Imaging the Schwarzschild-radius-scale Structure of M87
with the Event Horizon Telescope Using Sparse Modeling
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CitationAkiyama, Kazunori, Kazuki Kuramochi, Shiro Ikeda, Vincent L. Fish,
Fumie Tazaki, Mareki Honma, Sheperd S. Doeleman, et al. “Imaging
the Schwarzschild-Radius-Scale Structure of M87 with the Event
Horizon Telescope Using Sparse Modeling.” The Astrophysical
Journal 838, no. 1 (March 17, 2017): 1. © 2017 The American
Astronomical Society
As Publishedhttp://dx.doi.org/10.3847/1538-4357/aa6305
PublisherIOP Publishing
VersionFinal published version
Citable linkhttp://hdl.handle.net/1721.1/109549
Terms of UseArticle is made available in accordance with the publisher's
policy and may be subject to US copyright law. Please refer to the
publisher's site for terms of use.
Imaging the Schwarzschild-radius-scale Structure of M87 with the
Event Horizon Telescope Using Sparse Modeling
Kazunori Akiyama
1,2,3,4,15
, Kazuki Kuramochi
2,3
, Shiro Ikeda
5,6
, Vincent L. Fish
1
, Fumie Tazaki
2
, Mareki Honma
2,7
,
Sheperd S. Doeleman
1,4,8
, Avery E. Broderick
9,10
, Jason Dexter
11
, Monika Mościbrodzka
12
, Katherine L. Bouman
13
,
Andrew A. Chael
4,7
, and Masamichi Zaizen
14
1
Massachusetts Institute of Technology, Haystack Observatory, Route 40, Westford, MA 01886, USA; kazu@haystack.mit.edu
2
Mizusawa VLBI Observatory, National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan
3
Department of Astronomy, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
4
Black Hole Initiative, Harvard University, 20 Garden Street,Cambridge, MA 02138,USA
5
Department of Statistical Science, School of Multidisciplinary Sciences, Graduate University for Advanced Studies,
10-3 Midori-cho, Tachikawa, Tokyo 190-8562, Japan
6
Graduate University for Advanced Studies, 10-3 Midori-cho, Tachikawa, Tokyo 190-8562, Japan
7
Department of Astronomical Science, School of Physical Sciences, Graduate University for Advanced Studies, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan
8
Harvard Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA
9
Perimeter Institute for Theoretical Physics, 31 Caroline Street, North Waterloo, Ontario N2L 2Y5, Canada
10
Department of Physics and Astronomy, University of Waterloo, 200 University Avenue West, Waterloo, Ontario N2l 3G1, Canada
11
Max Planck Institute for Extraterrestrial Physics, Giessenbachstr. 1, D-85748 Garching, Germany
12
Department of Astrophysics/IMAPP, Radboud University Nijmegen, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands
13
Massachusetts Institute of Technology, Computer Science and Artificial Intelligence Laboratory, 32 Vassar Street, Cambridge, MA 02139, USA
14
Department of Astronomy, School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
Received 2016 November 1; revised 2017 February 17; accepted 2017 February 23; published 2017 March 17
Abstract
We propose a new imaging technique for radio and optical/infrared interferometry. The proposed technique
reconstructs the image from the visibility amplitude and closure phase, which are standard data products of short-
millimeter very long baseline interferometers such as the Event Horizon Telescope (EHT) and optical/infrared
interferometers, by utilizing two regularization functions: the ℓ
1
-norm and total variation (TV) of the brightness
distribution. In the proposed method, optimal regularization parameters, which represent the sparseness and
effective spatial resolution of the image, are derived from data themselves using cross-validation (CV).Asan
application of this technique, we present simulated observations of M87 with the EHT based on four physically
motivated models. We confirm that ℓ
1
+TV regularization can achieve an optimal resolution of ∼20%–30% of the
diffraction limit λ/D
max
, which is the nominal spatial resolution of a radio interferometer. With the proposed
technique, the EHT can robustly and reasonably achieve super-resolution sufficient to clearly resolve the black hole
shadow. These results make it promising for the EHT to provide an unprecedented view of the event-horizon-scale
structure in the vicinity of the supermassive black hole in M87 and also the Galactic center Sgr A
*
.
Key words: accretion, accretion disks – black hole physics – Galaxy: center – submillimeter: general –
techniques: interferometric
1. Introduction
Supermassive black holes (SMBHs) reside in the majority of
the galactic nuclei in the universe. In a subset of such galaxies,
accretion drives a highly energetic active galactic nucleus often
associated with powerful jets. Understanding the nature of
these systems has been a central goal in astronomy and
astrophysics. The SMBHs at the centersof our galaxy (Sgr A
*
)
and the giant elliptical galaxy M87 provide unprecedented
opportunities to directly image the innermost regions close to
the central black hole, since the angular size of the event
horizon is the largest among known black holes. The angular
size of the Schwarzschild radius (R
s
) is ∼10μas for Sgr A
*
for
a distance of 8.3 kpc and a mass of 4.3×10
6
M
e
(e.g.,
Chatzopoulos et al. 2015), and ∼3–7 μas for M87 with a
distance of16.7Mpc (Blakeslee et al. 2009) and a mass of
3–6×10
9
M
e
(e.g., Gebhardt et al. 2011; Walsh et al. 2013).
The apparent diameter of the dark silhouette of the black hole is
27
R
s
for the non-rotating black hole. It corresponds with
∼52 μas for Sgr A
*
and ∼16–36 μas for M87, which changes
by only 4% with the black hole spin and viewing orientation
(Bardeen
1973).
Very long baseline interferometric (VLBI) observations at
short/submillimeter wavelengths (λ1.3 mm, ν230 GHz)
can achieve a spatial resolution of a few tens of microarcseconds
and therefore are expected to resolve event-horizon-scale
structures, including the shadow of SMBHs. Indeed, recent
significant progress on 1.3mm VLBI observations with the
Event Horizon Telescope (EHT; Doeleman et al.
2009) has
succeeded in resolving compact structures of a few R
s
in the
vicinity of the SMBHs in both Sgr A
*
and M87 (e.g., Doeleman
et al.
2008, 2012;Fishetal.2011, 2016; Akiyama et al. 2015;
Johnson et al.
2015). Direct imaging of these scales will be
accessible in the next few years with technical developments and
the addition of new (sub)millimeter telescopes such as the
Atacama Large Submillimeter/millimeter Array (ALMA) to the
EHT (e.g., Fish et al. 2013).
Regardless of the observing wavelength, angular resolution
(often referred to as “beam size” in radio astronomy and
“diffraction limit” in optical astronomy ) is simply given by
θ≈λ/D
max
, where λ and D
max
are the observed wavelength
and the diameter of the telescope (or the longest baseline length
The Astrophysical Journal, 838:1 (13pp), 2017 March 20
https://doi.org/10.3847/1538-4357/aa6305
© 2017. The American Astronomical Society. All rights reserved.
15
JSPS Postdoctoral Fellow for Research Abroad.
1
for the radio interferometer ), respectively. A practical limit for
a ground-based, 1.3mm VLBI array like the EHT is ∼25 μas
(=1.3 mm/10,000 km), which is comparable to the radius of
the black hole shadow in M87 and Sgr A
*
. Hence, imaging
techniques with good imaging fidelity at a spatial resolution
smaller than λ/D
max
would be desirable, particularly for future
EHT observations of M87 and Sgr A
*
.
The imaging problem of interferometry is formulated as an
under-determined linear problem when reconstructing the
image from full-complex visibilities that are Fourier compo-
nents of the source image. In the context of compressed sensing
(also known as “compressive sensing”), it has been revealed
that an ill-posed linear problem may be solved accurately if the
underling solution vector is sparse (Candes & Tao
2006;
Donoho
2006). Since then, many imaging methods
16
have been
applied to radio interferometry after pioneering work by Wiaux
et al. (
2009a) and Wiaux et al. (2009b); (see Garsden
et al.
2015, and references therein). We call these approaches
“sparse modeling” since they utilize the sparsity of the ground
truth.
In Honma et al. (
2014), we applied LASSO (Least Absolute
Shrinkage and Selection Operator; Tibshirani
1996),a
technique of sparse modeling, to interferometric imaging.
LASSO solves under-determined ill-posed problems by utiliz-
ing the ℓ
1
-norm (see Section 2.2 for details). Minimizing the
ℓ
1
-norm of the solution reduces the number of non-zero
parameters in the solution, equivalent to choosing a sparse
solution. The philosophy of LASSO is therefore similar to that
of the traditional CLEAN technique (Högbom
1974), which
favors sparsity in the reconstructed image and has been
independently developed as Matching Pursuit (Mallat &
Zhang 1993) in statistical mathematics for sparse reconstruc-
tion. In Honma et al. (
2014), we found that LASSO can
potentially reconstruct structure ∼4 times finer than λ/D
max
.
Indeed, it works well for imaging the black hole shadow for
M87 with the EHT in simulations.
Our previous work (Honma et al. 2014) has three relevant
issues. The first issue is reconstructing the image only from the
visibility amplitudes and closure phases (see Section
2.1),
which have been the standard data products of EHT
observations (Lu et al.
2012, 2013; Akiyama et al. 2015;
Wagner et al.
2015; Fish et al. 2016) and optical/infrared
interferometry. The algorithm of Honma et al. (
2014) is
applicable only for full-complex visibilities, which are the
usual data products from longer-wavelength radio interferom-
eters. We have recently developed a fast and computationally
cheap method to retrieve the visibility phases from closure
phases (PRECL; Ikeda et al.
2016), which can reconstruct the
black hole shadow of M87 combined with LASSO in simulated
EHT observations. However, since the phase reconstruction in
PRECL adopts a different prior on visibilities than LASSO, the
resultant image may not be optimized well in terms of ℓ
1
-norm
minimization and sparse image reconstruction. Another
potential approach is to solve for the image and visibility
phases with ℓ
1
-norm regularization simultaneously, enabling us
to reconstruct the image with thefull advantage of the
regularization function.
The s econd i ssue is that the ℓ
1
-norm regularization might
not provi de a unique solutio n an d/or could reconstruct an
image that is too sparse of animage if the number of pixels
with non-zero brightness is not small enough compared to the
number of pixels. This violates a critical assumption in
technique s with ℓ
1
-norm regularization that the solution (i.e.,
the true image) should be sparse. Such a situation may occur
foranextendedsourceoralsoevenforacompactsourceifthe
imaging pixel size is set to b e much smaller than the size of
the emission structure. Pioneering work has made use of
transforms to wavelet or curvele t basesin which the image
can be represen te d spar sely (e.g., Li et al.
2011;Carrillo
et al.
2012, 2014; Dabbech et al. 2015 ; Garsden et al. 2015).
As another strategy to resolve this potential issue, in Honma
et al . (
2014), we proposed the addi tion ofanot he r regulariza-
tion, Total Variation (TV; e.g., Rudin et al. 1992),whichis
another p op ula r regularizati on in spars e modeling. T V is a
good i nd ica tor for spar si ty of the image in its gradi ent do main
instead of the image domai n (see Section
2.2 for details)
and it has been applied to astronomical imaging (e.g.,
Wiaux et al.
2010;McEwen&Wiaux2011;Carrillo
et al.
2012, 2014;Uemuraetal.2015; Chael et al. 2016)
that includes optical interferometric imaging without the
visibility phases (e.g., MiRA; Thi éba ut
2008;andalsosee
Thiébaut 2013 for a review). TV regul ar izat ion ge ner al ly
favors a smooth image (i.e., with larger effective resolution)
but with a sharp edge, in contrast with maximum entropy
methods (MEM; e.g., Narayan & Nityananda
1986),which
favor a smooth edge (see, e.g., Uemura et al. 2015,for
comparison between T V and MEM). Inclusion of TV
regularization enables reconstruction of an extended image
while preserving s harp emission features preferred by ℓ
1
-norm
regulariza tion , thereby extending the class of objects whe re
sparse modeling is applicable. Indeed, regularization with
both the ℓ
1
-norm and TV has been shown to be e ff ect ive for
imaging polarizat ion with full-complex visibilities in our
recent work (Akiyama et al. 2017).
An important detail is the determination of regularization
parameters (e.g., weights on regularization functions),which
is common in the vast majority of existing techniques. Since
one cannot know the true image of the source a priori, one
should evaluate goodness-of-fitting and select appropriate
regularization parameters from the data the msel ve s. In
well-posed problems, one can use statisti cal quanti ties
considering residuals between data and m odels as well as
model complexity to avoid over- fitting, such as reduced χ
2
,
the Akaike information criterion and the Bayesian inform-
ation criterion using the degrees of freedom to constrain
model complexity. However, for ill-posed problems like
interferometric imaging, degrees of freedom cannot be
rigorously defined, preventing the use of such statistical
quantities.
In this paper, we propose a new technique to reconstruct
images from interferometric data using sparse modeling. The
proposed technique directly solves the image from visibility
amplitudes and closure phases. In addition to the ℓ
1
-norm
(LASSO), we also utilize another new regularization term, TV,
so that a high-fidelity image will be obtained even with a small
pixel size and/or for extended sources. Furthermore, we
propose a method to determine optimal regularization para-
meters with cross-validation (CV; see Section
2.3), which can
be applied to many existing imaging techniques. As an
example, in this paper, we applied our new technique to data
obtained from simulated observations of M87 with the array of
the EHT expected in Spring 2017.
16
A list of papers are available in https://ui.adsabs.harvard.edu/#/public-
libraries/wmxthNHHQrGDS2aKt3gXow.
2
The Astrophysical Journal, 838:1 (13pp), 2017 March 20 Akiyama et al.
2. The Proposed Method
2.1. A Brief Introduction of the Closure Phase
A goal of radio and optical/infrared interferometry is to
obtain the brightness distribution I(x, y) of a target source at a
wavelength λ or a frequency ν, where (x, y) is a sky coordinate
relative to a reference position, so-called the phase-tracking
center. The observed quantity is a complex function called
visibility V(u, v), which is related to I(x, y) by two-dimensional
Fourier transform given by
ò
=
p-+
() () ()
()
Vuv dxdyIxye,,.1
iuxvy2
Here, the spatial frequency (u, v) corresponds to the baseline
vector (in units of the observing wavelength λ) between two
antennas projected to the tangent plane of the celestial sphere at
the phase-tracking center.
Observed visibilities are discrete quantities, and the sky
image can be approximated by a pixellated version where the
pixel size is much smaller than the nominal resolution of the
interferometer. The image can therefore be represented as a
discrete vector
I
, related to the Stokes visibilities
V
by a
discrete Fourier transform
F
:
= ()VFI.2
The sampling of visibilities is almost always incomplete. Since
the number of visibility samples
V
is smaller than the number
of pixels in the image, solving the above equation for the image
I
is an ill-posed problem.
Here, we consider that the complex visibility V
j
is obtained from
observation(s) with multiple antennas. Let us define its phase and
amplitude as f
j
and
¯
V
j
, respectively, denoted as follows
=
f
¯
()VVe,3
jj
i
j
where j is the index of the measurement. Each measurement
corresponds to a point
()
uv,
jj
in the (u, v)-plane and recorded at
time t
j
. In actual observations, some instrumental effects and the
atmospheric turbulence primary from the troposphere induce the
antenna-based errors in the visibility phase, leading to the observed
phase
f
˜
j
being offset from the true phase f
j
ofthetrueimage.In
particular, this is a serious problem in VLBI observations
performed at different sites (see Thompson et al.
2001).
However, the robust interferometric phase information can
be obtained through the measurements of the closure phase,
defined as a combination of triple phases on a closed triangle of
baselines recorded at the same time. It is known that the closure
phase is free from antenna-based phase errors (Jennison
1958),
which can be seen from the following definition of the closure
phase,
y ffffff=++=++
˜˜ ˜
(),4
m
jkl jkl
123
12 23 31
12 23 31
where m is the index of the closure phase, and upper numbers (1,
2, 3) mean the index of stations involved in the closure phase or
the visibility phase. The closure phase is also known as a phase
term of the triple product of visibilities on closed baselines
recorded at the same time,
V
VV
jkl
12 23 31
, known as the bi-
spectrum.
17
Closurephaseshavebeenusedtocalibratevisibility
phases in VLBI observations (e.g., Rogers et al.
1974).
In short/submillimeter VLBI or optical/infrared interfero-
metry, the stochastic atmospheric turbulence in the troposphere
over each station induces a rapid phase rotation in the visibility,
making it difficult to calibrate or even measure the visibility
phase reliably (e.g., see Rogers et al. 1995; Thiébaut 2013).
Thus, image reconstruction using more robust closure phases,
free from station-based phase errors, is useful for interfero-
metric imaging with such interferometers.
2.2. Image Reconstruction from Visibility Amplitudes and
Closure Phases
In this paper, we propose a method to solve the two-
dimensional image
= {}
I
I
ij,
by the following equations:
() ()ICImin subject to 0. 5
I
ij,
The cost function C(
I
) is defined as
ch h=+ +( ) ( ) ∣∣ ∣∣ ∣∣ ∣∣ ( )III IC ,6
lt
2
1tv
where
∣
∣∣∣I
p
is l
p
-norm of the vector
I
given by
åå
=>
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
∣∣ ∣∣ ∣ ∣ ( ) ( )I Ipfor 0 , 7
p
ij
ij
p
,
p
1
and
∣
∣∣∣I
t
v
indicates an operator of TV.
The first term of Equation (
6) is the traditional χ
2
term
representing the deviation between the reconstructed image and
observational data (i.e., the visibility amplitude
=
¯
{∣ ∣}V V
j
and
closure phase
yY = {}
m
),defined by
c Y=- +-() ∣∣
¯
( )∣∣ ∣∣ ( )∣∣ ( )IVAFI BFI,8
2
2
2
2
2
where
A
and
B
indicate operators to calculate the visibility
amplitude and closure phase, respectively. Deviations between
the model and observational data are normalized with the
errors. This form of the residual sum of squares (RSS) is
originally proposed in the Bi-spectrum Maximum Entropy
Method (Buscher
1994) and also for modeling EHT data (e.g.,
Lu et al.
2012, 2013). Note that it could be replaced with an
RSS term for bi-spectra (e.g., Bouman et al.
2015; Chael
et al.
2016).
The second term represents LASSO-like regularization using
the ℓ
1
-norm. Under the non-negative condition, ℓ
1
-norm is
equivalent to the total flux. η
l
is the regularization parameter for
LASSO, adjusting the degree of sparsity by changing the
weight of the ℓ
1
-norm penalty. In general, a large η
l
prefers a
solution with very few non-zero components, while η
l
=0
introduces no sparsity. In this paper, we use the normalized
regularization parameter
h
˜
l
defined by
hhº+
˜
()(
¯
)()VNN max , 9
ll
amp cphase
which is less affected by the number of visibility amplitude and
closure phase data points, N
amp
and N
cphase
, respectively, and
also by the total flux density of the target source.
17
Data products of visibility amplitudes and closure phases are also
sometimes named as “bi-spectrum” in the literature (e.g., Buscher
1994).In
this paper, we strictly distinguish them.
3
The Astrophysical Journal, 838:1 (13pp), 2017 March 20 Akiyama et al.
The third term is the TV regularization, defined by the sum
of all differences of the brightness between adjacent image
pixels. In this paper, we adopt a typical form for two-
dimensional images (Rudin et al.
1992) that has been used in
astronomical imaging, (e.g., Wiaux et al.
2010; McEwen &
Wiaux
2011; Uemura et al. 2015; Chael et al. 2016),defined as
åå
=-+-
++
∣∣ ∣∣ ∣ ∣ ∣ ∣ ( )I IIII.10
ij
i j ij ij ijtv 1, ,
2
,1 ,
2
TV is a good indicator of image sparsity in its gradient domain
instead of the image domain. TV is highly affected by the
effective spatial resolution of the image. For instance, an image
with a small TV value has blocks of image pixels whose
brightness are similar, since the image is sparse in the gradient
domain. The size of such blocks is equivalent to the effective
spatial resolution, getting smaller for images with higher TV
values. Thus, the regularization parameter η
t
adjusts the
effective spatial resolution of the reconstructed image. In
general, a larger (smaller) η
t
prefers asmoother (finer)
distribution of power with less (higher) discreteness, leading
to larger (smaller) angular resolution. In the present work, we
use the normalized regularization parameter
h
˜
t
defined by
hhº+
˜
()(
¯
)()VNN 4max , 11
tt
amp cphase
similar to the LASSO term. A factor of four is based on a
property of TV that displaysa difference in the brightness in all
four directions at each pixel. Note that a major difference to
maximum entropy methods, which also favor smooth images,
is that TV regularization has a strong advantage in edge-
preserving; strong TV regularization favors a piecewise smooth
structure, but with clear and often sharp boundaries between
non-emitting and emitting regions.
The problem described in Equations (
5) and (6) is nonlinear
minimum optimization. In this work, we adopt a nonlinear
programming algorithm L-BFGS-B (Byrd et al. 1995; Zhu
et al.
1997) that is an iterative method for solving bound-
constrained nonlinear optimization problems. L-BFGS-B is one
of the quasi-Newton methods that approximates the Broyden–
Fletcher–Goldfarb–Shanno (BFGS) algorithm using a limited
amount of computer memory. In L-BFGS-B, the cost function
and its gradient are used to determine the next model
parameters at each iterative process. We approximately set
partial derivatives to 0 at non-differentiable points for both the
ℓ
1
-norm and TV. The partial derivatives of χ
2
are calculated
numerically with central differences. We use the latest Fortran
implementation of L-BFGS-B (L-BFGS-B v3.0; Morales &
Nocedal
2011). We note that the problem is non-convex like
other imaging techniques using closure quantities (e.g.,
Buscher
1994; Thiébaut 2008; Bouman et al. 2015; Chael
et al.
2016), and a global solution is generally not guaranteed.
2.3. Determination of Imaging Parameters
In the proposed method, the most important tuning
parameters are the regularization parameters for the ℓ
1
-norm
(
h
˜
l
) and TV (
h
˜
t
), which determine the sparseness and effective
spatial resolution of the image, respectively. Smaller regular-
ization parameters generally favor images with larger numbers
of non-zero image pixels and more complex image structure,
which could give better χ
2
values by over-fitting. On the other
hand, large regularization parameters provide images that are
too simple and that do not fit the data well. To determine
optimal parameters, we need to evaluate the goodness-of-fit
using Occam’s razor to prevent over-fitting.
In this work, we adopt CV to evaluate goodness-of-fit. CV is
a measure of the relative quality of the models for a given set of
data. CV checks how the model will generalize to an
independent data set by using separate data sets for fitting the
model and for testing the fitted model. CV consists of three
steps: (1) randomly partitioning a sample of data into
complementary subsets, (2) performing the model fitting on
one subset (called the training set), and (3) validating the
analysis on the other subset (called the validation set).To
reduce variability, multiple rounds of CV are performed using
different partitions, and the validation results are averaged over
the rounds. If the regularization parameters are too small, the
established model from the training set would be over- fitted
and too complicated, resulting in a large deviation in the
validation set. On the other hand, if the regularization
parameters are too large, the established model would be too
simple and not well-fitted to the training set, also resulting in a
large deviation in the validation set. Thus, reasonable
parameters can be estimated by finding a parameter set that
minimizes deviations (e.g., χ
2
) of the validation set.
In this work, we adopted 10-fold CV for evaluating the
goodness-of-fit. The original data were randomly
18
partitioned
into 10 equal-sized subsamples. Nine subsamples were used in
the image reconstruction as the training set, and the remaining
single subsample was used as the validation set for testing the
model using χ
2
. We repeated the procedure by changing the
subsample for validation data 10 times, until all subsamples
were used for both training and validation. The χ
2
values of the
validation data were averaged and then used to determine
optimal tuning parameters.
An important advantage of this method compared with
previously proposed methods is that it is applicable to any type
of regularization functions and also imaging with multiple
regularization functions. For instance, Carrillo et al.
(
2012, 2014) and subsequent work solve images by utilizing
the ℓ
1
-norm on wavelet-transformed image or TV regulariza-
tion alone. In this case, the parameter can be uniquely
determined from the ℓ
2
-norm of the estimated uncertainties
on observational data (see Carrillo et al.
2012, for details).
However, it is not straightforward to extend the idea for the
problems with multiple regularization functions. For another
example, Garsden et al. (
2015) proposes another heuristic
method to determine the regularization parameters on ℓ
1
-
regularization on the wavelet/curvelet-transformed image by
estimating its noise level on each scale, which is successful.
However, the method would not work for all types of
regularization functions. On the other hand, CV is a general
technique that can be applied to imaging with any other
regularization functions or any combination of them, in
principle, which include MEM (e.g., Buscher
1994; Chael
et al.
2016) and patch priors (e.g., Bouman et al. 2015). This
advantage is particularly important for Sgr A
*
, which needs to
involve a regularization function to mitigate the interstellar
18
In this work, subsamples are obtained with a uniform probability regardless
of baselines following the most basic style of the CV. However, there could be
amore effective way of choosing subsamples for interferometric imaging. The
optimum partitioning for CV is in the scope of our next studies in thenear
future.
4
The Astrophysical Journal, 838:1 (13pp), 2017 March 20 Akiyama et al.