and orbits of parabolic Higgs bundles with parabolic Higgs field

$\Phi=\begin{pmatrix}0&-v_{2+m_{1}-m_{1}}\\\ w_{2-m_{1}+m_{2}}&0\end{pmatrix}$

in terms of factorizations $q=v_{2+m_{1}-m_{1}}w_{2-m_{1}+m_{2}}$,
parametrized by partitions $I\sqcup I^{\prime}=\\{1,2,3,4\\}$ induced by
fixing the divisors $(v_{2+m_{1}-m_{1}})$ and $(w_{2-m_{1}+m_{2}})$. For each
degree parity, there are exactly four partitions, and for any such partition,
we can assume without loss of generality that $I\in\mathcal{I}$. By corollary
8 and proposition 14, the induced isomorphism
$H^{0}\left(\mathbb{C}\mathbb{P}^{1},K^{2}_{\mathbb{C}\mathbb{P}^{1}}(D)\right)\backslash\\{0\\}\cong\mathrm{H}_{\\{I,I^{\prime}\\}}$
can be extended uniquely to a section $\sigma_{I}$ as
$\sigma_{I}(0):=\mathscr{K}^{+}_{I}$ (or $\sigma_{I}(0):=\mathscr{L}^{+}_{I}$
if that were the case for $I$). Since exterior wall-crossing leaves all basic
building blocks of the form (6.4) invariant, in the case of an exterior
chamber $\mathcal{C}_{I^{\prime\prime}}$, the corresponding holomorphic
sections can be defined to be the same as the corresponding sections of
$\mathscr{M}_{\mathcal{C}_{\mathcal{I}(I^{\prime\prime})}}$. ∎

We will refer to the previous holomorphic sections as the Hitchin sections of
the geometric model $\mathscr{M}_{\mathcal{C}}$. When $d$ is even, three of
the Hitchin sections are defined in the Zariski open Harder–Narasimhan stratum
of $\mathscr{M}_{\mathcal{C}}$, while the fourth one belongs to the
complementary stratum. This fourth section corresponds to the holomorphic
section constructed by Gothen-Oliveira [GO19]. In turn, when $d$ is odd, all
Hitchin sections always belong to the Zariski open Harder–Narasimhan stratum
of $\mathscr{M}_{\mathcal{C}}$.

######  Corollary 9.

For every open chamber
$\mathcal{C}\subset\mathcal{W}_{\mathbin{\vbox{\hbox{\scalebox{0.6}{$\bullet$}}}}}$,
the projection $\pi_{1}$ determines a biholomorphism

(7.4)
$\mathscr{M}_{\mathcal{C}}\Big{\backslash}\bigsqcup\sigma_{I}\left(H^{0}\left(\mathbb{C}\mathbb{P}^{1},K^{2}_{\mathbb{C}\mathbb{P}^{1}}(D)\right)\right)\cong
T^{*}\mathbb{C}\mathbb{P}^{1}.$

In view of the restrictions (7.2) and (7.4) and corollary 9, every geometric
model $\mathscr{M}_{\mathcal{C}}$ is the simultaneous extension of each of the
two complex surfaces

$T^{*}\mathbb{C}\mathbb{P}^{1}\qquad\text{and}\qquad\Sigma_{D}\times\mathbb{C}^{*}$

towards the Hitchin section loci and the nilpotent cone, respectively, in such
a way that the intersection of any elliptic fiber of $\pi_{2}$ and the Hitchin
section loci corresponds to the set of Weierstrass points of the former. In
the case when $\mathcal{C}$ is an interior chamber and
$\mathscr{N}_{\mathcal{C}}\neq\emptyset$, $\pi_{1}$ corresponds to the
dominant abelianization morphism constructed in [BNR89] for the only possible
spectral curve $\Sigma_{D}$, whose very stable locus equals
$\mathscr{N}_{\mathcal{B}_{\mathbin{\vbox{\hbox{\scalebox{0.6}{$\bullet$}}}}}}$.
The $\mathbb{C}^{*}$-action on $\mathscr{M}_{\mathcal{C}}$ corresponds to the
standard $\mathbb{C}^{*}$-action on both surfaces. The characterization of the
limits of this action follows from propositions 14 and 15 (figure 13). There
is always a bijective correspondence between $\mathbb{C}^{*}$-fixed points not
in the central sphere and points in the wobbly locus.

######  Corollary 10.

Under the hypotheses of proposition 15, the limit $c\to 0$ collapses the
Zariski open set
$\mathscr{M}_{\mathcal{C}}\Big{\backslash}\bigsqcup\sigma_{I^{\prime}}\left(H^{0}\left(\mathbb{C}\mathbb{P}^{1},K^{2}_{\mathbb{C}\mathbb{P}^{1}}(D)\right)\right)$
to the central sphere in $\pi_{2}^{-1}(0)$, and every component
$\mathrm{H}_{\\{I,I^{\prime}\\}}$ to the point $\mathscr{K}^{+}_{I}$ $($or
$\mathscr{L}^{+}_{I})$.

Figure 13. The topology of the union of the nilpotent cone and the four
Hitchin sections, indicating in red the four 0-dimensional and one
1-dimensional components of fixed points of the global $\mathbb{C}^{*}$-action
in the geometric model, to which $\mathscr{M}_{\mathcal{C}}$ collapses in the
limit $c\to 0$.

###  7.4. Weight degeneration to semi-stability walls

We will finish this section with a brief discussion of an interpretation of
the notion of parabolic $S$-equivalence for any choice of parabolic weights
along the intersection of any given semi-stability wall with the interior of
$[0,1]^{4}$. Strictly speaking, it is necessary to exclude the points in
$\partial[0,1]^{4}$, as they lead to a degeneration of the moduli problem by
definition, to a strictly smaller number of marked points in
$\mathbb{C}\mathbb{P}^{1}$. Recently, Godinho–Mandini [GM] have studied the
geometry of the special degeneration $\boldsymbol{\beta}=0$ of the Zariski
open set of holomorphically trivial parabolic Higgs bundles (in the case when
$d=0$), in terms of a correspondence with the so-called null hyperpolygons.

######  Proposition 16.

$(i)$ For any $\boldsymbol{\beta}\in\mathring{\mathcal{H}}_{I,|I|-3}$, all
points in the associated basic building block (6.3) and
$\mathscr{N}^{\mathrm{ev}}_{\mathcal{C}_{\mathcal{I}(I)}}$ are strictly semi-
stable. $(ii)$ For any
$\boldsymbol{\beta}\in\mathring{\mathcal{H}}_{I,|I|-2}=\mathring{\mathcal{H}}_{I^{\prime},|I^{\prime}|-2}$
and a corresponding pair
$\\{\mathcal{C}_{\mathcal{I}},\mathcal{C}_{\mathcal{I}^{\prime}}\\}$ of
neighboring interior chambers, all points in the associated pair (6.4), as
well as the pair $\\{\mathscr{N}_{I},\mathscr{N}_{I^{\prime}}\\}$, are
strictly semi-stable. Additionally, the orbits of
$\iota\left(\mathrm{B}_{I,j}(E)\cap\mathrm{B}_{I^{\prime},j}(E)\right)\subset\iota\left(\mathrm{X}(E)\right)$
are strictly semi-stable.

###### Proof.

The result is a straightforward consequence from lemma 5, proposition 11, and
the definition of strict semi-stability for parabolic Higgs bundles. ∎

######  Corollary 11 (Parabolic $S$-equivalence on geometric models).

For any $\boldsymbol{\beta}$ in the interior of a semi-stability wall, the
pointwise identification of strictly semi-stable components as in proposition
16 determines a surface biholomorphic to $\mathscr{M}_{\mathrm{toy}}$.

##  Appendix A Residue models

We will describe another approach to explicitly construct moduli spaces of
stable parabolic Higgs bundles based on residue evaluations. We will refer to
the resulting orbit spaces $\mathscr{M}^{\mathrm{ev}}_{\mathcal{C}}(E)$ as
residue models for Harder–Narasimhan strata.

The desingularization by blow-up of the cone of nilpotent endomorphisms
$\mathfrak{n}(V)$ of a 2-dimensional vector space $V$ is constructed as the
smooth complex hypersurface

$\widetilde{\mathfrak{n}\left(V\right)}:=\left\\{(\phi,F)\in\mathfrak{n}\left(V\right)\times\mathbb{P}(V)\;:\;F\subset\ker(\phi)\right\\}.$

The restriction to $\widetilde{\mathfrak{n}\left(V\right)}$ of the projection
onto the second factor is again a projection which we will denote by
$\mathrm{pr}$, and endows $\widetilde{\mathfrak{n}\left(V\right)}$ with the
structure of a holomorphic line bundle of degree -2, identifying its
exceptional divisor $\mathscr{E}\cong\mathbb{P}(V)$ with the zero section of
the latter. The fiber $\mathfrak{n}(F)$ over a given line
$F\in\mathbb{P}\left(V\right)$ is identified with the Lie algebra of nilpotent
endomorphisms of $V$ whose kernel contains $F$.
$\widetilde{\mathfrak{n}\left(V\right)}$ is the fiber product of
$\mathfrak{n}\left(V\right)\backslash\\{0\\}$ (viewed as a principal
$\mathbb{C}^{*}$-bundle) under the standard $\mathbb{C}^{*}$-representation
$\rho$ on $\mathbb{C}$,

$\widetilde{\mathfrak{n}\left(V\right)}=\left(\mathfrak{n}\left(V\right)\backslash\\{0\\}\right)\times_{\rho}\mathbb{C}.$

Our goal is to parametrize the spaces $\mathrm{QPH}(E)$ and their
stratifications in terms of the geometry of a natural evaluation map
associated to $D$. Consider the maps

$\mathrm{ev}_{z_{i}}:\mathrm{QPH}(E)\rightarrow\widetilde{\mathfrak{n}\left(E|_{z_{i}}\right)},\qquad(E_{\boldsymbol{\cdot}},\Phi)\mapsto\left(\operatorname{Res}_{z_{i}}\Phi,F_{i}\right),\qquad
i=1,2,3,4.$

We define the total evaluation to be the map
$\mathrm{Ev}=\prod_{i=1}^{4}\mathrm{ev}_{z_{i}}$. Letting $\mathrm{Pr}$ be the
projection map associated to the product of projections
$\prod_{i=1}^{4}\mathrm{pr}_{z_{i}}$, we obtain the following commutative
diagram

(A.1)
$\textstyle{\mathrm{QPH}(E)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Ev}}$$\scriptstyle{\mathrm{par}}$$\textstyle{\displaystyle\prod_{i=1}^{4}\widetilde{\mathfrak{n}\left(E|_{z_{i}}\right)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{Pr}}$$\textstyle{\mathrm{QP}(E)}$

which is equivariant with respect to the induced
$\mathbb{P}(\operatorname{Aut}(E))$-actions. In other words, the total
evaluation map is derived from the residue map on parabolic Higgs fields

$\operatorname{Res}_{E}:\mathrm{Int}(E)\rightarrow\prod_{i=1}^{4}\mathfrak{n}(E|_{z_{i}}),\qquad\Phi\mapsto(\operatorname{Res}_{z_{1}}\Phi,\operatorname{Res}_{z_{2}}\Phi,\operatorname{Res}_{z_{3}}\Phi,\operatorname{Res}_{z_{4}}\Phi),$

by taking simultaneous blow-ups on its domain and codomain along the loci
defined by the identities (4.2).

######  Remark 5.

The surface $\widetilde{\mathfrak{n}\left(V\right)}$ can be also described in
terms of explicit coordinate charts.666A choice of isomorphism
$V\cong\mathbb{C}^{2}$ determines an identification
$\mathfrak{n}\left(V\right)\cong\mathbb{Z}_{2}\backslash\mathbb{C}^{2}$ under
the explicit parametrization
$(Z_{0},Z_{1})\mapsto\begin{pmatrix}Z_{0}Z_{1}&-Z_{0}^{2}\\\
Z_{1}^{2}&-Z_{0}Z_{1}\end{pmatrix},$ which is invariant under the
$\mathbb{Z}_{2}$-action given by $(Z_{0},Z_{1})\mapsto(-Z_{0},-Z_{1})$, and
associates the kernel line
$[Z_{0}:Z_{1}]\in\mathbb{P}\left(\mathbb{C}^{2}\right)$ whenever
$(Z_{0},Z_{1})\neq(0,0)$. Thus, the introduction of the standard affine charts
$\mathscr{U}_{0},\mathscr{U}_{1}$ on $\mathbb{P}\left(\mathbb{C}^{2}\right)$
induces two affine charts $\mathscr{W}_{0},\mathscr{W}_{1}\cong\mathbb{C}^{2}$
for the blown-up surface $\widetilde{\mathbb{Z}_{2}\backslash\mathbb{C}^{2}}$,
defined on their domain intersections in the complement of the exceptional
divisor by the maps  $\left(Z_{0}/Z_{1},Z_{1}^{2}\right)\leftmapsto
Z_{1}^{2}\begin{pmatrix}Z_{0}/Z_{1}&-(Z_{0}/Z_{1})^{2}\\\
1&-Z_{0}/Z_{1}\end{pmatrix}=Z_{0}^{2}\begin{pmatrix}Z_{1}/Z_{0}&-1\\\
(Z_{1}/Z_{0})^{2}&-Z_{1}/Z_{0}\end{pmatrix}\mapsto\left(Z_{1}/Z_{0},Z_{0}^{2}\right).$
These coordinates determine another way to parametrize $\mathrm{QPH}(E)$ from
local trivializations on $E$, from which a projective model for
Harder–Narasimhan strata can be constructed repeating the ideas of this
section (cf. [KS19]). We do not pursue that possibility here, since a
coordinate-invariant approach is self-sufficient for our general purposes.

For every $z\in\mathbb{C}\mathbb{P}^{1}$, there is a natural action of the
group $\operatorname{Aut}(E)|_{z}$ on the space $\mathfrak{n}(E|_{z})$ by
conjugation. Since the action of central elements is trivial, there is an
induced action of $\mathbb{P}(\operatorname{Aut}(E|_{z}))$ on
$\widetilde{\mathfrak{n}\left(E|_{z}\right)}$ and $\mathbb{P}(E|_{z})$, making
the projection $\mathrm{pr}_{z}$ equivariant. When $m_{1}=m_{2}$ the induced
action of $\mathbb{P}(\operatorname{Aut}(E)|_{z})$ on $\mathbb{P}(E|_{z})$ is
transitive with stabilizer

$\mathbb{P}(\operatorname{Aut}(E|_{z}))_{F}=\mathbb{P}\left(\mathrm{P}(F)\right)$

for any $F\in\mathbb{P}(E|_{z})$. Moreover, since the action of
$\mathbb{P}\left(\operatorname{Aut}(E)|_{z}\right)$ on
$\mathfrak{n}\left(E|_{z}\right)\backslash\\{0\\}$ is also transitive, and
such that for any $\phi\in\mathfrak{n}\left(E|_{z}\right)\backslash\\{0\\}$,
its stabilizer is equal to the subgroup

$\mathbb{P}(\operatorname{Aut}(E|_{z}))_{\phi}=\mathbb{P}\left(\mathrm{R}(F)\right),$

this action partitions the surface
$\widetilde{\mathfrak{n}\left(E|_{z}\right)}$ into a total of two orbits,
corresponding to the exceptional divisor $\mathscr{E}_{z}$ and its complement,
which we will denote by $\mathscr{O}_{z}$, that is

$\widetilde{\mathfrak{n}\left(E|_{z}\right)}=\mathscr{O}_{z}\sqcup\mathscr{E}_{z},\qquad\mathscr{O}_{z}\cong\mathfrak{n}(E|_{z})\backslash\\{0\\}.$

When $m_{1}>m_{2}$, the $\mathrm{pr}_{z}$-inverse image of each of the two
orbits $\mathscr{V}_{z}(E)\sqcup\\{E_{1}|_{z}\\}=\mathbb{P}(E|_{z})$ is in
turn partitioned into the two $\mathbb{P}(\operatorname{Aut}(E)|_{z})$-orbits
which result from the intersection with $\mathscr{E}_{z}$ and its complement
$\mathscr{O}_{z}$. This way, the action of
$\mathbb{P}(\operatorname{Aut}(E)|_{z})$ on
$\widetilde{\mathfrak{n}\left(E|_{z}\right)}$ partitions it into a total of
four orbits (figure 14), that we will denote by $\mathscr{O}_{z,1}$,
$\mathscr{O}_{z,2}$, $\mathscr{O}_{z,3}$ and $\mathscr{O}_{z,4}$, where

$\mathscr{O}_{z,1}:=\mathrm{pr}_{z}^{-1}(\mathscr{V}_{z}(E))\backslash\mathscr{E}_{z}=\left(\mathscr{E}_{z}\cup\mathfrak{n}\left(E_{1}|_{z}\right)\right)^{\mathsf{c}}$

is Zariski open and a principal homogeneous space for
$\mathbb{P}(\operatorname{Aut}(E)|_{z})$. The remaining orbits

$\mathscr{O}_{z,2}:=\mathfrak{n}(E_{1}|_{z})\backslash\mathscr{O}_{z,4},\qquad\mathscr{O}_{z,3}:=\mathscr{E}_{z}\backslash\mathscr{O}_{z,4},\qquad\mathscr{O}_{z,4}:=\mathscr{E}_{z}\cap\mathfrak{n}\left(E_{1}|_{z}\right)$

are respectively isomorphic to $\mathbb{C}^{*}$, $\mathbb{C}$, and a single
point, since the stabilizer of any point
$(\phi,E_{1}|_{z})\in\mathscr{O}_{z,2}$ is equal to
$\mathbb{P}(\mathrm{R}(E_{1}|_{z}))$.

Figure 14. Orbit stratification of
$\widetilde{\mathfrak{n}\left(E|_{z}\right)}$ when $m_{1}>m_{2}$ $\forall
z\in\mathbb{C}\mathbb{P}^{1}$.

Given that $\mathrm{Ev}(\mathrm{Q}(E))=\prod_{i=1}^{4}\mathscr{E}_{z_{i}}$,
the inclusion $\iota:\mathrm{QP}(E)\rightarrow\mathrm{QPH}(E)$ satisfies
$\mathrm{Pr}\circ\mathrm{Ev}\circ\iota=\mathrm{id}$. Since the sum of an
arbitrary $\operatorname{End}(E)$-valued holomorphic differential and any
parabolic Higgs field is again a parabolic Higgs field, there is an additional
affine action of the (possibly trivial) vector space
$H^{0}\left(\mathbb{C}\mathbb{P}^{1},\operatorname{End}(E)\otimes
K_{\mathbb{C}\mathbb{P}^{1}}\right)$ on $\mathrm{QPH}(E)$. The orbit of any
$(E_{\boldsymbol{\cdot}},\Phi)\in\mathrm{QPH}(E)$ is the level set
$\mathrm{Ev}^{-1}\left(\mathrm{Ev}(E_{\boldsymbol{\cdot}},\Phi)\right)$, and
$\mathrm{Ev}$ is precisely the invariant map for this action, endowing
$\mathrm{QPH}(E)$ with the structure of an (possibly trivial) affine bundle
over its image.

######  Proposition 17.

$(i)$ If $E$ is evenly-split, then $\mathrm{Ev}$ is a
$\mathbb{P}\left(\operatorname{Aut}(E)\right)$-equivariant holomorphic
embedding, whose image is the smooth 5-dimensional variety of residue
constraints. $(ii)$ When $m_{1}-m_{2}=2$,
$\mathrm{QPH}_{\mathbb{C}^{*}}(E)\sqcup\mathrm{S}_{m-1}(E)$ and
$\mathrm{Q}(E)\sqcup\mathrm{R}(E)$ are
$H^{0}\left(\mathbb{C}\mathbb{P}^{1},\operatorname{End}(E)\otimes
K_{\mathbb{C}\mathbb{P}^{1}}\right)$-invariant,
$\mathrm{Ev}\left(\mathrm{QPH}_{\mathbb{C}^{*}}(E)\sqcup\mathrm{S}_{m-1}(E)\right)\subset\prod_{i=1}^{4}\mathscr{O}_{z_{i},1}$
and
$\mathrm{Ev}\left(\mathrm{Q}(E)\sqcup\mathrm{R}(E)\right)=\prod_{i=1}^{4}\mathscr{O}^{\mathsf{c}}_{z_{i},1}$.
$(iii)$ When $m_{1}-m_{2}=3$,
$\mathrm{Ev}\left(\mathrm{R}_{\emptyset}(E)\right)=\prod_{i=1}^{4}\mathscr{O}_{z_{i},2}$.

###### Proof.

$(i)$ $\mathrm{Ev}$ is injective if and only if $E$ is evenly-split. The
definition of $\mathrm{Ev}$ implies that, in general, $d\mathrm{Ev}$ has
constant rank at any point $(E_{\boldsymbol{\cdot}},\Phi)\in\mathrm{QPH}(E)$.
Consequently, $\mathrm{Ev}$ is a holomorphic embedding if $E$ is evenly-split.

($ii$) Since nonzero elements in
$H^{0}\left(\mathbb{C}\mathbb{P}^{1},\operatorname{End}(E)\otimes
K_{\mathbb{C}\mathbb{P}^{1}}\right)$ preserve $E_{1}$, and these exist if and
only if $m_{1}-m_{2}\geq 2$, while
$\mathrm{QPH}(E)=\mathrm{Q}(E)\sqcup\mathrm{R}(E)$ whenever $m_{1}-m_{2}\geq
3$, $H^{0}\left(\mathbb{C}\mathbb{P}^{1},\operatorname{End}(E)\otimes
K_{\mathbb{C}\mathbb{P}^{1}}\right)$-invariance is trivial unless
$m_{1}-m_{2}=2$, and then it follows from (4.4), since
$\mathrm{QPH}_{\mathbb{C}^{*}}(E)\sqcup\mathrm{S}_{0}(E)$ is characterized by
$w_{0}\neq 0$, which is unchanged under addition of
$\operatorname{End}(E)$-valued holomorphic differentials, and is independent
of the isomorphism $E\cong\mathcal{O}(m+1)\oplus\mathcal{O}(m-1)$. The second
claim is verified from (4.4) in analogy to the proof of proposition 4-$(ii)$
and $(iii)$.

$(iii)$ When $m_{1}-m_{2}=3$ by definition we have that
$\mathrm{Ev}\left(\mathrm{R}_{\emptyset}(E)\right)=\prod_{i=1}^{4}\mathscr{O}_{z_{i},2}$.
∎

######  Corollary 12.

Assume that $\mathrm{QPH}^{s}_{\mathcal{C}}(E)\neq\emptyset$. $(i)$ If $E$
evenly-split, there is an isomorphism
$\mathscr{M}_{\mathcal{C}}(E)\cong\mathscr{M}^{\mathrm{ev}}_{\mathcal{C}}(E)$,
where $\mathscr{M}^{\mathrm{ev}}_{\mathcal{C}}(E)$ is the
$\mathbb{P}(\operatorname{Aut}(E))$-orbit space of

$\mathrm{Ev}\left(\mathrm{QPH}^{s}_{\mathcal{C}}(E)\right)\subset\prod_{i=1}^{4}\widetilde{\mathfrak{n}\left(E|_{z_{i}}\right)}.$

$(ii)$ When $m_{1}-m_{2}=2$,
$\mathscr{M}^{\mathrm{ev}}_{\mathcal{C}}(E):=\mathrm{Ev}\left(\mathrm{QPH}^{s}_{\mathcal{C}}(E)\right)/\mathrm{R}(\operatorname{Aut}(E))\cong\mathscr{M}_{\mathcal{C}}(E)$.
$(iii)$ When $m_{1}-m_{2}=3$, $\mathrm{R}(\operatorname{Aut}(E))$ acts freely
and transitively on
$\mathrm{Ev}\left(\mathrm{QPH}^{s}_{\mathcal{C}}(E)\right)$ and
$\mathscr{M}_{\mathcal{C}}(E)\cong\mathbb{P}\left(H^{0}\left(\mathbb{C}\mathbb{P}^{1},\operatorname{End}(E)\otimes
K_{\mathbb{C}\mathbb{P}^{1}}\right)\right)$.

###### Proof.

$(i)$ The claim follows from the equivariance of the commutative diagram
(A.1), proposition 17-$(i)$, and theorem 1.

$(ii)$-$(iii)$ When $m_{1}-m_{2}\geq 2$, the affine
$H^{0}\left(\mathbb{C}\mathbb{P}^{1},\operatorname{End}(E)\otimes
K_{\mathbb{C}\mathbb{P}^{1}}\right)$-action on $\mathrm{QPH}(E)$ is nontrivial
and commutes with the action of the subgroup
$\mathrm{R}(\operatorname{Aut}(E))$. $\mathrm{R}(\operatorname{Aut}(E))$ acts
trivially on $H^{0}\left(\mathbb{C}\mathbb{P}^{1},\operatorname{End}(E)\otimes
K_{\mathbb{C}\mathbb{P}^{1}}\right)$, and the induced action of
$\mathbb{P}(\operatorname{Aut}(E))/\mathrm{R}(\operatorname{Aut}(E))$ on it
equals the square of its standard $\mathbb{C}^{*}$-action. When
$m_{1}-m_{2}=2$, it follows from the classification of nilpotent cone assembly
kits and proposition 17-$(ii)$ that the action of
$\mathbb{P}(\operatorname{Aut}(E))$ on the fibers of
$\mathrm{Ev}|_{\mathrm{QPH}^{s}_{\mathcal{C}}(E)}$ is transitive, and the
result follows. When $m_{1}-m_{2}=3$ there is a single case when
$\mathrm{QPH}^{s}_{\mathcal{C}}(E)\neq\emptyset$, namely
$\mathcal{C}=\mathcal{C}_{\emptyset}$, for which
$\mathrm{QPH}^{s}_{\mathcal{C}}(E)=\mathrm{R}_{\emptyset}(E)$. Every element
in $\mathrm{R}_{\emptyset}(E)$ can be uniquely expressed as the sum of an
element in $\iota(B_{\emptyset,m+2}(E))$ and a nonzero element in
$H^{0}\left(\mathbb{C}\mathbb{P}^{1},\operatorname{End}(E)\otimes
K_{\mathbb{C}\mathbb{P}^{1}}\right)$. The result follows from proposition
17-$(iii)$, since $\mathrm{R}(\operatorname{Aut}(E))$ acts freely and
transitively on $\prod_{i=1}^{4}\mathscr{O}_{z_{i},2}$. ∎

### Acknowledgments

I would like to thank Hartmut Weiß, whose encouragement and support where
crucial in prompting the appearance of the present manuscript, and Steven
Rayan for providing many insightful remarks. This work was supported by the
DFG SPP 2026 priority programme “Geometry at infinity”.

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# Exploring Exponential Parametrization of Neutrino Mass Matrix

Pralay Chakraborty<EMAIL_ADDRESS>Subhankar Roy
<EMAIL_ADDRESS>Department of Physics, Gauhati University, India

###### Abstract

We explore the exponential parameterization of the Majorana neutrino mass
matrix texture. In this regard, the elements of the mass matrix are expressed
in terms of their absolute values and arguments. We propose a neutrino mass
matrix texture that highlights four correlations among its elements. The
mixing scheme obtained from the proposed texture is consistent with
experimental observations. In addition, we derive the proposed texture from
the $SU(2)_{L}\times U(1)_{Y}\times A_{4}\times Z_{10}$ group in the light of
the seesaw mechanism.

††journal: Physics Letters B

##  1 Introduction

The neutrino oscillation suggests that neutrinos change their identity as they
traverse space [1]. This phenomenon can be understood from the quantum
mechanics framework which tells us that the three flavour states of neutrinos
($\nu_{l=e,\mu,\tau}$) are the mixture of three mass eigenstates
$(\nu_{i=1,2,3})$ having definite masses ($m_{i=1,2,3}$). The neutrino
oscillation predicts the existence of non-zero mass, which goes against the
predictions of the Standard Model (SM) [2, 3, 4]. In this regard, we need to
go beyond the SM framework to understand the non-zero neutrino masses. The
neutrino mass matrix ($M_{\nu}$) originates from the Yukawa Lagrangian, which
can be understood from the see-saw mechanism [5, 6, 7, 8, 9, 10, 11, 12, 13].
In general, a Majorana neutrino mass matrix contains the information of twelve
parameters: three mass eigenvalues ($m_{i=1,2,3}$), three mixing angles
($\theta_{12},\theta_{13},\theta_{23}$), three CP phases
($\delta,\alpha,\beta$), and three unphysical phases
($\phi_{1},\phi_{2},\phi_{3}$). Among them, nine are physical parameters
excluding the unphysical phases. Recent experiments predict three mixing
angles and two mass-squared differences ($\Delta\,m_{21}^{2}$ and
$\Delta\,m_{31}^{2}$) with much precision [14]. However, the octant of
$\theta_{23}$ and a precise bound of $\delta$ are yet to be determined. In
addition, the oscillation experiments do not measure three individual mass
eigenvalues and the Majorana phases. However, the $M_{\nu}$ carries the
information of all nine physical parameters, and by imposing certain
constraints on $M_{\nu}$, some of the physical parameters can be predicted.
Various constraints, such as texture zeroes [15], vanishing minors [16],
hybrid textures [17], and $\mu-\tau$ symmetry [18] are widely studied in the
literature. The experiments have ruled out the $\mu-\tau$ symmetry due to its
prediction of $\theta_{13}$ being zero. In this regard, there are several
attempts made to explore deviations from the exact $\mu-\tau$ symmetry [19,
20, 21]. In the present work, we explore different types of constraints,
deviating slightly from the existing literature. We adhere to the exponential
parametrization of the neutrino mass matrix, which was first discussed in Ref
[22]. We propose a neutrino mass matrix texture that highlights four
correlations based on the absolute and argument parts of the mass matrix
elements, thereby predicting four physical parameters. In addition, starting
from the symmetry framework, we shall try to derive the texture in its exact
form.

The plan of the paper is outlined as follows: In section (2), we discuss the
formalism of our work. In section (3), we study the predictions of the
proposed texture. The section (4) is devoted to the symmetry realization. In
section (5), we conclude.

##  2 Formalism

In principle, physics is independent of parametrization. The underlying
principles and phenomena do not alter regardless of the parametrization we
choose. However, a suitable parametrization may lead to understand the
underlying physics in a better way. The Majorana neutrino mass matrix is a
complex symmetric and the matrix elements can be visualized in the form of
$(re^{i\theta})$, where, $r$ represents the absolute value and $\theta$ stands
for argument. In the light of the exponential parametrization [22], the
neutrino mass matrix elements can be visualized in the following manner,

$M_{\nu}=\begin{bmatrix}|m_{11}|e^{i\,\text{arg}[m_{11}]}&|m_{12}|e^{i\,\text{arg}[m_{12}]}&|m_{13}|e^{i\,\text{arg}[m_{13}]}\\\
|m_{12}|e^{i\,\text{arg}[m_{12}]}&|m_{22}|e^{i\,\text{arg}[m_{22}]}&|m_{23}|e^{i\,\text{arg}[m_{23}]}\\\
|m_{13}|e^{i\,\text{arg}[m_{13}]}&|m_{23}|e^{i\,\text{arg}[m_{23}]}&|m_{33}|e^{i\,\text{arg}[m_{33}]}\end{bmatrix}.$
(1)

The said parameterization offers certain important phenomenological aspects,
where, the absolute version of the neutrino mass matrix can be visualized.
However, in the present work, we treat absolute and argument parts on the
equal footing. Under the said parametrization, we propose a neutrino mass
matrix texture highlighting four correlations among its elements as shown
below,

$M=\begin{bmatrix}A&|\textbf{B}|e^{i\,\text{{arg}}[\textbf{B}]}&|\textbf{B}|e^{-i\,\text{{arg}}[\textbf{B}]}\\\
|\textbf{B}|e^{i\,\text{{arg}}[\textbf{B}]}&F&G\\\
|\textbf{B}|e^{-i\,\text{{arg}}[\textbf{B}]}&G&\mathbf{2}\,|\textbf{B}|e^{i\,\text{{arg}}[\textbf{B}]}\end{bmatrix},$
(2)

where, $A$, $B$, $F$ and $G$ are complex parameters.

On the other hand, a complex number $z$ can also be expressed as $z=x+iy$,
where, $x$ and $y$ are real numbers. This parametrization scheme gives us the
liberty to visualize the real and imaginary parts on the same footing and
allows us to draw some fruitful correlations among them. We highlight the
importance of the said parametrization of the neutrino mass matrix in Ref
[22].

To draw information on the physical parameters, we need to diagonalize
$M:M^{d}=V^{T}.M.V$, where, $V$ is a $3\times 3$ unitary matrix known as
Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix [23]. In general, the PMNS
matrix is defined in flavour basis i.e., the basis where the charged lepton
mass matrix is diagonal. The Particle Data Group (PDG) [24] has adopted a
parametrization for $V$, where, $V$ is parametrized using three angles and six
phases as shown below,

$V=P_{\phi}.U.P_{M},$ (3)

where, $P_{\phi}=diag(e^{i\phi_{1}},e^{i\phi_{2}},e^{i\phi_{3}})$ and
$P_{M}=diag(e^{i\alpha},e^{i\beta},1)$. The matrix $U$ is depicted as shown in
the following,

$\begin{split}U&=\begin{bmatrix}1&0&0\\\ 0&c_{23}&s_{23}\\\
0&-s_{23}&c_{23}\end{bmatrix}\times\begin{bmatrix}c_{13}&0&s_{13}\,e^{-i\delta}\\\
0&1&0\\\ -s_{13}e^{i\delta}&0&c_{13}\end{bmatrix}\\\
&\qquad\qquad\times\begin{bmatrix}c_{12}&s_{12}&0\\\ -s_{12}&c_{12}&0\\\
0&0&1\end{bmatrix},\end{split}$ (4)

where, $s_{ij}=\sin\theta_{ij}$ and $c_{ij}=\cos\theta_{ij}$.

In general, the unphysical phases can be removed from $V$ by redefining the
charged lepton fields. In the present work, we shall proceed with the neutrino
mass matrix texture that excludes the unphysical phases.

##  3 Numerical Analysis

The proposed texture shelters four correlations in terms of absolute and
argument parts as shown below,

$\displaystyle|M_{12}|$ $\displaystyle=$ $\displaystyle|M_{13}|,$ (5)
$\displaystyle|M_{33}|$ $\displaystyle=$ $\displaystyle 2\,|M_{12}|,$ (6)
$\displaystyle\text{arg}\,[M_{12}]$ $\displaystyle=$
$\displaystyle-\,\text{arg}\,[M_{12}],$ (7)
$\displaystyle\text{arg}\,[M_{33}]$ $\displaystyle=$
$\displaystyle\text{arg}\,[M_{12}].$ (8)

Figure 1:  Shows the correlation plots between (a) $m_{1}$ vs
$m_{2}\,\text{and}\,m_{3}$. (b) $m_{1}$ vs $\Sigma m_{i}$. (c) $\theta_{23}$
vs $\delta$. (d) $\alpha$ vs $\beta$. (e) Lightest neutrino mass ($m_{1}$) vs
$m_{\beta\beta}$.

There exist two more correlations among the elements $M_{13}$ and $M_{33}$.
However, those correlations are not independent and can be derived from the
above four equations. The equations arising from (5)-(8) are transcendental
equations involving nine physical parameters as discussed earlier. We solve
Eq. (5)-(8) by the Newton-Raphson method to generate a sufficient number of
random numbers for the parameters $\alpha$, $\beta$, $\theta_{23}$ and
$\delta$. For this, we set the mixing angles $\theta_{12}$ and $\theta_{13}$
at their respective $3\sigma$ bound. The mass eigenvalues $m_{1}$, $m_{2}$ and
$m_{3}$ are solved from two mass squared differences $\Delta m_{21}^{2}$ and
$\Delta m_{31}^{2}$ in the light of normal ordering, such that $\Sigma m_{i}$
is consistent with the cosmological data [25]. The variation of the mass
parameters is shown in Figs. (1) and (1). We generate the correlation plots to
visualize the prediction of the studied parameters (see Figs. (1)-(1)). It is
seen that the prediction of $\theta_{23}$ is consistent with the experimental
observation [14]. In addition, the Dirac CP phase $\delta$ and the Majorana
phases are constrained to sharp bounds. The maximum and minimum values of the
studied parameters are listed in Table (1). It is important to mention that
the proposed texture is valid only for the normal ordering of neutrino masses.
For inverted ordering, the texture is ruled out as the prediction of
$\theta_{23}$ goes outside the experimental bound.

Parameters | Minimum Value | Maximum Value
---|---|---
$m_{1}/\text{eV}$ | 0.017 | 0.029
$m_{2}/\text{eV}$ | 0.019 | 0.031
$m_{3}/\text{eV}$ | 0.052 | 0.058
$\sum{m_{i}}/\text{eV}$ | 0.09 | 0.118
$\theta_{23}/^{\circ}$ | 42.09 | 47.60
$\delta/^{\circ}$ | 347.51 | 354.55
$\alpha/^{\circ}$ | 85.36 | 87.67
$\beta/^{\circ}$ | 77.32 | 80.74
$m_{\beta\beta}/\text{meV}$ | 16.10 | 28.50
Table 1:  Shows the maximum and minimum values of $m_{1}$, $m_{2}$, $m_{3}$,
$\Sigma m_{i}$, $\theta_{23}$, $\delta$, $\alpha$, $\beta$ and
$m_{\beta\beta}$.

The effective Majorana neutrino mass $m_{\beta\beta}$ is an observational
parameter and several experiments have provided the upper bounds of
$m_{\beta\beta}$: SuperNEMO(Se82) as $67-131$ meV, GERDA(Ge76) as $104-228$
meV, EXO-200(Xe136) as $111-477$ meV, CUORE(Te130) as $75-350$ meV and
KamLAND-Zen(Xe136) as $61-165$ meV [26, 27, 28, 29, 30, 31, 32]. In this
regard, we visualize the prediction of the parameter $m_{\beta\beta}$ from the
proposed texture (see Fig. (1)). It is important to mention that the
prediction of $m_{\beta\beta}$ from the proposed texture partially falls
within the sensitivity of future experiments such as LEGEND-1000 [33].

##  4 Symmetry Framework

To derive the proposed texture from the symmetry, we incorporate see-saw
mechanism in the light of $A_{4}$ group. In this regard, we extend the field
content of the SM by introducing right-handed neutrinos and extra scalar
fields. The transformation properties of the extended field content are
summarised in Table (2).

Fields |  $D_{l_{L}}$ |  $l_{R}$ |  $\nu_{l_{R}}$ |  $H$ |  $\Delta$ |  $\chi$ |  $\psi$ |  $\xi$ |  $\eta$ |  $\kappa$
---|---|---|---|---|---|---|---|---|---|---
$SU(2)_{L}$ |  2 |  1 |  1 |  2 |  3 |  1 |  1 |  1 |  1 |  1
$U(1)_{Y}$ |  -1 |  $(-2,-2,-2)$ |  $(0,0,0)$ |  $1$ |  -2 |  0 |  0 |  0 |  0 |  0
$A_{4}$ |  3 |  $(1,1^{\prime},1^{\prime\prime})$ |  $(1,1^{\prime},1^{{}^{\prime\prime}})$ |  3 |  3 |  1 |  1 |  1 |  $1^{\prime}$ |  1
$Z_{10}$ |  0 |  $(0,0,0)$ |  $(0,4,3)$ |  $0$ |  0 |  6 |  $7$ |  $0$ |  $2$ |  3
Table 2:  The transformation properties of various fields under
$SU(2)_{L}\times U(1)_{Y}\times A_{4}\times Z_{10}$.

The $SU(2)_{L}\times U(1)_{Y}\times A_{4}\times Z_{10}$ invariant Lagrangian
is constructed in the following way,

$\displaystyle-\mathcal{L}_{Y}$ $\displaystyle\supset$ $\displaystyle
y_{e}(\bar{D}_{l_{L}}H)e_{R}+y_{\mu}(\bar{D}_{l_{L}}H)\mu_{R}+y_{\tau}(\bar{D}_{l_{L}}H)\tau_{R}$
(9)
$\displaystyle+\,y_{1}(\bar{D}_{l_{L}}\tilde{H})\nu_{eR}+\,\frac{y_{2}}{\Lambda}(\bar{D}_{l_{L}}\tilde{H})\nu_{\mu_{R}}\chi+\,\frac{y_{3}}{\Lambda}(\bar{D}_{l_{L}}$
$\displaystyle\tilde{H})\nu_{\tau_{R}}\psi+\frac{y_{a}}{2}(\overline{\nu^{c}}_{e_{R}}\nu_{e_{R}})_{1}\xi+\frac{y_{b}}{2}[(\overline{\nu^{c}}_{\mu_{R}}\nu_{\tau_{R}})+$
$\displaystyle(\overline{\nu^{c}}_{\tau_{R}}\nu_{\mu_{R}})]\xi\,\kappa+\frac{y_{c}}{2}(\overline{\nu^{c}}_{\mu_{R}}\nu_{\mu_{R}})_{1}\eta\,+y_{T_{2}}(\bar{D}_{l_{L}}$
$\displaystyle D_{l_{L}}^{c})i\tau^{2}\Delta+h.c.$

The Lagrangian is constructed in leading order. The additional group $Z_{10}$
[34] is introduced to restrict some unwanted terms that are allowed by $A_{4}$
group. The vacuum expectation values (vev) for the scalar fields are chosen
as: $\langle H\rangle_{0}=v_{H}(1,1,1)^{T}$,
$\langle\Delta\rangle_{0}=v_{\Delta}(1,1,0)^{T}$,
$\langle\chi\rangle_{0}=v_{\chi}$, $\langle\psi\rangle_{0}=v_{\psi}$,
$\langle\xi\rangle_{0}=v_{\xi}$, $\langle\eta\rangle_{0}=v_{\eta}$ and
$\langle\kappa\rangle_{0}=v_{\kappa}$ respectively. The scalar potential for
the the chosen vacuum alignments is given in appendix A. The product rules
under $A_{4}$ group can be found in appendix B.

We derive the charged lepton mass matrix($M_{L}$) as shown in the following,

$M_{l}=v_{H}\begin{bmatrix}y_{e}&y_{\mu}&y_{\tau}\\\
y_{e}&\omega\,y_{\mu}&\omega^{2}\,y_{\tau}\\\
y_{e}&\omega^{2}\,y_{\mu}&\omega\,y_{\tau}\\\ \end{bmatrix},$ (10)

We diagonalize $M_{l}$ as $M_{l}^{diag}=U_{l_{L}}^{\dagger}M_{l}U_{l_{R}}$,
where $M^{diag}_{l}=\sqrt{3}\,v_{H}\,diag\,(y_{e},y_{\mu},y_{\tau})$. The
$U_{l_{L}}$ and $U_{l_{R}}$ are expressed as shown below,

$U_{l_{L}}=\frac{1}{\sqrt{3}}\begin{bmatrix}1&1&1\\\ 1&\omega&\omega^{2}\\\
1&\omega^{2}&\omega\\\ \end{bmatrix},\quad
U_{l_{R}}=diag\begin{bmatrix}1,&1,&1\end{bmatrix}.$ (11)

The Dirac neutrino mass matrix($M_{D}$) and the right-handed neutrino mass
matrix($M_{R}$) are derived as follows,

$M_{D}=\begin{bmatrix}y_{1}v_{H}&\frac{y_{2}}{\Lambda}v_{H}v_{\chi}&\frac{y_{3}}{\Lambda}v_{H}v_{\psi}\\\
y_{1}v_{H}&\omega\,\frac{y_{2}}{\Lambda}v_{H}v_{\chi}&\omega^{2}\,\frac{y_{3}}{\Lambda}v_{H}v_{\psi}\\\
y_{1}v_{H}&\omega^{2}\,\frac{y_{2}}{\Lambda}v_{H}v_{\chi}&\omega\,\frac{y_{3}}{\Lambda}v_{H}v_{\psi}\\\
\end{bmatrix},$ (12)

and

$M_{R}=\begin{bmatrix}\frac{y_{a}}{2}v_{\xi}&0&0\\\
0&\frac{y_{c}}{2}v_{\eta}&\frac{y_{b}}{2}v_{\xi}v_{\kappa}\\\
0&\frac{y_{b}}{2}v_{\xi}v_{\kappa}&0\\\ \end{bmatrix},$ (13)

where, $\omega=e^{2i\pi/3}$.

The Type-I seesaw contribution to the neutrino mass matrix comes as
$M_{T_{1}}=-M_{D}M_{R}^{-1}M_{D}^{T}$. We take the Type-II seesaw contribution
as shown in the following,

$M_{T_{2}}=\begin{bmatrix}0&0&y_{T_{2}}v_{\Delta}\\\
0&0&y_{T_{2}}v_{\Delta}\\\ y_{T_{2}}v_{\Delta}&y_{T_{2}}v_{\Delta}&0\\\
\end{bmatrix}.$ (14)

We construct the neutrino mass matrix by taking he both Type-I and Type-II
seesaw contributions. Now, we need to move to a basis where the charged lepton
mass matrix is diagonal i.e., flavour basis. In this regard, we incorporate
the certain transformation $(U_{l_{L}}^{T}M_{\nu_{s}}U_{l_{L}})$ to obtain the
effective neutrino mass matrix as shown below,

$M=\begin{bmatrix}A&|\textbf{B}|e^{i\,\text{{arg}}[\textbf{B}]}&|\textbf{B}|e^{-i\,\text{{arg}}[\textbf{B}]}\\\
|\textbf{B}|e^{i\,\text{{arg}}[\textbf{B}]}&F&G\\\
|\textbf{B}|e^{-i\,\text{{arg}}[\textbf{B}]}&G&\mathbf{2}\,|\textbf{B}|e^{i\,\text{{arg}}[\textbf{B}]}\end{bmatrix},$
(15)

where, $A$, $B$, $F$ and $G$ are complex parameters.

The texture parameters $A$, $B$, $F$ and $G$ can be expressed in terms of the
model parameters as shown below,

$\displaystyle A$ $\displaystyle=$
$\displaystyle\frac{4y_{T}v_{\Delta}}{3}-\frac{6v_{H}^{2}y_{1}^{2}}{y_{a}v_{\xi}},$
(16) $\displaystyle B$ $\displaystyle=$
$\displaystyle-\frac{1}{6}y_{T}v_{\Delta}-\frac{iy_{T}v_{\Delta}}{2\sqrt{3}},$
(17) $\displaystyle F$ $\displaystyle=$
$\displaystyle\frac{6v_{H}^{2}y_{3}^{2}y_{c}v_{\eta}v_{\psi}^{2}}{\Lambda^{2}y_{b}^{2}v_{\kappa}^{2}v_{\xi}^{2}}+\frac{y_{T}v_{\Delta}}{3}-\frac{iy_{T}v_{\Delta}}{\sqrt{3}},$
(18) $\displaystyle G$ $\displaystyle=$
$\displaystyle-\frac{6v_{H}^{2}y_{2}y_{3}v_{\chi}v_{\psi}}{\Lambda^{2}y_{b}v_{\kappa}v_{\xi}}-\frac{2y_{T}v_{\Delta}}{3}.$
(19)

It is important to mention that deriving a model-independent neutrino mass
matrix texture from the symmetry framework is quite challenging. To maintain
the independence of the texture parameters, we involve contributions from both
Type-I and Type-II seesaw mechanisms, which leads to an increase in the field
content. In addition, to cut short certain terms from the right-handed
Majorana neutrino mass matrix, we introduce some extra scalar fields in our
model. This step is necessary to enhance the productivity of the model.

##  5 Summary

In the present work, we explore the exponential parametrization of the
neutrino mass matrix. In this regard, we propose a neutrino mass matrix
texture highlighting four correlations among its elements. The texture
predicts four physical parameters in the light of normal ordering of neutrino
masses which are consistent with experimental observation. For inverted
ordering, the texture is ruled out. As a potential application, we study the
prediction of the observational parameter $m_{\beta\beta}$ from the proposed
texture. To understand the proposed texture from first principle, we adhere to
Type-I+II seesaw mechanism in association with $SU(2)_{L}\times U(1)_{Y}\times
A_{4}\times Z_{10}$ symmetry group.

It is important to mention that exponential parametrization can also be
applied to the quark sector. As a future prospect, we aim to generate
promising quark mass matrix textures in the light of exponential
parametrization and study their phenomenological aspects.

## Acknowledgements

The research work of PC is supported by Innovation in Science Pursuit for
Inspired Research (INSPIRE), Department of Science and Technology, Government
of India, New Delhi vide grant No. IF190651

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##  Appendix A The Scalar Potential

The $SU(2)_{L}\times U(1)_{Y}\times A_{4}\times Z_{10}$ invariant Lagrangian
is constructed in the following way,

$\displaystyle V$ $\displaystyle=$ $\displaystyle
V(H)+V(\Delta)+V(\chi)+V(\psi)+V(\xi)+V(\eta)+$ (20) $\displaystyle
V(\kappa)+V(H,\Delta)+V(H,\chi)+V(H,\psi)+V(H,\xi)$
$\displaystyle+V(H,\eta)+V(H,\kappa)++V(\Delta,\chi)+V(\Delta,\psi)+$
$\displaystyle
V(\Delta,\xi)+V(\Delta,\eta)+V(\Delta,\kappa)+V(\chi,\psi)+V(\chi,\xi)$
$\displaystyle+V(\chi,\eta)+V(\chi,\kappa)+V(\psi,\xi)+V(\psi,\eta)+V(\psi,\kappa)$
$\displaystyle+V(\xi,\eta)+V(\xi,\kappa)+V(\eta,\kappa),$

where,

$\displaystyle V(H)$ $\displaystyle=$
$\displaystyle\mu^{2}_{H}(H^{\dagger}H)+\lambda^{H}_{1}(H^{\dagger}H)(H^{\dagger}H)+\lambda^{H}_{2}$
(21)
$\displaystyle(H^{\dagger}H)_{1^{\prime}}(H^{\dagger}H)_{1^{\prime\prime}}+\lambda^{H}_{3}(H^{\dagger}H)_{3_{s}}(H^{\dagger}$
$\displaystyle
H)_{3_{s}}+\lambda^{H}_{4}(H^{\dagger}H)_{3_{s}}(H^{\dagger}H)_{3_{a}}+\lambda^{H}_{5}$
$\displaystyle(H^{\dagger}H)_{3_{a}}(H^{\dagger}H)_{3_{a}},$ $\displaystyle
V(\Delta)$ $\displaystyle=$
$\displaystyle\mu^{2}_{\Delta}Tr(\Delta^{\dagger}\Delta)+\lambda^{\Delta}_{1}Tr(\Delta^{\dagger}\Delta)Tr(\Delta^{\dagger}$
(22)
$\displaystyle\Delta)+\lambda^{\Delta}_{2}Tr(\Delta^{\dagger}\Delta)_{1^{\prime}}Tr(\Delta^{\dagger}\Delta)_{1^{\prime\prime}}+$
$\displaystyle\lambda^{\Delta}_{3}Tr(\Delta^{\dagger}\Delta)_{3_{s}}Tr(\Delta^{\dagger}\Delta)_{3_{s}}+\lambda^{\Delta}_{4}Tr$
$\displaystyle(\Delta^{\dagger}\Delta)_{3_{s}}Tr(\Delta^{\dagger}\Delta)_{3_{a}}+\lambda^{\Delta}_{5}Tr(\Delta^{\dagger}$
$\displaystyle\Delta)_{3_{a}}Tr(\Delta^{\dagger}\Delta)_{3_{a}},$
$\displaystyle V(\chi)$ $\displaystyle=$
$\displaystyle\mu^{2}_{\chi}(\chi^{\dagger}\chi)+\lambda^{\chi}(\chi^{\dagger}\chi)(\chi^{\dagger}\chi),$
(23) $\displaystyle V(\psi)$ $\displaystyle=$
$\displaystyle\mu^{2}_{\psi}(\psi^{\dagger}\psi)+\lambda^{\psi}(\psi^{\dagger}\psi)(\psi^{\dagger}\psi),$
(24) $\displaystyle V(\xi)$ $\displaystyle=$
$\displaystyle\mu^{2}_{\xi}(\xi^{\dagger}\xi)+\lambda^{\xi}(\xi^{\dagger}\xi)(\xi^{\dagger}\xi),$
(25) $\displaystyle V(\eta)$ $\displaystyle=$
$\displaystyle\mu^{2}_{\eta}(\eta^{\dagger}\eta)+\lambda^{\eta}(\eta^{\dagger}\eta)(\eta^{\dagger}\eta),$
(26) $\displaystyle V(\kappa)$ $\displaystyle=$
$\displaystyle\mu^{2}_{\kappa}(\kappa^{\dagger}\kappa)+\lambda^{\kappa}(\kappa^{\dagger}\kappa)(\kappa^{\dagger}\kappa),$
(27) $\displaystyle V(H,\Delta)$ $\displaystyle=$
$\displaystyle\lambda^{H\Delta}_{1}(H^{\dagger}H)Tr(\Delta^{\dagger}\Delta)+\lambda^{H\Delta}_{2}[(H^{\dagger}$
(28) $\displaystyle
H)_{1^{\prime}}Tr(\Delta^{\dagger}\Delta)_{1^{\prime\prime}}+(H^{\dagger}H)_{1^{\prime\prime}}Tr$
$\displaystyle(\Delta^{\dagger}\Delta)_{1^{\prime}}]+\lambda^{H\Delta}_{3}(H^{\dagger}H)_{3_{s}}Tr(\Delta^{\dagger}$
$\displaystyle\Delta)_{3_{s}}+\lambda^{H\Delta}_{4}[(H^{\dagger}H)_{3_{s}}Tr(\Delta^{\dagger}\Delta)_{3_{a}}$
$\displaystyle+(H^{\dagger}H)_{3_{a}}Tr(\Delta^{\dagger}\Delta)_{3_{s}}]+\lambda^{H\Delta}_{5}$
$\displaystyle(H^{\dagger}H)_{3_{a}}Tr(\Delta^{\dagger}\Delta)_{3_{a}}+\mu(H^{T}i\sigma_{2}$
$\displaystyle\Delta^{\dagger}H),$ $\displaystyle V(H,\chi)$ $\displaystyle=$
$\displaystyle\lambda^{H\chi}_{1}(H^{\dagger}H)(\chi^{\dagger}\chi),$ (29)
$\displaystyle V(H,\psi)$ $\displaystyle=$
$\displaystyle\lambda^{H\psi}_{1}(H^{\dagger}H)(\psi^{\dagger}\psi),$ (30)
$\displaystyle V(H,\xi)$ $\displaystyle=$
$\displaystyle\lambda^{H\xi}_{1}(H^{\dagger}H)(\xi^{\dagger}\xi),$ (31)
$\displaystyle V(H,\eta)$ $\displaystyle=$
$\displaystyle\lambda^{H\eta}_{1}(H^{\dagger}H)(\eta^{\dagger}\eta),$ (32)
$\displaystyle V(H,\kappa)$ $\displaystyle=$
$\displaystyle\lambda^{H\kappa}_{1}(H^{\dagger}H)(\kappa^{\dagger}\kappa),$
(33) $\displaystyle V(\Delta,\chi)$ $\displaystyle=$
$\displaystyle\lambda^{\Delta\chi}_{1}Tr(\Delta^{\dagger}\Delta)(\chi^{\dagger}\chi),$
(34) $\displaystyle V(\Delta,\psi)$ $\displaystyle=$
$\displaystyle\lambda^{\Delta\psi}_{1}Tr(\Delta^{\dagger}\Delta)(\psi^{\dagger}\psi),$
(35) $\displaystyle V(\Delta,\xi)$ $\displaystyle=$
$\displaystyle\lambda^{\Delta\xi}_{1}Tr(\Delta^{\dagger}\Delta)(\xi^{\dagger}\xi),$
(36) $\displaystyle V(\Delta,\eta)$ $\displaystyle=$
$\displaystyle\lambda^{\Delta\eta}_{1}Tr(\Delta^{\dagger}\Delta)(\eta^{\dagger}\eta),$
(37) $\displaystyle V(\Delta,\kappa)$ $\displaystyle=$
$\displaystyle\lambda^{\Delta\kappa}_{1}Tr(\Delta^{\dagger}\Delta)(\kappa^{\dagger}\kappa),$
(38) $\displaystyle V(\chi,\psi)$ $\displaystyle=$
$\displaystyle\lambda^{\chi\psi}_{1}(\chi^{\dagger}\chi)(\psi^{\dagger}\psi),$
(39) $\displaystyle V(\chi,\xi)$ $\displaystyle=$
$\displaystyle\lambda^{\chi\xi}_{1}(\chi^{\dagger}\chi)(\xi^{\dagger}\xi),$
(40)

$\displaystyle V(\chi,\eta)$ $\displaystyle=$
$\displaystyle\lambda^{\chi\eta}_{1}(\chi^{\dagger}\chi)(\eta^{\dagger}\eta),$
(41) $\displaystyle V(\chi,\kappa)$ $\displaystyle=$
$\displaystyle\lambda^{\chi\kappa}_{1}(\chi^{\dagger}\chi)(\kappa^{\dagger}\kappa),$
(42) $\displaystyle V(\psi,\xi)$ $\displaystyle=$
$\displaystyle\lambda^{\psi\xi}_{1}(\psi^{\dagger}\psi)(\xi^{\dagger}\xi),$
(43) $\displaystyle V(\psi,\eta)$ $\displaystyle=$
$\displaystyle\lambda^{\psi\eta}_{1}(\psi^{\dagger}\psi)(\eta^{\dagger}\eta),$
(44) $\displaystyle V(\psi,\kappa)$ $\displaystyle=$
$\displaystyle\lambda^{\psi\kappa}_{1}(\psi^{\dagger}\psi)(\kappa^{\dagger}\kappa),$
(45) $\displaystyle V(\xi,\eta)$ $\displaystyle=$
$\displaystyle\lambda^{\xi\eta}_{1}(\xi^{\dagger}\xi)(\eta^{\dagger}\eta),$
(46) $\displaystyle V(\xi,\kappa)$ $\displaystyle=$
$\displaystyle\lambda^{\xi\kappa}_{1}(\xi^{\dagger}\xi)(\kappa^{\dagger}\kappa),$
(47) $\displaystyle V(\eta,\kappa)$ $\displaystyle=$
$\displaystyle\lambda^{\eta\kappa}_{1}(\eta^{\dagger}\eta)(\kappa^{\dagger}\kappa).$
(48)

Without loss of generality, the following equations are valid from the
minimisation conditions of the scalar potential for the chosen vacuum
alignments: $\langle H\rangle_{0}=v_{H}(1,1,1)^{T}$,
$\langle\Delta\rangle_{0}=v_{\Delta}(1,1,0)^{T}$,
$\langle\chi\rangle_{0}=v_{\chi}$, $\langle\psi\rangle_{0}=v_{\psi}$,
$\langle\xi\rangle_{0}=v_{\xi}$, $\langle\eta\rangle_{0}=v_{\eta}$ and
$\langle\kappa\rangle_{0}=v_{\kappa}$,

$\displaystyle\frac{\partial V}{\partial H_{1}}$ $\displaystyle=$
$\displaystyle v_{H}(\mu_{H}^{2}+2\mu
v_{\Delta}v_{\xi}+(2\lambda_{1}^{H\Delta}+\lambda_{2}^{H\Delta}+2\lambda_{3}^{H\Delta}-$
(49) $\displaystyle
2\lambda_{4}^{H\Delta})v_{\Delta}^{2}+\lambda_{1}^{H\eta}v_{\eta}^{2}+\lambda_{1}^{H\kappa}v_{\kappa}^{2}+\lambda_{1}^{H\xi}v_{\xi}^{2}+(6\lambda_{1}^{H}$
$\displaystyle+8\lambda_{3}^{H})v_{H}^{2}+\lambda_{1}^{H\chi}v_{\chi}^{2}+\lambda_{1}^{H\psi}v_{\psi}^{2})=0,$
$\displaystyle\frac{\partial V}{\partial H_{2}}$ $\displaystyle=$
$\displaystyle
v_{H}(\mu_{H}^{2}+(2\lambda_{1}^{H\Delta}+\lambda_{2}^{H\Delta}+2(\lambda_{3}^{H\Delta}+\lambda_{4}^{H\Delta}))v_{\Delta}^{2}$
(50)
$\displaystyle+\lambda_{1}^{H\eta}v_{\eta}^{2}+\lambda_{1}^{H\kappa}v_{\kappa}^{2}+\lambda_{1}^{H\xi}v_{\xi}^{2}+(6\lambda_{1}^{H}+8\lambda_{3}^{H})v_{H}^{2}$
$\displaystyle+\lambda_{1}^{H\chi}v_{\chi}^{2}+\lambda_{1}^{H\psi}v_{\psi}^{2})=0,$
$\displaystyle\frac{\partial V}{\partial H_{3}}$ $\displaystyle=$
$\displaystyle v_{H}(\mu_{H}^{2}+2\mu
v_{\Delta}v_{\xi}+2(\lambda_{1}^{H\Delta}-\lambda_{2}^{H\Delta})v_{\Delta}^{2}+\lambda_{1}^{H\eta}$
(51) $\displaystyle
v_{\eta}^{2}+\lambda_{1}^{H\kappa}v_{\kappa}^{2}+\lambda_{1}^{H\xi}v_{\xi}^{2}+(6\lambda_{1}^{H}+8\lambda_{3}^{H})v_{H}^{2}+$
$\displaystyle\lambda_{1}^{H\chi}v_{\chi}^{2}+\lambda_{1}^{H\psi}v_{\psi}^{2})=0,$
$\displaystyle\frac{\partial V}{\partial\Delta_{1}}$ $\displaystyle=$
$\displaystyle
v_{H}^{2}((3\lambda_{1}^{H\Delta}+2\lambda_{3}^{H\Delta}-2\lambda_{4}^{H\Delta})v_{\Delta}+2\mu
v_{\xi})+v_{\Delta}$ (52)
$\displaystyle(\mu_{\Delta}^{2}+(4\lambda_{1}^{\Delta}+(1-i\sqrt{3})\lambda_{2}^{\Delta}+4\lambda_{3}^{\Delta}-2\lambda_{4}^{\Delta})v_{\Delta}^{2}$
$\displaystyle+\lambda_{1}^{\Delta\eta}v_{\eta}^{2}+\lambda_{1}^{\Delta\kappa}v_{\kappa}^{2}+\lambda_{1}^{\Delta\xi}v_{\xi}^{2}+\lambda_{1}^{\Delta\chi}v_{\chi}^{2}+\lambda_{1}^{\Delta\psi}$
$\displaystyle v_{\psi}^{2})=0,$ $\displaystyle\frac{\partial
V}{\partial\Delta_{2}}$ $\displaystyle=$ $\displaystyle
v_{\Delta}(\mu_{\Delta}^{2}+2(2\lambda_{1}^{\Delta}-\lambda_{2}^{\Delta}+2\lambda_{3}^{\Delta}+\lambda_{4}^{\Delta})v_{\Delta}^{2}+$
(53)
$\displaystyle\lambda_{1}^{\Delta\eta}v_{\eta}^{2}+\lambda_{1}^{\Delta\kappa}v_{\kappa}^{2}+\lambda_{1}^{\Delta\xi}v_{\xi}^{2}+\lambda_{1}^{\Delta\chi}v_{\chi}^{2}+\lambda_{1}^{\Delta\psi}$
$\displaystyle
v_{\psi}^{2})+v_{H}^{2}((3\lambda_{1}^{H\Delta}+2(\lambda_{3}^{H\Delta}+\lambda_{4}^{H\Delta}))v_{\Delta}+$
$\displaystyle 2\mu v_{\xi})=0,$ $\displaystyle\frac{\partial
V}{\partial\Delta_{3}}$ $\displaystyle=$ $\displaystyle
2v_{H}^{2}(2\lambda_{3}^{H\Delta}v_{\Delta}+\mu v_{\xi})=0,$ (54)
$\displaystyle\frac{\partial V}{\partial\chi}$ $\displaystyle=$ $\displaystyle
v_{\chi}(\mu_{\chi}^{2}+2\lambda_{1}^{\Delta\chi}v_{\Delta}^{2}+\lambda_{1}^{\chi\eta}v_{\eta}^{2}+\lambda_{1}^{\chi\kappa}v_{\kappa}^{2}+2\lambda^{\chi}v_{\chi}^{2}$
(55)
$\displaystyle+\lambda_{1}^{\chi\xi}v_{\xi}^{2}+\lambda_{1}^{\chi\psi}v_{\psi}^{2}+3\lambda_{1}^{H\chi}v_{H}^{2})=0,$

$\displaystyle\frac{\partial V}{\partial\psi}$ $\displaystyle=$ $\displaystyle
v_{\psi}(\mu_{\psi}^{2}+2\lambda_{1}^{\Delta\psi}v_{\Delta}^{2}+\lambda_{1}^{\psi\eta}v_{\eta}^{2}+\lambda_{1}^{\psi\kappa}v_{\kappa}^{2}+2\lambda^{\psi}v_{\psi}^{2}$
(56)
$\displaystyle+\lambda_{1}^{\psi\xi}v_{\xi}^{2}+\lambda_{1}^{\chi\psi}v_{\chi}^{2}+3\lambda_{1}^{H\psi}v_{H}^{2})=0,$
$\displaystyle\frac{\partial V}{\partial\xi}$ $\displaystyle=$ $\displaystyle
v_{H}^{2}(3\lambda_{1}^{H\xi}v_{\xi}+4\mu
v_{\Delta})+v_{\xi}(\mu_{\xi}^{2}+2\lambda_{1}^{\Delta\xi}v_{\Delta}^{2}$ (57)
$\displaystyle+\lambda_{1}^{\xi\eta}v_{\eta}^{2}+\lambda_{1}^{\xi\kappa}v_{\kappa}^{2}+2\lambda^{\xi}v_{\xi}^{2}+\lambda_{1}^{\chi\xi}v_{\chi}^{2}+$
$\displaystyle\lambda_{1}^{\psi\xi}v_{\psi}^{2})=0,$
$\displaystyle\frac{\partial V}{\partial\eta}$ $\displaystyle=$ $\displaystyle
v_{\eta}(\mu_{\eta}^{2}+2\lambda_{1}^{\Delta\eta}v_{\Delta}^{2}+2\lambda^{\eta}v_{\eta}^{2}+\lambda_{1}^{\eta\kappa}v_{\kappa}^{2}+\lambda_{1}^{\xi\eta}$
(58) $\displaystyle
v_{\xi}^{2}+\lambda_{1}^{\chi\eta}v_{\chi}^{2}+\lambda_{1}^{\psi\eta}v_{\psi}^{2}+3\lambda_{1}^{H\eta}v_{H}^{2})=0,$
$\displaystyle\frac{\partial V}{\partial\kappa}$ $\displaystyle=$
$\displaystyle
v_{\kappa}(\mu_{\kappa}^{2}+2\lambda_{1}^{\Delta\kappa}v_{\Delta}^{2}+\lambda_{1}^{\eta\kappa}v_{\eta}^{2}+2\lambda^{\kappa}v_{\kappa}^{2}+\lambda_{1}^{\xi\kappa}v_{\xi}^{2}$
(59)
$\displaystyle+\lambda_{1}^{\chi\kappa}v_{\chi}^{2}+\lambda_{1}^{\psi\kappa}v_{\psi}^{2}+3\lambda_{1}^{H\kappa}v_{H}^{2})=0.$

##  Appendix B Product Rules of $A_{4}$

The discrete group $A_{4}$ is a subgroup of $SU(3)$ and has 12 elements. The
group has four irreducible representation out of which three are one
dimensional and one is three dimensional. In S-basis, the product of two
triplets gives,

$3\times 3=1+1^{\prime}+1^{\prime\prime}+3_{S}+3_{A},$ (60)

where,

$\displaystyle 1$ $\displaystyle=$
$\displaystyle(a_{1}\bar{b_{1}}+a_{2}\bar{b_{2}}+a_{3}\bar{b_{3}})$ (61)
$\displaystyle 1^{\prime}$ $\displaystyle=$
$\displaystyle(a_{1}\bar{b_{1}}+\omega^{2}a_{2}\bar{b_{2}}+\omega
a_{3}\bar{b_{3}})$ (62) $\displaystyle 1^{\prime\prime}$ $\displaystyle=$
$\displaystyle(a_{1}\bar{b_{1}}+\omega
a_{2}\bar{b_{2}}+\omega^{2}a_{3}\bar{b_{3}})$ (63) $\displaystyle(3\times
3)_{S}$ $\displaystyle=$ $\displaystyle\begin{bmatrix}a_{2}b_{3}+a_{3}b_{2}\\\
a_{3}b_{1}+a_{1}b_{3}\\\ a_{1}b_{2}+a_{2}b_{1}\\\ \end{bmatrix},$ (64)
$\displaystyle(3\times 3)_{A}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}a_{2}b_{3}-a_{3}b_{2}\\\ a_{3}b_{1}-a_{1}b_{3}\\\
a_{1}b_{2}-a_{2}b_{1}\\\ \end{bmatrix}.$ (65)

The trivial singlet can be obtained from the following singlet product rules,

$1\times 1=1,\,\,1^{\prime}\times
1^{\prime\prime}=1,\,\,1^{\prime\prime}\times 1^{\prime}=1.$ (66)


11institutetext:  1 PAII Inc., Bethesda, Maryland, USA
2 University of Maryland College Park, USA
3 Chang Gung Memorial Hospital, Linkou, Taiwan, ROC
4 Ping An Technology, Shenzhen, China

# Opportunistic Screening of Osteoporosis Using Plain Film Chest X-ray

Fakai Wang1,2 This work was done when Fakai Wang interned at PAII Inc. Kang
Zheng1 Yirui Wang1 Xiaoyun Zhou1 Le Lu1 Jing Xiao4  Min Wu2  Chang-Fu Kuo3
Shun Miao1

###### Abstract

Osteoporosis is a common chronic metabolic bone disease that is often under-
diagnosed and under-treated due to the limited access to bone mineral density
(BMD) examinations, _e.g._ , Dual-energy X-ray Absorptiometry (DXA). In this
paper, we propose a method to predict BMD from Chest X-ray (CXR), one of the
most common, accessible and low-cost medical image examinations. Our method
first automatically detects Regions of Interest (ROIs) of local and global
bone structures from the CXR. Then a multi-ROI model is developed to exploit
both local and global information in the chest X-ray image for accurate BMD
estimation. Our method is evaluated on 329 CXR cases with ground truth BMD
measured by DXA. The model predicted BMD has a strong correlation with the
gold standard DXA BMD (Pearson correlation coefficient 0.840). When applied
for osteoporosis screening, it achieves a high classification performance (AUC
0.936). As the first effort in the field to use CXR scans to predict the spine
BMD, the proposed algorithm holds a strong potential in enabling early
osteoporosis screening through routine chest X-rays and contributing to the
enhancement of public health.

###### Keywords:

Bone Mineral Density Estimation Chest X-ray Multi-ROI model

##  1 Introduction

Osteoporosis is the most common chronic metabolic bone disease, characterized
by low bone mineral density (BMD) and decreased bone strength to resist
pressure or squeezing force. With an aging population and longer life span,
osteoporosis is becoming a global epidemic, affecting more than 200 million
people worldwide [8]. Osteoporosis increases the risk of fragility fractures,
which is associated with reduced life quality, disability, fatality, and
financial burden to the family and the society. While with an early diagnosis
and treatment, osteoporosis can be prevented or managed, osteoporosis is often
under-diagnosed and under-treated among the population at risk [4]. More than
half of insufficiency fractures occur in individuals who have never been
screened for osteoporosis [7]. The under-diagnosis and under-treatment of
osteoporosis are mainly due to 1) low osteoporosis awareness and 2) limited
accessibility of Dual-energy X-ray Absorptiometry (DXA) examination, which is
the currently recommended modality to measure BMD.

Opportunistic screening of osteoporosis is an emerging research field in
recent years [1, 2, 3, 6]. It aims at using medical images done for other
indications to screen for osteoporosis, which offers an opportunity to
increase the screening rate at no additional cost and time. Previous attempts
mainly focused on using the CT attenuation (i.e., Housefield Unit) of the
vertebrae to correlate with BMD and/or fracture risk. Plain films have much
greater accessibility than CT scans. Its excellent spatial resolution permits
the delineation of fine bony microstructure that may contain information that
correlates well with the BMD. We hypothesize that specific regions of interest
(ROI) in the standard chest x-rays (CXR), the most common medical imaging
prescribed clinically, may help the opportunistic screening for osteoporosis.

In this work, we introduce a method to estimate the BMD from CXR and screen
osteoporosis. Our method first locates anatomical landmarks of the patient’s
bone structures and extracts multiple ROIs that may provide imaging biomarkers
for osteoporosis. We proposed a novel network architecture that jointly
processes the ROIs to accurately estimate the BMDs. We evaluate our method on
1651 CXRs from 1638 patients with ground truth BMDs measured using DXA. The
results demonstrate that the BMDs estimated from CXR achieve a high
correlation with the DXA BMD (_i.e._ , r-value 0.840) and a high osteoporosis
diagnosis AUC of 0.936.

In summary, our contributions are three-fold: 1) we develop the first method
to estimate BMD from CXR, leading to a practical solution for opportunistic
screening of osteoporosis; 2) we propose a clinical knowledge-driven method to
combine global and local information in CXR to accurately estimate lumbar
spine BMD. 3) we demonstrate that our method has significant performance gain
over the baseline methods and achieves clinically useful osteoporosis
screening performance.

##  2 Methodology

###  2.1 Problem Overview

Our task is to predict the BMDs of the patient’s lumbar spine from a frontal
chest X-ray. Formulated as a regression problem, the input is a CXR, and the
output is the predicted BMD value. We choose to predict the BMDs of the lumbar
spine because it is one of the standard region for BMD examination and is
often affected by insufficiency fracture. Since the lumbar spine is not
visible in CXR, a major challenge in this task is to identify the regions and
visual patterns in the CXR that are correlated with the lumbar spine BMD. To
overcome this challenge, our method comprises two steps: 1) extraction of
Regions of Interest (ROIs) that are potentially correlated with the lumbar
spine BMDs (described in Section 2.2); 2) training a multi-task regression
neural network that jointly analyzes the ROIs to predict BMDs (described in
Section 2.3).

Figure 1: An overview of the training workflow, with landmark detection and
ROI extraction in the upper part, and a multi-ROI model in the lower part. The
landmark detector localizes 16 red dots on CXR images, to anchor local bones.
Each dashed yellow box is normalized to the same orientation, height/width
ratio. The cropped ROIs are used by all models. In the multi-ROI model,
independent inputs include 6 local ROIs and the whole chest image. During
training, Mean Square Error loss (MSE) is applied on 7 individual feature BMD
predictions $y_{0},..,y_{6}$, and on the concatenated feature BMD prediction
$y_{c}$.

###  2.2 Automatic ROI Extraction

Since it is unclear which local regions within the chest X-ray are the most
informative for our task, we need to explore the correlations between the
lumbar spine BMD and different regions in the CXR by extracting multiple ROIs.
To exploit both local textures and global structures, the extracted ROIs
should contain different scales. A clinical prior knowledge is that
osteoporosis is a metabolic bone disease that affects all bones in the human
body. Therefore, we hypothesize that bone regions in the CXR provide visual
cues of the lumbar spine BMDs and extract ROIs from the following regions:
left/right clavicle bones, cervical spine, left/right rib-cage area, T12
vertebra. Besides these 6 ROIs of local bone structures/patterns, we also
include the whole CXR as another ROI to provide global structural information.
To automatically extract the ROIs, we employ a graph-based landmark detection
method, DAG [5], and detect 16 landmarks which include 1) 3 points on the
left/right clavicles, 2) 4 points along the left/right rib cages, 3) 1 point
on the cervical spine and 1 point on the T12 vertebra. The ROIs are placed
based on the detected landmarks. Examples of the CXR with 16 landmarks and
extracted ROIs are shown in Figure 1.

###  2.3 BMD Estimation via Joint Analysis of the ROIs

Given the 7 extracted ROIs, we proposed a multi-task learning network
architecture, termed multi-ROI model, to jointly analyze the ROIs to estimate
the lumbar BMDs. By jointly analyze the global and local ROIs, the network is
capable of extracting visual patterns at different scales that are informative
for predicting lumbar BMDs. In particular, we employ VGG-16 as the backbone
network to produce a feature map encoding of the input ROI and apply Global
Average Pooling (GAP) to aggregate the feature map to a single feature vector.
The 7 ROIs go through the same VGG-16 network independently, generating 7
feature vectors. These feature vectors are then decoded both separately and
jointly. First, they are individually processed by a shared fully connected
(FC) layer to produce BMD estimations. Second, the 7 features are connected to
obtain a joint feature vector, which is decoded by an FC layer to produce BMD
estimation. During training, the mean squared error (MSE) is calculated on
both the BMDs produced from the features of individual ROI and on the BMDs
produced from the concatenated features. The average MSE is used as the
training loss. During testing, the BMDs produced from the concatenated feature
is taken as the output.

##  3 Experiments

###  3.1 Experiment Setup

Dataset. We collected 1651 frontal view CXR scans with paired BMD scores (on
four lumbar vertebrae L1 - L4) from anonymous hospital after de-identification
of the patient information. All experiments use the same data split, with
1087, 265 and 329 cases for training, validation and testing, respectively.
The scans of the same patient only go to the same set.

Figure 2: Multi-ROI model BMD prediction on different lumbar vertebra (L1-L4).
X-axis is the ground truth BMD, and Y-axis represent the predictions.

Metrics. We adopt two commonly used evaluation metrics: Pearson Correlation
Coefficient (R-value) and Area Under Curve. R-value measures the linear
relationship between the predicted BMD and the ground truth BMD. It only
considers the correlation between two sequences regardless their absolute
values. While Area Under Curve (AUC) measures accumulated true positive
(osteoporosis) rate under different judging threshold for binary
classification problem. We calculate the AUC score under the T-score range
instead of the BMD values as T-score is a widely adopted measurement in the
clinical practices. We convert the BMD scores to the T-score by checking the
transforming table coming with the DXA machine. To evaluate osteoporosis
diagnosis performance, we label the cases with ground truth T-scores below
$-2.5$ as positive osteoporosis (clinical definition of osteoporosis).

###  3.2 Implementation Details

We trained our model on a workstation with Intel Xeon CPU E5-2650 v4 CPU @ 2.2
GHz, 132 GB RAM, and 4 NVIDIA TITAN V GPUs. Our models are implemented with
PyTorch. X-ray inputs are resized to (256, 256) , and repeated 3 times to form
the RGB channels. During training, we apply augmentations such as scaling,
rotation, translation, random flip. The SGD optimizer has a learning rate of
$0.0001$, a weight decay of 4e-4. A mini-batch size of 64 is used to train the
model for $100$ epochs.

Table 1: Comparison of different deep learning backbones. They all use the whole chest X-ray images as input, and use the baseline working flow. The only difference is convolutional backbone. Backbone | L1 | L2 | L3 | L4 | Average
---|---|---|---|---|---
R-value | AUC | R-value | AUC | R-value | AUC | R-value | AUC | R-value | AUC
Vgg11 | 0.843 | 0.919 | 0.838 | 0.916 | 0.832 | 0.917 | 0.772 | 0.881 | 0.821 | 0.908
Resnet18 | 0.759 | 0.875 | 0.799 | 0.863 | 0.774 | 0.852 | 0.717 | 0.840 | 0.762 | 0.858
Resnet34 | 0.787 | 0.903 | 0.787 | 0.866 | 0.768 | 0.887 | 0.723 | 0.861 | 0.766 | 0.879
Resnet50 | 0.799 | 0.894 | 0.823 | 0.888 | 0.795 | 0.880 | 0.741 | 0.858 | 0.789 | 0.880
Vgg16 | 0.865 | 0.923 | 0.863 | 0.913 | 0.824 | 0.918 | 0.804 | 0.901 | 0.839 | 0.914

###  3.3 Comparison of Backbone Models

We evaluate the performance of different backbone models using a baseline
method. The baseline method takes the whole CXR as input. It employs a
backbone network as encoder, followed by a GAP layer and a FC layer for
regression. We evaluated multiple popular backbone networks and found that
VGG16 delivers the best performance, as summarized in table 1. Resnet provides
complex feature computation and works better on classification tasks. The bone
texture encoding task involves only low-level pattern recognition, where VGG
models fit better.

Table 2: Comparison of different modeling schemes. The fist 5 rows are baseline experiments on different ROIs (ROI models), the 6th is the multi-ROI scheme fed on all input modalities (multi-ROI model). Modality | L1 | L2 | L3 | L4 | Average
---|---|---|---|---|---
R-value | AUC | R-value | AUC | R-value | AUC | R-value | AUC | R-value | AUC
cervi | 0.823 | 0.915 | 0.817 | 0.930 | 0.802 | 0.937 | 0.738 | 0.841 | 0.795 | 0.906
clavi | 0.793 | 0.893 | 0.778 | 0.917 | 0.745 | 0.867 | 0.709 | 0.857 | 0.756 | 0.883
lumbar | 0.750 | 0.829 | 0.791 | 0.889 | 0.754 | 0.859 | 0.711 | 0.841 | 0.751 | 0.855
ribcage | 0.748 | 0.868 | 0.768 | 0.876 | 0.730 | 0.850 | 0.703 | 0.830 | 0.737 | 0.856
chest | 0.865 | 0.923 | 0.863 | 0.913 | 0.824 | 0.918 | 0.804 | 0.901 | 0.839 | 0.914
Prop. | 0.875 | 0.936 | 0.857 | 0.944 | 0.843 | 0.941 | 0.783 | 0.924 | 0.840 | 0.936

Using the same VGG16 backbone network, we compare different ROI in Table 2. In
the first 5 rows, single ROIs are used as input to the regression network to
predict the BMDs. The first 4 rows are results using local ROIs, and the 5th
row, chest, uses the whole chest image (global ROI model). The cervical models
results in the best performance among the the local ROIs, reporting an R-value
of 0.795 and an AUC of 0.906. The chest model performs better than all local
ROI models since the whole CXR image contains the overall bone information.

###  3.4 Comparison with Individual ROIs

As there are many bones in the CXR scan, the BMD of one bone may not be
consistent with another. ROI models lack the mechanism to solve conflicts of
inconsistent BMD from different bones. The key idea of multi-ROI model is to
allow the network to automatically identify the informative regions for
estimating lumbar spine BMD.

The multi-ROI model in Figure 1 takes 7 input modalities all at once. They
share the same encoding backbone. Each encoded feature vector is of size 512,
the concatenated feature has a shape of 512*7. Individual ROI features share
the same fully connected layer, while the concatenated vector goes through a
stand-along FC layer.

We plot the predicted BMDs and ground truth BMDs in Figure 2. Compared to
prediction distribution of other ROI models, the feature concatenation and its
following FC layer in the multi-ROI model tightens the prediction range,
reducing predicting error variation. The shared FC layer enforces
regularization on individual feature vector, reducing the absolute predicting
error.

The multi-ROI model achieves an average R-value of 0.840 and an average AUC
score of 0.936, surpassing all individual ROI schemes in Table 2. Compared
with the chest model, the multi-ROI model encodes additional 6 local ROI
modalities, which allows for detailed local texture computation. The feature
concatenation retains encoded information from both local ROIs and the global
chest, involving regional information interactions. The multi-ROI model even
yields a higher AUC score than model ensembles in Table 3.

Table 3: Ensemble_roi calculate the average of 5 ROI model predictions as result, Ensemble_all use all individual model predictions as result. Ensemble | L1 | L2 | L3 | L4 | Average
---|---|---|---|---|---
R-value | AUC | R-value | AUC | R-value | AUC | R-value | AUC | R-value | AUC
Ensm_roi | 0.879 | 0.935 | 0.876 | 0.946 | 0.850 | 0.929 | 0.812 | 0.894 | 0.854 | 0.926
Ensm_all | 0.881 | 0.938 | 0.880 | 0.949 | 0.851 | 0.933 | 0.816 | 0.903 | 0.857 | 0.931

###  3.5 Evaluating Model Ensemble Performance

The individual model performance is influenced by the data distribution, input
modality choice, and the training process. Overfitting is also a challenge due
to the limited amount of labeled data. A common strategy to boost model
performance is ensemble learning. In Table 2, we have 6 models using different
ROIs. The mean of the BMD predictions from the these models is used as
ensemble predictions, listed in Table 3. Ensemble_roi is the ensemble of all
single-ROI models (cervical, clavicle, lumbar, ribcage, chest models), and
Ensemble_all is the ensemble of all models including the multi-ROI model.

Compared to Ensemble_roi, Ensemble_all has the additional multi-ROI model and
it performs stronger in all the metrics. Although Ensemble_roi has information
from local modalities and the whole chest X-ray, the simple average of single-
ROI model predictions lack the adaptive feature fusion process. In the multi-
ROI model, the concatenated feature and the following FC layer exploit all
regional information. The different importance of various regions is learned
in the FC layer weights.

Even though both ensemble schemes work better in terms of averaged R-value,
they do not exceed the multi-ROI model in terms of averaged AUC score. A
higher R-value indicates high consistency of sequential order with ground
truth, while a higher AUC score means better osteoporosis decision around the
T-score threshold. multi-ROI model has a substantially higher AUC score than
any individual ROI model, proving its reliability to judge osteoporosis
status. The Ensemble_all could not improve multi-ROI model upon the AUC score,
meaning that averaged prediction from ROI models are less accurate around the
T-score threshold.

###  3.6 Evaluation of CXR BMD Prediction Task

It is observed that adjacent bones have more consistent BMD scores, while the
correlation decrease as bone distances increase. For example, in the lumbar
BMD ground truth, L1 has a R-value of 0.907 to L2, and has a R-value of 0.883
to L3, but only has a R-value of 0.825 to L4. So the model performance may not
exceed the neighboring lumbar BMD R-values (such as L1 and L3 ).

Another cause of the performance bottleneck is that there exist physiological
differences between lumbar bones and chest bones. The accuracy for cross-
region BMD prediction would have an upper limit due to osteoporosis
distribution and bone differences. This can be observed in Table2, where the
cervical model perform best among local ROI models, and ribcage model performs
worst. The cervical vertebra has a more similar role to the lumbar in spine
activity, while bones in the rib cage ROIs stay static from body activities.
The lumbar model’s low performance is due to X-ray overexposure and noise in
the lumbar region of frontal-view CXR scan.

By analyzing the lumbar BMD ground truth, we find that L1 BMD is generally
smaller than L2’s, L2 BMD is generally smaller than L3’s and L3 BMD is
generally smaller than L4’s. In Figure 2, the L1 BMD distribution has a much
lower range than L4’s. The multi-ROI model get a R-value of 0.875 and an AUC
score of 0.936 on L1 BMD prediction, which is already close to the R-value
between the ground truth BMDs of L1 and L3 (0.883). This further proves the
promising applicability of our proposed algorithm.

##  4 Conclusion

In this paper, we studied the task of BMD prediction from chest X-ray images.
We introduced a multi-ROI model to jointly analyze multiple local and global
ROIs from the CXR to predict the patient’s lumbar spine BMDs. Experimental
results and analysis show that by incorporating multiple ROIs, our model is
able to produce significant improved performance compared to using individual
ROIs or the whole CXR. Our method shows great potential to be applied to
opportunistic screening of osteoporosis.

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