# Companion Shocking Fits to the 2018 ZTF Sample of SNe Ia Are Consistent with
Single-Degenerate Progenitor Systems

J. Burke Las Cumbres Observatory, 6740 Cortona Dr, Suite 102, Goleta, CA
93117-5575, USA  Department of Physics, University of California, Santa
Barbara, CA 93106-9530, USA  D. A. Howell Las Cumbres Observatory, 6740
Cortona Dr, Suite 102, Goleta, CA 93117-5575, USA  Department of Physics,
University of California, Santa Barbara, CA 93106-9530, USA  D. J. Sand
Steward Observatory, University of Arizona, 933 North Cherry Avenue, Tucson,
AZ 85721-0065, USA  G. Hosseinzadeh Steward Observatory, University of
Arizona, 933 North Cherry Avenue, Tucson, AZ 85721-0065, USA

(Received XXX; Revised XXX; Accepted XXX)

###### Abstract

The early lightcurves of Type Ia supernovae (SNe Ia) can be used to test
predictions about their progenitor systems. If the progenitor system consists
of a single white dwarf in a binary with a Roche-lobe-overflowing non-
degenerate stellar companion, then the SN ejecta should collide with that
companion soon after the explosion and get shock-heated, leaving an early UV
excess in the lightcurve. This excess would only be observable for events with
favorable viewing angles, $\sim$10% of the time. We model the 2018 ZTF sample
of 127 SNe Ia using companion shocking models, and recover an observed early
excess rate of $12.0\pm 3.6\%$, consistent both with several other rates
calculated throughout the literature, and with the expectation that SNe Ia
predominantly occur in single-degenerate systems. We observe early excesses
only in spectroscopically normal SNe Ia, in contradiction to the claim that
such excesses occur more frequently in overluminous SNe Ia. We also show that
the detection of early excesses can be methodology-dependent. We encourage the
observation of large samples of SNe Ia with high-cadence multiwavelength early
data in order to test the statistical predictions of SN Ia progenitor models,
and we also encourage the refinement of existing models.

supernovae

††journal: ApJL††facilities: Zwicky Transient Facility (Bellm et al.,
2019)††software:  astropy  (Astropy Collaboration et al., 2013; The Astropy
Collaboration et al., 2018), SNooPy (Burns et al., 2011), lightcurve_fitting
(Hosseinzadeh, 2019), emcee (Foreman-Mackey et al., 2013), extinction
(Barbary, 2016)

##  1 Introduction

Type Ia supernovae (SNe Ia) come from exploding white dwarfs (WDs). This
statement is uncontroversial and has been understood for decades (Hoyle &
Fowler, 1960), but almost every detail of the progenitor system and explosion
mechanism is the subject of active research. What mass are the WDs when they
explode? Do they need to accrete mass up to the Chandrasekhar limit, or, as
some models predict (e.g. Polin et al., 2019; Shen et al., 2021), can they
explode at sub-Chandrasekhar masses? Where does the explosion start? Does it
begin roughly at the center of the WD (Khokhlov, 1991), or does it begin in a
surface layer of accreted He which causes the underlying WD to detonate (Polin
et al., 2019)? How do the WDs gain enough mass to explode? Are they in a
binary system with a Roche-lobe-overflowing nondegenerate stellar companion
(referred to as the “single-degenerate” case; Hoyle & Fowler, 1960), or does
the primary WD tidally disrupt a less massive secondary WD (the “double-
degenerate” case; Iben & Tutukov, 1984)? These and many other questions have
yet to be definitively answered.

SNe Ia have standardizable lightcurves, especially around peak (see e.g. Riess
et al., 1998; Perlmutter et al., 1999; Phillips et al., 1999). But their early
lightcurves, within a few days of explosion, are much less homogeneous and can
contain observational signatures which reveal information about their
progenitor systems. One such signature was predicted in Kasen (2010): in the
single-degenerate case, as the SN ejecta collide with a nondegenerate
companion they will get shock-heated, resulting in early UV excesses which
should be observable for binaries with favorable viewing angles ($\sim$10% of
events). After these early UV excesses were predicted they (or similar
effects) have subsequently been observed in a small number of objects: SN
2012cg (Marion et al., 2016), iPTF14atg (Cao et al., 2015), SN 2016jhr (aka
MUSSES1604D, Jiang et al., 2017), iPTF16abc (Miller et al., 2018), SN 2017cbv
(Hosseinzadeh et al., 2017), SNe 2017erp and 2018yu (Burke et al., 2022), SN
2019yvq (Miller et al., 2020a; Siebert et al., 2020; Tucker et al., 2021;
Burke et al., 2021), and SN 2021aefx (Ashall et al., 2022; Hosseinzadeh et
al., 2022). Some other objects have early excesses, but without the color
information needed to determine their temperature, such as SN 2018oh (Li et
al., 2019; Dimitriadis et al., 2019; Shappee et al., 2019) and SN 2020hvf
(Jiang et al., 2021), and still others (SN 2018aoz; Ni et al., 2022a, b) have
early color evolution which can differ by more than a magnitude from a typical
SN Ia.

Most of the above objects were modeled with companion interaction models, but
there are other physical models which can produce early excesses. “Double-
detonation” models, where the WD builds up a layer of He on its surface until
the He detonates, driving a shockwave into the WD causing it to detonate in
turn, can also produce a range of early lightcurve behavior due to the
presence of extra radioactive products in the outer ejecta (see Sim et al.,
2012; Polin et al., 2019). Models which vary the distribution of 56Ni, which
powers SN Ia lightcurves (Pankey, 1962), can also produce a range of early
behavior, including early bumps (Magee et al., 2018; Magee & Maguire, 2020;
Magee et al., 2020). Both classes of models produce extra radioactive material
in the outer ejecta, resulting in some similar effects (e.g. “red bumps” at
early times), which makes them potentially difficult to distinguish for near-
Chandrasekhar-mass WDs.

Recently, Burke et al. (2022) examined a sample of 9 SNe Ia with exemplary
high-cadence multiwavelength early data. Overall the paper favored companion
interaction models to explain both objects which exhibit an early excess, and
those that do not. The sample was constructed using a set of criteria to make
it as unbiased as possible, and contained one object with a strong early
excess and two others with weaker excesses. Based on the distributions of
early excess strengths and best-fit viewing angles, that paper concluded that
there was not enough evidence to disprove the null hypothesis, i.e. that all
SNe Ia come from single-degenerate progenitor systems.

In this paper we will be focusing solely on companion shocking models to
explain early excesses. We do not claim that the models can perfectly explain
every aspect of the dataset: we refer to the discussion section of Burke et
al. (2022) for an explicit list of their pros and cons. Perhaps the strongest
point against them is the lack of H observed in the nebular spectra of SNe Ia
(see e.g. Sand et al., 2019; Tucker et al., 2020), even though the ejecta-
companion interaction should strip H from a non-degenerate companion
(Botyánszki et al., 2018; Dessart et al., 2020). They also struggle to fit UV
data: even though the process of shock heating should make the early excess
strongest in the UV , the models consistently overpredict the UV flux (see
e.g. Hosseinzadeh et al., 2017, 2022). Keeping these caveats in mind (and
again, for a more in-depth discussion we refer to Burke et al., 2022), we
nevertheless favor the models and use them to investigate the sample of SNe Ia
presented here.

This paper follows very similar methodology to Burke et al. (2022), but
applied to a different sample of SNe Ia. The sample here has an order of
magnitude more objects but limited multi-band information when compared to
Burke et al. (2022), making this a complementary analysis. In Section 2 we
briefly describe our data and sample, and in Section 3 we describe the
companion interaction models we use to fit the data. We show and discuss the
results of the fits in Section 4, before concluding in Section 5.

##  2 Data and Sample

The full dataset in this paper comes from Yao et al. (2019), so we refer to
that paper for details of its acquisition and reduction. The data come from
the ZTF survey (Bellm et al., 2019), and the dataset has many advantages: the
large number of objects (127); the uniform inclusion of non-detections; and
the high cadence of coverage, with some objects being observed not only
nightly, but revisited six times per night (thrice in $g$ and thrice in $r$).
The primary downside of the dataset is that it is in only two filters, $g$ and
$r$, which strongly limits the number of filter combinations which can be used
to determine color/temperature (compare e.g. the $UBVgri$ data used in Burke
et al., 2022).

To obtain the data, we simply downloaded the digital version of Table 5 of Yao
et al. (2019), which is a FITS file containing the full dataset in flux space.
We convert to magnitude space following Equations 7–10 of that paper.

The sample was introduced in a series of three papers: Paper I (Yao et al.,
2019) described the overall data reduction method and the properties of the
sample itself; Paper II (Miller et al., 2020b) modeled the rising lightcurves
of the sample to infer rise times; and Paper III (Bulla et al., 2020) analyzed
the sample’s color evolution. Again, we refer to Yao et al. (2019) for a full
description of the sample. In short, the sample contains 127 SNe Ia. All
objects have observations earlier than at least 10 rest-frame days before
maximum light, and 50 of the objects have detections earlier than 14 rest-
frame days before peak. The median redshift of the sample is 0.076. Though we
focus on photometry here, Yao et al. (2019) also discusses the spectroscopic
classifications of the objects: 107 objects are “normal” SNe Ia (including the
25 SNe listed in the paper as “normal*,” i.e. tentatively classified as
normal), 10 objects are 99aa-likes (including three tentative
classifications), four objects are super-Chandrasekhar SNe Ia (including two
tentative classifications), three are 91T-likes (including two tentative
classifications), one is an 86G-like, one is an 02cx-like, and one is a Ia-
CSM. We again refer to Yao et al. (2019) for details of the classifications,
and also to Taubenberger (2017) for a discussion of the various different
subtypes of SNe Ia.

##  3 Models

###  3.1 Initial Parameter Measurements

Following Burke et al. (2022), we utilize the Python package SNooPy (Burns et
al., 2011) to measure several necessary quantities for the objects. We do
SNooPy fits using the default EBV_model and the fitMCMC() procedure, enforcing
$R_{V,\rm{host}}=3.1$. We impose a uniform prior on EBVhost ranging from 0 to
1. We limit the data to the epochs relevant for SNooPy models, i.e. $-10$ to
$+50$ days from peak (using $\rm{MJD}_{\rm{peak}}$ as measured by Yao et al.,
2019). We visually inspect fits to ensure that they are reasonable. We adopt
the SNooPy values of distance moduli and host extinction for all objects
throughout our analysis. We convert each object’s $E(B-V)$ (Milky Way $+$
host) to per-filter extinction values using the Schlafly & Finkbeiner (2011)
recalibration of the Schlegel et al. (1998) dust maps, accessed by the Python
package extinction (Barbary, 2016).

We also use SNooPy to measure K-corrections for the data, similar to Bulla et
al. (2020). The way that SNooPy fits SN Ia lightcurves already involves
K-corrections, making them straightforward to extract from the fits. This
method of K-correction is the largest deviation from the methodology of Burke
et al. (2022), which otherwise this paper follows quite closely. The sample of
Burke et al. (2022) was low redshift ($z<0.01$), so any K-correction would
both be small and (as is inherent in doing K-corrections) would rely on
assumptions about the underlying SED, even though those assumptions could be
incorrect for objects with early excesses since their early spectra differ
from SNe Ia without excesses (see e.g. Marion et al., 2016; Hosseinzadeh et
al., 2017; Burke et al., 2021). Due to the much higher redshift of the sample
here (median redshift 0.076) we deemed K-corrections necessary, since the
models described in Section 3.2 rely on accurate absolute magnitudes both to
compare to the rest-frame $g$ and $r$ template lightcurves and to compare to
the strengths of any early excesses.

###  3.2 Companion Shocking Models

####  3.2.1 Description of models

Figure 1:  Corner plot of the model described in Section 3.2 for ZTF18aaxsioa,
one of the objects we identify as having an early excess due to the fact that
its posterior distribution of the $\theta$ parameter is peaked to
$\theta=0\degree$. The inset data show the extremely high cadence and useful
non-detections (first epoch shown, unfilled circles) typical of the sample.

Our models are similar to the ones used in Burke et al. (2022) Hosseinzadeh et
al., 2017, along with and Hosseinzadeh et al., 2022, and to some extent those
in Dimitriadis et al., 2019 Miller et al., 2020a, and. They make use of the
lightcurve_fitting Python package (Hosseinzadeh, 2019), which performs MCMC
fits using the emcee package (Foreman-Mackey et al., 2013). The model consists
of two components: a template lightcurve from SiFTO (Conley et al., 2008)
which is scaled and stretched to maximize overlap with the data, to which is
added a blackbody component (which can dominate at early times) representing a
companion shock interaction based on the analytical formulae from Kasen
(2010). As in Burke et al. (2022) we add a parameter to represent the viewing
angle, implemented as a multiplicative factor on the shock component following
the semi-analytic formulation of Brown et al. (2012).

This results in a total of six parameters:

  1. 1.

$a$, the companion separation of the shock component

  2. 2.

$t_{\rm{max}}$, the time of $B$-band maximum light for the stretch component

  3. 3.

$s$, the stretch applied to the stretch component

  4. 4.

$r_{r}$, a factor on the $r$-band flux of the stretch component

  5. 5.

$\theta$, the viewing angle (which determines a multiplicative factor on the
shock component)

  6. 6.

$\sigma$, a multiplicative factor on the errors of the data to account for
error underestimation.

The models are identical to those used in Burke et al. (2022), except for the
fact that two parameters used in that paper are excluded because they affect
only non-$gr$ data and are thus irrelevant for the dataset here. We refer to
that paper for a more detailed description of the models – the two differences
here are the slightly fewer parameters and the fact that we do the fits on the
K-corrected data. We also refer to the discussion section of that paper for an
explicit list of the pros and cons of the model, including various simplifying
assumptions it makes.

A corner plot for an object we identify as having an early excess is shown in
Figure 1.

####  3.2.2 Is It Valid to Use Just $gr$ Data in Companion Shocking Models?

Figure 2:  Joint viewing angle posteriors for the sample modeled in Burke et
al. (2022), which markedly change from the result presented in that paper (the
pink line) when the lightcurves are limited to just $gr$ data (the purple
line). We conclude that these models can still be used to detect early
excesses in $gr$ data, as objects exhibiting early excesses converge to low
viewing angles, but the models shouldn’t be used to measure physical
parameters. It also means that the K-S test method used in Burke et al. (2022)
is no longer applicable for $gr$ data alone.  Figure 3:  The per-object
posteriors of the sample presented in Burke et al. (2022), which change
depending on whether we use the object’s $UBVgri$ dataset (as done in that
paper) or just its $gr$ data (as we do for the sample presented here). The
three objects identified in that paper as having early excesses (SNe 2017cbv,
2017erp, and 2018yu) have best-fit viewing angles which shift towards
$\theta=0$ when the data are limited to $gr$.

Most of the literature using this style of modeling has access to highly
multiwavelength datasets Burke et al., 2022, see Cao et al., 2015;
Hosseinzadeh et al., 2017; Dimitriadis et al., 2019; Miller et al., 2020a;
Burke et al., 2021; Hosseinzadeh et al., 2022, for the most direct comparison,
but also. The wavelength coverage is often $UBVgri$, but sometimes extends
further into the UV with data from the Neil Gehrels Swift Observatory (Swift;
Gehrels et al., 2004). The temperature of the early data, accessed through
different filter/color combinations, is a critical parameter implicitly
measured in the models, making it natural to ask: is it valid to use only $gr$
data with these models?

To summarize the answer to that question: these models can still detect early
excesses using only $gr$ data, but they should not be used to measure physical
parameters.

To see why this is the case, we reexamine the objects modeled in Burke et al.
(2022), limiting the data from $UBVgri$ down to $gr$. One result from that
paper had to do with the joint MCMC posteriors of the $\theta$ parameter from
the nine objects in that sample, where, when comparing the joint posterior to
the expected distribution of viewing angles if all SNe Ia arose from single-
degenerate systems, there was not enough evidence to disprove that null
hypothesis.

Figure 2 shows the joint viewing angle posterior both as it was presented in
that paper (the pink line), and when the data are limited to just $gr$ (the
purple line). This behavior is shown per-object in Figure 3, for both the
$\theta$ and $a$ posteriors. Looking at the figures, it’s clear that objects
which have no detectable excess have still converged to having their maximum
likelihood at $\theta=180\degree$, as they did in the $UBVgri$ models (see SNe
2017fgc, 2018gv, and 2018xx in Figure 3). However, the three objects with
detectable excesses (SNe 2017cbv, 2017erp, and 2018yu) now have a maximum
likelihood closer to $\theta=0\degree$ in the $gr$ fits.

ZTF18 | $a\textrm{ (R}_{\odot}\textrm{)}$ |  $t_{\rm{max}}$ (MJD) | $s$ | $r_{r}$ |  $\theta$ $(\degree)$ | $\sigma$ | IAU | z | Subtype
---|---|---|---|---|---|---|---|---|---
Gold tier
aaxsioa | $15.8^{+2.7}_{-2.3}$ | $58285.78^{+0.05}_{-0.05}$ | $0.888^{+0.007}_{-0.007}$ | $1.006^{+0.004}_{-0.005}$ | $10^{+13}_{-7}$ | $1.97^{+0.13}_{-0.12}$ | cfa | 0.0315 | normal*
abcflnz | $9.4^{+1.5}_{-1.4}$ | $58304.78^{+0.08}_{-0.08}$ | $0.962^{+0.008}_{-0.007}$ | $0.969^{+0.004}_{-0.004}$ | $6^{+9}_{-5}$ | $2.03^{+0.08}_{-0.07}$ | cuw | 0.0273 | normal
Silver tier
abssuxz | $34^{+17}_{-11}$ | $58376.82^{+0.20}_{-0.19}$ | $0.860^{+0.024}_{-0.018}$ | $0.987^{+0.014}_{-0.014}$ | $31^{+47}_{-23}$ | $1.15^{+0.13}_{-0.11}$ | gfe | 0.0649 | normal
abxxssh | $47^{+28}_{-15}$ | $58396.54^{+0.22}_{-0.19}$ | $1.077^{+0.027}_{-0.022}$ | $0.902^{+0.013}_{-0.014}$ | $27^{+34}_{-19}$ | $1.49^{+0.12}_{-0.11}$ | gvj | 0.0782 | normal
aavrwhu | $31.5^{+8.7}_{-7.5}$ | $58275.29^{+0.13}_{-0.13}$ | $1.090^{+0.019}_{-0.017}$ | $0.967^{+0.006}_{-0.007}$ | $15^{+22}_{-11}$ | $1.42^{+0.13}_{-0.11}$ | bxo | 0.062 | normal
abfhryc | $15.9^{+6.9}_{-4.2}$ | $58322.39^{+0.09}_{-0.08}$ | $0.980^{+0.010}_{-0.009}$ | $1.009^{+0.005}_{-0.005}$ | $25^{+27}_{-18}$ | $1.93^{+0.13}_{-0.12}$ | dhw | 0.0323 | normal
Bronze tier
aawjywv | $38^{+19}_{-12}$ | $58270.85^{+0.18}_{-0.17}$ | $0.855^{+0.024}_{-0.020}$ | $0.962^{+0.015}_{-0.016}$ | $30^{+40}_{-22}$ | $2.02^{+0.19}_{-0.16}$ | ccj | 0.0509 | normal*
aaqcozd | $170^{+67}_{-25}$ | $58253.02^{+0.11}_{-0.11}$ | $0.825^{+0.020}_{-0.015}$ | $0.863^{+0.019}_{-0.015}$ | $37^{+40}_{-25}$ | $0.92^{+0.09}_{-0.08}$ | bjc | 0.0732 | normal
abdfazk | $94^{+53}_{-23}$ | $58306.60^{+0.22}_{-0.20}$ | $0.906^{+0.032}_{-0.025}$ | $0.924^{+0.019}_{-0.017}$ | $43^{+49}_{-29}$ | $1.17^{+0.11}_{-0.10}$ | dbe | 0.084 | normal
abimsyv | $55^{+28}_{-19}$ | $58333.03^{+0.19}_{-0.18}$ | $1.003^{+0.021}_{-0.020}$ | $0.915^{+0.012}_{-0.012}$ | $41^{+31}_{-26}$ | $1.16^{+0.07}_{-0.06}$ | eni | 0.088 | normal*
aazsabq | $12.1^{+4.8}_{-3.3}$ | $58293.40^{+0.10}_{-0.10}$ | $0.891^{+0.012}_{-0.011}$ | $1.006^{+0.006}_{-0.006}$ | $21^{+26}_{-15}$ | $1.38^{+0.09}_{-0.08}$ | crn | 0.06 | normal
Table 1:  Values of the best-fit parameters and some object properties for the
11 SNe we identify as having early excesses. The “ZTF18” column is the ZTF
name (with the “ZTF18-” excluded for space). The next six columns are the
best-fit parameters from the models described in Section 3.2. Lastly we give
the IAU name (again excluding the “SN 2018-” for space), the redshift, and the
classification as listed in Yao et al. (2019). We have split the objects into
tiers based on visual inspection of the fits and their residuals (see Figure 5
and surrounding discussion).

This has three implications when the data are limited to $gr$. One is that the
best-fit values of the two relevant physical parameters (i.e. $\theta$ and
$a$) should not fully be believed: for example, even though the viewing angles
seem to have converged nicely for the $gr$ models, there is a systematic
offset from the best-fit $UBVgri$ values for the three objects with early
excesses (SNe 2017cbv, 2017erp, and 2018yu). Their best-fit companion
separations are also systematically lower, though at lower significance.
However, even though the $gr$ models shouldn’t be used to measure physical
parameters, they can still be used to detect early excesses. Early excesses
are still present in the data, regardless of filter (see Figure 3 of Burke et
al., 2022), and will manifest as $\theta$ posteriors which peak towards low
viewing angles. Lastly, it also means that the K-S test done in Burke et al.
(2022) to compare the joint viewing angle posterior to the null hypothesis is
no longer applicable, since the per-object posteriors have changed
significantly.

##  4 Results

###  4.1 Early Excess Rate

In Burke et al. (2022), SNe were classified as having an early excess if they
met three criteria: (1) if the object had data within five rest-frame days of
inferred first light (the typical epochs for an early excess in Kasen, 2010),
(2) if the best-fit viewing angle was less than 90$\degree$, and (3) if the
residuals with respect to the stretch template showed a systematic which was
representative of an early-but-fading shock component (i.e. if they showed an
initial discrepancy of $>$5$\sigma$ which decreased over time). We use similar
criteria here: we keep criterion (1) unchanged; following the results of
Section 3.2.2 we change criterion (2) to a best-fit $\theta\leq 45\degree$;
and we loosen the requirement of criterion (3) due to the lower S/N of the
dataset, although such a pattern is still obvious for some objects.

Using the above criteria, we identify 11 objects with early excesses in our
sample. Table 1 lists the best-fit parameters and some object properties for
these 11 SNe. We have split the objects into tiers based on visual inspection
of their fits and residuals (see Figure 5 and surrounding discussion). Since
the sample contains 127 SNe in total, these 11 objects naïvely represents 8.7%
of the total sample, although as we will discuss more shortly, the intrinsic
rate of early excesses is slightly higher due to the low S/N of early data
potentially obscuring some excesses.

Figure 4:  The best-fit model and residuals for ZTF18aaxsioa, an object with
an early excess. The left panels show the per-filter data and the two
components of the model: the stretched template (dotted line) and the shock
component (dashed), which are added together to the full model. The right
panels show the residuals with respect to both the full model (filled squares)
and the stretched template (empty diamonds), where the signature of the early
excess is obvious (initial discrepancy which vanishes over time). “Pull” is
defined as the number of standard deviations a data point is away from the
relevant model, divided by the best-fit $\sigma$ value – the majority of
square points should cluster within one standard deviation of zero (the grey
shaded box), which they do.

Figure 4 shows the best-fit model and residuals for a single object which we
identify as having an early excess (ZTF18aaxsioa, the same object shown in
Figure 1). The distinctive pattern in the residuals with respect to the
stretch template (diamonds), where their discrepancy diminishes over time, is
characteristic of an early-but-fading shock component (see Figure 3 of Burke
et al., 2022).

The sample modeled in Burke et al. (2022) consisted of nine objects, three of
which had early excesses: that paper therefore calculated an early excess rate
from simple Poissonian statistics of $33\pm 19\%$. However, the objects in
that paper were extremely low-redshift ($z<0.01$), and all objects were
detected well above the telescopes’ limiting magnitudes within five days of
inferred first light. (“First light” in this context being the time when the
SN flux is first detectable, which is not necessarily the same as when the
explosion occurred due to “dark phases” in SNe Ia when the radiation has not
yet diffused out of the ejecta – see Piro & Nakar, 2013, where this is
discussed in more detail.) Burke et al. (2022) assumed that any potential
early excesses would be revealed by the data. The sample here is significantly
different: it is much higher redshift (median redshift of 0.076), resulting in
lower average signal-to-noise (S/N) in the crucial earliest epochs.

S/N cut ($\sigma$) | $n_{\rm{objects}}$ | $n_{\rm{EEx}}$ | Rate
---|---|---|---
N/A | 127 | 11 | $8.7\pm 2.6\%$
3 | 103 | 11 | $10.7\pm 3.2\%$
5 | 92 | 11 | $12.0\pm 3.6\%$
8 | 68 | 11 | $16.2\pm 4.9\%$
10 | 49 | 9 | $18.4\pm 6.1\%$
15 | 26 | 5 | $19.2\pm 8.6\%$
20 | 18 | 3 | $16.7\pm 9.6\%$
30 | 9 | 2 | $22\pm 16\%$
50 | 4 | 2 | $50\pm 35\%$
Table 2:  The effect on the early excess (EEx) rate of imposing different S/N
cuts on the early data, e.g. requiring that at least one epoch within five
rest-frame days of inferred first light has a $>$10$\sigma$ detection. The two
early-excess SNe that survive the $50\sigma$ cut are the two gold-tier objects
(see Table 1). We quote an overall early excess rate of $12.0\pm 3.6\%$, the
value from the $5\sigma$ cut, since the rates calculated for more stringent
S/N cuts are all consistent (although systematically higher) than that value.

This lower S/N could serve to obscure weak early excesses hiding in the data.
For instance, in this sample only 81% of the objects (103 out of 127) are
detected at all within 5 rest-frame days of inferred first light. Table 2
shows the results of imposing different S/N cuts on the early data of the
sample, e.g. only considering objects which have a $>$10$\sigma$ detection
with 5 days of first light. Each S/N cut leaves some total number of objects,
and also some subset of the early-excess objects, from which an early excess
rate can be calculated. The $30\sigma$ cut leaves a sample extremely similar
to the one in Burke et al. (2022): that paper had an early excess rate of 3
out of 9 objects ($33\pm 19\%$), and limiting the sample here to SNe with a
$30\sigma$ detection in the epochs where an early excess could be detected
yields a rate of 2 out of 9 objects ($22\pm 16\%$). We thus believe that the
sample here is identical to the sample of Burke et al. (2022), except at
higher redshift and observed in fewer bands. As in that paper, the
distribution of early excess strengths here is consistent with SNe Ia
predominantly arising in single-degenerate systems.

Which rate should we quote as the single rate of early excesses in SNe Ia? As
is obvious in Table 2, using a less restrictive S/N cut results in a smaller
formal uncertainty due to the larger sample size, but it also means that lower
intensity early excesses could be hidden in the data, undetected. We use the
value calculated from the S/N$>$5 cut, since that S/N is sufficient to detect
strong early excesses, and since rates calculated with higher S/N cuts are all
consistent (although systematically higher) than that value. It is possible
that the higher rates derived from stricter S/N cuts are consistent with the
prediction in Kasen (2010), i.e. that higher S/N data could detect weaker
excesses, leading to a higher overall rate of excesses, but the uncertainties
on the rates make it difficult to state this with confidence. We thus quote
the rate of early excesses in SNe Ia as $12.0\pm 3.6\%$.

The rate of early excesses in SNe Ia which we measure here is consistent with
the rate calculated in Deckers et al. (2022) of $18\pm 11\%$ (calculated from
the same dataset), and is also consistent with the limit of $\lesssim$30%
based on the non-detections in Miller et al. (2020b), again calculated from
the same dataset. It is also consistent with the rate calculated in Burke et
al. (2022) of $33\pm 19\%$. Magee et al. (2020) find 5 excesses in a sample of
23 SNe, corresponding to a rate of $22\pm 10\%$, which we are also consistent
with. The theoretically expected rate quoted in Kasen (2010) is $\sim$10% for
strong excesses: our $5\sigma$ cut should be able to detect such excesses, and
we are again consistent with this value. The rates of early excesses from all
these studies, with their widely varying methodologies, are consistent with
the null hypothesis that SNe Ia predominantly arise from single-degenerate
systems.

Additionally, as seen in Table 1, all 11 early-excess SNe are classified as
normal SNe Ia. The sample contains 13 overluminous SNe Ia (i.e. 99aa-like or
91T-like), and Jiang et al. (2018) suggested that such overluminous SNe Ia
uniformly have early excesses based on six out of six such objects in their
sample exhibiting early excesses. We do not see this effect here: most of the
overluminous objects in this sample do have relatively low early S/N due to
their higher redshift, but two of them (ZTF18abgmcmv and ZTF18abauprj) have
early $\rm{S/N}>30$ and still have no detectable early excess. The early-
excess objects are not even clustered to the luminous end of normal SNe Ia,
with only 4 of the 11 having $\Delta m_{15}<1.0$ (as measured by SNooPy).

###  4.2 Methodology-dependent Early Excess Detections

Table 3: Which SNe in this sample have early excesses, as identified by
different papers

[1] | [2] | [3] | [4] | [5]
---|---|---|---|---
|  | aapqwyv |  |
|  |  |  | aaqcozd
|  |  | aaqqoqs |
aavrwhu |  |  |  | aavrwhu
|  |  |  | aawjywv
|  |  |  | aaxsioa
|  |  | aayjvve |
|  |  |  | aazsabq
|  | abcflnz |  | abcflnz
|  | abckujq |  |
|  | abcrxoj |  |
|  |  | abdfazk | abdfazk
|  |  | abdfwur |
|  |  |  | abfhryc
|  | abgxvra |  |
|  |  |  | abimsyv
|  |  | abpamut |
|  |  |  | abssuxz
abxxssh |  | abxxssh | abxxssh | abxxssh

Note. — The “ZTF18” has been omitted from each object name to save space. As
is apparent in the table and as discussed in the text, the statement “this SN
does/does not have an early excess” is methodology-dependent. References: 1:
Yao et al. (2019), 2: Miller et al. (2020b), 3: Bulla et al. (2020), 4:
Deckers et al. (2022), 5: this work.

As noted in Section 2, this sample of objects was introduced in a series of
three papers, and an additional paper (Deckers et al., 2022) examined the same
dataset through the lens of Ni models. Each of these four previous studies,
and now the one presented here, have independently looked for objects with
early excesses. Table 3 therefore lists the early-excess objects identified by
each paper, each of which approached the identification with a distinct
methodology: Yao et al. (2019) identified two early-excess objects based on
photometric comparison to other objects with extremely early data (more than
17 days before peak); Miller et al. (2020b) identified zero early-excess
objects, since all rising lightcurves were consistent with their general
power-law fits; Bulla et al. (2020) identified six objects which had red bumps
at early times in their $g-r$ color evolutions; and Deckers et al. (2022)
identified six objects with early excesses, based on quantitative measures of
whether they could be fit by any Ni-mixing models in the model grids from
Magee et al. (2018) and Magee et al. (2020). As the table makes apparent, each
paper (i.e. each methodology) identified mostly mutually exclusive sets of
SNe. As stated above, we identify 11 objects with early excesses based on
their fits to companion shocking models, adding another mostly-mutually-
exclusive set of early-excess SNe Ia from this sample.

Even though they might seem contradictory, two things are both true: this work
and each of the papers mentioned above have (different) quantitative measures
to detect early excesses, and also they all mostly disagree on which objects
have early excesses. These statements do not invalidate each other – we merely
want to stress that even though the final cuts for which objects have early
excesses can be quantitative (i.e. objective), the assumptions undergirding
the methodologies introduce (subjective) biases which end up with different
results.

Figures 5, 6, and 7 show the fits and residuals for three different sets of
SNe drawn from the sample. Figure 5 shows the 11 objects which we identify as
having an early excess, in the order of their tier as listed in Table 3. The
first two objects (ZTF18aaxsioa and ZTF18abcflnz) are the two “gold” objects,
because their residuals show a clear and temporally resolved shock component,
manifesting as an initial discrepancy with respect to the stretch template,
which then fades over time. ZTF18aaxsioa was already plotted in Figure 4,
since its residuals have such a clear systematic. The next four objects
(ZTF18abssuxz through ZTF18abfhryc) are in the silver tier, since either the
shock component is not as well resolved temporally (e.g. ZTF18abssuxz) or only
the initial epoch or two are significantly discrepant with the stretch
template. ZTF18abxxssh was classified as an early-excess object by most papers
studying this sample (see Table 3), primarily due to its first two epochs with
detections. The last five objects (ZTF18aawjywv through ZTF18aazsabq) are in
the bronze tier: their residuals don’t display systematics as obvious as the
above objects, but nevertheless they are poorly fit by the stretch template
alone and the models have unambiguously converged to have some shock
component.

Figure 5:  Fits and residuals for objects we identify as having an early
excess. Objects are ordered by tier (see Table 1): the first two SNe are gold-
tier (i.e. the residuals show a strong well-resolved shock component), the
next four are silver-tier (the shock is not as resolved), and the last five
are bronze-tier.  Figure 6:  Identical to Figure 5, but for the objects which
other papers identify as having an early excess (see Table 3). Panels are
labelled with the object name, sans ZTF18-. See the text for discussion of
different fits.  Figure 7:  Identical to Figures 5 and 6, but for objects
which have especially good early data, and show no signatures of an early
excess. We include this figure to show that the stretch template by itself can
be an excellent fit to some objects, validating its use.

The next set of objects (Figure 6) are the eight objects which other papers
identify as having an early excess, but we do not. Some objects (ZTF18aapqwyv,
ZTF18aaqqoqs, ZTF18abdfwur) have converged to a similar region of parameter
space as SN 2019np did in Burke et al. (2022), with a large best-fit companion
separation seen at a high viewing angle needed to explain the discrepancy with
the stretch template – their high viewing angles have excluded them from being
listed among our early-excess SNe (see the criteria listed at the start of
Section 4.1). This matches the analysis of SN 2019np, where another study did
identify it as having an early excess (Sai et al., 2022), even though Burke et
al. (2022) did not. Others (ZTF18aayjvve, ZTF18abgxvra) have no signs of
disagreement with our stretch template – in the case of ZTF18abgxvra only
Bulla et al. (2020) identified it as having an early excess, and since that
paper relied on color evolution to identify excesses, the object’s early
peculiarities may not be obvious when looking filter-by-filter.

The last set of objects (Figure 7) was selected because the objects have
exemplary early data, and also showed no signs of any early excesses (i.e. the
two sets of residuals are on top of each other). We include this set primarily
to show that some objects have early lightcurves which are fit excellently by
the stretch template alone, validating its use.

Figure 8:  Color evolution of the sample. The 11 SNe we identify as having an
early excess are shown in warm colors, and the 8 SNe other papers identify as
having an early excess are shown in cool colors (see Table 3 for early-excess
classifications). The rest of the sample is shown in grey. Colors have been
extinction- and K-corrected. Compare Figures 1 and 4 of Bulla et al. (2020).

Lastly, we show the color evolution of the sample in Figure 8, highlighting
early-excess objects. In Burke et al. (2022), the SN with the clearest early
excess (SN 2017cbv) also had the bluest $g-r$ colors at early times. That is
not quite true here, where the two SNe with the clearest early excesses in our
fits (ZTF18aaxsioa and ZTF18abcflnz) are not the bluest at early times;
however, ZTF18abxxssh is among the bluest SNe within three days of first
light, and it is the object that most papers identify as having an early
excess (see Table 3). Generally, SNe Ia seem to show lower dispersion in their
early $g-r$ evolution compared to other colors (such as $B-V$ or UV filter
combinations, see Figures 9 and 10 from Burke et al., 2022), but we refer to
Bulla et al. (2020) for a much more thorough investigation of this sample’s
color properties.

##  5 Conclusion

We have reexamined the 2018 ZTF sample of early SNe Ia (Yao et al., 2019;
Miller et al., 2020b; Bulla et al., 2020), using models from Kasen (2010) to
search for signatures of nondegenerate companion interaction in the form of
early excesses in the lightcurves. We found 11 such objects with signatures of
companion interaction: this naïvely represents 8.7% of the sample, but when
compensating for the S/N of the early data we calculate an overall early
excess rate of $12.0\pm 3.6\%$. This rate is consistent with several others
calculated throughout the literature, using different methodologies and/or
samples to detect early excesses. This rate is also consistent with the
expectation that SNe Ia predominantly arise from progenitor systems with a
Roche-lobe-overflowing nondegenerate companion. However, early excesses only
occur in this sample in normal SNe Ia and not in any of the 13 overluminous
objects, which is inconsistent with the claim in Jiang et al. (2018) that
overluminous SNe Ia have ubiquitous early excesses.

In addition to this result, we also showed that the detection of early
excesses can be methodology-dependent. For instance, despite the fact that
this same sample had been analyzed by four previous papers, each of which
looked for objects with early excesses, we identify seven SNe Ia as having
early excesses which none of the previous papers identified as such. This
meta-analysis shows that different methodologies make a variety of assumptions
about the early lightcurves of SNe Ia, which can lead to different results.

Companion interaction models have been shown for several years to be excellent
(if imperfect) fits to SNe Ia with early excesses (see e.g. Hosseinzadeh et
al., 2017; Dimitriadis et al., 2019; Burke et al., 2021; Hosseinzadeh et al.,
2022; Burke et al., 2022). We strongly encourage the observation of large
samples of SNe Ia with high-cadence multiwavelength early data (such as the
ones in this paper and in Burke et al., 2022) so that models’ predictions
might be tested and the the uncertainty around the progenitor systems of SNe
Ia might be gradually reduced. We also encourage continued theoretical work on
companion interaction models, since the state of the art is now more than a
decade old (Kasen, 2010).

J.B. and D.A.H. are supported by NSF grants AST-1911151 and AST-1911225, as
well as by NASA grant 80NSSC19kf1639. Time domain research by D.J.S. and G.H.
is supported by NSF grants AST-1821987, 1813466, 1908972, & 2108032, and by
the Heising-Simons Foundation under grant #2020-1864.

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# NBD-GAP: Non-Blind Image Deblurring Without Clean Target Images

###### Abstract

In recent years, deep neural network-based restoration methods have achieved
state-of-the-art results in various image deblurring tasks. However, one major
drawback of deep learning-based deblurring networks is that large amounts of
blurry-clean image pairs are required for training to achieve good
performance. Moreover, deep networks often fail to perform well when the
blurry images and the blur kernels during testing are very different from the
ones used during training. This happens mainly because of the overfitting of
the network parameters on the training data. In this work, we present a method
that addresses these issues. We view the non-blind image deblurring problem as
a denoising problem. To do so, we perform Wiener filtering on a pair of blurry
images with the corresponding blur kernels. This results in a pair of images
with colored noise. Hence, the deblurring problem is translated into a
denoising problem. We then solve the denoising problem without using explicit
clean target images. Extensive experiments are conducted to show that our
method achieves results that are on par to the state-of-the-art non-blind
deblurring works.

Index Terms—  Non blind deblurring, Wiener Deconvolution, No-reference.

##  1 Introduction

Motion blur is a common and prominent problem that occurs in hand-held
photography. It destroys the aesthetics of the image and adversely affects the
performance of many computer vision applications [1, 2]. In this work, we
focus on the case of uniform blur where the blurred image is represented by,

$y=k*x+n,$ (1)

where $y$ is the blurry image, $x$ is the latent clean image, $k$ is the blur
kernel, $n$ is the additive white Gaussian noise, and $*$ denotes the
convolution operation. Existing deblurring methods that restore the clean
image $x$ can be grouped into blind and non-blind deblurring methods. Non-
blind deblurring methods take both the blurry image $y$ and the blur kernel
$k$ as input to restore $x$. On the other hand, blind deblurring is a more
difficult problem that requires only the blurry image $y$ as input to restore
$x$. In this paper, we mainly focus on the non-blind deblurring problem.

Fig. 1: Sample output from our restoration network. (a) Blurry input image.
(b) Wiener filtered image. Note the artifacts and noise after Wiener
filtering. (c) Restored Image.

Early non-blind deblurring algorithms used statistical techniques to derive
minimum mean square error (MMSE) solutions such as the Wiener filter [3] and
Richardson-Lucy algorithm [4]. Although these algorithms work well for very
low noise levels, they add a significant amount of colored noise when the
noise level increases. Also, these methods suffer from serious ringing
artifacts and cannot deal with degradations caused by large motions. Another
line of deconvolution algorithms focus on developing effective image priors
[5, 6, 7, 8] using natural image statistics. But, these priors are very
reliant on the distribution of the natural images and often lead to highly
non-convex optimization problems, hence requiring expensive computational
power.

Recently neural networks have been used for various image deblurring tasks [9,
10, 11] and have achieved state-of-the-art results in different deblurring
scenarios [12, 13, 10]. There are two major techniques by which deep learning
has been adopted for non-blind deblurring. One line of works develop a model
which uses both the blurred image and the kernel as input to address the
deblurring problem [14, 15]. Another line of works decomposes the problem into
a denoising problem using a deconvolution algorithm or deep neural network.
Then address the deconvolution and denoising problems separately [16, 9, 17].
These techniques require large amounts of paired data, i.e., blurred images
and their corresponding clean target pairs, to train the deep networks. These
algorithms also fail to generalize well when the blur kernels during testing
differ from the training kernels. Non-blind deblurring methods are often used
as a black-box module for solving the blind deblurring algorithms [18, 19, 20,
21] where the kernels and the latent images are found iteratively until the
latent clean image is restored. Hence, there exists a need to develop non-
blind deblurring algorithms that generalize over unseen images and kernels and
learn with less data. To this end, we propose an approach that can be trained
without clean target images and can generalize well to unseen data and
kernels.

Fig. 2: An overview of the proposed network. During training, two input
blurred images are processed through Wiener filtering to produce the
corresponding two noisy images. Then these two noisy images are used as input
and targetto train a network. During testing, for restoration, a single image
is Wiener filtered and passed through the denoising network to obtain the
restored image.

In this work, we follow the second line of techniques as mentioned above and
propose a Non-blind image deblurring technique. Our method doesn’t require the
actual clean target data corresponding to the blurry input images.
Specifically, we first perform Wiener filtering to deblur two different blurry
images of the same scene and get a pair of images with colored noise. Hence,
we decompose the deblurring problem into a denoising problem. The key idea
behind our denoising network is motivated by the work of Lehtinan et al. [22]
where the authors have performed denoising of images corrupted by both
Gaussian as well as colored noise without explicit training using clean target
images. The main idea is that when a network is trained with multiple pairs of
noisy images of the same latent scene, the expected loss between the network
output and the noisy target is approximately equal to the expected loss
between the network output and the corresponding clean target. The network
hence learns the distribution of noise and can perform denoising. Inspired by
this, we utilize Wiener deconvolution to produce a pair of images with zero
centered colored noise. We then treat one of the outputs of Wiener
deconvolution as source and the other as the target to train our network.

##  2 Proposed method

###  2.1 Deconvolution as a denoising problem

Given $y$ and $k$ in (1), our objective is to restore $x$. A simple solution
to this problem is finding an inverse filter $g$ where,

$\hat{x}=g*y,$ (2)

such that we can make an estimate of the clean latent image represented by
$\hat{x}$. The ideal solution for $\hat{x}_{ideal}$ minimizes the expected
mean square error between the clean latent image $x$ and the estimate

$\hat{x}_{ideal}=\min\hskip 2.84526ptE|x-\hat{x}|^{2}.$ (3)

This inverse filtering problem can be easily solved in the Fourier domain and
the solution for the inverse filter $g$ is the Wiener filter defined by

$G=\frac{K^{H}}{|K|^{2}+\frac{E|X|^{2}}{E|N|^{2}}},$ (4)

where $G,K,X,N$ denote the variables in the Fourier domain and $H$ denotes the
Hermitian operator (detailed derivation is given in the supplementary
document). The corresponding solution turns out to be
$\hat{x}_{ideal}=x+n_{c},$ where

$n_{c}=IFFT\left(\frac{K^{H}N}{|K|^{2}+\frac{E|N|^{2}}{E|X|^{2}}}\right).$ (5)

Here, $n_{c}$ represents the inverse Fourier transform of the colored noise
present along with the image after Wiener filtering. Now, since
$\frac{E|N|^{2}}{E|X|^{2}}$ is an even function, the Fourier transform of
$n_{c}$ is odd, and the inverse Fourier transform is also odd. Hence, its mean
is zero. Thus the output of Wiener filtering contains the latent image
corrupted with zero mean colored random noise. Hence by performing Wiener
filtering, the deblurring problem converts into a denoising problem. One
crucial aspect of Wiener filtering is estimating the noise to signal ratio
(NSR) term defined by $\frac{E|N|^{2}}{E|X|^{2}}$. In our work, similar to
[17], we first find a median filtered estimate of the blurred image and
compute its variance with the original blurred image to estimate NSR. Multiple
works have previously taken this approach to simplify the deconvolution
problem to a denoising problem by utilizing various transformations such as
shearlet [23], and wavelet [24].

Given blurry images $y$ with the corresponding blur kernels $k$, we first
convert the deconvolution task into a denoising by applying Wiener filtering
as follows,

$\hat{x}=F^{-1}\left(\frac{\overline{F(k)}F(y)}{|F(k)|^{2}+\frac{E|N|^{2}}{E|X|^{2}}}\right),$
(6)

where $F$ denotes the Fourier transform. In practical scenarios, the output of
Wiener filtering is often corrupted with ringing artifacts along with the
coloured noise. The colored noise could be removed to an extent by training
the network using another set of noisy images as targets as the output like in
the case of Lehtinan et al[22].

Table 1: Quantitative evaluation on Levin et al[25] dataset consisting of 32
images in terms of PSNR, SSIM. Please note that all the deep learning based
methods used for comparison ad achieves better result than us in the
$\sigma=12.75$ dataset is a supervised technique. MD-Motion deblurring, NBD-
Non-Blind Deblurring.

Noise | Metrics | Supervised-MD | Supervised-NBD | No clean target training-NBD
---|---|---|---|---
Kupyn[26] |  Gong [27] |  Son [17] |  IRCNN[28] |  Zhang[9] |  Kruse[15] |  EPLL [29] | OURS
$\sigma=2.55$ | PSNR | 24.52 | 32.32 | 31.56 | 32.94 | 32.87 | 33.68 | 33.72 | 34.97
SSIM | 0.712 | 0.913 | 0.896 | 0.912 | 0.919 | 0.922 | 0.929 | 0.950
$\sigma=7.65$ | PSNR | 23.97 | 29.22 | 28.95 | 30.51 | 29.55 | 29.80 | 29.05 | 30.21
SSIM | 0.641 | 0.852 | 0.837 | 0.875 | 0.860 | 0.857 | 0.845 | 0.883
$\sigma=12.75$ | PSNR | 22.21 | 27.13 | 27.35 | 27.92 | 27.77 | 27.95 | 26.53 | 26.91
SSIM | 0.542 | 0.791 | 0.802 | 0.821 | 0.814 | 0.812 | 0.778 | 0.822

###  2.2 Network architecture

The overall pipeline of our network is shown in Figure 2. During the training
process, we create two different blurry images using different blur kernels
and random noise signals. Then, each of these blurred images are Wiener
filtered to generate two images of the same latent scene with different random
coloured noise. We use a simple U-net based architecture as our base network.
We then use one of these images as the input and the other image as target
output of the network.

Noisy Image EPLL [29] Son [17] IRCNN [28] Kruse [15] NBD-GAP(ours) Ground-
Truth

Fig. 3: Qualitative evaluations on the Levin et al dataset [25] for
$\sigma=2.55$ noise level.

###  2.3 Loss function

Let $I_{n}$ and $\hat{I_{2}}$, denote two noisy images of dimensions $C\times
H\times W$. The loss function used to train our network is

$\mathcal{L}_{2}=\frac{1}{CHW}\sum_{i,j,k}\lVert I_{1}-\hat{I_{2}}\rVert^{2}.$
(7)

##  3 Experiments

We use a batch size of 8 and use the AdamW optimizer with parameters
$\beta_{1}=0.9$ and$\beta_{2}=0.999$. We set the initial learning rate to
$10^{-4}$ and reduce it by half every 10 epochs. We train our network for a
total of 60 epochs. ALl the experiments are run in

###  3.1 Training Dataset

For creating our training dataset, we follow the approach that has been
followed in multiple non-blind deblurring works [11, 9, 28, 17]. We create a
dataset of 1,000 images consisting of 600 images from the Waterloo exploration
dataset [30] and 400 images from the Berkeley Segmentation dataset [31]. We
synthesize kernels of random sizes varying from $11\times 11$ to $35\times 35$
according to the method proposed by [32] and convolve them with a patch of
size $256\times 256$ cropped from a clean image. We finally apply additive
Gaussian noise of standard deviation $\\{2.55,7.65,12.75\\}$ to form blurry
images.

###  3.2 Testing Dataset

For testing our network, we use three different testing datasets. Similar to
the approaches [11, 9, 17], we use the popular benchmark datasets from Levin
et al. [25]. Levin et al. [33] consists of 4 grayscale images respectively. We
utilize the 8 standard kernels released by [25] and blur these images and add
noise of different variances in the range of $\\{2.55,7.65,12.75\\}$ to create
blurry images. We also create a synthetic dataset using the 100 test images of
the BSD dataset [31] and blur it with a random blur kernel generated using
[32]. Note that the kernels used for the testing are different from the ones
used for training.

Table 2: Quantitative evaluation on 500 blurry images generated from the
BSD500 dataset [31] in terms of PSNR, SSIM. Please note that all the deep
learning-based methods used for comparison achieve better results than us in
the $\sigma=7.65$ noise case are supervised techniques. MD denotes Motion
deblurring and NBD denotes Non-Blind Deblurring.

Noise | Metrics | Supervised-MD | Supervised-NBD | Self-supervised-NBD
---|---|---|---|---
Kupyn[26] |  Gong [27] |  Son [17] |  IRCNN[28] |  Zhang[9] |  Kruse[15] |  EPLL [29] | OURS
$\sigma=2.55$ | PSNR | 23.25 | 27.61 | 27.40 | 27.85 | 27.55 | 27.22 | 27.49 | 30.27
SSIM | 0.712 | 0.841 | 0.831 | 0.854 | 0.840 | 0.825 | 0.839 | 0.862
$\sigma=7.65$ | PSNR | 23.97 | 25.73 | 25.51 | 25.63 | 26.70 | 25.61 | 25.62 | 26.81
SSIM | 0.641 | 0.751 | 0.729 | 0.758 | 0.802 | 0.738 | 0.746 | 0.761
$\sigma=12.75$ | PSNR | 22.21 | 24.35 | 24.42 | 24.66 | 24.60 | 24.53 | 24.71 | 24.82
SSIM | 0.542 | 0.649 | 0.641 | 0.660 | 0.652 | 0.677 | 0.688 | 0.685

Noisy Image EPLL [29] Zhang [28] Son [17] Gong [27] NBD-GAP(ours) Ground-Truth

Fig. 4: Qualitative evaluations on the BSD500 dataset [25] for $\sigma=7.65$
noise level and $\sigma=12.75$ noise level. The First two figures from top
represent images of $\sigma=7.65$ noise level and the third figure has
$\sigma=12.75$ noise level. The corresponding absolute error map from the
ground truth is shown below each cropped patch.

###  3.3 Results

We evaluate the performance of our network by comparing qualitatively and
quantitatively with different supervised works in non-blind deblurring. For
comparison, we use a combination of classical non-blind deblurring algorithms
[29], supervised deep networks designed for non-blind deblurring [9, 28, 17],
methods which utilize a combination of supervised deep learning and
optimization algorithms [27] and one supervised motion deblurring network
[26]. Peak signal to noise ratio (PSNR) and structural similarity index
measure (SSIM) are used to measure the performance of different methods
quantitatively.

####  3.3.1 Results on the Levin et al.[25] dataset

Table.1 shows the quantitative results for different noise levels in the
dataset prepared using images and kernels released by Levin et al. [25]. We
can from Figure 3 that conventional patch-based method EPLL [29] but creates a
small amount of unwanted artifacts. Son et al. [17] converts the deblurring
problem to a denoising problem, similar to us. But we can see that some noise
is still retained in the output. Note that the deep learning-based method
IRCNN [28] achieves better results than us in the $\sigma=7.65$ noise case.
However, IRCNN is a supervised method and is trained with a lot of paired
training data.

####  3.3.2 Results on the BSD500 dataset [31]

Since both Levin et al.[25] and Sun et al. [34] consist of gray scale images,
to evaluate the performance on coloured images, we generate a test set using
100 test images from [31] using kernels from [32] for three different noise
levels. As we can see in Table 2 and Fig. 4, our method is able to perform
well and produce results with fewer artifacts and noise compared to [17]. Our
method is even able to produce good results at higher levels of noise. The
residual maps show that the absolute error is much lesser in our method when
compared to the other techniques.

##  4 Conclusion

In this paper, we proposed the first work in non-blind deblurring, where we
train a deep network without clean targets. We achieve this by converting the
deblurring problem to a denoising problem using Wiener filtering. We perform
extensive experiments on multiple datasets to show the relevance of our
network and show that our network can generalize well over multiple datasets.
Furthermore, it is shown that our network is able to produce good results even
at higher levels of noise.

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