∎

11institutetext: J.E. Horvath
<EMAIL_ADDRESS>
1 Universidade de São Paulo, Department of Astronomy IAG-USP
R. do Matão 1226, 05508-090, Cidade Universitária, São Paulo SP, Brazil
Tel.: +55-11-30912710
Fax: +55-11-30912806

# A precursor interpretation for the Crab supernova 1054 A.D. very early light
curve

J.E. Horvath1

(Received: date / Accepted: date)

###### Abstract

In spite that the Crab supernova in 1054 A.D. was studied over the years, it
is still not clear what type of event produced the explosion. The most
detailed and reliable source of the observed light curve is recorded in the
Chinese and Japanese chronicles, and suggests a quick dimming after a very
bright initial peak. We shall show in this work that the Crab event can be
well explained by a type of precursor, a phenomenon emerging from large
supernova sampling and quite expected from Stellar Evolution considerations of
low-mass progenitors. This very early bright transient is followed by the
explosion itself, likely a low-luminosity supernova from “small iron core”
type instead of an electron-capture event. The latter conclusion stems from
recent simulation work, which predict that an electron-capture supernova would
render the magnitude to be much brighter for $\sim 3$ months, hence visible
during daytime, and would not match the Chinese records.

###### Keywords:

supernovae: individual: SN 1054 A.D. Optical transients supernovae: general

##  1 Introduction

The supernova 1054 A.D., which originated the Crab nebula and its associated
pulsar are one of the most widely known and studied events in Astronomy. The
connection of the Crab nebula with the historical supernova observed in 1054
A.D., with records in Chinese, Japanese and Western sources provided the best
example of Baade and Zwicky’s (1934) suggestion of the formation of neutron
stars in events generally seen in core-collapse supernovae. However, this
paradigm led to a series of questions which arise because the observed
supernova contain some puzzling features, not observed in any other case in
History.

Among the latter, we can count the lack of clear evidence of the supernova
remnant itself, precluding a more direct identification of the type of event,
and the reconstructed early light curve (mainly from Chinese sources) which
has been difficult to understand in general. On the theoretical side, since
the work of Nomoto and collaborators (1982), the Crab has been seriously
considered as a candidate to be a case of electron-capture supernova (ECSN),
in which a super AGB star explodes after the capture of electrons in an $O-Mg-
Ne$ degenerate core. The progenitors of these super-AGB stars are expected to
have $8-10\,M_{\odot}$ in the MS, although the exact values depend on
metallicity (Doherty et al. 2017). A neutron star formation is expected in
many cases from these events too. Alternatively, low-mass iron cores collapses
(CCSNs) could lead to similar explosions and leave a neutron star as observed,
the latter featuring masses below the “fixed” value $\sim\,1.25\,M_{\odot}$
attributed for the ECSN events for the compact remnant (Horvath et al. 2022).

It is now known that the lightcurves of explosive events have revealed a
variety, and a database on the long-term behavior of the lightcurves is better
known because of detailed observations and theoretical work along the last
decades. Nevertheless, several so-called “supernova impostors”, transients and
events that release $\geq 10^{51}erg$ but are not actual supernovae have been
recognized and studied over the years. Eta Carina is one of the well-known
cases that could have been identified as a supernova, although it belongs to a
different class of transients (Davidson and Humphreys 2012).

In the particular case of the SN 1054 A.D., a total energy of the Crab $\leq
7\times 10^{49}\,erg$ as an upper limit (see Hester 2008) has been reported.
The optical lightcurve would have been similar to the cases of SN2005cs,
SN2016ov and SN2018zd (Spiro et al. 2014, Hiramatsu et al. 2021). A sudden
large drop of the brightness around $\sim 120-130\,d$ should have been
occurred in the SN 1054 event according to model simulations, although there
are no reports of that in the available records. The supernova should have
decreased by additional $\sim 6-7\,mag$ after vanishing during daytime, to
become completely invisible, even at night, after $653\,d$ (Table 1). Note
that these features are quite independent of the exact nature of the event,
either a core-collapse of a small iron core progenitor or an electron-capture
event, and any model must comply with the temporal behavior.

In either of these scenarios, ECSN or CCSN, the features of the initial
lightcurve reconstructed for the Crab explosion are quite difficult to obtain.
A bright maximum ($M_{V}\geq-18$) and rapid decay were observed, while the
ECSNs and CCSNs scenarios predict in turn low explosion energies. We shall
argue below that the very early lightcurve can be associated to a precursor of
the type that are being seen in many supernova surveys (Section 3), and that
the subsequent behavior of the light curve is the one expected from a low-mass
iron core CCSN, not of an ECSN type (Section 4).

##  2 Historical records

Several Chinese records refer to the “guest star” of 1054 A.D.. A comparative
discussion by Breen and McCarthy (1995) concluded that the most likely date
for its appearance in the 4th July with a fast (but not precisely determined)
rise time $\leq\,1\,d$. The Chief of the Astronomical Bureau at K’Ai-Feng
writings provide accurate times for the appearance of the object, comparing
its brightness to Venus, which has $m_{V}=-4.47$, interpreted as its maximum
brightness (it should be noted that in spite of being able to distinguish
$\sim 0.1\,mag$, ancient Chinese astronomers have not registered the
brightness variation in time of Venus, exceeding $0.5\,mag$, Filipović et al.
2022). After the classical papers of Lundmark (1921), Mayall and Oort (1942)
and Shkilovsky (1968), among others, the common assumption is that, after
applying the extinction correction $A_{V}=1.6$ (Miller 1973), the peak
absolute magnitude should have been $M_{V}=-18$ or higher. However, the
Japanese chronicles compared the event with a dimmer Jupiter ($m_{V}=-2.2$,
Green and Stephenson 2003), although the proximity of the Sun in the sky and
some uncertainty in the dating made these reports not as solid. Therefore, it
is possible that the brightness of the object at peak was slightly lower than
Venus. We shall adopt the accepted value $M_{V}=-18$ hereafter as a lower
limit. The precise value is impossible to refine, but fortunately not crucial
here.

The same Chinese chronicles (Breen and McCarthy 1995) provide another
important clue to the early lightcurve, saying that the guest star was no
longer visible during the day after $23\,d$. This can be quantified by
observing that the $m_{V}=-0.24$ Saturn is often visible during the day, and
leads after extinction correction to a value $M_{V}\,\sim\,-14$. This
estimation in uncertain by about $0.5$ mag, but overall it is fair to state
that the transient faded by $\Delta M_{V}\sim\,4$ in about three weeks. This
temporal feature is reliable from the Chinese imperial astronomers records and
a prime feature to be understood.

The last temporal benchmark is related to the late lightcurve, i.e. the
disappearance of the guest star from the night sky. Again, it is difficult to
pin down exactly the sky background brightness, but this “disappearance” has
been quantified as $M_{V}\sim-7$ after $653\,d$ (Breen and McCarthy 1995). It
is unfortunate that no intermediate values are known, since supernova often
present interesting and revealing behavior at these times.

The Western chronicles are scarce, more vaguely described (to the point of
being doubtful that they describe a “guest star” event at all), and not due to
astronomers. Some have considered the image of the Holy Roman Emperor Henry
III considered as a proof that the 1054 A.D. was seen in Italy even though no
written account of this specific observation remains (Hoffman and Gudrun
2021). A Middle East observation performed by Ibn Butlan is also known
(Brecher 1978) but it does not contain substantial information either. An
argument to associate a Flemish writing addressing the day of the Pope Leo IX
death would need to bring the explosion to a much earlier date (Guidoboni et
al. 1994). Because the death of the Pope Leo IX and the Great Schism of the
Church, declared by the Patriarch of Constantinople on July 16th 1054, a heavy
“contamination” (in the sense that some phenomena were reported to reinforce
the holiness of the Pope, but could be completely unreal) of the reports is
believed to have taken place, quite unrelated to the 1054 A.D. event itself,
or in any case difficult to disentangle from the former. Certainly there is no
consensus for a much earlier date for the explosion of the Crab supernova,
which would also create big problems to sustain very high luminosities over a
period of $>\,3\,months$ (April-July 1054) if accepted, thus it has been
considered untenable (Breen and McCarthy 1995).

##  3 The very early SN 1054 A.D. light curve as a precursor

Precursors (i.e. transient events) preceding supernovae have been reported in
the literature (see, for instance Ofek et al. 2014 for a study of a Type IIn
supernova sample). The statistics is very incomplete since large surveys are
only now reaching the point when an evaluation becomes feasible. In the case
of the Crab as a confirmed low-mass explosion, a super-AGB character of the
progenitor would enhance the expectations for episodic mass ejections or
super-Eddington winds.

Precursors can be originated in a couple of different ways in this context.
The first is the ejection of mass, whose kinetic energy is converted into
luminosity, with an efficiency factor $\epsilon$. The simplest estimation for
the ejected mass stems from energetic considerations and reads

$M_{ej}\approx\epsilon{{2L_{p}{\delta
t}}\over{v^{2}}}\,=\,10^{-2}M_{\odot}\times{{\biggl{(}{\epsilon\over{0.1}}\biggr{)}}}{{\biggl{(}{Lp\over{10^{9}L_{\odot}}}\biggr{)}}}{{\biggl{(}{\delta
t\over{23\,d}}\biggr{)}}}{{\biggl{(}{1000\,kms^{-1}\over{v}}\biggr{)}}}^{2}$
(1)

where $L_{p}$ is the precursor luminosity, $\delta t$ its duration and $v$ the
velocity of the ejecta.

A second mechanism, studied by Shaviv (2000, 2001) and applied to novae and
other problems, is a super-Eddington wind outflow accelerating the matter,
yielding a simplified expression

$M_{ej,wind}\approx W{{L_{p}{\delta
t}}\over{c_{s}c}}\,=\,10^{-2}M_{\odot}\times{{\biggl{(}{W\over{10}}\biggr{)}}}{{\biggl{(}{Lp\over{10^{9}L_{\odot}}}\biggr{)}}}{{\biggl{(}{\delta
t\over{23\,d}}\biggr{)}}}{{\biggl{(}{{60kms^{-1}}\over{c_{s}}}\biggr{)}}}$ (2)

with $W\approx 5-10$ an empirical constant derived for each particular problem
(Shaviv 2000, 2001), and the sound velocity $c_{s}$ has been scaled to its
expected value at the base of the wind $\approx 60\,km\,s^{-1}$.

Once we apply these expressions to reproduce the very early light curve of the
Crab, we obtain in both cases $M_{ej}\simeq 10^{-2}\,M_{\odot}$, provided the
velocity $v$ is high enough or the sound velocity $c_{s}$ is not too small.
This mass is very small, and although it can produce a bright transient, and
more importantly fast rising ($t_{rise}\leq 1\,d$) and decay
($t_{decay}\sim\,weeks$) as required by observational constraints, its effect
on the later shock breakout of the supernova itself is small. On the other
hand, if the whole early lightcurve has to be sustained by circumstellar
material (CSM), the mass must be much higher ($\sim 0.3M_{\odot}$, Smith
2013), and an initial rapid decay of at least 4 magnitudes in just three weeks
appears much more extreme.

Table 1: Variation in the absolute magnitude $\Delta M_{V}$ of the remnant with time (after Breen and McCarthy 1995) $|\Delta M_{V}|$ |  $\Delta t$ (days) |
---|---|---
$\geq 4$ | 23 |
$\sim 6-7$ | 653 |
$\sim 4$ | $\gg 300$ |

A possibly related phenomenon are the rapidly evolving optical transients
reported in the last decade (Poznanski et al. 2010, Drout et al. 2014, Arcavi
et al. 2016). These events form a growing group, generally characterized by
fast rising times ($\sim$ days), absolute magnitudes comparable to Type Ia
supernovae and decay within a month. They have been also associated to the
effect of a shock wave interacting with CSM or energy injection by a magnetar
birth, both models implying a stellar explosion as an original trigger. The
super-Eddington scenario would also be a suitable model for these transients
on general grounds (Shaviv 2000, Ofek et al. 2014), although a detailed
discussion has not been presented for this specific case.

A case study of the optical transient KSN2015K recently reported by K2/Kepler
(Rest et al. 2018) serves to exemplify the possible relation with the early
lightcurve of the Crab supernova. This transient raised in $\sim 2\,d$ and
decayed on a $\sim\,1\,month$ timescale, with a peak luminosity a factor of
$\sim\,5$ higher that the reconstructed for the Crab. However, as stated the
peak magnitude of the Crab could have been brighter that $-18$ when first
spotted the morning of July 4th (Green and Stephenson 2003, Smith 2013),
making the KSN2015K almost a perfect fit to the lightcurve without any
modification. If a transient like this happened in the SN 1054 event, the time
to decay to a magnitude that would have rendered the transient unobservable
during daytime, as reported by the Chinese, come out automatically right (Fig.
1) and is consistent with the historical observations (Murdin and Murdin 1985,
Breen and McCarthy 1995).

Figure 1: The very early ($t\leq 23$ d) and early ($t\geq 23$ d) light curves
of the 1054 A.D. explosion. Asterisks are the reconstructed values according
to the Chinese records. The solid curve corresponds to the reported KSN2015K
transient, with the maximum of the latter matched to the time of explosion.
The squares are the $V$-band calculation of Kozyreva et al. (2021)
corresponding to the s9.0 model, assumed to explode $\sim 3$ weeks after the
precursor rise. According to the Chinese records, the latter curve should drop
below $M_{V}=7.5$ in April, 1056, 21 months after the first sight (Table 1).

In the CSM breakout scenario, a rapid rise $<1$ day due to shock breakout
diffusion $t_{d}\sim 30R/c$ could be achieved if the CSM is standing at $\sim
10^{13}\,cm$. A compact progenitor of the carbon/oxygen type would have a much
smaller radius. Only a small mass fraction $\leq 0.1\,M_{\odot}$ would have to
be ejected on a timescale $t_{csm}=R/v_{csm}$ over the last years before the
explosion to explain the very early light curve (Rest et al. 2018).

A super-Eddington model will rise even faster and require a somewhat smaller
ejected mass. This model seems consistent with the evidence available the
observed rise and decay in the KSN2015K too. The decay of the optical
transient suggests that no other energy source like ${}^{56}Ni$ or continuous
injection by a central magnetar (not expected for the “normal” magnetic field
associated to the Crab pulsar (Allen and Horvath 2004, Kasen et al. 2016,
Shukbold and Woosley 2016) is involved in the transient (Rest et al. 2018). To
be sure, the growing group of transients include short events which are not
consistent with shock breakout, and that can be interpreted within an ejected
mass with radioactive decays as well (Ofek et al. 2021). Without any
photometry and a scarce temporal information, we may never know which kind of
precursor happened in SN 1054.

##  4 The subsequent and long-term behavior of the lightcurve

An extensive study of low-luminosity explosions (Kozyreva et al. 2021) shows
that it would be very difficult to understand the very early lightcurve from
the theoretical point of view invoking just a “bare” explosion.It remains to
be seen whether this feature stands with other calculations with varying
physical content.

If the calculations of Kozyreva et al. (2021) are taken as a guide, we further
conclude that the ECSN lightcurve based on the e8.8 model does not fit the
Crab event, because it will render SN 1054 observable during the daytime for
$\sim 4$ months. Explosions based on the more compact progenitors z9.6 and
s9.0 will be acceptable, for a range of metallicities and explosion energies
$few\times 10^{49}erg$. However, and guided by the simulations, this would
mean that the Crab was not a ECSN, but rather a low-mass iron core CCSN.
Moreover, this conclusion is actually independent of the precursor
interpretation of the very early light curve, and relies on the Chinese report
on the supernova becoming invisible during the day after $23\,d$. An ECSN
would be very visible during the day because it would be about 2 magnitudes
brighter (looking like Jupiter) according to the detailed models of Kozyreva
et al. (2021). As a final corollary, the Crab pulsar mass is predicted to be
below $1.25M_{\odot}$ to achieve consistency within the low-mass iron core
CCSN model.

In addition, it should be noted that the observed analogues SN2005cs and
related events seem too dim at peak magnitude to be compared to the Crab case.
However, the small overall mass $\sim 5\,M_{\odot}$ in the Crab Nebula, the
chemical evidence for a progenitor $\simeq 8\,M_{\odot}$ and the low
${}^{56}Ni$ present (Smith 2013) remain as strong indications of a low-energy
explosion, which also produced a neutron star. This is why we believe that a
bright precursor interpretation, followed by the explosion itself, is a good
fit to the whole picture.

##  5 Conclusions

In summary, we have interpreted the initial phase of the supernova 1054 A.D.
as a bright transient precursor, followed by the explosion itself after $\sim
3\,weeks$. This interpretation has been motivated by the expected features of
a low-mass progenitor near the explosion, which also predicts a plateau-type
visual magnitude lasting 3-4 months but at a level that will not re-brighten
the event (which was not reported). Bright optical transients such as KSN2015K
possess all the features of the very early SN 1054 light curve and have been
brought as comparison examples, although their actual relevance to the event
remains to be proved.

In fact, it is quite remarkable that a lightcurve from a recent FELT event
(KSN2015K) can fit accurately the reconstructed historical time behavior with
just a slight (if any) brightness scale-down amounting to a small numerical
factor. This can not, of course, taken as a proof, but certainly suggests a
kinship of optical transients and SN precursors, a central thesis for our
model.

Interestingly, the idea that CSM may be involved in supernovae with an early
luminosity excess was also present in Goldberg and Bildsten (2016), Morozova
et al. (2020) and Moriya et al. (2020). Smith (2013) has stressed the apparent
incompatibility of the low-energy hypothesis of the Crab supernova with the
high-luminosity early lightcurve and offered a detailed discussion of a CSM
hypothesis. His view is different from our precursor transient interpretation
that would explain the very early phase, while the well-known ordinary
expansion of the supernova would have taken over after $\sim 3\,weeks$ and be
responsible for the long-term behavior, but this will be problematic if the
calculations of Kozyreva et al. (2021) stand for the ECSN cases due to the
non-visibility of SN 1054 A.D. supernova during daytime after $23\,d$.

There are a few observational tests that may resolve the issue of the type of
event. The most likely one would be the detection of “light echoes” of the
event, which have the potential to reveal the spectrum and possibly its time
evolution (Rest et al. 2008). However, in spite of the efforts over the years,
the light echoes of the SN 1054 A.D. have not been detected yet. Therefore,
other evidences such as the nucleosynthesis yields and structure of the
remnant should be analyzed. We have remarked above the difficulties to study
the latter. In fact, it is not clear that any other event of the “SN 1054
A.D.-type” has been observed, although this would not be so surprising because
the derived number of rapidly evolving luminous transients is $\sim$ a few
percent of the whole core-collapse SN (Drout et al. 2014). The follow-up of
these fast transients could revel a weak SN after a few weeks, provided they
are close enough and a strategy to identify and locate them quickly is
developed (Inserra 2019). This would resemble the cases in which a long GRB is
followed by a supernova, mutatis mutandis, and would constitute a crucial
observation to assess the precursor scenario discussed above.

According to our picture, around $\sim\,1\,month$ the light curve would have
leveled off due to the underlying supernova emergence, and decayed very slowly
until $120-130\,d$ (Fig. 1). Although the suggestions of a Type IIn-P from a
small iron core explosion (Kozyreva et al. 2021) and ECSN (Nomoto et al. 1982)
would be difficult to distinguish as low-energy events at late times in
historical supernovae, the theoretical models of Kozyreva et al. (2021)
disfavor the ECSN explosions as a model for the SN 1054 because they would be
far too bright (by two magnitudes) to match the Chinese temporal records.
Therefore, an agreement of present models and the SN 1054 reconstruction
suggests a core-collapse SN, not an electron-capture one, independently of the
origin of the very early bright lightcurve and related to the disappearance
during the daytime only. Since the available models are scarce, and there are
caveats which apply to a handful of points, we are not claiming anything
definitive, but rather point out an alternative interpretation of the Crab
event that could tie optical transients to the exploding stars, and
tentatively identify which one produces the observed phenomenology. It may be
that the confirmed diversity of supernova events and associated
precursor/transients could be crucial to understand a millennium-old puzzle in
a paradigmatic case.

## Acknowledgements and declarations

* The author wish to acknowledge the financial support of the Fapesp Agency (São Paulo) through the grant 2020/08518-2 and the CNPq (Federal Government, Brazil) for the award of a Research Fellowship.The members of the GARDEL Group at USP are acknowledged for their encouragement and scientific advise in these topics. An anonymous Referee helped to improve the first version with several remarks and suggestions.

* Informed consent does not apply to this work.

* Data availability does not apply, no data beyond the one publicly available in the cited works has been employed.

* The author declares no competing interests.

* All the conception, execution and writing of the paper was performed by the author.

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# Ultra-Sparse View Reconstruction for Flash X-Ray Imaging using Consensus
Equilibrium

###### Abstract

A growing number of applications require the reconstruction of 3D objects from
a very small number of views. In this research, we consider the problem of
reconstructing a 3D object from only 4 Flash X-ray CT views taken during the
impact of a Kolsky bar. For such ultra-sparse view datasets, even model-based
iterative reconstruction (MBIR) methods produce poor quality results.

In this paper, we present a framework based on a generalization of Plug-and-
Play, known as Multi-Agent Consensus Equilibrium (MACE), for incorporating
complex and nonlinear prior information into ultra-sparse CT reconstruction.
The MACE method allows any number of agents to simultaneously enforce their
own prior constraints on the solution. We apply our method on simulated and
real data and demonstrate that MACE reduces artifacts, improves reconstructed
image quality, and uncovers image features which were otherwise indiscernible.

Index Terms—  Sparse View Tomography, In-Situ Tomography, Consensus
Equilibrium, Inverse Problems

##  1 Introduction

High speed volume visualization of dynamic processes is an emerging
application of computed tomography (CT). In particular, Flash X-ray has been
used to study explosive and ballistic events [1, 2], material deformation [3],
and production line quality control [4], where space and time constraints
necessitate sparse view sampling. Even in the absence of such constraints,
sparse view CT presents benefits in the form of reduced computation and memory
requirements, as well as lower radiation exposure in medical CT.

Conventional CT reconstruction, with a few thousand views, is often performed
using analytical reconstruction methods such as Filtered Back Projection
(FBP). However, these analytical methods typically generate severe artifacts
for sparse view CT problems in which the number of views is subsampled by a
factor of 2 to 10. For sparse-view CT, methods such as model based iterative
reconstruction (MBIR) must be used to achieve high quality results [5].

When the number of views is less than 10, we enter the domain of ultra sparse
CT, and it becomes very challenging to reconstruct high quality CT images. In
this case, even established methods such as MBIR tend to generate severe
artifacts with reduced contrast and resolution. Recently, Henzler et. al [6]
and Shen et. al [7] have independently developed methods for inferring 3D
reconstructions from a single medical view using deep learning methods.
However, Flash X-ray CT events are typically destructive and stochastic in
nature. This makes it difficult to collect the data that would be required to
train the machine learning methods. In other recent work, Moser et. al [3, 2]
used the algebraic reconstruction technique (ART) to reconstruct 3D volumes
from as few as 6 views. However, ART does not allow for the explicit modeling
of prior distributions or even the sensor forward model.

Traditional MBIR formulates the reconstruction problem as the solution to a
maximum a posteriori (MAP) or regularized maximum likelihood optimization
problem [5]. More recently, Plug and Play (PnP) methods [8], originally based
on ADMM optimization algorithms [9, 10, 11], have expanded MBIR by allowing
the prior distribution to be replaced with advanced denoising operations [12,
13]. Notably, BM3D [14] and its higher dimensional variants are among the best
known denoisers that do not require training, and have been found to work
particularly well with PnP.

Most recently, Multi-Agent Consensus Equilibrium, or MACE [15] is a
generalization of PnP that is based on the solution of a set of equilibrium
equations. The MACE equilibrium has the interpretation of being the consensus
solution reached by multiple agents each operating in a way that enforces its
own preference or constraint. A discussion of practical implementation
considerations for fast execution is presented in [16]. Multi-Slice Fusion
[17, 18] is a variant of MACE, where multiple lower dimensional denoisers are
fused to form a higher dimensional prior model.

In this paper, we introduce a new algorithm, multi-slice fusion with
rotational invariance (MSF-RI) for the reconstruction of ultra-sparse
tomographic data; and we apply the MSF-RI algorithm to the reconstruction of 4
view CT data from a flash X-ray in-situ imaging system used in a Kolsky bar
impact experiment [19]. Our algorithm works by integrating the constraints of
four distinct agents using the MACE framework. The first three agents are
based on 2D regularization using BM3D, and the fourth agent enforces partial
rotational invariance using a rotational smoothing operator. Our experiments
compare MSF-RI to FBP, traditional MBIR, and PnP using a BM4D prior on both
synthetic and real data sets. Our proposed MSF-RI method achieves better RMSE
and SSIM values than previous methods and reveals structures in our volumetric
reconstructions that were previously indistinguishable from streak artifacts.

##  2 Problem Formulation

A CT scan is a collection of X-ray measurements of an object taken at
different angles. Reconstructing the object from the scan is an inverse
problem with a forward model of the form

$y=Ax+noise$

where $y\in\mathbb{R}^{M}$ is the measurement vector, $A\in\mathbb{R}^{M\times
N}$ is the scanner system matrix, and $x\in\mathbb{R}^{N}$ is the latent image
vector we wish to recover. The measurements contain additive Gaussian noise
distributed as $N(0,\alpha\Lambda^{-1})$. The MAP cost function then has the
form

$\hat{x}_{MAP}=\arg\min_{x}\\{f(x)+h(x)\\}$ (1)

with a forward model term, $f(x)=-\log p(y|x)$, to enforce data fidelity, and
a prior model term, $h(x)=-\log p(x)$ to impose regularity. In this case, the
forward model term can be written within a constant as

$f(x)=\frac{1}{2\alpha}||y-Ax||_{\Lambda}^{2}\ .$ (2)

In practice, it is often difficult to express very complex prior distributions
as a single tractable functions $h(x)$. Therefore, the remainder of this
section outlines how we will replace this prior with one tailored for our
application using MACE.

###  2.1 MACE framework

Consider the MAP cost function with $K$ regularizers

$\hat{x}_{MAP}=\arg\min_{x}\left\\{f(x)+\sum_{k=1}^{K}\beta_{k}\,h_{k}(x)\right\\}$
(3)

where $\beta_{k}\geq 0$ controls the amount of regularization applied by
$h_{k}$. The MACE algorithm splits this MAP estimation problem into pieces,
with each piece enforced using an agent. For the exact MAP estimation problem,
the forward model agent is the proximal map given by

$F(w)=\arg\min_{z\in\mathbb{R}^{N}}\left\\{f(z)+\frac{1}{2\sigma^{2}}||z-w||^{2}\right\\}$
(4)

and the prior model agents are given by

$H_{k}(w)=\arg\min_{z\in\mathbb{R}^{N}}\left\\{h_{k}(z)+\frac{1}{2\sigma^{2}}||z-w||^{2}\right\\}$
(5)

where $\sigma>0$ controls the convergence speed of the algorithm.

Like PnP, MACE allows us to replace the proximal maps $H_{1},\dots,H_{K}$ with
more general denoising agents that typically are not proximal maps. Moreover,
MACE is a generalization of PnP [15] because it allows for the case of
multiple agents, which we will use in this work. Additionally, MACE specifies
the solution to the problem using an equilibrium condition, rather than as a
solution to an optimization problem. This is important since, in general, the
PnP or MACE solution is no longer the solution to a MAP style optimization
problem when the agents are not proximal maps.

In order to write the MACE equilibrium condition, define the following
notation. Let
$\mathbf{w}=[w_{0},w_{1},\dots,w_{K}]\in\mathbb{R}^{N\times(K+1)}$ be the
concatenation of $K+1$ states where each state will serve as the input to its
corresponding agent. Next we define the agent operator,

$L(\mathbf{w})=[F(w_{0}),H_{1}(w_{1}),\dots,H_{K}(w_{K})]\ ,$

to denote the parallel application of the $K+1$ agents on their respective
state inputs. We also define the averaging operator
$G(\mathbf{w})=[\bar{w},\dots,\bar{w}]$, where
$\bar{w}=\sum_{k=0}^{K}\mu_{k}w_{k}$. The vector $\mu\in\mathbb{R}^{K+1}$
controls the influence each agent exerts on the equilibrium solution and is
given by

$\mu=\frac{1}{1+\sum_{k=1}^{K}\beta_{k}}\left[1,\beta_{1},\dots,\beta_{K}\right]$
(6)

Algorithm 1 MACE algorithm

0: Initial Reconstruction: $x^{(0)}\in\mathbb{R}^{N}$

0: Final Reconstruction: $x^{*}$

1: $\mathbf{w}\leftarrow[x^{(0)},\dots,x^{(0)}]$

2: while not converged do

3: $\mathbf{x}\leftarrow L(\mathbf{w})$

4: $\mathbf{z}\leftarrow G(2\mathbf{x}-\mathbf{w})$

5: $\mathbf{w}\leftarrow\mathbf{w}+2\rho(\mathbf{z}-\mathbf{x})$

6: end while

7: return $x^{*}\leftarrow\sum_{k=1}^{K+1}\mu_{k}x_{k}$

Using this framework, the solution to our problem is given by
$\hat{x}=\sum_{k=0}^{K}\mu_{k}w_{k}^{*}$, where $w^{*}$ is the solution to the
consensus equilibrium (CE) equation given by

$L(\mathbf{w}^{*})=G(\mathbf{w}^{*})\ .$ (7)

We note that when the agents are the proximal maps of (4) and (5), then
$\hat{x}=\hat{x}_{MAP}$ is exactly the MAP estimate of (3).

In [15], it is shown that the CE equations can be solved by using the Douglas-
Rachford algorithm to solve for the fixed point of the operator
$T=(2G-I)(2F-I)$. Algorithm 1 shows the general method for doing this. When
$K=1$, and $\rho=0.5$, this algorithm corresponds exactly to PnP implemented
with the consensus ADMM algorithm [15], and a sufficient condition for
convergence is that $T$ is non-expansive.

###  2.2 MACE Agents

Our reconstruction algorithm uses five agents. The first agent, $F$, enforces
data fidelity and is defined in (4) and solved using Iterative Coordinate
Descent.

The agents $H_{1}$ through $H_{4}$ are chosen to enforce spatial regularity
and together act as a prior model for reconstruction. More specifically,
agents $H_{1}$, $H_{2}$, and $H_{3}$ are 2D BM3D denoisers applied to the
$(\overrightharp{x},\overrightharp{y})$,
$(\overrightharp{y},\overrightharp{z})$, and
$(\overrightharp{z},\overrightharp{x})$ planes respectively. The $H_{1}$ agent
independently applies the BM3D denoiser to each
$(\overrightharp{x},\overrightharp{y})$ slice of the 3D volume. $H_{2}$ and
$H_{3}$ perform similar functions along slices in
$(\overrightharp{y},\overrightharp{z})$, and
$(\overrightharp{z},\overrightharp{x})$. We refer to this integration of three
2D denoising algorithms as multi-slice fusion (MSF) as described in more
detail in [17].

Agent $H_{4}$ is designed to enforce weak rotational invariance in the
reconstruction. By weak rotational invariance we mean that the reconstructed
object should be approximately invariant to small rotations around its axis of
symmetry. This is physically reasonable since both the sample under test and
the experimental forces are approximately rotationally invariant. Nonetheless,
the actual sample will exhibit asymmetric cracks of importance, so $H_{4}$
acts to weakly regularize the solution, rather than to enforce strict
rotational symmetry.

The weak rotational invariance agent is defined by

$\displaystyle H_{4}(x)=\sum_{n=-J}^{J}\gamma_{n}\,R_{n(\theta/J)}(x)$

where $R_{\phi}(x)$ rotates the 3D volume, $x$, around its axis of rotational
symmetry by an angle of $\phi$. Intuitively, $H_{4}$ performs a weighted
average of rotations over an angle of $2\theta$ degrees, with weights,
$\gamma_{j}$, drawn from a scaled Hamming window.

##  3 RESULTS

In this section, we present both simulated and real data reconstruction
results for a Flash X-ray CT scanner that acquires four views oriented at
$18^{\circ}$, $162^{\circ}$, $234^{\circ}$, and $306^{\circ}$ [19] using the
following five reconstruction algorithms:

Algorithm | Description
---|---
FBP | Filtered Back Projection
qGGMRF | MBIR reconstruction is a qGGMRF prior
PnP-BM4D | PnP prior using the 3D denoiser BM4D
MSF | MACE with agents $H_{1}$, $H_{2}$, and $H_{3}$
MSF-RI | MACE with agents $H_{1}$, $H_{2}$, $H_{3}$, and $H_{4}$

For the simulated cylinder, results and parameters are chosen to minimize RMSE
scores achieved using the different priors. For the experimental case, the
most visually pleasing results are reported. In all instances of MACE, we set
$\rho=0.4$, and $\theta=8^{\circ}$ for $H_{4}$, where applicable. The block
matching modules use the implementation in [20] and [21] with default
parameters. The PnP and MACE algorithms are all initialized using a qGGMRF
reconstruction.

Convergence plots are shown for both datasets in Figures 2 and 5. Distance to
convergence is quantified as the RMSE between $G(\mathbf{w})$ and
$L(\mathbf{w})$ normalized over $\mathbf{w}$. This value reflects how closely
equation (7) is satisfied. We see that all PnP and MACE algorithms are able to
converge to within less than $5\%$ of the final solution.

(a) Ground Truth

(b) FBP

(c) qGGMRF

(d) PnP-BM4D

(e) MSF

(f) MSF-RI

Fig. 1:  Simulated results. (a) sample cross section of ground truth for
comparison with CT reconstructions (b) through (f) with various methods. (d),
(e), and (f) are initialized with the qGGMRF result in (c).

Fig. 2: Convergence rates for phantom reconstruction with PnP-BM4D, MSF, and
MSF-RI

(a) FBP

(b) qGGMRF

(c) PnP-BM4D

(d) MSF

(e) MSF-RI

Fig. 3: Reconstructions of slice 21 from experimental dataset. Features of
interest are numbered in the result of the proposed method for cross
referencing with measurement data.

###  3.1 Simulated dataset

Figure 1(a) shows the simulated cylinder phantom with transverse, and radial
cracks, typically present in concrete specimens that have been loaded in a
Kolsky bar experiment [22]. The volume is $121\times 121$ voxels by 100
slices. Cracks numbered 2 and 4 lie along the view angle at $234^{\circ}$,
while 1 and 3 are oriented at $90^{\circ}$. The concentric crack, labelled 5,
has been added to further test the model’s capabilities. The 3D phantom was
forward projected with the parallel beam $A$ matrix and corrupted with AWGN at
$20\%$ of the signal strength.

Table 1: Metrics for simulated dataset reconstruction. Method | NRMSE | SSIM
---|---|---
FBP | 0.8678 | 0.275052
qGGMRF | 0.2389 | 0.625456
PNP-BM4D | 0.2044 | 0.778083
MSF | 0.2006 | 0.778822
MSF-RI | 0.1909 | 0.789262

Figures 1(b) through (g) show CT reconstructions of the sample made using the
five algorithms, and Table 1 lists the associated normalized root mean squared
error (NRMSE) and SSIM values reported for the entire volume.

Notice that FBP fails, and even traditional MBIR performs poorly on this very
underdetermined dataset. PnP-BM4D shows improved results, but generates blocky
artifacts. This is likely due to the lack of similar patches in three
dimensions as opposed to two dimensions in such a small volume. Consequently,
the multi-slice fusion used in MSF is more effective than the BM4D prior, and
the result shows more detail. The best result, both visually, and numerically
is the MACE-RI algorithm, which reveals the most detail, including the
concentric crack 5, and the transverse cracks 2 and 4. Notice that all the
methods are effectively blind to crack 1, and crack 3 appears as a blur.

(a) View at $18^{\circ}$

(b) View at $162^{\circ}$

(c) View at $234^{\circ}$

(d) View at $306^{\circ}$

Fig. 4: Experimental view data, where the location of slice 21 is marked with
a white line and features of interest are numbered for comparison in
reconstructions.

Fig. 5: Convergence rates for reconstruction from of real dataset with PnP-
BM4D, MSF, and MSF-RI

###  3.2 Experimental dataset

The real sandstone ring dataset is imaged at the moment of impact in the
Kolsky bar experiment. The measurement data consists of four views of 478
detector channels per view and 198 slices. After a $2\times 2$ binning factor,
the reconstructed volume is $229\times 229$ by 89 slices with voxels of width
0.097 mm. The reconstructions in Figure 3 each depict slice 21 in the
$(\overrightharp{x},\overrightharp{y})$ plane, which is also marked in the
preprocessed view data in Figure 4 by a white line.

Notice that the FBP and qGGMRF results again show severe streaking artifacts
that appear along the view angles due to the presence of noise in the
projection data. Once again, PnP-BM4D produces blocky regions and cannot
resolve the finer cracks. MACE-RI appears to show the most detail with
transverse cracks sharpened. The object appears to be delaminating under
impact. Cross referencing the cracks with the view data confirms that these
cracks are real and not residual streak artifacts.

##  4 CONCLUSION

In this paper, we demonstrated the versatility of MACE by tailoring an
advanced 3D prior model to address the specific needs of the sparse view flash
X-ray CT system. We envision a tunable system where end users can combine
different agents as modules into the MACE framework to build expressive priors
that can exploit geometries of specimens beyond rotational invariance.

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