Roundness of the ample cone and existence of double Lagrangian fibrations on
hyperkähler manifolds

Ljudmila Kamenova111Partially supported by a grant from the Simons
Foundation/SFARI (522730, LK), Misha Verbitsky222Partially supported by the
Russian Academic Excellence Project ’5-100’, CNPq Process 313608/2017-2 and
FAPERJ E-26/202.912/2018.

Abstract. Let $M$ be a hyperkähler manifold of maximal holonomy (that is, an
IHS manifold), and let $K$ be its Kähler cone, which is an open, convex subset
in the space $H^{1,1}(M,{\mathbb{R}})$ of real (1,1)-forms. This space is
equipped with a canonical bilinear symmetric form of signature $(1,n)$
obtained as a restriction of the Bogomolov-Beauville-Fujiki form. The set of
vectors of positive square in the space of signature $(1,n)$ is a disconnected
union of two convex cones. The “positive cone” is the component which contains
the Kähler cone. We say that the Kähler cone is “round” if it is equal to the
positive cone. The manifolds with round Kähler cones have unique bimeromorphic
model and correspond to Hausdorff points in the corresponding Teichmüller
space. We prove thay any maximal holonomy hyperkähler manifold with $b_{2}45$
has a deformation with round Kähler cone and the Picard lattice of signature
(1,1), admitting two non-collinear integer isotropic classes. This is used to
show that all known examples of hyperkähler manifolds admit a deformation with
two transversal Lagrangian fibrations, and their Kobayashi metric vanishes,
unless the Picard rank is maximal.

##  1 Introduction

This paper gives a simple solution for a construction problem of hyperkähler
geometry. We construct a hyperkähler manifold with rank 2 hyperbolic Picard
lattice and maximal possible Kähler cone.

For our purposes, “a hyperkähler manifold” is a compact, holomorphic
symplectic compact manifold $M$ of Kähler type, which satisfies “the maximal
holonomy condition”, that is, $\pi_{1}(M)=0$, $\dim H^{2,0}(M)=1$. This
condition is also known as IHS (“irreducible holomorphic symplectic”).

The shape of the Kähler cone of a hyperkähler manifold is more or less
understood by now (see [AV3]). However, finding examples of manifolds with
prescribed shape of their Kähler cone is a complicated task. Such
constructions are either very explicit or based on convoluted arguments from
number theory. The automorphism group of a hyperkähler manifold and its set of
Lagrangian fibrations can be described explicitly in terms of its periods and
the shape of the Kähler cone ([AV4]). Therefore, finding manifolds with
prescribed Kähler cones has many practical applications.

Recall that the second cohomology of a maximal holonomy hyperkähler manifold
is equipped with a bilinear symmetric form of signature $(3,b_{2}-3)$, which
is essentially of topological origin ([Bea, Bo, F]). This form, denoted by $q$
further on, is called the Bogomolov-Beauville-Fujiki form; it is positive on
the Kähler cone, and has signature $(1,b_{2}-3)$ on $H^{1,1}(M,{\mathbb{R}})$.

This implies that the set of “positive vectors” (the vectors with positive
square) in $H^{1,1}(M,{\mathbb{R}})$ has two connected components, both of
them convex cones. However, only one of these two components may contain
Kähler forms. We call this component “the positive cone” of a hyperkähler
manifold.

Every hyperkähler manifold is equipped with a collection ${\mathfrak{S}}$ of
primitive integer cohomology classes in $H^{2}(M,{\mathbb{Z}})$ with negative
squares, called the MBM classes ([AV1]). This set is invariant on deformations
of $M$ and under the action of the monodromy group $\Gamma$ generated by the
monodromy of the Gauss-Manin connection for all deformations of $M$. The group
$\Gamma$ (originally defined by E. Markman, [Mar1], who called it the
monodromy group) is mapped to the orthogonal lattive
$O(H^{2}(M,{\mathbb{Z}}))$ with finite kernel, and its image is a finite index
sublattice in $O(H^{2}(M,{\mathbb{Z}}))$ ([V1]).

In [AV2], it was shown that the monodromy group $\Gamma$ acts on the set of
MBM classes with a finite number of orbits, which were computed for some
deformational classes of hyperkähler manifolds in [BM], [HT3], [HT4], [AV5].

As shown in [AV3] (the result is essentially due to E. Markman, [Mar2]), the
positive cone $\operatorname{Pos}(M,I)$ of a hyperkähler manifold is cut into
pieces by hyperplanes orthogonal to the MBM classes which lie in
$H^{1,1}(M,I)$, and each of the connected components of this complement can be
realized as a Kähler cone of a certain hyperkähler birational model of
$(M,I)$. In other words, the Kähler cone is a connected component of the set

$\operatorname{Pos}^{\circ}(M,I):=\operatorname{Pos}(M,I)\left\backslash\bigcup_{\alpha\in{\mathfrak{S}}\cap
H^{1,1}(M,I)}\alpha^{\bot}\right.,$ (1.1)

where ${\mathfrak{S}}$ is the set of all MBM classes in
$H^{2}(M,{\mathbb{Z}})$, and all connected components are realized as Kähler
cones for birational models of $(M,I)$

The authomorphism group of a hyperkähler manifold $(M,I)$ is expressed in
terms of its Kähler cone and the monodromy group as follows. Let
$\Gamma_{I}\subset\Gamma$ be the subgroup of the monodromy group preserving
the Hodge decomposition on $H^{2}(M)$. Then $\operatorname{Aut}(M,I)$ is a
subgroup of all elements in $\Gamma_{I}$ which preserve the Kähler cone.

We say that a manifold $M$ has round Kähler cone if
$\operatorname{Kah}(M)=\operatorname{Pos}(M)$, or, equivalently, when the set
of MBM classes in $H^{1,1}(M,I)$ is empty.

To construct a manifold with round Kähler cone and rank 2 Picard lattice, we
use Kneser’s orbit theorem, claiming that for any non-degenerate quadratic
lattice $\Lambda$, the orthogonal group $O(\Lambda)$ acts on the set of non-
degenerate sublattices with given discriminant and rank with finitely many
orbits (3).

A rank 2 integer quadratic lattice is called hyperbolic if it is generated by
2 isotropic vectors,111A vector $x\in\Lambda$ in a lattice $(\Lambda,q)$ is
isotropic if $q(x,x)=0$. in other words, if its intersection lattice in an
appropriate basis has the form $\begin{pmatrix}0&N\\\ N&0\end{pmatrix}$. Such
a hyperbolic lattice is denoted by $U(N)$.

An MBM bound for a hyperkähler manifold is the number

$C:={\rm max}\\{-q(x,x)\ \ |\ \ x\in{\mathfrak{S}}\\},$

where ${\mathfrak{S}}$ denotes the set of MBM classes. Since $\Gamma\subset
O(H^{2}(M,{\mathbb{Z}}))$ acts on ${\mathfrak{S}}$ with finitely many orbits
(by [AV2]), this number is finite.

Clearly, for all vectors $x$ in $U(N)$, the square $(x,x)$ is divisible by
$N$. Therefore, any primitive sublattice $U(N)\subset H^{2}(M,{\mathbb{Z}})$
with $N>C$ contains no MBM classes and has round Kähler cone. However, if
$H^{2}(M,{\mathbb{Z}})$ does not contain primitive lattices isomorphic to
$U(N)$ for all $N>C$, then the set of $O(H^{2}(M,{\mathbb{Z}}))$-classes of
primitive hyperbolic sublattices in $H^{2}(M,{\mathbb{Z}})$ is finite, by
Kneser’s theorem. This easily leads to a contradiction (3).

##  2 Kobayashi metric on hyperkähler manifolds

We apply the results of the current paper to the vanishing of the Kobayashi
pseudometric on hyperkähler manifolds. In this section we summarize some of
our results in [KLV] joint with S. Lu. The aim of the current paper is to
imrove some of the bounds imposed on the Betti numbers, and also to show
vanishing of the Kobayashi pseudometric for all of the known compact
hyperkähler examples.

Definition 2.1: An ergodic complex structure is a complex structure $I$ on $M$
such that for any complex structure $I^{\prime}$ in the same deformation class
there exists a sequence of diffeomorphisms $\nu_{i}\in\operatorname{Diff}(M)$
such that $\lim_{i}\nu_{i}(I)=I^{\prime}$, where the limit of
$\nu_{i}(I)\in\operatorname{End}(TM)$ is taken with respect to the
$C^{\infty}$-topology on the space of tensors. We denote the space of all
integrable complex structures with this topology by $\operatorname{Comp}$.

Theorem 2.2: Any complex structure of hyperkähler type on a hyperkähler
manifold with $b_{2}\geqslant 5$ with $(H^{2,0}(M)\oplus H^{0,2}(M))\cap
H^{2}(M,{\mathbb{Q}})=0$ is ergodic.

Proof: [V2, V2bis]).

We will need another diffeomorphism orbit, which is smaller than the maximal
one, but has many of the same properties.

Theorem 2.3:  Let $(M,I)$ be a hyperkähler manifold such that
$(H^{2,0}(M)\oplus H^{0,2}(M))\cap H^{2}(M,{\mathbb{Q}})$ is a rank 1 space
generated by a class $\alpha\in H^{2}(M,{\mathbb{Q}})$, and
$\operatorname{Teich}_{\alpha}$ the Teichmüller space of all complex structure
with $\alpha\in(H^{2,0}(M)\oplus H^{0,2}(M))\cap H^{2}(M,{\mathbb{Q}})$ and
deformationally equivalent to $I$. Then $\operatorname{Diff}(M)\cdot I$ is
dense in $\operatorname{Teich}_{\alpha}$.

Proof: [V2bis, Theorem 2.5, Theorem 3.1].

Theorem 2.4:  [KLV] Let $(M,I)$ be a complex manifold with vanishing Kobayashi
pseudometric. Then the Kobayashi pseudometric vanishes for all ergodic complex
structures in the same deformation class. Moreover, for each complex structure
$I_{1}$ such that the closure of $\operatorname{Diff}(M)I_{1}$ in
$\operatorname{Comp}$ contains $I$, the pseudometric on $(M,I_{1})$ also
vanishes.

Proof: The proof follows easily from semicontinuity of the diameter of the
Kobayashi pseudometric, considered as a function on $\operatorname{Comp}$
([KLV]).

Theorem 2.5:  [KLV] Let $M$ be a hyperkähler manifold admitting two Lagrangian
fibrations associated with two non-proportional parabolic classes. Then the
Kobayashi pseudometric on $M$ vanishes.

To prove that a given hyperkähler manifold admits a deformation with two
distinct Lagrangian fibrations, in [KLV] we used an argument based on [AV4].

Theorem 2.6:  [KLV] Let $M$ be a maximal holonomy hyperkähler manifold with
$b_{2}(M)>13$. Then $M$ admits a projective deformation with Picard lattice of
signature $(1,2)$, with round Kähler cone (that is, with the Kähler cone equal
to the positive cone), and its automorphism group has finite index in the
arithmetic group $SO(\operatorname{Pic}(M))$ of orthogonal automorphisms of
its Picard lattice.

Proof: From [AV4, Theorem 3.11] it follows that there exists a projective
deformation with Picard rank 3, isotropic classes in $H^{1,1}(M)\cap
H^{2}(M,{\mathbb{Q}})$ and without MBM classes of type (1,1). From [AV4,
Theorem 2.10] it follows that for such a manifold the Kähler cone is equal to
the positive cone, and from [AV4, Theorem 2.6, Theorem 2.7, Corollary 2.12] it
follows that its automorphism group has finite index in the arithmetic group
$SO(\operatorname{Pic}(M))$.

Theorem 2.7:  [KLV] Let $M$ be a projective, maximal holonomy hyperkähler
manifold with two non-collinear isotropic rational classes in $H^{1,1}(M)$ and
with round Kähler cone. Assume that $M$ satisfies the SYZ conjecture, that is,
any nef bundle on $M$ is semiample. Then $M$ admits infinitely many
transversal holomorphic Lagrangian fibrations. In particular, the Kobayashi
pseudometric on $M$ vanishes.

Proof: Since the Kähler cone of $M$ is round, there exist rational vectors on
the boundary of the Kähler cone of $M$. These points correspond to rational
points in the real quadric $\\{l\in{\mathbb{P}}H^{2}(M,{\mathbb{Q}})\ \ |\ \
q(l,l)=0\\}$.

Each of such points corresponds to a Lagrangian fibration, because we assume
that the SYZ conjecture holds.

Theorem 2.8: [KLV] Let $(M,I)$ be a maximal holonomy, compact hyperkähler
manifold with non-maximal Picard rank. Suppose that it has a deformation which
has two transversal Lagrangian fibrations. Then the Kobayashi pseudometric on
$(M,I)$ vanishes.

Proof: The vanishing of the Kobayashi pseudometric then immediately follows
from 2 and 2. Indeed, there exists a complex structure $I^{\prime}$ with
vanishing Kobayashi pseudometric, and a sequence of diffeomorphisms such that
$\lim_{i}\nu_{i}(I)=I^{\prime}$. Then the Kobayashi pseudometric of $(M,I)$
vanishes by semicontinuity properties of the diameter of the Kobayashi
pseudometric.

Theorem 2.9: Let $M$ be a compact, maximal holonomy hyperkähler manifold with
$b_{2}(M)\geqslant 6$. Suppose that all its deformations satisfy the SYZ
conjecture. Then the Kobayashi pseudometric on $M$ vanishes.

Proof: See 2, 2 and 2.

Remark 2.10: All known examples of hyperkähler manifolds have
$b_{2}(M)\geqslant 7$ and satisfy the SYZ conjecture. By the results above,
the Kobayashi pseudometric of all known manifolds vanishes, unless their
Picard rank is maximal.

Remark 2.11:  The SYZ conjecture is true for all known hyperkähler examples,
i.e., for deformations of Hilbert schemes of points on K3 surfaces (Bayer-
Macrì [BM]; Markman [Mar3]), for deformations of the generalized Kummer
varieties (Yoshioka [Y]), for O’Grady’s sixfolds (Mongardi-Rapagnetta [MR]),
and for O’Grady’s tenfolds (Mongardi-Onorati, [MO]).

##  3 Main results

Here is a general result about lattices found in [KS, Satz 30.2], known as
Kneser’s orbit theorem.

Theorem 3.1:  Let $\Lambda$ be a non-degenerate integer lattice, and
$O(\Lambda)$ its isometry group. Then for any $D>0$, the group $O(\Lambda)$
acts with finitely many orbits on the set of non-degenerate sublattices
$\Lambda^{\prime}\subset\Lambda$ with discriminant $\operatorname{\sf
disc}(\Lambda^{\prime})\leqslant D$.

The main technical theorem of this note is the following.

Theorem 3.2:  Fix a natural number $A$. Let $(\Lambda,q)$ be a non-degenerate
indefinite lattice containing an isotropic vector. Then $\Lambda$ contains a
primitive sublattice $\Lambda_{1}$ isomorphic to $U(N)=\begin{pmatrix}0&N\\\
N&0\end{pmatrix}$ for some integer $N>A$.

Proof. Step 1: Let $\Lambda$ be a lattice which does not contain primitive
hyperbolic lattices with $U(N)\subset\Lambda$ with $N>A$. By 3 it follows that
$O(\Lambda)$ acts with finitely many orbits on the set of all hyperbolic
lattices in $\Lambda$. We will prove that this is impossible.

Step 2: Let $S_{1}$ be the set of primitive hyperbolic lattices in a
hyperbolic lattice $\Lambda$, and $S_{2}$ the set of pairs $(x,y)$ of non-
orthogonal primitive isotropic vectors up to a permutation and a sign change.
The group $O(U(N))$ of isometries of $U(N)$ is
$({\mathbb{Z}}/2{\mathbb{Z}})^{2}$. Indeed, a primitive isotropic vector can
be mapped only to another primitive isotropic vector, and there are only 4 of
them. From this observation it follows that the natural map
$S_{1}{\>\longrightarrow\>}S_{2}$ mapping a lattice to its primitive isotropic
vectors is bijective. Therefore, the set of equivalence classes
$\frac{S_{1}}{O(\Lambda)}$ is finite if and only if $\frac{S_{2}}{O(\Lambda)}$
is finite.

Step 3: Let $V:=\Lambda\otimes_{\mathbb{Z}}{\mathbb{R}}$ and let
$Q\subset{\mathbb{P}}V$ be the quadric defined by the equation $q=0$. The
primitive isotropic vectors in $\Lambda$ are determined uniquely up to a sign
by the rational points in $Q$. To prove that $O(\Lambda)$ does not act with
finitely many orbits on the set of pairs $(x,y)$ of non-orthogonal primitive
isotropic vectors, it would suffice to show that $O(\Lambda)$ does not act
with finitely many orbits on the set of pairs $(u,v)\in Q\times Q$ of
rational, non-orthogonal points.

Step 4: In this step we prove that $O(\Lambda)$ does not act with finitely
many orbits on the set of pairs $(u,v)\in Q\times Q$, where $u\in Q$ is fixed.
In this case, the set of such $v$ is in bijective correspondence with the
rational lines passing through $u$, and not orthogonal to $u$ (indeed, any
rational line transversally intersecting quadric intersects it twice).
Therefore, it would suffice to show that the stabilizer
$\operatorname{St}_{u}(O(\Lambda))$ does not act with finitely many orbits on
the set of rational projective lines
$l\subset{\mathbb{P}}(\Lambda\otimes_{\mathbb{Z}}{\mathbb{Q}})$ passing
through $u$ and not orthogonal to $u$.

Step 5: Let $\Lambda_{1}:=\frac{u^{\bot}}{u}$ be the quotient of the lattice
$u^{\bot}\subset\Lambda$ by $\langle u\rangle$. Then
$\operatorname{St}_{u}(O(\Lambda))$ acts on $\Lambda_{1}$, and each line
$l\subset{\mathbb{P}}(\Lambda\otimes_{\mathbb{Z}}{\mathbb{Q}})$ produces a
rank 1 rational subspace in $\Lambda_{1}\otimes_{\mathbb{Z}}{\mathbb{Q}}$. The
natural action of $\operatorname{St}_{u}(O(\Lambda))$ factorizes through
$O(W)$.

To prove that $\operatorname{St}_{u}(O(\Lambda))$ acts with infinitely many
orbits on the set of projective lines passing through $u$, it would suffice to
prove that $O(W)$ acts with infinitely many orbits on
${\mathbb{P}}(W\otimes_{\mathbb{Z}}{\mathbb{Q}})$. This is clear, because each
1-dimensional subspace in $W$ has an integer invariant - the square of the
smallest primitive vector - and this square can be arbitrary large.

Here is the main application of 3.

Theorem 3.3:  Let $M$ be a compact maximal holonomy hyperkähler manifold with
$b_{2}(M)\geqslant 4$, satisfying the SYZ conjecture. Assume that
$H^{2}(M,{\mathbb{Q}})$ has non-zero isotropic vectors. Then $M$ admits a
deformation with two distinct Lagrangian fibrations. If, in addition, $M$
satisfies one of the two assumptions

(a)

$(H^{2,0}(M)\oplus H^{0,2}(M))\cap H^{2}(M,{\mathbb{Q}})=0$,

(b)

$b_{2}(M)\geqslant 6$, and $M$ has Picard lattice of non-maximal rank,

then the Kobayashi pseudometric on $M$ vanishes.

Proof: Consider a primitive lattice $\Lambda\subset H^{2}(M,{\mathbb{Z}})$ of
signature $(p,q)$, $p\leqslant 1,q\leqslant b_{2}-3$. Using the global Torelli
theorem [V1], we can find a deformation $(M,I_{1})$ of $M$ with Picard lattice
$\Lambda$.

Since the rank of the indefinite lattice $H^{2}(M,{\mathbb{Z}})$ is at least
$5$, by Meyer’s theorem [Me] there exists an isotropic vector $x\in
H^{2}(M,{\mathbb{Z}})$. Applying 3, we can find a primitive hyperbolic
sublattice $U(N)\subset H^{2}(M,{\mathbb{Z}})$ with $N>|q(v,v)|$ for all
integer $(1,1)$-classes. Choose the complex structure $I$ such that $U(N)$ is
the Picard lattice $H^{1,1}_{I}(M,{\mathbb{Z}})$ of $(M,I)$. Then $(M,I)$ has
round Kähler cone, hence both integer isotropic generators of $U(N)$ are nef.

These classes correspond to Lagrangian fibrations since $(M,I)$ satisfies SYZ,
hence Kobayashi metric of $(M,I)$ vanishes. This takes care of the first
statement of 3.

Applying 2, we obtain that the Kobayashi metric vanishes for all ergodic
complex structures, that is, for all complex structures $I_{1}$ such that
$(H^{2,0}(M)\oplus H^{0,2}(M))\cap H^{2}(M,{\mathbb{Q}})=0$. This proves the
case (a) of 3.

It remains to prove 3 (b). Suppose that $(H^{2,0}(M,I)\oplus H^{0,2}(M,I))\cap
H^{2}(M,{\mathbb{Q}})$ has rank one and is generated by $\alpha$, Since
$\alpha^{\bot}$ has rank $\geqslant 5$, it contains isotropic vectors.
Applying 3 again, we find a deformation $(M,I^{\prime})$ of $M$ which
satisfies $H^{1,1}_{I^{\prime}}(M,{\mathbb{Z}})=U(N)$ and
$(H^{2,0}(M,I^{\prime})\oplus H^{0,2}(M,I^{\prime}))\cap
H^{2}(M,{\mathbb{Q}})=\langle\alpha\rangle$. For an appropriate choice of
diffeomorphisms $\pi_{i}\in\operatorname{Diff}(M)$, the sequence
$\nu_{i}(I^{\prime})$ converges to $I$ (by 2), hence the Kobayashi metric on
$(M,I)$ also vanishes.

Corollary 3.4: All known compact hyperkähler examples have vanishing Kobayashi
pseudometric.

Proof: See 2 and 3.

Acknowledgments. We are grateful to Christian Lehn for pointing out a gap in
an argument from our paper [KLV]. We also thank Giovanni Mongardi for his
suggestions and references.

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Ljudmila Kamenova
SCGP and Department of Mathematics, 3-115
Stony Brook University
Stony Brook, NY 11794-3651, USA,
<EMAIL_ADDRESS>

Misha Verbitsky
Instituto Nacional de Matemática Pura e Aplicada (IMPA)
Estrada Dona Castorina, 110
Jardim Botânico, CEP 22460-320
Rio de Janeiro, RJ - Brasil
also:
Laboratory of Algebraic Geometry,
Faculty of Mathematics, HSE University,
6 Usacheva Str. Moscow, Russia
<EMAIL_ADDRESS>


# Supernova Early Warning in Daya Bay Reactor Neutrino Experiment

Hanyu Wei for the Daya Bay Collaboration  Department of Engineering Physics,
Tsinghua University, Beijing, China<EMAIL_ADDRESS>
###### Abstract

Providing an early warning of a galactic supernova using neutrino signals is
of importance in studying both supernova dynamics and neutrino physics. The
Daya Bay reactor neutrino experiment, with a unique feature of multiple liquid
scintillator detectors separated in space, is sensitive to the full energy
spectrum of supernova burst electron-antineutrinos. By deploying 8
Antineutrino Detectors (ADs) in three different experimental halls, we obtain
a more powerful and prompt rejection of muon spallation background than
single-detector experiments. A dedicated supernova online trigger system
embedded in the data acquisition system has been installed to allow the
detection of a coincidence of neutrino signals within a 10-second window, thus
providing a robust early warning of a supernova occurrence within the Milky
Way.

##  1 Motivation

The Daya Bay reactor neutrino experiment is specifically designed for
measuring the neutrino mixing angle $\theta_{13}$ with a sensitivity down to
the 1% level [1]. However, the deployment of 8 electron-antineutrino detectors
(8 ADs) in three different experimental halls (Daya Bay near site, Ling Ao
near site, and Far site) motivates studies for a supernova online trigger
without complicated reconstruction and offline analysis. The three
experimental halls are more than 1 km apart from each other, which enables a
powerful and prompt rejection of muon-induced and accidental backgrounds
superior to that of a single-detector. In addition, a relatively low energy
threshold of 0.7 MeV enhances the detection of the full energy spectrum of
supernova burst neutrinos (SN$\nu$) since the energy spectrum may vary
according to the supernova core collapse model. A supernova online trigger
system is installed and in preparation to join the Supernova Early Warning
System (an international organization abbreviated to SNEWS), providing the
astronomical community with a prompt alert of the occurrence of a galactic
core collapse event [2] with the false alarm rate $<$ 1/month.

##  2 Detection of electron-antineutrinos in Daya Bay

The ADs in Daya Bay are designed to detect the $\bar{\nu}_{e}$ via inverse
beta decay (IBD) interactions $\bar{\nu}_{e}+p\rightarrow n+e^{+}$. Each
single AD has 22 tons of liquid scintillator (LS) and 20 tons of liquid
scintillator doped with gadolinium (Gd-LS), giving a total target mass of
$\sim$330 tons in 8 ADs. The coincidence of the prompt scintillation from the
$e^{+}$ and the delayed gamma emission of the neutron capture provides a
distinctive $\bar{\nu}_{e}$ signature against the background. The dominant
backgrounds are accidentals, cosmogenically produced fast neutrons, 9Li/8He
decays and the neutrons from the retracted 241Am-13C calibration source. The
average delay of the gamma emission of the neutron capture is 28 $\mu$s for
gadolinium and 200 $\mu$s for hydrogen. [3, 4]

##  3 Neutrino emission from supernovae

Supernova burst neutrinos (consisting of
$\nu_{e},\bar{\nu}_{e},\nu_{\mu},\bar{\nu}_{\mu},\nu_{\tau},\bar{\nu}_{\tau}$)
play a role in the study of both supernova dynamics and neutrino physics,
because

  * •

$\sim$99% of the stellar collapse gravitational binding energy is converted to
neutrinos which arrive at the earth a few hours before the visual supernova
explosion (SNe). So, it is believed that neutrino emission and interaction are
a key diagnostic for the dynamics of core collapse and supernova explosion
[5];

  * •

Supernova burst neutrinos can serve as probes of neutrino properties, e.g.
neutrino mixing, neutrino mass, neutrino lifetime, magnetic moment of
neutrino, electric charge of neutrino, radiative decay of neutrino, etc. [6]
and also the mass hierarchy [7] ;

  * •

Joint analysis with gravitational waves can provide deep insight into the core
collapse of supernovae [8].

The expected SN explosion rate is $\sim$0.01/year [9] within kilo-parsec (kpc)
distances and is around once per 50 years in the Milky Way. Within Mpc
distances, the rate is $\sim$1/year [9], but the neutrino flux is much
smaller. The SN$\nu$ energy spectrum within the first 10 seconds of the
supernova exlosion [10] for different flavor components implies the energy
range of electron-antineutrinos is up to $\sim$60 MeV with average energy
12$\sim$15 MeV. The explosion timescale is $\sim$10 s with $\sim$98% of the
$\bar{\nu}_{e}$ luminosity emitted [11]. This timing feature is exploited to
form an online trigger for SN$\nu$ in all the experiments listed in Tab. 1,
where the main features are summarized. Based on the target mass, Tab. 1 shows
that about 12 SN$\nu$ events in one AD and 100 events in all for 8-ADs are
expected at Daya Bay and a SN$\nu$ event is defined as the detection of one
neutrino from a single SN explosion. Even though other experiments may have
higher expected SN$\nu$ events mainly due to the target mass, it is emphasized
that the unique feature of Daya Bay in contrast is that it is not a single-
detector. This paper explains this advantage and shows that the Daya Bay
experiment is sensitive to all the 1987A-type (referring to the luminosity and
average energy of $\bar{\nu}_{e}$) SNe in the Milky Way which can be seen in
Fig. 4.

Table 1: SN$\nu$ sensitive detectors and expected events for a SN at 10 kpc, emission of $5\times 10^{52}$ erg in $\bar{\nu}_{e}$, average energy 12 MeV, compatible with SN1987A. [5] Detector | Type | Location | Mass[kt] | Events | Status
---|---|---|---|---|---
IceCube | Ice Cherenkov | South Pole | 0.6/OM | $10^{6}$ | Running
Super-K IV | Water | Japan | 32 | 7000 | Running
LVD | Scintillator | Italy | 1 | 300 | Running
KamLAND | Scintillator | Japan | 1 | 300 | Running
SNO+ | Scintillator | Canada | 1 | 300 | Commissioning 2013
MiniBOONE | Scintillator | USA | 0.7 | 200 | Running
Daya Bay | Scintillator | China | 0.33 | 100 | Running
Borexino | Scintillator | Italy | 0.3 | 80 | Running
BST | Scintillator | Russia | 0.2 | 50 | Running
HALO | Lead | Canada | 0.079 | tens | Almost ready
ICARUS | Liquid argon | Italy | 0.6 | 200 | Running
|  |  |  |  |

##  4 Background sources and the supernova burst neutrino event

The supernova online trigger system in Daya Bay is embedded in the Data
Acquisition System (DAQ), online looking for increase in multi-AD signals in
10s-time-window and sending prompt alarms. According to this task, all the
study of supernova online trigger is for online prompt trigger judgment and
not so precise as offline analysis. The purpose of the background study on one
hand is to have a good understanding of the background coincidences in multi-
AD, thus allowing to set a precise false alarm rate threshold. The false alarm
happens frequently as the detectable SN explosion to the earth is so rare and
the selected events are always backgrounds. On the other hand, the background
study contributes to the event selection criteria establishment which has to
be simpler than that of the offline analysis so as to be prompt and not to
bring much workload to DAQ. A data sample from Dec. 24, 2011 to Jul. 28, 2012
is used to train our online trigger algorithm to give the event selection
criteria and study the backgrounds since no observation of SN$\nu$ was
declared during the period of the data sample by all detectors including Daya
Bay. In addition, the supernova burst neutrinos that undergo an IBD in the
detector volume are simulated aiming to study the detection efficiency of
SN$\nu$.

###  4.1 Background sources

In the Daya Bay ADs (Section 2), referring to Fig. 1, the delayed signal of an
IBD event is either an 8 MeV $\gamma$ cascade from neutron capture on Gd, or a
2.2 MeV $\gamma$ from neutron capture on H. It is observed that the large
amount of accidental backgrounds in the low energy region significantly affect
the background event rate, therefore we set the online energy threshold at 2
MeV for the prompt signal associated with the 8 MeV $\gamma$ cascade and 8 MeV
for that associated with single 2.2 MeV $\gamma$ in which case the majority of
accidental backgrounds are removed. Using optimized selection criteria for
SN$\nu$, the prompt vs. delayed signal energy plot is shown in the red box in
Fig. 1. Along the Y-axis, the Gd neutron capture peak is seen around 8 MeV and
the hydrogen neutron capture peak is around 2.2 MeV where many fast neutrons
are present in the high energy range along the X-axis and reactor neutrino
signals are present below 10 MeV.

Notice that the data for trigger algorithm training are offline reconstructed
while the supernova online trigger can only access the raw data. A simple but
relatively effective reconstruction is applied online for real SN$\nu$
selection in which the average PMT gain and energy scale calibration constants
are used for energy reconstruction and a charge-weighted method is used for
prompt vertex reconstruction. The resulting online, measured single AD event
rates are 0.019, 0.013 and 0.0013 Hz/AD at the Daya Bay near site, Ling Ao
near site and far site, respectively.

Figure 1: Prompt signal energy vs. delayed signal energy 2-D plot for
backgrounds. In the red box is the selection region for SN$\nu$, suggesting
the prompt and delayed energy cut.

###  4.2 Supernova burst neutrino event

Assuming that the spectrum of supernova burst neutrinos follows a quasithermal
distribution [13]

$f_{\nu}(E)\propto E^{\alpha}e^{-(\alpha+1)E/E_{av}}$

where $E_{av}$ is the average energy and $\alpha$ a numerical parameter
describing the amount of spectral pinching. The value $\alpha$ = 2.30
corresponds to a Fermi-Dirac distribution with zero chemical potential. In our
simulation $\alpha$ is $>$ 2.30 and varies with the three main phases of the
detectable supernova neutrino signals: prompt $\nu_{e}$ burst phase, accretion
phase and cooling phase [5].

SN$\nu$s have been simulated separately in both the Gd-LS and LS region of a
single AD, whose results after selection cuts are shown in Fig. 2. With these
simulation results, the detection efficiency of SN$\nu$ that undergo an IBD in
the detector volume is estimated to be $\sim 70\%$ and used to determine the
expected number of SN$\nu$ events in each AD at Daya Bay.

Figure 2: The plots are after selection cut with respect to one single AD.
Top: Simulation 2-D plot of supernova neutrino selection for delayed signal
against prompt signal. Bottom: Prompt signal energy projection of the
corresponding 2-D plot above, which indicates the shape of the supernova burst
neutrinos. Left: For Gd-LS volume. Right: For LS volume.

##  5 Supernova online trigger judgment

An approach is developed to investigate the background coincidence rate, e.g.
false alarm rate, by combining all 8 ADs’ SN$\nu$ candidate events in the
10s-time-window. The SN$\nu$ candidates are always backgrounds as so rare SN
explosion can be observed by neutrino detection in the earth. Every one
second, the SN$\nu$ candidates in the previous 10s-time-window are counted in
each AD, forming a combination to judge whether to trigger or not.

###  5.1 Trigger table and trigger cut

A trigger table is generated to list the AD background combination cases in
order of their corresponding false alarm rate for the sliding 10s-time-window.
Utilizing this table, it is convenient to set the cut for the combination
cases due to a certain false alarm rate threshold according to SNEWS
requirement. Below (Tab. 2), part of the trigger table for online test is
shown as an example where the contents are all for backgrounds.

In Tab. 2, the number under each AD is the background event number counted in
the 10s-time-window. The first two columns correspond to the detectors in the
Daya Bay near site, the next two columns correspond to the Ling Ao near site
and the four remaining columns correspond to the Far site. The column “False
Alarm Rate” is defines not as the trigger rate relative to the combination in
that row but as the total trigger rate of all the AD background combination
cases below. Before the false alarm rate calculation, the AD background
combination cases are firstly in descending order with respect to their
trigger rates. Then for each combination case, the total trigger rate of those
below it and itself is calculated serving as the corresponding quantity “False
Alarm Rate”. Obviously, a trigger cut can be determined easily due to the
false alarm rate threshold. For example, a 1/34s (0.0293111 Hz) false alarm
rate threshold is required and then the last row of Tab. 2 is where to place
the cut below which all the AD background combination cases have a smaller
“False Alarm Rate” and are supposed to trigger a supernova early warning.

This table is for background false alarm control and SN$\nu$ events are
expected to have higher probability for coincidence in 8-ADs than muon-induced
fast neutrons or reactor neutrinos, etc. In detailed detection probability for
SN explosion, please see Section 6. It is also emphasized here that the “False
Alarm Rate” in Tab. 2 is predicted rather than measured. This will be
explained in the next subsection.

Table 2: Part of the trigger table for supernova online judgment. AD1 to AD8 indicates the 8 antineutrino detectors in the three experimental halls in Daya Bay. AD1 | AD2 |  | AD3 | AD4 |  | AD5 | AD6 | AD7 | AD8 |  | False Alarm Rate (Hz)
---|---|---|---|---|---|---|---|---|---|---|---
0 | 0 |  | 0 | 0 |  | 0 | 0 | 0 | 0 |  | 1
0 | 1 |  | 0 | 0 |  | 0 | 0 | 0 | 0 |  | 0.499092
1 | 0 |  | 0 | 0 |  | 0 | 0 | 0 | 0 |  | 0.404098
⋮ | ⋮ |  | ⋮ | ⋮ |  | ⋮ | ⋮ | ⋮ | ⋮ |  | ⋮
0 | 0 |  | 0 | 1 |  | 0 | 1 | 0 | 0 |  | 0.0317780
0 | 0 |  | 1 | 0 |  | 0 | 1 | 0 | 0 |  | 0.0305445
0 | 1 |  | 0 | 2 |  | 0 | 0 | 0 | 0 |  | 0.0293111
⋮ | ⋮ |  | ⋮ | ⋮ |  | ⋮ | ⋮ | ⋮ | ⋮ |  | ⋮
|  |  |  |  |  |  |  |  |  |  |

###  5.2 Background rate prediction

The reason we use the predicted background rate is that the data sample used
for the supernova online trigger study is only about 120 days, which provides
insufficient statistics to set a false alarm rate threshold like 1/year.
However, the prediction has a challenge – the overlap in the sliding 10s-time-
window – every one second, the SN$\nu$ candidates of each AD are combined for
judgment and the 10s-time-window is overlapped by a few adjacent ones.

For a single AD, it is verified with the numerical simulation that the rate
(Hz) (here the probability is numerically equal to the rate as every one
second there is a combination) of the event count in the sliding 10s-time-
window still follows the Poisson distribution with the mean value
$10~{}seconds~{}\times~{}single~{}AD~{}event~{}rate$. This is the fundament of
the combination calculation.

In terms of the combination of multiple ADs, assuming different experimental
halls are mutually independent for backgrounds, the correlation between ADs in
the same site is considered and measured using the data. The correlation
between ADs in the same site originates from the muon-induced fast neutrons
which cause several consecutive signals in detectors of the same experiment
hall. The trigger rate for each combination case is predicted using several
unknown independent Poisson variables that formulate the event rate of each AD
and some of which are shared by the correlated ADs in the same experimental
halls representing the correlation part. These unknown Poisson variables can
be calculated eventually based on the measured single AD event rates and
correlation between ADs.

In addition, given the trigger rate of each combination case, the statistical
error can be derived utilizing some statistic skills in which case the data
sample has to be split into 10 parts according to the time and each of the 10
parts is 1s delay or earlier than the adjacent one. To verify the prediction,
the rates measured on data are compared to the prediction and 82% are within
1$\sigma$, 98.4% are within 2$\sigma$, and 99.7% are within 3$\sigma$
consistent with the prediction. Therefore, the prediction of background
combination rate is plausible to replace the measured one. Notice that the
systematic error is negligible compared with the statistical error when the
threshold is set too small such as 1/year, or even 1/month.

###  5.3 Supernova online trigger diagram

The scheme of the supernova online trigger system in Daya Bay is shown in Fig.
3. It includes several software applications implemented in the DAQ of Daya
Bay.

The online part is able to get access to all the raw data and make a simple
reconstruction. The IBD selection program for each AD provides the information
of SN$\nu$ candidates to a combination server with the function of combination
and trigger judgment according to the trigger table mentioned above. There are
two levels of trigger, silent trigger (1/month) and golden trigger (1/year),
which are related to different offline responses. In case of a golden trigger,
an e-mail alert is immediately sent and information of those SN$\nu$
candidates is written into an offline database with about 10 seconds time
latency. A pure offline analysis would cross check both the golden and silent
triggers with less than 40 min latency. The shaded area in the diagram has
been tested and officially installed, while the offline analysis/cross-check
is being developed based on the Performance Quality Monitoring System (PQM) of
Daya Bay. Daya Bay is negotiating to join the SNEWS and the e-mail alert is
presently sent to Daya Bay collaborators who are interested.

To exclude unexpected trigger bursts (e.g. electronic noise) in one detector
or one experimental hall, a simple but effective uniformity cut based on the
$\chi^{2}$ method is applied with less than 1% detection probability lost for
supernova explosions. This $\chi^{2}$ is the minimum value of
$\sum_{i}\frac{(n_{i}-\lambda)^{2}}{n_{i}}$ where $n_{i}$ is the event counts
in the combination for each AD and $\lambda$ is the best fit value of event
counts for all ADs considering SN$\nu$ events are distributed uniformly among
ADs. Detection probability of a supernova explosion is explained next section.

Figure 3: Diagram of supernova online trigger system in Daya Bay. It is the
framework of the software applications on the basis of the existing DAQ system
and on-site host.

##  6 Detection probability of a supernova explosion

According to the target mass of the Daya Bay detectors, the detection
efficiency of SN$\nu$ obtained based on MC and the relation between supernova
neutrino time-integrated flux and distance to the earth [5], single AD’s
expected SN$\nu$ event counts can be determined below,

$N_{AD}=N_{0}\times\frac{L_{\bar{\nu}_{e}}}{5\times
10^{52}erg}\times(\frac{10kpc}{D})^{2}$

where $L_{\bar{\nu}_{e}}$ is the luminosity of electron-antineutrino emission
and $D$ is the SN explosion distance to earth. $N_{0}$ is the single AD’s
expected SN$\nu$ event number in 10s-time-window corresponding to $5\times
10^{52}erg$ luminosity and $10~{}kpc$ distance. Detection efficiency of
SN$\nu$ is considered in $N_{0}$.

Here, the supernova model for the detection probability calculation is set to
SN1987A-type and a typical value for $N_{AD}$ is $\sim$8 at a distance of 10
kpc to earth. Based on the expected SN$\nu$ events of each AD, the detection
probability of a supernova explosion is calculated by summing up the
probabilities of the combination cases that pass the trigger threshold. Notice
that the single AD event rate increases simultaneously during a supernova
explosion and coincidence signals in multiple ADs occur more frequently. As a
result, the detection probability of the SN explosion has been calculated as a
function of distance to the earth. The result is shown in Fig. 4. From the
“8-AD Golden Trigger” line, the Milky Way center is around 8.5 kpc from the
earth with a 100% detection probability and the most distant edge of the Milky
Way is 23.5 kpc from the earth with a 94% detection probability. Moreover, the
silent trigger will add a potential 5% to 10% detection probability of SN
explosion.

Particularly, the “Single Detector” line is comparable to the “8-AD Golden
Trigger” which obviously indicates the gain in sensitivity of the 8-AD
configuration over a single detector. A rough estimation implies the Daya Bay
is equivalent to a single 0.7 kton liquid scintillator detector with respect
to the detection probability of SN explosion as a consequence of the multi-AD
configuration. The background rate level per target mass is the average
background rate per unit of the target mass of Daya Bay ADs.

Figure 4: The X-axis is SN explosion distance from the earth and the Y-axis is
the corresponding detection probability. “8-AD golden trigger” corresponds to
the result with false alarm rate $<$1/year, and “8-AD silent trigger”
corresponds to that with false alarm rate $<$1/month. “Single Detector” is the
scenario also with false alarm rate $<$1/year in which the 8-AD target mass is
combined into a single detector with the background rate level per target mass
of Daya Bay ADs.

The detection probability has two elements in reality: one is the probability
for a SN explosion can be detected, the other one is the corresponding “false
alarm rate” threshold (defined in Subsection 5.1), for example, 1/month or
1/year here which is for background false alarm control. Based on this, the
difference between single-detector and multi-detector can be explained. In the
scenario of single-detector, the total number of SN$\nu$ events is exploited
for trigger cut setting, for example, 10 SN$\nu$ events in 10s-time-window
corresponding to 1/month false alarm rate threshold. While in the scenario of
multi-detector, the background combination case is exploited for trigger cut
setting, for example, combination 0-0-2-3-1-1-0-0 corresponding to 1/month
false alarm rate threshold. Obviously, the total number of events in multi-
detector here is 7 which is smaller than the single-detector, thus providing a
higher detection probability.

##  7 Summary

The supernova online trigger system in Daya Bay has been officially installed
after several pretests. The extra workload to the current CPU consumption of
DAQ is around 8% and is far from the computing maximum workload online.
Moreover, the time latency from electronics triggers to an alarm is around 10
s (20 s considering the duration of 10s-time-window). In the future, the pure
offline cross check will be added and joining the SNEWS is underway. With a
relatively low energy threshold, superior energy resolution and separated 8-AD
deployment, the online detection probability for a SN1987A-type SN explosion
could be larger than 94% within the Milky Way.

This work is supported in part by the Ministry of Science and Technology of
China and the National Natural Science Foundation of China (Grants
No.11235006). In addition, the author also wishes to acknowledge the Daya Bay
Reactor Neutrino Experiment Collaboration, particularly Shaomin Chen, Zhe
Wang, Logan Lebanowski and Fei Li for precious information, useful discussion
and selfless help.

## References

## References

  * [1] Daya Bay Collaboration, arXiv: hep-ex/0701029
  * [2] SuperNova Early Warning System, http://snews.bnl.gov
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  * [5] Raffelt G, arXiv: 1201.1637v2 [astro-ph. SR]
  * [6] Mohapatra R and Pal P 2004 Massive Neutrinos in Physics and Astrophysics (Singapore: World Scientific Printers) Chapter 17
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11institutetext: Spanish National Centre for Biotechnology (CNB–CSIC), Madrid,
Spain 22institutetext: Faculty of Technical Sciences, University of Novi Sad,
Serbia 33institutetext: Faculty of Information Technology and Electrical
Engineering, NTNU, Gjøvik, Norway 44institutetext: Centre for Image Analysis,
Uppsala University, Sweden 55institutetext: Mathematical Institute of the
Serbian Academy of Sciences and Arts, Belgrade, Serbia

# Ensemble of Convolutional Neural Networks for Dermoscopic Images
Classification

Tomáš Majtner These two authors contributed equally to this work.11 Buda
Bajić††footnotemark:  22 Sule Yildirim 33 Jon Yngve Hardeberg 33 Joakim
Lindblad 4455 Nataša Sladoje 4455

###### Abstract

In this report, we are presenting our automated prediction system for disease
classification within dermoscopic images. The proposed solution is based on
deep learning, where we employed transfer learning strategy on VGG16 and
GoogLeNet architectures. The key feature of our solution is preprocessing
based primarily on image augmentation and colour normalization. The solution
was evaluated on Task 3: Lesion Diagnosis of the ISIC 2018: Skin Lesion
Analysis Towards Melanoma Detection.

###### Keywords:

s

kin cancer; CNN; melanoma classification; skin lesion recognition

##  1 Introduction

Malignant melanoma, which is commonly known as melanoma, is the most dangerous
type of skin cancer [8]. It is developed in melanocytes, which are cells
producing melanin and and they are giving skin its colour. The risk of
melanoma has increasing trend, especially for women under 40. The positive
fact is that this form of cancer can be treated successfully with high rate,
when it is detected and recognized early. However, the examination of
suspicious skin lesions is time-consuming and it requires the knowledge of an
expert. It is therefore natural that much research is put to automation of the
melanoma recognition process.

The International Skin Imaging Collaboration (ISIC) is an international effort
to improve melanoma diagnosis, sponsored by the International Society for
Digital Imaging of the Skin (ISDIS). The ISIC Archive contains the largest
publicly available collection of quality controlled dermoscopic images of skin
lesions. In 2018, ISIC organized already third grand challenge focused on
early melanoma detection. This challenge was divided into three separate
tasks, where the third tasks was disease classification.

We are presenting here our framework for automated predictions of disease
classification within dermoscopic images. Data for this task was extracted
from the ”ISIC 2018: Skin Lesion Analysis Towards Melanoma Detection” (ISIC
2018) grand challenge datasets [2, 12]. The disease categories recognized in
this challenge include: Melanoma (MEL), Melanocytic nevus (NV), Basal cell
carcinoma (BCC), Actinic keratosis / Bowen’s disease (AKIEC), Benign keratosis
(BKL), Dermatofibroma (DF), and Vascular lesion (VASC). See Fig. 1 for
illustration.

Figure 1: Illustration of disease categories recognized in this study. Image
source: https://challenge2018.isic-archive.com/task3/

##  2 Proposed Method

Balancing Training Set: The official training set for ISIC 2018 Task 3
competition consists of 10,015 images. As can be seen from the Table 1, the
number of samples in different categories is strongly unbalanced. Moreover, in
some particular classes, there is also very small number of images, which
could result in inefficient training of automated convolutional neural network
(CNN). Therefore, with the aim to increase the number of training images, and
at the same time to increase the robustness of our system, we performed data
augmentation by horizontal flipping of training samples. This operation double
the number of training images in each class.

The class balancing was achieved by sample rotation. Here, we considered the
fact that position of skin lesion during acquisition process is arbitrary and
the training data should reflect that property. Rotation was performed around
the image center by various rotation factors which are specified in Table 1.
Rotation factor $i$ implies that each image is rotated $i-1$ times. This means
that for $i=10$, each images is rotated by angle 36°$\cdot j$, where
$j\in\\{1,2,\dots,9\\}$.

Presented augmentation leads to more balanced image classes and it also
increases rotation invariance of the CNN. Bicubic interpolation is used when
the angle was not a multiple of 90°. As was shown by Goodfellow et al. [4],
such data augmentation by flipping and rotation compensates for variations
between the training and the test sets and has a positive impact on the CNN
performance. Total number of training images after balancing step is 96,274.

Table 1: Total number of training images before and after augmentation. | AKIEC | BCC | BKL | DF | MEL | NV | VASC
---|---|---|---|---|---|---|---
Training data | 327 | 514 | 1,099 | 115 | 1,113 | 6,705 | 142
After flipping | 654 | 1,028 | 2,198 | 230 | 2,226 | 13,410 | 284
Rotation factor | 17 | 28 | 60 | 6 | 62 | 0 | 7
After rotation | 14,388 | 13,364 | 13,188 | 13,800 | 13,356 | 13,410 | 14,768

Colour Normalization: Because the images are from different sources, various
lightning conditions change the visual appearance of the skin lesions.
Traditional conversion to grayscale images would lead to significant lost of
input information, therefore we aimed for colour constancy algorithms. This
topic was already covered in number of papers [7]. In our work, we tested
several methods and decided to use max-RGB [6], which was also used in a
recent study [1]. Max-RGB method is based on the assumption that the
reflectance, which is achieved for each of the three channels, is equal [13].
The illustration of this method is presented in Fig. 2.

|  |  |  |  |
---|---|---|---|---|---
|  |  |  |  |

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

Figure 2: Examples of image samples after colour normalization. First row
corresponds to original images and second row to max-RGB normalization.

Transfer Learning: Our classification is based on fine-tunning the VGG16
network [9] and the GoogLeNet network [10]. As it was observed in number of
previous works [11], training a deep convolutional neural network from scratch
is difficult because it requires a large amount of labeled training data which
in case of medical applications is rarely available. A promising alternative
is to fine-tune of a CNN that has been pre-trained using, for instance, a
large set of labeled natural images such as ImageNet dataset.

The usage of transfer learning for skin lesion classification in this work was
additionally motivated by recent skin cancer classification study [3]. It was
performed by retraining the weights of the last three layers, where our
version has seven neurons in its output layer. For training purposes, we used
colour normalized augmented training dataset. We utilize stochastic gradient
descent (SGD) for optimization, with a momentum factor of 0.9, and $\ell_{2}$
regularization level of $10^{-4}$. The learning rate of 0.0001 was used in a
mini-batch scheme of 8 images and employing 50 epochs for retraining. Our
solution was implemented using Matlab R2018a.

Ensembling: As a last step, we employed ensemble of both our VGG16 and
GoogLeNet networks fine-tuned with preprocessed images. The ensemble typically
leads to higher performance, when compared with results of a single network
[5]. In our case, the ensemble weights were selected manually to be 0.5 for
both networks.

##  3 Results

The proposed solution was evaluated using the official validation set for ISIC
2018 Task 3 competition. This set consists of 193 images and it was evaluated
automatically via submission webpage. The achieved balanced accuracy was 0.801
for the VGG16 architecture, 0.797 for the GoogLeNet architecture, and 0.815
for their ensemble.

## References

  * [1] Barata, C., Celebi, M., Marques, J.: Improving Dermoscopy Image Classification Using Color Constancy. IEEE Journal of Biomedical and Health Informatics 19(3), 1146–1152 (2015)
  * [2] Codella, N., Gutman, D., Celebi, M., Helba, B., Marchetti, M., Dusza, S., Kalloo, A., Liopyris, K., Mishra, N., Kittler, H., et al.: Skin lesion analysis toward melanoma detection: A challenge at the 2017 international symposium on biomedical imaging (ISBI), hosted by the international skin imaging collaboration (ISIC). In: IEEE 15th International Symposium on Biomedical Imaging (ISBI 2018). pp. 168–172. IEEE (2018)
  * [3] Esteva, A., Kuprel, B., Novoa, R., Ko, J., Swetter, S., Blau, H., Thrun, S.: Dermatologist-Level Classification of Skin Cancer with Deep Neural Networks. Nature 542(7639), 115 (2017)
  * [4] Goodfellow, I., Bengio, Y., Courville, A.: Deep Learning, vol. 1. MIT press Cambridge (2016)
  * [5] Kumar, A., Kim, J., Lyndon, D., Fulham, M., Feng, D.: An Ensemble of Fine-Tuned Convolutional Neural Networks for Medical Image Classification. IEEE J. of Biomed. and Health Inf. 21(1), 31–40 (2017)
  * [6] Land, E.: The Retinex Theory of Color Vision. Scientific American 237(6), 108–129 (1977)
  * [7] Madooei, A., Drew, M.: Incorporating Colour Information for Computer-Aided Diagnosis of Melanoma from Dermoscopy Images: A Retrospective Survey and Critical Analysis. International Journal of Biomedical Imaging 2016 (2016)
  * [8] Mermelstein, R., Riesenberg, L.: Changing knowledge and attitudes about skin cancer risk factors in adolescents. Health Psychology 11(6), 371 (1992)
  * [9] Simonyan, K., Zisserman, A.: Very Deep Convolutional Networks for Large-Scale Image Recognition. arXiv preprint arXiv:1409.1556 (2014)
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