth. The fact that the cloud component is localized in these
patchy pixels represents a limitation of the current method,
which assumes that all of the components have a static
distribution over the observation period. Further improvement
is needed so that non-static components can be included to the
model.
5. Summary and Discussion
In this paper, we constructed a unified retrieval model for
spectral unmixing and spin–orbit tomography (spin–orbit
unmixing)using nonnegative matrix factorization and L2 and
volume regularization. The spin–orbit unmixing works on the
cloudless toy model and real multicolor light curves by
DSCOVR. Here, we raise several remaining issues that we
did not consider in this study.
The simultaneous estimate of the axial tilt parameters gis
first. For simple two-dimensional mapping, Schwartz et al.
(2016)analyzed how the axial tilt parameters are inferred from
amplitude modulation, and Farr et al. (2018)constructed a
Bayesian framework to estimate the parameters. Similar work
should also be done in the spin–orbit unmixing. However, the
computational cost will be an issue that would need to be
addressed, as the optimization of NMF requires a high
numerical cost.
Moreover, the clock setting problem still remains. Thus far,
all of the works done on two-dimensional mapping assume that
we know the exact phase of the geometric kernel. The spin
rotation period, derived by the auto-correlation function, was
assumed to be used (Fujii & Kawahara 2012). However, as
Kawahara (2016)pointed out, the apparent periodicity of the
photometric variability is not identical to the spin rotation
period. The frequency modulation analysis provides the spin
rotation period. For instance, we need to check if the spin–orbit
unmixing works well when we use an inferred spin rotation
period from the frequency modulation. Otherwise, a technique
with a simultaneous estimate of the spin might be required.
The next challenge is how non-static compositions such as
clouds can be included in the model (see Luger et al. 2019,as
an attempt of the time-dependent mapping). This will become
vitally important when we apply this technique to gaseous
planets.
Another challenge is the dependency on results of various
types of regularization. For instance, Aizawa et al. (2020)
reported that the L1+TSV regularization provided better results
than the Tikhonov regularization. Furthermore, several other
types of volume regularization have been proposed in the field
of remote sensing (e.g., Ang & Gillis 2019); therefore, a
comparative study of regularization is required.
Additionally, a more quantitative criterion is needed to select
the optimal number of surface components and the regulariza-
tion parameters. Because of the high computational cost, cross
validation is unrealistic for the current scheme. An objective
criterion to select these parameters will help us to apply the
technique to unknown exoplanets where the ground truth is not
known.
The author is grateful to the DSCOVR team for making the
data publicly available. I deeply appreciate Siteng Fan and Yuk
L. Yung for providing the processed light curves and their
geometric kernel from the DSCOVR data set. I would also like
to thank Masataka Aizawa, Kento Masuda, and Nick Cowan
for their insightful discussions. I would also like to thank the
anonymous reviewer for a careful reading and constructive
suggestions. This work was supported by JSPS KAKENHI
grant Nos. JP17K14246, JP18H04577, JP18H01247, and
JP20H00170. This work was also supported by the JSPS
Core-to-Core Program Planet2 and SATELLITE Research
from Astrobiology center (AB022006).
Appendix A
Geometric Kernel of the Spin–Orbit Tomography
A.1. Disk-integrated Scattered Light
Here, we summarize the computation of the reflection light
from a planet to an observer. The outward energy from a facet
d
Ato a direction with a solid angle
Wd
(the left panel in
Figure A1)is expressed as
Jl=W
dE L dAd dcos , A1
1()
where L
↑
is the upward radiance, and
J
1is a zenith angle
between a direction and a normal vector. Let us assume that we
observe flux from a planet at a distance of dusing a telescope
with an effective area A
tel
, then light in a cone with a solid
angle W=
d
dA d
tel 2contributes to the flux. Therefore, the
flux from a facet
d
Aon a plane to the telescope area
d
Atel can be
written as
JJD= W=
EdA L d dA L
ddAdAcos cos . A2
tel 1 21tel ()
10
The Astrophysical Journal, 894:58 (14pp), 2020 May 1 Kawahara
Thus, we obtain the total flux from a planet as
òò J=D=
fEdA
L
dcos . A3
pplanet planet 21()
The BRDF of the surface element sis defined by the ratio of
the outward radiance to the inward irradiance,
JjJj p Jj
Jj
º
RE
,,, L,
,A4
s0011
11
00
()
()
() ()
where
J
0and
j0
are the solar zenith angle and azimuth angle,
respectively, and
j
1is the azimuth angle to an observer (see
Figure A1).
7
For most surface types, the BRDF almost solely
depends on a relative azimuth angle
j
jj=-
1
0
instead of
each azimuth angle as
JjJj JJj=RR,,, ,,. A5
ss
001101
()()()
The stellar irradiance is expressed as
JpJJ==
EL
a
fd
a4cos cos , A6
020
2
20
() ( )
where ais the star–planet distance, and
L
and
fare the stellar
luminosity and flux, respectively. The flux from a planet is
expressed as
ò
ò
J
pJJj J
pJJj J J
=
=W
fdA
E
dR
fR
adR
,, cos
,, coscos, A7
ps
ps
IV
0
201 1
2
2IV 101 0 1
() ()
() ()
where IV is the illuminated and visible region as shown in the
right panel of Figure A1.
Assuming an isotropic reflection JJj qf=
R
m,, ,
s01
()()
,
we obtain,
òqf qf=WfdWtm,, , A8
g
p1()() ()
where qf
W
t,,
g()
is the geometric kernel for the Lambert
approximation.
qf JJ J J
=>
p
Wt,, cos cos for cos , cos 0
0otherwise,
A9
g
fR
a01 0 1
p
2
2
()
()
⎪
⎪
⎧
⎨
⎩
Here, we define the three fundamental vectors, ee,,
SO
and eR,
which are the unit vector from the planet center to the stellar
center, from the planet center to the observer, and the normal
unit vector at the planet surface, respectively. Using them, we
can rewrite J=eecos R0S
·and J=eecos R1O
·. Using the
orbital phase Θand an orbital inclination i, we obtain
=Q-Q Q-Qecos , sin , 0 , A10
T
Seqeq
(( ) ( )) ( )
=Q-Qeiiisin cos , sin sin , cos , A11
T
Oeqeq
()()
where
Q
e
q
is the orbital phase at equinox.
We also define the spherical coordinate fixed on the planet
surface,
fq fq fq q
¢=e, cos sin , sin sin , cos . A12
RT
()( ) ()
Applying a spin rotation along Φand a rotation matrix z
()
as a function of a planet’s obliquity ζ, we get
zf q
fq
zf q zq
zf q zq
=¢+F
=
+F
+F +
-+F+
ee,
cos sin
cos sin sin sin cos
sin sin sin cos cos
.A13
RR
() ( )
()
()
()
()
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
The geometric weight is given by
qf=>
p
eeee ee ee
Wt,, for 0,
0otherwise.
A14
g
fR
aRR R RSOS O
p
2
2
() (·)(· ) · ·
()
⎪
⎪
⎧
⎨
⎩
In addition, we consider the case where the reflectivity is
constant and isotropic over the surface, JJj=
RR
,,
s01
()
(the
Lambert approximation). We take =e1, 0, 0 T
O()
and define
the phase angle b=ee
SO
·, i.e., bb=ecos , sin , 0 T
O()
.
These definitions yield Equation (A7):
òò
pfqqf
bfq bfq
=
´+
pb
pp
-+
ffRR
addsin cos
cos cos sin sin sin sin A15
p
p
2
22
2
0
2
()()
Figure A1. Left panel: incoming light and outcoming light of a small facet
d
Aon the surface of a planet. Right panel: the visible and illuminated (IV)region of a
planet surrounded by the orange curve.
7
We inserted a factor of πso that the BRDF becomes identical to the
reflectivity when the scattering is isotropic.
11
The Astrophysical Journal, 894:58 (14pp), 2020 May 1 Kawahara
fb=
RR
af
2
3,A16
p
p2
() ( )
⎛
⎝
⎜⎞
⎠
⎟
where
fb pbpb bº+-
1sin cos , A17
p() [ ( ) ] ( )
is the Lambert phase function.
Appendix B
Optimization of the Weighted NMF by a Block Coordinate
Descent
The block coordinate descent (e.g., Zhou et al. 2011; Kim
et al. 2014; Ang & Gillis 2019)consists of the following two
subproblems:
1. QP(A): optimization of a quadratic form for
a
k(the
column vector of A)
2. QP(X): optimization of a quadratic form for xk(the row
vector of X)
The block coordinate descent solves these quadratic problems
(QP(A)and QP(X)) iteratively using a nonnegative least square
(NNLS)scheme. In this appendix, we derive the quadratic
forms and then explain the projected gradient descent and its
accelerated versions as the NNLS solver.
B.1. Quadratic Programming
B.1.1. Quadratic Form for ak
A(log)likelihood term for the weighted NMF can be
rewritten in the quadratic form
åå å
-= D-DWAX WAX
1
2
1
2B1
F
il
il
j
ij jk kl
2
2
∣∣ ∣∣ ( )
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
åå å
åå å
=
-D +D
XX WA
WA X
1
2
1
2
B2
il
kl lk
T
j
ij jk
il
li
T
j
ij jk kl F
2
2
∣∣ ∣∣
()
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
åå å
åå å
=
-D+D
¢¢¢
XA WWA
XWA
1
2
1
2B3
l
kl
jj
kj
T
i
ji
Tij jk
jl
kl
i
li
Tij jk F
2
,
2
∣∣ ∣∣ ( )
⎛
⎝
⎜⎞
⎠
⎟
⎡
⎣
⎢
⎢
⎛
⎝
⎜⎞
⎠
⎟
⎤
⎦
⎥
⎥
=-+aala
1
2const B4
k
TAk A
Tk()
where
ºxxWW B5
Ak
TkT()
ºDlxW,B6
ATk()
and
D
=Dk()
is defined by
D
º-
åå
¹
DWAX
il il sk j ij js sl.
The penalty of the Tikhonov regularization (L2 term)is
l=+aaA
1
2
1
2const B7
AFk
TAk
2
∣∣ ∣∣ ( )
lºI.B8
AA ()
Thus, the quadratic programming for the weighted NMF with a
spatial Tikhonov regularization minimizes
=+-aalaq1
2.B9
Ak
TAAk
A
Tk
() ()
B.1.2. Quadratic Form for xk
Likewise, we obtain the (log)likelihood term as a quadratic
form of xkfrom Equation (B3)as
-= -+xxlxDWAX
1
2
1
2const. B10
Fk
TXk X
Tk
2
∣∣ ∣∣ ( )
ºaWI B11
Xk
2
2
∣∣ ∣∣ ( )
ºDlaW.B12
XTk()
The volume regularization of the Gram determinant term can
be written in the quadratic form of xk
l=xxXX
1
2det 1
2B13
XTk
TXk
() ()
lº-
-
XX I X XX Xdet , B14
XX k
k
T
k
T
kk
T
k
1
()[ ()] ()
where Xk
is a submatrix of Xwhen we remove the k-th row of
X. The derivation of Equation (B13)is given in Zhou et al.
(2011).
In Table B1, we summarize the quadratic terms for different
regularization types. This list also includes the log-determinant
type of the volume regularization (Ang & Gillis 2018)and a
simple L2 term for xk.
B.2. Projected Gradient Descent
The projected gradient descent (PG)-based methods to solve
a quadratic problem,
=-xxbxqB15
TT ()
are described. The gradient descent with a nonnegative
condition is given by
hh=-=- -
+xx xxbq,B16
tt tt1[][()]()
( ) () () ()
where the projection operator on a nonnegative orthant is
defined by =
xxmax , 0
k
[] { ( )}
. We obtain the PG algorithm
by adopting the inverse of the Lipschitz constant Lto η.
As the Lipschitz constant, one can use the 2-norm of
=xxmax
222
∣
∣ ∣∣ (∣∣ ∣∣ ∣∣ ∣∣ ) for ιxx,0
mor a Frobe-
nius norm of =åå =tr
Fji ij T
2
∣
∣∣∣ ( )
. Although
the 2-norm is more efficient than the Frobenius norm (i.e.,
F2
∣
∣∣∣ ∣∣∣∣
), the computational cost of the 2-norm is much
higher than that of the Frobenius norm, especially for a large
matrix.
8
Projected Gradient Descent (PG)
Minimization of =-xxbxqTT
Initialization: =- =sb xTI L L,,
0
whileCondition do
=+
+
x
xsT
tt1[
]
() ()
end while
The convergence rate of the PG algorithm is relatively slow.
The PG algorithm with Nesterov’s acceleration (Nesterov 1983)
8
Therefore, we use a 2-norm for Xand a Frobenius norm for A.
12
The Astrophysical Journal, 894:58 (14pp), 2020 May 1 Kawahara
is called the accelerated projected gradient descent. The APG
algorithm is summarized as follows.
Accelerated Projected Gradient Descent (APG)
Minimization of =-xxbxqTT
Initialization: a=- = = =sb xy xTI L L,,, ,0.9
00 00
() () ()
while Condition do
=+
+
x
ysT
tt1[
]
() ()
aaaa=+-
+4
2
tttt
1422
()
baaaa=- +
++
1
tttt t
11
2
()(
)
b=+ -
++ ++
y
xxx
tt
ttt11 11
(
)
() () () ()
end while
A residual curve as a function of iteration using Nesterov’s
acceleration is not monotonic. Restarting Nesterov’s accelera-
tion when the residual increases significantly improves the
convergence rate (O’donoghue & Candes 2015).
APG+restart
Minimization of =-xxbxqTT
Initialization: a=- = = =sb xy xTI L L,,, ,0.9
00 00
() () ()
while Condition do
=+
+
x
ysT
tt1[
]
() ()
=-
++ + +
xxbxqttTtTt11 1 1
()
() () () ()
aaaa=+-
+4
2
tttt
1422
()
baaaa=- +
++
1
tttt t
11
2
()(
)
b=+ -
++ ++
y
xxx
tt
ttt11 11
(
)
() () () ()
iF>
+
qq
tt1() ()
then
=+
+
x
xsT
tt1[
]
() ()
aa==
++
+
y
x,
tt
t
11
1
0
() ()
end if
end while
Figure C1 shows a comparison of the above three algorithms
for a randomly generated matrix Aand a vector
p
as a quadratic
problem (-xxbx
TT
)for =
A
A
T(100×100 matrix)and
=
b
pAT. The residual after the t-th iteration is defined by
-xbAt2
2
∣
∣∣∣
() , where xt() is the estimated value after titerations.
The drawback of the PG and APG method is that
convergence is sensitive to the initial point. When all of the
components of h-D
xQ
0
[
]
are zero, the algorithm fails.
Appendix C
Optimization of Weighted NMF by Multiplicative Update
The multiplicative iterative algorithm (Lee & Seung 1999)is
often used to minimize the cost function of the standard NMF,
given in Equation (17). It can be directly derived from the cost
function and the Karush–Kuhn–Tucker first-order optimal
conditions (Cichocki et al. 2009). We need to extend the
standard multiplicative iterative algorithm to include the
geometric kernel Win Equation (27).
First, let us explain the algorithm for the weighted NMF with
no regularization term (=
R
A0() ). Following the derivation of
the multiplicative iterative algorithm, we compute the deriva-
tive of the cost function of Equation (27)as
= -Q W WAXX W DX C1
ATTTT ()
= -QAWWAXAWD.C2
XTT TT ()
To ensure the nonnegativity, we divide the derivative of the
cost function into the positive terms and negative terms
= - =
+-
QQ Q0, C3[] [] ()
where
-+
QQ0, 0
[
][]
. The multiplicative update is
an operation that multiplies
-+
QQ
[
][ ]
by Aor X. This
procedure can be interpreted as the steepest gradient descent
h¬- AA Q C4
AA()
h=
+
AQ C5
AA
[] ()
Table B1
Terms in Quadratic Problems
Term Cost Function (
´2
)
Aor
X
b
Aor
b
X
Likelihood for
a
k-DWAX
F
2
∣
∣∣∣ =
xxWW
Ak
TkT=D
l
xW
ATk
Tikhonov (L2)term for
a
k
l
A
AF
2
∣∣ ∣∣ l=
I
AA
J
L
Likelihood for
x
k-DWAX
F
2
∣
∣∣∣ =
aWI
XkL
2
2
∣∣ ∣∣ =D
l
aW
XTk
Volume Regularization (Det)
l
XXdet
XT
(
)
l=-
-
XX I X XX Xdet
XX k
k
T
Lk
T
kk
T
k
1
()[ ()]
L
Volume Regularization (Logdet)
l
d+XX Ilog det
XTK
[( )
]
lm=-
I
XX L
min
1L
Tikhonov (L2)term for
x
k
l
X
XF
2
∣∣ ∣∣ l=
I
XXL L
Note. Xk
is a submatrix of Xby removing the k-th row of X,D= -
åå
¹
DWAX
il il sk j ij js sl ,I
J
(or I
L
,I
K
)is an identity matrix δ
NN
jj
(or ´
NN
ll
,´
NN
kk
), and δis a
small number (we adopt 10
−6
). The minimum eigenvalue of d=+
E
XX Idet TK
(
)
is denoted by
m
min. In practice, we use Xin the previous iteration to compute
X
(Ang & Gillis 2018).
Figure C1. Residual of various PG solvers as a function of the number of
iterations.
13
The Astrophysical Journal, 894:58 (14pp), 2020 May 1 Kawahara
and
h¬- XX Q C6
XX()
h=
+
XQ C7
XX
[] ()
where eindicates the Hadamard product (the element-wise
product of two matrices), and %is the element-wise division.
The multiplicative iterative algorithm for the weighted NMF
with no regularization is given by
¬+
+
AA
WDX
W WAXX C8
jk jk
TT
jk
TT
jk
[]
[] ()
¬+
+
XX
AW D
AW WAX ,C9
kl kl
TTkl
TT kl
[]
[] ()
where òis a small value to prevent division by zero. To include
the regularization, the derivative of
R
AX,()
by Aor Xis
needed. For the dual-L2 type, we obtain
l=RA X A,C10
AA
() ()
l=RA X X,. C11
XX
() ()
Because these values remain positive when we take positive
values for the initial state, the multiplicative update for the
dual-L2-type regularization is expressed as
l
¬+
++
AA WDX
W WAXX A
UA: C12
jk jk
TT
jk
TT
Ajk
() []
[]
()
l
¬+
++
XX AW D
AW WAX X
UX: . C13
kl kl
TTkl
TT Xkl
() []
[]
()
Appendix D
On the Additional Constraints
In remote sensing, an additional constraint is sometimes applied.
For instance, the normalization for a spectrum is expressed by
å=X1. D1
l
kl ()
We found that the constraint of Equation (D1)functions as a
form of regularization if we combine the constraint with the
volume-regularization term. When we use the constraint of
Equation (D1)with the volume-regularization term, the effect
of the volume-regularization vanishes. We do not recommend
the use of the constraint of Equation (D1)in our case.
ORCID iDs
Hajime Kawahara https://orcid.org/0000-0003-3309-9134
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