# COLIN implies LIN for emergent algebras

Marius Buliga
mbuliga.github.io

(15.10.2021)

##  1 Emergent algebras

Emergent algebras come from sub-riemannian geometry [8], [11], [12], where
they have the name ”dilation structure” or ”dilatation structure”, introduced
in [10]. At the end of this article there are references for emergent algebras
and dilation structures. Mind that in all references I used consistently wrong
the right quasigroup name, it should be left quasigroup everywhere. This will
be fixed in the present exposition of the subject.

Emergent algebras, first time introduced in [9], are families of quasigroup
operations indexed by a commutative group, which satisfy some algebraic
relations and also topological (convergence and continuity) relations. Besides
sub-riemannian geometry, they appear as a semantics of a family of graph-
rewrite systems related to interaction combinators [5], or lambda calculus
[7]. In [6] there is a lambda calculus version of emergent algebras.

Emergent algebras need three ingredients: a commutative group $\Gamma$ with an
invariant topological filter, an uniform topological space $X$ and a family of
operations on $X$, indexed with the elements of $\Gamma$. The operations
satisfy algebraic axioms and also some continuity and convergence axioms which
are the reason behind the denomination of these algebras as ”emergent”.

Let’s take the ingredients one by one.

### The group of scalars.

This is a commutative group $\Gamma$ with an absolute, i.e. a topological
filter on $\Gamma$ which is invariant with respect to the group
multiplication.

The typical example is the group $\Gamma=(0,+\infty)$ with the multiplication
operation, where $0$, which is not an element of the group, but it is seen as
the topological filter generated by the intervals $(0,a)$, with $a>0$. For any
$x\in\Gamma$ and any interval $(0,a)$ the set of elemements from that interval
multiplicated by $x$ (at left or at right, it does not matter because the
group is commutative) is the interval $(0,ax)$, which implies that $0$, as a
topological filter, is invariant with respect of the group operation. Such a
topological filter is called an absolute. We shall use $0$ to write limits as
$a\in\Gamma$ converges to $0$.

Other examples of commutative groups with absolutes are $(\mathbb{Z},+)$ with
$+\infty$ as absolute, or $\displaystyle\mathbb{C}^{*}$ with complex number
multiplication and $0$ as absolute.

### The uniform space.

An uniform space $X$ is a nonempty set endowed with an uniformity. An
uniformity is a topological filter $\mathcal{U}$ over $X\times X$ which
satisfies the properties:

  1. (a)

the diagonal of $X$ $\displaystyle\left\\{(x,x)\mid x\in X\right\\}$ is a
subset of any $A\in\mathcal{U}$,

  2. (b)

for any $A\in\mathcal{U}$ we have $\displaystyle A^{-1}\in\mathcal{U}$ where

$\displaystyle A^{-1}=\left\\{(y,x)\mid(x,y)\in A\right\\}$

  3. (c)

for any $A,B\in\mathcal{U}$ we have $AB\in\mathcal{U}$, where

$\displaystyle AB=\left\\{(x,w)\mid\exists y\in X\mbox{ s.t. }(x,y)\in A\mbox{
and }(y,w)\in B\right\\}$

The typical example of an uniform space is a metric space $(X,d)$, where the
uniformity filter is generated by sets $A\subset X\times X$ defined for any
$a>0$ as

$A=\left\\{(x,y)\mid d(x,y)<a\right\\}$

Then (a) is implied by $d(x,x)=0$ for any $x\in X$, (b) is implied by
$d(x,y)=d(y,x)$ for any $x,y\in X$ and (c) is implied by $d(x,z)\leq
d(x,y)+d(y,z)$ for any $x,y,z\in X$.

We shall use the absolute $0$ of the group of scalars $\Gamma$ with the
uniformity on $X$ in statements which contain ”$\lim{a\rightarrow
0}f(a,x)=g(x)$ uniformly with respect to x”, for some functions $\displaystyle
f:\Gamma\times X\rightarrow X$.

### Definition 1.

An emergent algebra over a set $X$ is a family of idempotent left quasigroup
operations over X, indexed by the group $\Gamma$, which satisfy the following
algebraic and topological axioms (R1), (R2), (act) and (em).

For $\displaystyle a\in\Gamma$ let’s denote the left quasigroup operations
indexed by $a$ with $\displaystyle\circ_{a}$ and $\displaystyle\bullet_{a}$.
Therefore $\displaystyle(X,\circ_{a},\bullet_{a})$ is an idempotent left
quasigroup:

### (R1)

$\displaystyle x\circ_{a}x=x$

### (R2)

$\displaystyle x\circ_{a}(x\bullet_{a}y)=x\bullet_{a}(x\circ_{a}y)=y$

### (act)

algebraic axioms which relate the operations:

$\displaystyle x\circ_{a}(x\circ_{b}y)=x\circ_{ab}y$

$\displaystyle x\circ_{1}y=y$

$\displaystyle x\circ_{1/a}y=x\bullet_{a}y$

On $X$ we define the following operations which we will use later:

\- the approximate difference

$\displaystyle\Delta^{x}_{a}(y,z)=(x\circ_{a}y)\bullet_{a}(x\circ_{a}z)$

\- the approximate sum

$\displaystyle\Sigma^{x}_{a}(y,z)=x\bullet_{a}((x\circ_{a}y)\circ_{a}z)$

\- the approximate inverse

$\displaystyle inv^{x}_{a}y=(x\circ_{a}y)\bullet_{a}x$

Finally we have the topological axioms.

### (em)

$X$ is an uniform space; as $a$ converges to $0$

$\displaystyle(x,y)\mapsto x\circ_{a}y$ converges uniformly to $(x,y)\mapsto
x$

$\displaystyle(x,y,z)\mapsto\Delta^{x}_{a}(y,z)$ converges uniformly to a
function $(x,y,z)\mapsto\Delta^{x}(y,z)$

$(\displaystyle x,y,z)\mapsto\Sigma^{x}_{a}(y,z)$ converges uniformly to a
function $(x,y,z)\mapsto\Sigma^{x}(y,z)$

### Topological axioms, graph rewrites and ”emergence”.

The reason for topological axioms is to allow passage to the limit in any
order. The axioms (R1), (R2), (act) can be reformulated into graph rewrites on
trivalent graphs with typed nodes and numbered ports, which is the start of
the chemlambda project, as described in [5]. The axiom (em), translated into
graph rewrites, says that some patterns (corresponding to the graphs of the
approximate difference, sum and inverse) can be replaced with new nodes (for
the limit operations) in any order. From here ”emerge” new algebraic
identities, or equivalently new graph rewrites, involving the new operations.
The same effect can be obtained, but without reference to uniform structures
and limits, in the term rewriting formalism ”em” [6], which is an enhancement
of lambda calculus.

##  2 LIN, COLIN and SHUFFLE

Here are some other relevant notations, along with the more usual names from
the field of quasigroups.

### Definition 2.

An emergent algebra is linear if it is left distributive:

###  (R3) or (LIN)

$\displaystyle x\circ_{a}(y\circ_{b}z)=(x\circ_{a}y)\circ_{b}(x\circ_{a}z)$

An emergent algebra is co-linear if it is right distributive:

### (COLIN)

$\displaystyle(x\circ_{a}y)\circ_{b}z=(x\circ_{b}z)\circ_{a}(y\circ_{b}z)$

An emergent algebra has the shuffle trick if it is medial:

### (SHUFFLE)

$\displaystyle(x\circ_{a}y)\circ_{b}(u\circ_{a}v)=(x\circ_{b}u)\circ_{a}(y\circ_{b}v)$

An important example of a linear emergent algebra comes from conical groups.
This is an important class of topological groups which can be seen as a
generalizaton of vector spaces.

### Definition 3.

A conical group $\displaystyle(X,\cdot,e)$ is a topological group endowed with
a multiplication by scalars operation $\displaystyle(a,x)\in\Gamma\times
X\mapsto ax\in X$ such that the following algebraic relations are true:

$\displaystyle a(bx)=(ab)x$

$\displaystyle a(x\cdot y)=ax\cdot ay$

$\displaystyle a(x^{-1})=(ax)^{-1}$

$ae=e$
and there is a neighbourhood $U\subset X$ of the neutral element $e\in X$ and
a neighbourhood $V\subset\Gamma$ of the filter $0$ such that

$\displaystyle(a,x)\in V\times U\mapsto ax\in U$

$\displaystyle x\in U\mapsto ax$ converges uniformly to $x\mapsto e$, as $a$
converges to $0$.

We have the following structure theorem, see theorem 10 [10] or theorem 6.2
[6].

### Theorem 1.

For an emergent algebra $X$, fix an element $e\in X$ and define the
operations:

(addition) $\displaystyle(x,y)\mapsto x\cdot y=\Sigma^{e}(x,y)$

(inverse) $\displaystyle x\mapsto x^{-1}=inv^{e}x=\Delta^{e}(x,e)$

(scalar multiplication) $(a,x)\mapsto ax=e\circ_{a}x$

Then $\displaystyle(X,\cdot)$ is a conical group with neutral element $e$
which is conical.

Conversely, for any conical group $X$ and for any $a\in\Gamma$ define:

$\displaystyle x\circ_{a}y=x\cdot a(x^{-1}\cdot y)$

With this operation $X$ becomes a linear emergent algebra.

### Curvature as deviation from linearity.

For an emergent algebra, the following term represents the deviation from
linearity:

$\displaystyle
LIN_{a,b}(x,y,z)=y\bullet_{b}(x\bullet_{a}((x\circ_{a}y)\circ_{b}(x\circ_{a}z)))$

Indeed, (LIN) is equivalent with $\displaystyle LIN_{a,b}(x,y,z)=z$. But in
the realm of sub-riemanian geometry, more precisely for a dilation structure,
this term is related to curvature, as explained in [8], section 2.5
”Curvdimension and curvature”. For an arbitrary element $x\in X$ and for
$c\in\Gamma$ define

$\displaystyle
R^{a}_{b,c}(x,u,v,w)=x\bullet_{a}LIN_{b,c}((x\circ_{a}u,x\circ_{a}v,x\circ_{a}w)$

We know then, by Theorem 1, that $\displaystyle R^{a}_{b,c}(x,u,v,w)$
converges to $w$, as $a$ converges to $0$. In particular, for a riemannian
manifold $X$, there is an associated emergent algebra given by

$\displaystyle x\circ_{a}exp_{x}(y)=exp_{x}(ay)$

where $exp$ is the geodesic exponential. For this emergent algebra, we
recognize the construction called Schild’s ladder in the term

$\displaystyle
r^{a}_{x}(v,w)=log_{x}\left(R^{a}_{\frac{1}{2},\frac{1}{2}}(x,x,exp_{x}(av),exp_{x}(aw))\right)$

The distance, measured in the tangent space at $x$, between $w$ and
$\displaystyle r^{a}_{x}(v,w)$ is controlled by the absolute value of
$\displaystyle a^{2}\langle R_{x}(v,w)v,w\rangle$, where $\displaystyle R_{x}$
is the Riemann curvature tensor at $x$.

### Commutator as deviation from co-linearity.

We can measure the deviation from co-linearity with the term

$\displaystyle
COLIN_{a,b}(x,y,z)=(x\circ_{a}y)\bullet_{b}((x\circ_{b}z)\circ_{a}(y\circ_{b}z))$

For conical groups, the computation of this term shows that it is related with
the commutator. One can prove that for conical groups (COLIN) is equivalent
with (SHUFFLE).

The following theorem is a sort of Bruck-Murdoch-Toyoda theorem [1], [2], [3].

### Theorem 2.

Let $X$ be a conical group. The following are equivalent:

  1. -

$X$ is commutative

  2. -

The associated emergent algebra is medial, i.e. it satisfies (SHUFFLE)

  3. -

The associated emergent alegbra is right distributive, i.e. satisfies (COLIN).

### Proof.

In a conical group with neutral element $e$ we compute

$COLIN_{a,b^{-1}}(e,e\bullet_{a}y,z)=[y\circ_{b}e,z\circ_{a}e]z$

where $\displaystyle[x,y]=xyx^{-1}y^{-1}$ is the commutator with respect to
the group operation. But in conical groups the function $\displaystyle
x\mapsto x\circ_{a}e$ is surjective for any $a\not=1$. (COLIN) is therefore
equivalent with commutativity of the group operation. We already know that
(SHUFFLE) is equivalent with commutativity.

It is therefore natural to ask:

### Question.

Are there emergent algebras which satisfy (COLIN) but not (LIN)?

##  3 An useful tool: geometric series

The geometric series is

$\sum_{n=0}^{\infty}\varepsilon^{n}=\frac{1}{1-\varepsilon}$

for any $\varepsilon\in(0,1)$.

With the notation for dilations:

$\delta^{x}_{\varepsilon}y=x+\varepsilon(-x+y)$

we can rephrase the geometric series as an existence result. Namely that the
equation:

$\delta^{S}_{\varepsilon}0=x$

has the solution

$S=\sum_{n=0}^{\infty}\left(\delta^{0}_{\varepsilon}\right)^{n}x$

for any $\varepsilon\in(0,1)$.

The non-commutative version of this result is given in [11] proposition 8.4.
In that article is proposed a non-commutative affine geometry, where usual
affine spaces (over a vector space) are replaced with their non-commutative
versions over a conical group. The work uses dilation structures, which are
metrical versions of emergent algebras. In the mentioned article the proof
uses the existence of a metric on the (non-commutative affine) space. We can
do a simple proof without it.

Of course that the condition $\varepsilon\in(0,1)$ will be reformulated as
$\varepsilon^{n}\rightarrow 0$ as $n\rightarrow\infty$.

Pick a base point $e$ , which will play the role of the $0$. Then we want to
solve the equation in $S$

$\delta^{S}_{\varepsilon}e=x$

for a given, arbitrary $x$ and for an $\varepsilon$ with the property that
$\varepsilon^{n}\rightarrow 0$ as $n\rightarrow\infty$.

We want to prove that

(*)

$S=\sum_{n=0}^{\infty}\left(\delta^{e}_{\varepsilon}\right)^{n}x$

where the sum is with respect to the non-commutative addition operation based
at $e$. More precisely this operation is ”emergent”:

$\Sigma^{e}(v,w)=\lim_{\varepsilon\rightarrow 0}\Sigma^{e}_{\varepsilon}(v,w)$

where the approximate sum is

$\Sigma^{e}_{\varepsilon}(v,w)=\delta^{e}_{\varepsilon^{-1}}\delta^{\delta^{e}_{\varepsilon}v}_{\varepsilon}w$

Recall that the dilations satisfy the (LIN) property:

(LIN)

$\delta^{e}_{\varepsilon}\delta^{x}_{\mu}y=\delta^{\delta^{e}_{\varepsilon}x}_{\mu}\delta^{e}_{\varepsilon}y$

The conclusion (*) can be reformulated as: define $S_{0}=x$ and

$S_{n+1}=\Sigma^{e}(x,\delta^{e}_{\varepsilon}S_{n})$

Then $S=\lim_{n\rightarrow\infty}S_{n}$.

But this is simple, due to the following identities coming from (LIN).

The first identity uses the fact that, once we defined the addition from
dilations and a passage to the limit, then we can prove that dilations
themselves express via addition. This gives the first identity:

$\Sigma^{e}_{\varepsilon}(v,w)=\Sigma^{e}(\delta^{v}_{\varepsilon}e,w)$

The second identity is easier, just use (LIN), there is no passage to the
limit involved.

$\Sigma^{e}_{\varepsilon}(v,\delta^{e}_{\varepsilon}w)=\delta^{v}_{\varepsilon}w$

With these identities, the recurrence relation of the non-commutative
geometric series becomes:

$S_{n+1}=\Sigma^{e}(x,\delta^{e}_{\varepsilon}S_{n})=\Sigma^{e}_{\varepsilon}(S,\delta^{e}_{\varepsilon}S_{n})=\delta^{S}_{\varepsilon}S_{n}$

therefore

$S_{n}=\left(\delta^{S}_{\varepsilon}\right)^{n}S_{0}$

and the proof ends by recalling that $\varepsilon^{n}\rightarrow 0$ as
$n\rightarrow\infty$, which implies

$\lim_{n\rightarrow\infty}S_{n}=\lim_{n\rightarrow\infty}\delta^{S}_{\varepsilon^{n}}S_{0}=\lim_{\varepsilon\rightarrow
0}\delta^{S}_{\varepsilon}S_{0}=S$

The proof is new even in the commutative case.

Here is a non-commutative example. We are in the group $N$ of real $n\times n$
upper triangular matrices, with 1’s on the diagonal. This is a subgroup of
real linear group.

For any scalar $e\in(0,+\infty)$ we define a diagonal matrix

$\mathbf{e}_{ij}=e^{i}\delta_{ij}$

(here $\delta_{ij}=1$ if $i=j$, otherwise $\delta_{ij}=0$; please don’t make a
confusion with the dilation, denoted also with the letter $\delta$, which is
introduced in the following).

Conjugation with diagonal matrices is an automorphism of $N$, in particular
for any $e\in(0,+\infty)$ and any $\mathbf{x}\in N$ we have

$\mathbf{e}^{-1}\mathbf{x}\mathbf{e}\in N$

This allows us to define the dilation:

$\delta^{\mathbf{x}}_{e}\mathbf{y}=\mathbf{x}\mathbf{e}^{-1}\mathbf{x}^{-1}\mathbf{y}\mathbf{e}$

We leave to the reader to prove the following:

  1. (a)

this defines an emergent algebra over $N$

  2. (b)

that this is a linear emergent algebra.

We can define now the geometric series with respect to the ”addition”
operation which is simply the matrix multiplication in the group $N$: (here
$I$ is the identity matrix, the neutral element of the group)

$\sum_{k=0}^{\infty}\delta_{e^{k}}^{I}\mathbf{x}$

and by the previous post, it does converge if $e\in(0,1)$.

Let’s compute the finite sums. First notice that

$\delta^{\mathbf{x}}_{e}\mathbf{y}=[\mathbf{x},\mathbf{e}^{-1}]\mathbf{e}^{-1}\mathbf{y}\mathbf{e}$

where the square bracket denotes the commutator.

We then have:

$\sum_{k=0}^{m}\delta_{e^{k}}^{I}\mathbf{x}=\mathbf{x}\mathbf{e}^{-1}\mathbf{x}\mathbf{e}\mathbf{e}^{-2}\mathbf{x}\mathbf{e}^{2}...\mathbf{e}^{-m}\mathbf{x}\mathbf{e}^{m}$

or equivalently:

$\sum_{k=0}^{m}\delta_{e^{k}}^{I}\mathbf{x}=(\mathbf{x}\mathbf{e}^{-1})^{m}\mathbf{x}\mathbf{e}^{m}$

All in all, our convergence of the geometric series result says that:

if $e\in(0,1)$ and

$[\mathbf{y},\mathbf{e}^{-1}]=x$

then

$\mathbf{y}=\lim_{m\rightarrow\infty}(\mathbf{x}\mathbf{e}^{-1})^{m}\mathbf{x}\mathbf{e}^{m}$

### Remark

that for $n=2$, i.e. for the case of $2\times 2$ upper triangular matrices
with 1’s on the diagonal, the convergence result is the classical geometric
series convergence. Compute by hand the partial sums in tis case to convince
yourself that indeed the usual geometric (partial) sum appears in the $(1,2)$
position in the matrices.

##  4 (COLIN) implies (LIN)

In this section we give a negative answer to the question formulated in the
first section. At first sight this is surprising, because (COLIN) and (LIN)
are symmetric one to the other. Therefore it would be expected that, as we
have examples of emergent algebras which satisfy (LIN) but not (COLIN), which
are just the noncommutative conical groups, there should be examples of
emergent algebras which satisfy (COLIN) but not (LIN).

At closer examination we notice that the axiom (R2), satisfied in any emergent
algebra, breaks the symmetry between (LIN) and (COLIN). However, in itself
this remark is not sufficient to answer the question if there are emergent
algebras which satisfy (COLIN) but not (LIN).

### Theorem 3.

For an emergent algebra with the group $\Gamma=(0,\infty)$ the condition
(COLIN) implies the condition (LIN)

### Proof.

### Part 1.

Recall the (COLIN) condition: for any $a,b\in\Gamma$ and for any $x,y,z\in X$
we have

$(x\circ_{a}y)\circ_{b}z=(x\circ_{b}z)\circ_{a}(y\circ_{b}z)$

Fix an element $e\in X$, otherwise arbitrary. (COLIN) is then equivalent with

$y\circ_{b}z=(e\circ_{b}z)\bullet_{a}((e\circ_{b}y)\circ_{a}z)$

If we replace $z$ with $e\circ_{a}z$ and then we use (R2) and some groupings
of terms then we obtain the following relation equivalent with (COLIN):

$y\circ_{b}(e\circ_{a}z)=\Delta_{a}^{e}(e\circ_{b}z,E)$

where $E$ is a relative operation, namely

$E=e\bullet_{a}((e\circ_{a}y)\circ_{b}(e\circ_{a}z))$

We pass now with $a$ to $0$ by using the topological axiom (em) and we obtain
in the limit the following

$y\circ_{b}e=\Delta^{e}(e\circ_{b}z,E_{0})$

where $E_{0}$ is the limit of $E$, therefore the infinitesimal dilation of
coefficient $b$, based at $e$. We write it like this

$E_{0}=y\circ_{b}^{e}z$

### Part 2.

The relation (COLIN) passes to the infinitesimal level. Indeed, if we replace
the operations with the infinitesimal operations (dilations) based at $e$ then
(COLIN) remains true. This is true because for an arbitrary $c\in\Gamma$ we
deduce from (COLIN) the relation

$(x\circ_{a,c}^{e}y)\circ_{b,c}^{e}z=(x\circ_{b,c}^{e}z)\circ_{a,c}^{e}(y\circ_{b,c}^{e}z)$

where

$x\circ_{a,c}^{e}y=e\bullet_{c}((e\circ_{c}x)\circ_{a}(e\circ_{c}y))$

We can then pass to the limit with $c$ to $0$ and we get the ”infinitesimal”
(COLIN) relation

$(x\circ_{a}^{e}y)\circ_{b}^{e}z=(x\circ_{b}^{e}z)\circ_{a}^{e}(y\circ_{b}^{e}z)$

But the infinitesimal emergent algebra based at $e$ satisfies (LIN) [11]
proposition 7.11. Therefore now we know that it also satisfy (COLIN). As a
consequence we get that it comes from a commutative conical group. We denote
with a dot this conical group operation.

Because the group is conical and commutative it follows that

$(e\circ_{a}y)\cdot(e\circ_{b}y)=e\circ_{a+b}y$

via techniques explained in [6]. Therefore the conical group is a vector
space, with group operation being vector addition and $e\circ_{a}x$ equal to
the scalar $a$ which multiplies $x$ (up to an arbitrary exponential).

### Part 3.

The last two relations from Part 1 are then rewritten as

$y\circ_{b}e=(e\circ_{b}z^{-1})\cdot y\cdot(e\circ_{b}(y^{-1}\cdot z))$

Commutativity of the group operation $\cdot$ and (LIN) for the infinitesimal
level gives us the equivalent

$y\circ_{b}e=y\circ_{b}^{e}e=e\circ_{1-b}y$

(where in the last equality we used Part 2 and we make a slight abuse of
notation for the multiplication by the scalar $1-b$)

Now we come back to the initial (COLIN) and we remark that with the new
knowledge we can rewrite it as

$e\circ_{1-a}(x\circ_{b}y)=(e\circ_{1-a}x)\circ_{b}(e\circ_{1-a}y)$

which is equivalent with

$x\circ_{b}y=e\bullet_{1-a}((e\circ_{1-a}x)\circ_{b}(e\circ_{1-a}y))$

We pass to the limit with $a$ to $1$ this time and we obtain that

$x\circ_{b}y=x\circ_{b}^{e}y$

therefore the emergent algebra is identical with the infinitesimal emergent
algebra based at $e$. Therefore it satisfies (LIN).

We know more, actually, namely that (COLIN) is equivalent with (SHUFFLE).
Indeed, we proved that (COLIN) implies that the emergent algebra is the one of
a conical commutative group, or we already know that (SHUFFLE) is true if and
only if we are in a conical commutative group.

### Acknowledgements.

Thanks to D. Stanovsky, who points to his survey article [4], section 3, where
some examples of non-medial but right-distributive quasigroups are given.

## References

  * [1] K. Toyoda, On axioms of linear functions, Proc. Imp. Acad. Tokyo 17 (7), (1941), 221-226
  * [2] D.C. Murdoch, Structure of abelian quasigroups, Trans. Amer. Math. Soc. 49 (3), (1941), 392-409
  * [3] R.H. Bruck, Some results in the theory of quasigroups, Trans. Amer. Math. Soc. 55 (1), (1944), 19-52
  * [4] D. Stanovsky, A guide to self-distributive quasigroups, or latin quandles, Quasigroups and Related Systems 23 (1), (2015), 91-128, arXiv:1505.06609
References for emergent algebras:

  * [5] M. Buliga, Graph rewrites, from graphic lambda calculus, to chemlambda, to directed interaction combinators, arXiv:2007.10288.1520
  * [6] M. Buliga, The em-convex rewrite system, arXiv:1807.02058
  * [7] M. Buliga, Graphic lambda calculus. Complex Systems 22, 4 (2013), 311-360.
arXiv:1305.5786

  * [8] M. Buliga, Sub-riemannian geometry from intrinsic viewpoint, Course notes, Ecole de recherche CIMPA : Geometrie sous-riemannienne, Jan (2012), Beyrouth, Lebanon, arXiv:1206.3093
  * [9] M. Buliga, Emergent algebras, arXiv:0907.1520
  * [10] M. Buliga, Dilatation structures I. Fundamentals, J. Gen. Lie Theory Appl., 1 (2007), 2, 65-95.
arXiv:math/0608536

  * [11] M, Buliga, Infinitesimal affine geometry of metric spaces endowed with a dilatation structure, Houston Journal of Mathematics, 36, 1 (2010), 91-136.
arXiv:math/0608536

  * [12] M. Buliga, Braided spaces with dilations and sub-riemannian symmetric spaces, in: Geometry. Exploratory Workshop on Differential Geometry and its Applications, eds. D. Andrica, S. Moroianu, Cluj-Napoca 2011, 21-35.
arXiv:0804.0135


# Sato-Tate Distribution of $p$-adic hypergeometric functions

Sudhir Pujahari  School of Mathematical Sciences, National Institute of
Science Education and Research, Bhubaneswar, An OCC of Homi Bhabha National
Institute, P. O. Jatni, Khurda 752050, Odisha, India<EMAIL_ADDRESS>and  Neelam Saikia  Faculty of Mathematics, University of Vienna, Austria
1010<EMAIL_ADDRESS>
###### Abstract.

Recently Ono, Saad and the second author [21] initiated a study of value
distribution of certain families of Gaussian hypergeometric functions over
large finite fields. They investigated two families of Gaussian hypergeometric
functions and showed that they satisfy semicircular and Batman distributions.
Motivated by their results we aim to study distributions of certain families
of hypergeometric functions in the $p$-adic setting over large finite fields.
In particular, we consider two and six parameters families of hypergeometric
functions in the $p$-adic setting and obtain that their limiting distributions
are semicircular over large finite fields. In the process of doing this we
also express the traces of $p$th Hecke operators acting on the spaces of cusp
forms of even weight $k\geq 4$ and levels 4 and 8 in terms of $p$-adic
hypergeometric function which is of independent interest. These results can be
viewed as $p$-adic analogous of some trace formulas of [1, 2, 6].

###### Key words and phrases:

$p$-adic hypergeometric functions; Sato-Tate distributions; Trace of Frobenius

###### 2000 Mathematics Subject Classification:

11G20, 11T24, 33E50

##  1\. Introduction

Let $p$ be an odd prime and let $\mathbb{F}_{q}$ denote the finite field with
$q=p^{r}$ elements. In [11], Greene introduced Gaussian hypergeometric
functions over finite fields using Jacobi sums. One of the important fact
about these functions is that they satisfy many analogous hypergeometric type
identities. Recently Ono, Saad and the second author [21] initiated a study of
value distributions of certain families of these functions over random large
finite fields $\mathbb{F}_{q}.$ More precisely, they investigated the
distributions of the normalized values of the following two Gaussian
hypergeometric functions over $\mathbb{F}_{q}$ that are defined by

${}_{2}F_{1}(\lambda)_{q}:={{}_{2}F_{1}}\left(\begin{matrix}\phi,&\phi\\\
~{}&\varepsilon\end{matrix}\mid\lambda\right)_{q}:=\frac{q}{q-1}\sum\limits_{\chi}{\phi\chi\choose\chi}{\phi\chi\choose\chi}\chi(\lambda)$

and

${}_{3}F_{2}(\lambda)_{q}:={{}_{2}F_{1}}\left(\begin{matrix}\phi,&\phi,&\phi\\\
~{}&\varepsilon,&\varepsilon\end{matrix}\mid\lambda\right)_{q}:=\frac{q}{q-1}\sum\limits_{\chi}{\phi\chi\choose\chi}{\phi\chi\choose\chi}{\phi\chi\choose\chi}\chi(\lambda),$

where $\phi$ and $\varepsilon$ are quadratic and trivial characters of
$\mathbb{F}_{q}^{\times}$ and the sums on the extreme rights run over all
multiplicative characters 111$\chi(0):=0$ for all multiplicative characters of
$\mathbb{F}_{q}.$ of $\mathbb{F}_{q}.$ Moreover, the symbol ${A\choose
B}:=\frac{B(-1)}{q}\sum\limits_{x\in\mathbb{F}_{q}}A(x)\overline{B}(1-x)$ is
the normalized Jacobi sum. Ono et al. [21] showed that for the
${}_{2}F_{1}(\lambda)_{q}$ function the limiting distribution is semicircular
whereas for the ${}_{3}F_{2}(\lambda)_{q}$ function the distribution is
Batman. In this paper our goal is to study similar questions for $p$-adic
hypergeometric functions over random large finite fields.

McCarthy [18, 17] defined $p$-adic hypergeometric functions using $p$-adic
gamma functions extending Greene’s hypergeometric functions [11] for wider
classes of primes. These functions appear in the study of Frobenius trace of
elliptic curves [17], Fourier coefficients of Hecke eigen forms [22], and
proofs of supercongruence type identities [10].

Let $\Gamma_{p}(\cdot)$ be the Morita’s $p$-adic gamma function. Let $\omega$
be the Teichmüller character 222$\omega(t)\equiv t\pmod{p}$ for all
$t\in\mathbb{F}_{p}$ and $\omega(0):=0.$ of $\mathbb{F}_{p}$ and
$\overline{\omega}$ denote its character inverse. For $x\in\mathbb{Q}$ let
$\lfloor x\rfloor$ denote the greatest integer less than or equal to $x$ and
$\langle x\rangle$ denote the fractional part of $x$, satisfying $0\leq\langle
x\rangle<1$. Using these notation $p$-adic hypergeometric function is defined
as follows.

######  Definition 1.1.

[17, Definition 5.1] Let $p$ be an odd prime and $t\in\mathbb{F}_{p}$. For
positive integer $n$ and $1\leq k\leq n$, let $a_{k}$, $b_{k}$
$\in\mathbb{Q}\cap\mathbb{Z}_{p}$. Then

(1.3) ${}_{n}{G}_{n}\left[\begin{array}[]{cccc}a_{1},&a_{2},&\ldots,&a_{n}\\\
b_{1},&b_{2},&\ldots,&b_{n}\end{array}\mid
t\right]_{p}:=\frac{-1}{p-1}\sum_{j=0}^{p-2}(-1)^{jn}~{}~{}\overline{\omega}^{j}(t)$
$\displaystyle\times\prod\limits_{k=1}^{n}(-p)^{-\lfloor\langle
a_{k}\rangle-\frac{j}{p-1}\rfloor-\lfloor\langle-
b_{k}\rangle+\frac{j}{p-1}\rfloor}\frac{\Gamma_{p}(\langle
a_{k}-\frac{j}{p-1}\rangle)}{\Gamma_{p}(\langle
a_{k}\rangle)}\frac{\Gamma_{p}(\langle-
b_{k}+\frac{j}{p-1}\rangle)}{\Gamma_{p}(\langle-b_{k}\rangle)}.$

In this paper we aim to investigate the value distributions of certain
families of these functions over large finite fields $\mathbb{F}_{p}$. Namely,
we study the value distributions of the following families of
${}_{n}G_{n}$-functions for $n=2,6.$ If $\lambda\in\mathbb{F}_{p}^{\times}$
and $\psi_{6}=\omega^{\frac{p-1}{6}}$ is a character of order 6, then for
$\lambda\neq-1$ we define

${}_{2}G_{2}(\lambda)_{p}:=p\psi_{6}(2)\phi(1+\lambda)\cdot{{}_{2}G_{2}}\left[\begin{array}[]{cc}\frac{2}{3},&\frac{2}{3}\vspace{.1
cm}\\\
\frac{5}{12},&\frac{11}{12}\end{array}\mid\frac{4\lambda}{(1+\lambda)^{2}}\right]_{p}.$

and

${}_{6}G_{6}(\lambda)_{p}:=\phi(1+\lambda)\cdot{{}_{6}G_{6}}\left[\begin{array}[]{cccccc}\frac{1}{3},&\frac{1}{3},&\frac{2}{3},&\frac{2}{3},&0,&0\vspace{.1
cm}\\\
\frac{1}{12},&\frac{1}{4},&\frac{5}{12},&\frac{7}{12},&\frac{3}{4},&\frac{11}{12}\end{array}\mid\frac{2^{6}\lambda^{3}}{(1+\lambda)^{6}}\right]_{p}.$

As our first theorem we obtain the moments of values of
${{}_{2}G_{2}}(\lambda)_{p}.$

######  Theorem 1.2.

Let $m$ be a fixed positive integer and $p\equiv 1\pmod{3}$ be a prime. Then
as $p\rightarrow\infty$

$\sum\limits_{\lambda\in\mathbb{F}_{p}}{{}_{2}G_{2}}(\lambda)_{p}^{m}=\begin{cases}o_{m}(p^{\frac{m}{2}+1})&\text{if}\
\ m\ is\ \text{odd}\vspace{.2 cm}\\\
\frac{(2n)!}{n!(n+1)!}p^{n+1}+o_{m}(p^{n+1})&\text{if}\ \ m=2n\ is\
\text{even}.\end{cases}$

We use these moments to conclude the limiting behaviour of
${{}_{2}G_{2}(\lambda)_{p}}$ as $p\rightarrow\infty.$ If we view the
normalized values $p^{-1/2}\cdot{{}_{2}G_{2}(\lambda)_{p}}\in[-2,2]$ as random
variables over $\mathbb{F}_{p}$ then we obtain the limiting distribution of
this function. More precisely we have the following distribution.

######  Corollary 1.3.

If $-2\leq a<b\leq 2$ then

$\lim\limits_{p\rightarrow\infty}\frac{|\\{\lambda:\ \
p^{-1/2}{{}_{2}G_{2}}(\lambda)_{p}\in[a,b]\\}|}{p}=\frac{1}{2\pi}\int_{a}^{b}\sqrt{4-t^{2}}~{}dt.$

We also consider these problems for the ${{}_{6}G_{6}(\lambda)_{p}}$
functions.

######  Theorem 1.4.

Let $m$ be a fixed positive integer and $p\equiv 2\pmod{3}$ be a prime. Then
as $p\rightarrow\infty$

$\sum\limits_{\lambda\in\mathbb{F}_{p}}{{}_{6}G_{6}}(\lambda)_{p}^{m}=\begin{cases}o_{m}(p^{\frac{m}{2}+1})&\text{if}\
\ m\ is\ \text{odd}\vspace{.2 cm}\\\
\frac{(2n)!}{n!(n+1)!}p^{n+1}+o_{m}(p^{n+1})&\text{if}\ \ m=2n\ is\
\text{even}.\end{cases}$

Similarly as in Corollary 1.3, we conclude the limiting distribution of
$p^{-1/2}\cdot{{}_{6}G_{6}(\lambda)_{p}}\in[-2,2]$ in the following Corollary.

######  Corollary 1.5.

If $-2\leq a<b\leq 2$ then

$\lim\limits_{p\rightarrow\infty}\frac{|\\{\lambda:\ \
p^{-1/2}\cdot{{}_{6}G_{6}}(\lambda)_{p}\in[a,b]\\}|}{p}=\frac{1}{2\pi}\int_{a}^{b}\sqrt{4-t^{2}}~{}dt.$

######  Remark.

It is important to note that these results can be extended to
${}_{2}G_{2}(\lambda)_{q}$ and ${}_{6}G_{6}(\lambda)_{q}$ over finite fields
$\mathbb{F}_{q}$ where $q=p^{r}$ using similar arguments. For simplicity we
choose $q=p.$

##  2\. Traces of Hecke operators and hypergeometric functions

It turns out that the hypergeometric functions ${}_{2}G_{2}(\lambda)_{p}$ and
${}_{6}G_{6}(\lambda)_{p}$ can be related to the traces of Hecke operators
acting on the spaces of cusp forms. More precisely, for positive integers $N$
and $k$, let $S(N,k)$ be the space of cusp forms of weight $k$ with respect to
the congruence subgroup $\Gamma_{0}(N)$ and let
${\mathrm{Tr}_{k}}(\Gamma_{0}(N),p)$ denote the trace of the $p$th Hecke
operator acting on the space $S(N,k)$. In a series of papers Ahlgren [1],
Ahlgren-Ono [2] and Frechette-Ono-Papanikolas [6] studied the Eichler-Selberg
type trace formulas for the $p$th Hecke operators acting on the spaces
$S(N,k)$ for $N=2,4,8$ and established trace formulas using Gaussian
hypergeometric functions over finite fields. Here we obtain the following
results expressing the trace formulas ${\mathrm{Tr}_{k}}(\Gamma_{0}(4),p)$ and
${\mathrm{Tr}_{k}}(\Gamma_{0}(8),p)$ using $p$-adic hypergeometric functions
${}_{2}G_{2}(\lambda)_{p}$ and ${}_{6}G_{6}(\lambda)_{p}$. Note that both
these functions vanish at $\lambda=-1,$ therefore, we define refinements of
these functions. To this end we define the following two functions.

${{}_{2}\widetilde{G}_{2}}(\lambda)_{p}:={{}_{2}G_{2}}(\lambda)_{p}-\delta(1+\lambda)\cdot\Delta(p)\cdot
A_{4}(-1)\mathrm{Re}\left(\frac{g(\phi
A_{4})g(\overline{A}_{4})}{g(\phi)}\right)$

and

${{}_{6}\widetilde{G}_{6}}(\lambda)_{p}:={{}_{6}G_{6}}(\lambda)_{p}-\delta(1+\lambda)\cdot\Delta(p)\cdot
A_{4}(-1)\mathrm{Re}\left(\frac{g(\phi
A_{4})g(\overline{A}_{4})}{g(\phi)}\right),$

where $A_{4}$ is a character of order 4,

$\delta(x):=\begin{cases}1,&\text{if}\ x=0\\\ 0,&\text{if}\ x\neq
0,\end{cases}$ and $\Delta(p):=\begin{cases}1,&\text{if}\ p\equiv 1\pmod{4}\\\
0,&\text{if}\ p\equiv 3\pmod{4}.\end{cases}$

######  Theorem 2.1.

If $p\equiv 1\pmod{3}$ is an odd prime and $k\geq 4$ is even then

${\mathrm{Tr}_{k}}(\Gamma_{0}(4),p)=-3-\sum_{\lambda=2}^{p-1}P_{k}(_{2}\widetilde{G}_{2}(\lambda)_{p},p)$

and

${\mathrm{Tr}_{k}}(\Gamma_{0}(8),p)=-4-\sum_{\lambda=2}^{p-1}P_{k}(_{2}\widetilde{G}_{2}(\lambda^{2})_{p},p).$

######  Theorem 2.2.

If $p\equiv 2\pmod{3}$ is an odd prime and $k\geq 4$ is even then

${\mathrm{Tr}_{k}}(\Gamma_{0}(4),p)=-3-\sum_{\lambda=2}^{p-1}P_{k}(_{6}\widetilde{G}_{6}(\lambda)_{p},p)$

and

${\mathrm{Tr}_{k}}(\Gamma_{0}(8),p)=-4-\sum_{x=2}^{p-1}P_{k}(_{6}\widetilde{G}_{6}(\lambda^{2})_{p},p).$

##  3\. Gauss sums and $p$-adic gamma function

In this section we discuss some important theorems namely Gross-Koblitz
formula and Davenport-Hasse relation. We also recall some basic results in
Gauss sums and $p$-adic gamma functions. We begin by recalling the
orthogonality relations satisfied by multiplicative characters. Let
$\widehat{\mathbb{F}_{p}^{\times}}$ denote the cyclic group of multiplicative
characters of $\mathbb{F}_{p}^{\times}.$

######  Lemma 3.1.

_([13, Chapter 8])._ If $\chi$ is a multiplicative character 333For all
multiplicative characters $\chi$ including the trivial character $\chi(0):=0.$
of $\mathbb{F}_{p}^{\times},$ then

(1)
$\sum\limits_{\chi\in\widehat{\mathbb{F}_{p}^{\times}}}\chi(x)~{}~{}=\left\\{\begin{array}[]{ll}p-1&\hbox{if~{}~{}
$x=1$;}\\\ 0&\hbox{if ~{}~{}$x\neq 1$.}\end{array}\right.$

(2)
$\sum\limits_{x\in\mathbb{F}_{p}}\chi(x)~{}~{}=\left\\{\begin{array}[]{ll}p-1&\hbox{if~{}~{}
$\chi=\varepsilon$;}\\\ 0&\hbox{if
~{}~{}$\chi\neq\varepsilon$.}\end{array}\right.$

Let $\zeta_{p}$ be a primitive $p$th roots of unity $\chi$ be a multiplicative
character ${\mathbb{F}_{p}^{\times}}.$ Then the Gauss sum is defined by

$g(\chi):=\sum\limits_{x\in\mathbb{F}_{p}}\chi(x)\zeta_{p}^{x}.$

######  Theorem 3.2.

_([3, Davenport-Hasse Relation])._ Let $n$ be a positive integer and let $p$
be a prime such that $p\equiv 1\pmod{n}$. For multiplicative characters
$\chi,\psi\in\widehat{\mathbb{F}_{p}^{\times}}$, we have

$\displaystyle\prod\limits_{\chi^{n}=\varepsilon}g(\chi\psi)=-g(\psi^{n})\psi(n^{-n})\prod\limits_{\chi^{n}=\varepsilon}g(\chi).$

Let $\mathbb{Z}_{p}$ and $\mathbb{Q}_{p}$ denote the ring of $p$-adic integers
and the field of $p$-adic numbers, respectively. Let
$\overline{\mathbb{Q}_{p}}$ be the algebraic closure of $\mathbb{Q}_{p}$ and
$\mathbb{C}_{p}$ be the completion of $\overline{\mathbb{Q}_{p}}$. For a
positive integer $n$ the $p$-adic gamma function $\Gamma_{p}(n)$ is defined as

$\Gamma_{p}(n):=(-1)^{n}\prod\limits_{0<j<n,p\nmid j}j.$

The domain of definition can be extended to all $x\in\mathbb{Z}_{p}$ by simply
setting $\Gamma_{p}(0):=1$ and for $x\neq 0$

$\Gamma_{p}(x):=\lim_{x_{n}\rightarrow x}\Gamma_{p}(x_{n}),$

where $x_{n}$ is any sequence of positive integers $p$-adically approaching
$x.$ The product formula given in (3.1) can be understood as a $p$-adic
analogue of Davenport-Hasse relation. If $n\in\mathbb{Z}^{+}$, $p\nmid n$ and
$x=\frac{r}{p-1}$ with $0\leq r\leq p-1$, then

(3.1)
$\displaystyle\prod_{h=0}^{n-1}\Gamma_{p}\left(\frac{x+h}{n}\right)=\omega(n^{(1-x)(1-p)})\Gamma_{p}(x)\prod_{h=1}^{n-1}\Gamma_{p}\left(\frac{h}{n}\right).$

To this end we recall Gross-Koblitz formula which relates Gauss sums to
$p$-adic gamma functions. Let $\pi\in\mathbb{C}_{p}$ be the fixed root of the
polynomial $x^{p-1}+p$, which satisfies the congruence condition
$\pi\equiv\zeta_{p}-1\pmod{(\zeta_{p}-1)^{2}}$.

######  Theorem 3.3.

_[12, Gross-Koblitz]._ For $a\in\mathbb{Z}$,

$\displaystyle
g(\overline{\omega}^{a})=-\pi^{(p-1)\langle\frac{a}{p-1}\rangle}\Gamma_{p}\left(\left\langle\frac{a}{p-1}\right\rangle\right).$

######  Lemma 3.4.

_[17, Lemma 4.1]._ Let $p$ be a prime and $0\leq j\leq p-2$. For $t\geq 1$
with $p\nmid t$, we have

$\displaystyle\omega(t^{tj})\Gamma_{p}\left(\left\langle\frac{tj}{p-1}\right\rangle\right)\prod\limits_{h=1}^{t-1}\Gamma_{p}\left(\left\langle\frac{h}{t}\right\rangle\right)=\prod\limits_{h=0}^{t-1}\Gamma_{p}\left(\left\langle\frac{h}{t}+\frac{j}{p-1}\right\rangle\right),$

and

$\displaystyle\omega(t^{-tj})\Gamma_{p}\left(\left\langle\frac{-tj}{p-1}\right\rangle\right)\prod\limits_{h=1}^{t-1}\Gamma_{p}\left(\left\langle\frac{h}{t}\right\rangle\right)=\prod\limits_{h=0}^{t-1}\Gamma_{p}\left(\left\langle\frac{h}{t}-\frac{j}{p-1}\right\rangle\right).$

##  4\. proofs of theorems

For $\lambda\neq 0,1$ let $E_{\lambda}^{\mathrm{Leg}}:\ \
y^{2}=x(x-1)(x-\lambda)$ be the Legendre normal form of elliptic curve over
$\mathbb{F}_{p}$ and

$a_{p}(\lambda):=p+1-|E_{\lambda}^{\mathrm{Leg}}(\mathbb{F}_{p})|=-\sum\limits_{x\in\mathbb{F}_{p}}{\phi(x(x-1)(x-\lambda))}$

be the trace of Frobenius of the elliptic curve $E_{\lambda}^{\mathrm{Leg}}.$
Our main idea is to express the functions ${}_{2}G_{2}(\lambda)_{p}$ and
${}_{6}G_{6}(\lambda)_{p}$ in terms of the traces of Frobenius
$a_{p}(\lambda).$ Therefore, to obtain the asymptotic formulas of power
moments of these functions it is sufficient if we have asymptotic formulas for
the power moments of $a_{p}(\lambda).$ In [21] Ono et al. computed the
asymptotic formulas for power moments of the functions
${}_{2}F_{1}(\lambda)_{p}.$ Here we reformulate their formulas in terms of
power moments of $a_{p}(\lambda)$ in the following theorem.

######  Theorem 4.1.

Let $m$ be a fixed positive integer and $p>3$ be a prime. Then as
$p\rightarrow\infty$

(4.1) $\sum\limits_{\lambda\neq
0,1}a_{p}(\lambda)^{m}=\begin{cases}o_{m}(p^{\frac{m}{2}+1})&\text{if}\ \ m\
is\ \text{odd}\vspace{.2 cm}\\\
\frac{(2n)!}{n!(n+1)!}p^{n+1}+o_{m}(p^{n+1})&\text{if}\ \ m=2n\ is\
\text{even}.\end{cases}$

###### Proof.

The proof follows easily by making use of Theorem 1 of [20], Theorem 1.1 of
[21] and the value ${}_{2}F_{1}(1)_{p}=-\phi(-1)/p.$ ∎

Theorem 4.1 is a refinement of a classical theorem of Birch [4, Theorem 1]
(when restricted to Legendre normal elliptic curves) that established even
moments of all elliptic curves over finite fields. Recently Bringmann, Kane,
and the first author [14, 5] have refined Birch’s result to arithmetic
progressions.

###### Proof of Theorem 1.2.

By the definition we can write

$\displaystyle{{}_{2}G_{2}}(\lambda)_{p}$
$\displaystyle=\frac{p\cdot\psi_{6}(2)\phi(1+\lambda)}{(1-p)\Gamma_{p}(\frac{1}{12})\Gamma_{p}(\frac{7}{12})\Gamma_{p}(\frac{2}{3})^{2}}\sum_{j=0}^{p-2}\overline{\omega}^{j}\left(\frac{4\lambda}{(1+\lambda)^{2}}\right)$
$\displaystyle\times(-p)^{-\left\lfloor\frac{1}{12}+\frac{j}{p-1}\right\rfloor-\left\lfloor\frac{7}{12}+\frac{j}{p-1}\right\rfloor-2\left\lfloor\frac{2}{3}-\frac{j}{p-1}\right\rfloor}~{}\Gamma_{p}\left(\left\langle\frac{1}{12}+\frac{j}{p-1}\right\rangle\right)\Gamma_{p}\left(\left\langle\frac{7}{12}+\frac{j}{p-1}\right\rangle\right)$
$\displaystyle\times\Gamma_{p}\left(\left\langle\frac{2}{3}-\frac{j}{p-1}\right\rangle\right)^{2}.$

Using (3.1) we obtain
$\Gamma_{p}(\frac{1}{12})\Gamma_{p}(\frac{7}{12})\Gamma_{p}(\frac{2}{3})^{2}=\phi(-2).$

Then substituting this value and taking the transformation $j\rightarrow
j+\frac{p-1}{3}$ in the above sum we have

$\displaystyle{{}_{2}G_{2}}(\lambda)_{p}$
$\displaystyle=\frac{p\cdot\phi(-2)\psi_{6}(2)\phi(1+\lambda)}{1-p}\overline{\psi}_{3}\left(\frac{\lambda}{(1+\lambda)^{2}}\right)\sum_{j=0}^{p-2}\overline{\omega}^{j}\left(\frac{4\lambda}{(1+\lambda)^{2}}\right)$
$\displaystyle\times(-p)^{-\left\lfloor\frac{5}{12}+\frac{j}{p-1}\right\rfloor-\left\lfloor\frac{11}{12}+\frac{j}{p-1}\right\rfloor-2\left\lfloor\frac{1}{3}-\frac{j}{p-1}\right\rfloor}~{}\Gamma_{p}\left(\left\langle\frac{5}{12}+\frac{j}{p-1}\right\rangle\right)\Gamma_{p}\left(\left\langle\frac{11}{12}+\frac{j}{p-1}\right\rangle\right)$
(4.2)
$\displaystyle\times\Gamma_{p}\left(\left\langle\frac{1}{3}-\frac{j}{p-1}\right\rangle\right)^{2},$

where $\psi_{3}=\omega^{\frac{p-1}{3}}$ is a character of order 3. If we put
$n=2$ and

$x=\langle\frac{5}{6}+\frac{2j}{p-1}\rangle$ in (3.1), then we have

$\Gamma_{p}\left(\left\langle\frac{5}{6}+\frac{2j}{p-1}\right\rangle\right)=\overline{\omega}^{j}(4)\psi_{6}(2)\frac{\Gamma_{p}(\langle\frac{5}{12}+\frac{j}{p-1}\rangle)\Gamma_{p}(\langle\frac{11}{12}+\frac{j}{p-1}\rangle)}{\Gamma_{p}(\frac{1}{2})}.$

Also it is easy to verify that

$\left\lfloor\frac{5}{6}+\frac{2j}{p-1}\right\rfloor=\left\lfloor\frac{5}{12}+\frac{j}{p-1}\right\rfloor+\left\lfloor\frac{11}{12}+\frac{j}{p-1}\right\rfloor.$

Using these two facts in (4) we obtain

$\displaystyle{{}_{2}G_{2}}(\lambda)_{p}$
$\displaystyle=\frac{p\Gamma_{p}(\frac{1}{2})\phi(1+\lambda)}{1-p}\phi(-2)\overline{\psi}_{3}\left(\frac{\lambda}{(1+\lambda)^{2}}\right)\sum_{j=0}^{p-2}\overline{\omega}^{j}\left(\frac{\lambda}{(1+\lambda)^{2}}\right)$
(4.3)
$\displaystyle\times(-p)^{-\left\lfloor\frac{5}{6}+\frac{2j}{p-1}\right\rfloor-2\left\lfloor\frac{1}{3}-\frac{j}{p-1}\right\rfloor}~{}\Gamma_{p}\left(\left\langle\frac{5}{6}+\frac{2j}{p-1}\right\rangle\right)\Gamma_{p}\left(\left\langle\frac{1}{3}-\frac{j}{p-1}\right\rangle\right)^{2}.$

Now, Gross-Koblitz formula allow us to write

${{}_{2}G_{2}}(\lambda)_{p}=\frac{\phi(2)\phi(1+\lambda)}{(1-p)}\overline{\psi}_{3}\left(\frac{\lambda}{(1+\lambda)^{2}}\right)\sum_{j=0}^{p-2}\overline{\omega}^{j}\left(\frac{\lambda}{(1+\lambda)^{2}}\right)\frac{g(\psi_{6}^{5}\overline{\omega}^{2j})g(\psi_{3}\omega^{j})^{2}}{g(\phi)}.$

Replacing $\overline{\omega}^{j}$ by $\overline{\omega}^{j}\psi_{3}$ we obtain

(4.4) $\displaystyle{{}_{2}G_{2}}(\lambda)_{p}$
$\displaystyle=\frac{\phi(2)\phi(1+\lambda)}{(1-p)}\sum_{j=0}^{p-2}\overline{\omega}^{j}\left(\frac{\lambda}{(1+\lambda)^{2}}\right)\frac{g(\phi\overline{\omega}^{2j})g(\omega^{j})^{2}}{g(\phi)}$
$\displaystyle=\frac{\phi(2)\phi(1+\lambda)}{(1-p)g(\phi)}\sum_{u,v,y\in\mathbb{F}_{p}}\phi(u)\zeta_{p}^{u+v+y}\sum_{j=0}^{p-2}\omega^{j}\left(\frac{yv(1+\lambda)^{2}}{u^{2}\lambda}\right).$

Using the orthogonality of multiplicative characters and then replacing $u$ by
$-(1+\lambda)uy$ we deduce that

${{}_{2}G_{2}}(\lambda)_{p}=-\phi(-2)\sum\limits_{u\in\mathbb{F}_{p}}\phi(u(1-u)(1-\lambda
u)).$

Again transforming $u$ by $1/u$ we obtain

(4.5)
$\displaystyle{{}_{2}G_{2}}(\lambda)_{p}=-\phi(-2)\sum\limits_{u\in\mathbb{F}_{p}}\phi(u(u-1)(u-\lambda)).$

Therefore, for $\lambda\neq 0,\pm 1$ we have

(4.6) $_{2}G_{2}(\lambda)_{p}=\phi(-2)a_{p}(\lambda).$

Using this relation we write

(4.7)
$\displaystyle\sum\limits_{\lambda\in\mathbb{F}_{p}}{{}_{2}G_{2}}(\lambda)_{p}^{m}={{}_{2}G_{2}}(1)_{p}^{m}-\phi(-2)^{m}a_{p}(-1)^{m}+\phi(-2)^{m}\sum\limits_{\lambda\neq
0,1}a_{p}(\lambda)^{m}.$

By (4.5) and second part of Lemma 3.1 it is easy to see that

(4.8) $\displaystyle{{}_{2}G_{2}}(1)_{p}=\phi(-2).$

Using [11, eqn (2.10)] and the orthogonality relation Lemma 3.1 one can verify
that

(4.9) $\displaystyle a_{p}(-1)^{m}=\begin{cases}0,&\text{if}\ p\equiv
3\pmod{4}\\\ (-1)^{m}A_{4}^{m}(-1)\left(\frac{g(\phi
A_{4})g(\overline{A}_{4})}{g(\phi)}+\frac{g(\phi\overline{A}_{4})g(A_{4})}{g(\phi)}\right)^{m},&\text{if}\
p\equiv 1\pmod{4},\end{cases}$

where $A_{4}$ is a character of order 4. For any multiplicative character
$\chi$ we have $|g(\chi)|=\sqrt{p}.$ Using this in the above identity we
obtain

(4.10) $\displaystyle a_{p}(-1)^{m}=o(p^{m/2+1}).$

Finally, applying Theorem 4.1, (4.8) and (4.10) in (4.7) we conclude the
result.

∎

###### Proof of Theorem 1.4.

Let
$H(\lambda)={{}_{6}G_{6}}\left[\begin{array}[]{cccccc}\frac{1}{12},&\frac{1}{4},&\frac{5}{12},&\frac{7}{12},&\frac{3}{4},&\frac{11}{12}\vspace{.1
cm}\\\
\frac{1}{3},&\frac{1}{3},&\frac{2}{3},&\frac{2}{3},&0,&0\end{array}\mid\frac{(1+\lambda)^{6}}{2^{6}\lambda^{3}}\right]_{p}.$
Using definition we write

$\displaystyle H(\lambda)$
$\displaystyle=\frac{1}{1-p}\sum_{j=0}^{p-2}{\omega}^{j}\left(\frac{2^{6}\lambda^{3}}{(1+\lambda)^{6}}\right)\pi^{(p-1)S_{j}}$
(4.11)
$\displaystyle\times\prod_{h=1;~{}h~{}odd}^{11}\frac{\Gamma_{p}(\langle\frac{h}{12}-\frac{j}{p-1}\rangle)}{\Gamma_{p}(\langle\frac{h}{12}\rangle)}\times\frac{\Gamma_{p}(\langle\frac{1}{3}+\frac{j}{p-1}\rangle)^{2}~{}\Gamma_{p}(\langle\frac{2}{3}+\frac{j}{p-1}\rangle)^{2}~{}\Gamma_{p}(\langle\frac{j}{p-1}\rangle)^{2}}{\Gamma_{p}(\langle\frac{1}{3}\rangle)^{2}~{}\Gamma_{p}(\langle\frac{2}{3}\rangle)^{2}},$

where
$S_{j}=-2\left\lfloor\frac{j}{p-1}\right\rfloor-2\left\lfloor\frac{1}{3}+\frac{j}{p-1}\right\rfloor-2\left\lfloor\frac{2}{3}+\frac{j}{p-1}\right\rfloor-\sum\limits_{h=1;~{}h~{}odd}^{12}\left\lfloor\frac{h}{12}-\frac{j}{p-1}\right\rfloor.$

Now applying Lemma 3.4 we deduce that

$H(\lambda)=\frac{1}{1-p}\sum\limits_{j=0}^{p-2}{\omega}^{j}\left(\frac{\lambda^{3}}{2^{12}(1+\lambda)^{6}}\right)\pi^{(p-1)S_{j}}\frac{\Gamma_{p}(\langle\frac{-12j}{p-1}\rangle)\Gamma_{p}(\langle\frac{3j}{p-1}\rangle)\Gamma_{p}(\langle\frac{3j}{p-1}\rangle)}{\Gamma_{p}(\langle\frac{-6j}{p-1}\rangle)},$

For $1\leq j\leq p-2$ it is easy to very that
$\left\lfloor\frac{-12j}{p-1}\right\rfloor=1+\sum\limits_{h=0}^{11}\left\lfloor\frac{h}{12}-\frac{j}{p-1}\right\rfloor$
and

$\left\lfloor\frac{-6j}{p-1}\right\rfloor=1+\sum\limits_{h=0}^{5}\left\lfloor\frac{h}{6}-\frac{j}{p-1}\right\rfloor.$
Moreover,
$\left\lfloor\frac{-3j}{p-1}\right\rfloor=\left\lfloor\frac{j}{p-1}\right\rfloor+\left\lfloor\frac{1}{3}+\frac{j}{p-1}\right\rfloor+\left\lfloor\frac{2}{3}+\frac{j}{p-1}\right\rfloor.$

If we use of these three relations in the exponent of $\pi$ present in
$G(\lambda)$ then the Gross-Koblitz formula allows to write

$H(\lambda)=\frac{1}{(1-p)}\sum_{j=0}^{p-2}\frac{g({\omega}^{12j})g(\overline{\omega}^{3j})^{2}}{g({\omega}^{6j})}{\omega}^{j}\left(\frac{\lambda^{3}}{2^{12}(1+\lambda)^{6}}\right).$

By making use of Davenport-Hasse relation one can have

(4.12)
$g({\omega}^{6j})g(\phi{\omega}^{6j})=g({\omega}^{12j}){\omega}^{6j}(2^{-2})g(\phi).$

Therefore, using (4.12) we have

$H(\lambda)=\frac{1}{(1-p)}\sum_{j=0}^{p-2}\frac{g(\phi{\omega}^{6j})g(\overline{\omega}^{3j})^{2}}{g(\phi)}{\omega}^{j}\left(\frac{\lambda^{3}}{(1+\lambda)^{6}}\right).$

Since $p\equiv 2\pmod{3}$ therefore transforming $j\rightarrow-j/3$ we obtain

$H(\lambda)=\frac{1}{(1-p)}\sum_{j=0}^{p-2}\frac{g(\phi\overline{\omega}^{2j})g(\omega^{j})^{2}}{g(\phi)}\overline{\omega}^{j}\left(\frac{\lambda}{(1+\lambda)^{2}}\right).$

Now observe that the above summation is equal to the summation involved in
(4.4) up to a scalar multiple. Therefore following similar steps we obtain

$\phi(1+\lambda)H(\lambda)=\phi(-1)\cdot a_{p}(\lambda).$

Therefore, if $\lambda\neq 0,\pm 1$ then

(4.13) $_{6}G_{6}(\lambda)_{p}=\phi(-1)\cdot a_{p}(\lambda).$

The rest of the proof relies on similar arguments as in the proof of Theorem
1.2.

∎

###### Proof of Theorem 2.1.

If $\lambda\neq 0,\pm 1,$ then (4.6) gives
${}_{2}\widetilde{G}_{2}(\lambda)_{p}=a_{p}(\lambda).$ Also by (4.9) we have
${}_{2}\widetilde{G}_{2}(-1)_{p}=a_{p}(-1).$ Therefore, for $\lambda\neq 0,1$

${}_{2}\widetilde{G}_{2}(\lambda)_{p}=a_{p}(\lambda).$

If $k\geq 4$ is even then Proposition 2.1 of [6] gives

(4.14) $\displaystyle{\mathrm{Tr}_{k}}(\Gamma_{0}(4),p)$
$\displaystyle=-3-\sum_{\lambda=2}^{p-1}P_{k}(a_{p}(\lambda),p),$ (4.15)
$\displaystyle{\mathrm{Tr}_{k}}(\Gamma_{0}(8),p)$
$\displaystyle=-4-\sum_{\lambda=2}^{p-1}P_{k}(a_{p}(\lambda^{2}),p).$

Therefore, we conclude the results by replacing $a_{p}(\lambda)$ by
${}_{2}\widetilde{G}_{2}(\lambda)_{p}$ in the above two trace formulas. ∎

###### Proof of Theorem 2.2.

The proof follows similarly as in the proof of Theorem 2.1. The only
requirement is to show for $\lambda\neq 0,1$

${}_{6}\widetilde{G}_{6}(\lambda)_{p}=a_{p}(\lambda).$

This is straightforward by revisiting (4.13) and the definition of
${}_{6}\widetilde{G}_{6}(-1)_{p}.$

∎

##  5\. Acknowledgements

The second author thanks the Austrian Science Fund FWF grant P32305 for the
support.

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# Lumbar Spine Tumor Segmentation and Localization in T2 MRI Images Using AI

Rikathi Pal1, Sudeshna Mondal1, Aditi Gupta1 Priya Saha2, Somoballi Ghoshal1,
Amlan Chakrabarti1, and Susmita Sur-Kolay3
1 A. K. Choudhury School of Information Technology, University of Calcutta,
Kolkata, India.
2 Human Physiology, Rammohan College, University of Calcutta, Kolkata. India.
3 ACM Unit, Indian Statistical Institute. Indian Statistical Institute,
Kolkata, India.

###### Abstract

In medical imaging, segmentation and localization of spinal tumors in three-
dimensional (3D) space pose significant computational challenges, primarily
stemming from limited data availability. In response, this study introduces a
novel data augmentation technique, aimed at automating spine tumor
segmentation and localization through AI approaches. Leveraging a fusion of
fuzzy c-means clustering and Random Forest algorithms, the proposed method
achieves successful spine tumor segmentation based on predefined masks
initially delineated by domain experts in medical imaging. Subsequently, a
Convolutional Neural Network (CNN) architecture is employed for tumor
classification. Moreover, 3D vertebral segmentation and labeling techniques
are used to help pinpoint the exact location of the tumors in the lumbar
spine. Results indicate a remarkable performance, with 99% accuracy for tumor
segmentation, 98% accuracy for tumor classification, and 99% accuracy for
tumor localization achieved with the proposed approach. These metrics surpass
the efficacy of existing state-of-the-art techniques, as evidenced by superior
Dice Score, Class Accuracy, and Intersection over Union (IOU) on class
accuracy metrics. This innovative methodology holds promise for enhancing the
diagnostic capabilities in detecting and characterizing spinal tumors, thereby
facilitating more effective clinical decision-making.

###### Index Terms:

3D tumor segmentation, Data augmentation, Fuzzy C-Means, Random Forest, CNN

##  I Introduction

Aspinal tumor, whether intradural or extradural, poses significant diagnostic
challenges. While manual segmentation is prone to variability, existing
automatic models are not tailored for spinal tumors. We propose an
architecture capable of segmenting, localizing, and classifying spinal tumors
with 99% accuracy, addressing these limitations [1]. These tumors can be
intramedullary (within the spinal cord), extradural (outside the dura), or
intradural extramedullary (within the dura but outside the spinal cord), as
shown in Figure 1.

Segmenting spinal tumors is complex due to variations in size, intensity, and
location, as well as differences in image resolution and dimensions. Our
approach combines fuzzy c-means clustering for segmentation and Random Forest
for classification. fuzzy c-means handles data points belonging to multiple
clusters, accommodating the uncertainty in spine imaging. Additionally, we
utilize a Convolutional Neural Network (CNN) for tumor type classification.

Our proposed method overcomes the aforementioned challenges through the
following steps:

  * •

Data augmentation by a novel approach to augment MRI lumbar spine images by
inserting tumors along the cerebrospinal fluid (CSF), thereby doubling the
dataset size and facilitating automated tumor segmentation;

  * •

Advanced segmentation by combined fuzzy c-means clustering with Random Forest
classification to provide detailed insights into tumor characteristics and
spatial distribution; and

  * •

Enhanced precision for localization and classification of spinal tumors by
considering their position within the CSF and spatial relationship to
vertebrae, thus improving diagnostic accuracy and treatment planning in a
significant manner.

Our method presents a robust and automated solution for spine tumor
segmentation, marking a notable advancement in the field. It enhances clinical
decision-making, leading to improved patient outcomes in spinal tumor
management by delivering accurate tumor information.

(a) Intramedullary

(b) Intradural Extra-
medullary

(c) Extra Dural

Figure 1: Types of Spine Tumor.

##  II Related Works

For medical image segmentation, a variety of techniques have been explored to
accurately delineate anatomical structures and pathological regions. Kermi et
al. [2] proposed a comprehensive method that amalgamates brain symmetry
analysis with region-based and boundary-based methods, offering a holistic
approach to segmentation. For brain tumor segmentation, several methodologies
have been investigated. Nimeesha et al. [3] and Demirhan et al. [4] leveraged
K-means along with fuzzy c-means clustering for tumor detection, while Pereira
et al. [5] introduced a CNN-based approach tailored specifically for gliomas,
achieving commendable accuracy metrics. Yin et al. [6] introduced a 3D
reconstruction technique for breast tumor segmentation. Agbley et al. [7]
proposed a novel method for breast tumor segmentation.

For accurate spinal cord tumor segmentation [8], [9], [10], and [11], various
models, such as ATS-CDM and SNCDM, are presented. Chang et al. [12] developed
the MCLN for automated vertebrae segmentation and tumor diagnosis, achieving
notable accuracy in segmentation and diagnosis. These efforts reflect ongoing
advancements in medical image segmentation techniques to address specific
clinical needs.

##  III Proposed Methodology

Our methodology involves data augmentation for diversity, followed by 3D tumor
segmentation, classification, and localization. We use 2D MRI slices,
reconstruct them using the method in [13], apply denoising, and cluster with
fuzzy c-means. Random Forest is then used for segmentation, to successfully
delineate tumor regions. Tumors are localized within CSF and are relative to
vertebra in 3D MRI Spine images. Classification refines analysis, categorizing
lesions. This integrated approach aids diagnosis and treatment planning in
clinical settings.

###  III-A Tumor Data Augmentation

As public data for spine tumors is sparse, automating the process of spine
tumor/lesion segmentation becomes a challenge. The first step is to increase
the size of the dataset.

We used a novel data augmentation process shown in Figure 2. First, we
extracted the tumors from each slice to collect the different types of tumors.
Then, we identify the cerebrospinal fluid in each slice based on a predefined
mask. Next, we add a tumor of each type in each slice starting from T11 and
slide it one slice at a time, along the cerebrospinal fluid from T11-L5. The
size of the tumor in each slice is decided by the size that was originally in
a corresponding slice. Thus we have 2D MRI scan slices of the lumbar spine,
each depicting a tumor. To account for potential slight rotation during image
capture, we deliberately rotated the images by an angle lying between 1 and 10
degrees to the left and the right. This strategy effectively doubled our
dataset, providing us with an increased abundance of 3D MRI data for the
lumbar spine.

Suppose $X$, a 2D T2 weighted MRI has a tumor $x$, then at first we separate
the tumor $x$ from $X$ and generate two images, namely $I_{x}$ which has only
$x$, and $I_{\setminus x}$ which has $X$ without $x$. Note that in
$I_{\setminus x}$, we have some missing data as the tumor has been removed, so
the missing data is regenerated using bicubic interpolation [13]. Then, we
extract the cerebrospinal fluid in each image as marked by professionals. We
glide $x$ in each spine image without tumor along the cerebrospinal fluid
(CSF) starting from T11, shift it by 1 mm, i.e., 3 pixels, and generate
another image. While gliding the tumor on the cerebrospinal fluid, we check
for the curvature of the CSF and align the tumor accordingly, i.e., rotate the
tumor in such a way to match the curvature of the CSF and fit the tumor in the
CSF. We continue the process till L5 and generate approximately 20 images from
one image. This process is continued for each type of tumor for each slice.
The size of the tumor in each slice increases with the increase in slice
number moving towards the center and decreases again as it moves towards the
right for the data of each patient, considering the sagittal axis is moving
from left to right in the image capture.

We developed the augmented images solely to test the effectiveness of the
proposed model. Since we worked with the data for sagittal MRI slices of the
lumbar spine taken with a slice gap of (3-5) mm, we reconstructed the missing
data using a combination of bilinear and bicubic interpolation [14], [13] and
extracted the data for consecutive intermediate slices. This process also
helps in data augmentation by increasing the number of 2D slices. After data
augmentation, we executed automated tumor segmentation as the next step of our
method.

Figure 2: Flowchart for Proposed Data Augmentation Technique.

###  III-B Tumor Segmentation

For automated tumor detection and characterization in lumbar spine images as
shown in Figure 3, we adopted a comprehensive approach that includes advanced
techniques in unsupervised clustering and supervised learning. This
methodology is specifically tailored for three-dimensional (3D) magnetic
resonance imaging (MRI) data, enabling precise analysis of lumbar spine
abnormalities.

Figure 3: Overview of Proposed Tumor Segmentation Method.

Before any analysis, 3D MRI slices of the lumbar spine undergo preprocessing
steps aimed at enhancing image quality and reducing noise through shearlet
transform [15].

The study collected lumbar spine MRI data from IPGMER, Kolkata, and augmented
it using sagittal slices. The images were sliced out consecutively, and the 2D
sagittal slices were used for tumor segmentation. The 3D tumor segmentation
output was generated using an unsupervised fuzzy c-means algorithm and a
supervised random forest with masks.

The fuzzy c-means (FCM) clustering algorithm [16] is used to identify putative
areas of interest (ROIs) in 2D sagittal slices from 3D MRI volumes of the
lumbar spine. FCM divides image data into clusters, making segmenting easier
and identifying different tissue types or structures. This targeted strategy
improves tumor identification accuracy, reduces processing time, and optimizes
resources for later model building and refining, leading to better diagnostic
results in lumbar spine MRI analysis. After completing unsupervised clustering
using the fuzzy c-means (FCM) algorithm, the next critical step in our
methodology involves identifying the relevant MRI slices containing tumor
information with the help of domain experts. This process is essential for
pinpointing areas of interest that may harbor potential tumors within the
lumbar spine images. We determine the relevance of MRI slices by examining the
presence of clustered regions that exhibit characteristics indicative of
tumors, such as abnormal intensities or irregular shapes. The study optimizes
MRI slice analysis by strategically allocating resources to areas with the
highest tumor likelihood, improving analysis efficiency and accuracy. This
approach reduces processing time and computational burden, enhancing
diagnostic outcomes and patient care.

In the supervised training phase of our methodology, we use MRI slices and
tumor masks to train Random Forest [17], a supervised learning algorithm, to
identify tumors in lumbar spine images accurately. The algorithm learns from
features like shape, size, and color, enabling accurate classification and
localization. The iterative training process refines the algorithm’s
performance, resulting in reliable automated tumor detection and
characterization in clinical settings. This approach enhances the accuracy of
tumor detection and characterization.

Feature Integration and Model Development:

In this stage, the amalgamation of insights from both unsupervised clustering
and supervised training phases is instrumental in constructing a comprehensive
machine-learning model catered specifically for tumor detection and
characterization in lumbar spine images. It blends representations from
different pipeline stages and implements hierarchical models. It includes
feature selection, judicious architecture design, and strategic optimization
strategies.

Figure 4: Proposed CNN Architecture for Tumour Classification.

###  III-C Tumour Classification

For tumor classification into one of the three types of tumor as in Figure 1,
we have devised a robust Convolutional Neural Network (CNN) architecture
comprising six convolutional layers, interlaced with four strategically placed
max-pooling layers, and capped off with two fully connected dense layers. This
configuration outlined in Figure 4, embodies a meticulous arrangement
meticulously crafted to enhance the network’s ability to discern subtle
intricacies within medical images, particularly those concerning lumbar spine
tumors.

In essence, each convolutional layer acts as a discerning filter, applying a
set of filters to the input image to produce feature maps that capture
distinct aspects of the composition of the image. The number of filters within
each layer dictates the depth of these feature maps, thereby influencing the
capacity of the network to grasp nuanced details. Mathematically, the output
size of a convolutional layer is determined by a formula that factors in the
input size, filter dimensions, padding, and stride, effectively delineating
the spatial dimensions of the resulting feature maps.

$\text{Output size}=\left(\frac{{\text{input size}-\text{filter
size}+2\times\text{padding}}}{{\text{stride}}}\right)+1$ (1)

Similarly, our architecture incorporates max-pooling layers strategically
positioned to reduce the spatial dimensions of the feature maps, thereby
facilitating efficient information compression while preserving salient
features. The output size of each max-pooling layer is calculated analogously
to convolutional layers, similar to spatial reduction.

This meticulously structured CNN architecture operates progressively,
systematically extracting hierarchical representations of input data. Such a
methodology enables the network to discern intricate patterns and nuances
inherent within lumbar spine tumor images, culminating in effective
classification outcomes. The CNN structure works in the format:

$P(L)=\frac{1}{{|L-c|+1}}$ (2)

where $P(L)$ is the performance of the CNN model,$L$ is the number of CNN
layers, and $c$ is a constant that depends on the type and size of the
dataset.

Through experimentation and iterative refinement, we have validated the
efficacy of this specific CNN arrangement for lumbar spine tumor
classification. By adeptly extracting and discerning pertinent features, our
modified CNN architecture demonstrates a heightened ability to accurately
classify lumbar spine tumors, thereby holding significant promise in medical
image analysis and diagnosis.

###  III-D Localization of Tumor

Localizing spine tumors poses a significant challenge in medical practice,
often requiring the identification of both the layer of origin and the
specific vertebrae affected. However, our study focuses exclusively on
pinpointing the vertebrae impacted by the tumor [18] along with the exact
origin of the tumor in the lumbar spine vertebrae. To achieve this, we employ
a method that involves 3D segmentation and labeling of the lumbar vertebrae,
followed by the integration of this labeled image with the detected tumor
image. This fusion process enables us to precisely determine which vertebrae
are affected by the tumor, offering a clear visualization of the impacted
areas. Through our approach, illustrated in Figure 5, we aim to improve the
accuracy and efficiency of spine tumor localization, ultimately aiding in more
effective clinical intervention and treatment planning.

(a)

(b)

Figure 5: Illustration of Tumor Localization in Lumbar Spine.

The block diagram illustrating the complete process of tumor segmentation and
localization is detailed in Figure 6.

Figure 6: Flowchart for Tumor Segmentation & Localization.

##  IV Results

This section briefly presents the results of tumor segmentation, localization,
and classification obtained from various steps in our proposed methodology.

###  IV-A Dataset Description

We collected 7 types of spine MRI lesion/tumor data from IPGMER, Bangur
Institute of NeuoroSciences in 2018 with patients’ consent after review by the
ethical committee of IPGMER, and healthy spine images are collected from [19]
dataset. We consider only the T2 weighted MRI as the tumor is most prominent
in this modality [20]. From these, 7 types of data we have applied the
proposed data augmentation technique to generate 30000 data of 2D spine MRI,
and 350 sets of 3D spine MRI as in Figure 7.

Figure 7: Assessing the Impact of Novel Augmentation Technique on Data
Variability: Before and After Analysis.

###  IV-B Experimental Configuration and Parameterization

A comparative study was conducted on an Intel i5 processor (4.90 GHz, 16GB
RAM) and an NVIDIA Tesla K80 GPU (12GB VRAM). Training utilized an Intel Xeon
CPU (2.20 GHz, 13GB RAM) with 12GB GDDR5 VRAM. Initial training used 0.05% of
unlabelled data, with $\alpha$ set to 0.6. Different $k$-neighborhood values
were applied to datasets for analysis. Results highlight the effectiveness of
the hardware and parameter settings on system performance.

###  IV-C Performance Evaluation Metrics

We have chosen three separate metrics for the training process. For the
prediction of correct masking topology, Intersection-over-Union (IoU) [21] has
been chosen. As we are also dealing with a 3D multi-class classification, we
have taken class-based accuracy and used dice scores to find out the
similarity between the original and predicted masks.

In our case, if an image contains a tumor correctly identified, it counts as a
true positive ($TP$). If a tumor is present in the image but not identified at
all, it counts as a false negative ($FN$). If another tumor or some other
tissue is falsely identified in place of a tumor, it counts as a false
positive ($FP$). If there is no tumor in the image and it is not identified,
it counts as a true negative ($TN$).

The Intersection over Union (IoU) is expressed as:

$IoU=\frac{TP}{TP+FP+FN}$ (3)

It is computed as the mean over all the training samples in a batch. During
training, the step mean IoU is calculated over a range, providing feedback to
the model during a single step and correcting it over the next batch. An IOU
Score above 0.5 is considered a good score.

3D multi-class accuracy refers to the accuracy metric used to evaluate the
performance of machine learning or deep learning models that deal with three-
dimensional data (such as images or volumetric data) and involve
classification tasks with multiple classes. It measures how often the model
correctly predicts the class labels for the given 3D data across all classes.
In short, it assesses the model’s ability to classify 3D data accurately
across multiple categories. The 3D multi-class accuracy [22] is calculated by:

${ClassAccuracy}=\frac{TN+TP}{TN+TP+FP+FN}$ (4)

The Dice score [23], also known as the Dice coefficient or Dice similarity
coefficient, is a common metric used in image segmentation tasks to evaluate
the similarity between two binary images. It is particularly useful when
dealing with imbalanced datasets, where the number of foreground (positive)
pixels vastly differs from the number of background (negative) pixels. The
Dice score is defined as:

$\text{Dice}=\frac{2\times|X\cap Y|}{|X|+|Y|}$ (5)

where:

  * •

$X$ and $Y$ are the sets of pixels in the predicted segmentation mask and the
ground truth mask, respectively.

  * •

$|X\cap Y|$ denotes the number of pixels where both the predicted and ground
truth masks have positive values (i.e., true positives).

  * •

$|X|$ and $|Y|$ represent the total number of positive pixels in the predicted
and ground truth masks, respectively.

The Dice score ranges from 0 to 1, where a score of 1 indicates a perfect
overlap between the predicted and ground truth masks, while a score of 0
indicates no overlap at all.

###  IV-D Comparative Study for Tumour Segmentation

TABLE I: Evaluating the Effectiveness of Our Proposed Tumor Segmentation Method Against Existing Approaches. Method | Accuracy | Dice Score
---|---|---
Z Zhuo et al. [24] | 96% | 0.94
Lemay et al. [9] | 87% | 0.86
Liu et al. [25] | 82% | 0.79
Ito et al. [26] | 93% | 0.89
Jung et al. [27] | 63.4% | 0.59
Orringer et al. [28] | 90% | 0.85
Jakubicek et al. [29] | 87% | 0.83
Nam KH et al. [30] | 92% | 0.9
Karhade et al. [31] | 92% | 0.87
Proposed Algorithm | 99% | 0.97

The comparative analysis presented in Table I underscores the limited
availability of research on spine tumor segmentation. Within this sparse
landscape, our model stands out among existing methodologies. Specifically,
the discernible discrepancies in Dice scores and accuracy metrics underscore
the robustness and superiority of our methodology for spine tumor
segmentation. This notable achievement underscores the potential impact and
significance of our contributions to this specialized field of medical imaging
analysis. Additionally, we have evaluated mask quality using Intersection over
Union (IOU) Scores, as depicted in Figure 8. Notably, all IOU scores of the
predicted masks surpass the threshold of 0.9, serving as compelling evidence
of the robust performance of our model across all tested tumor images. This
comprehensive assessment further solidifies the efficacy and reliability of
our methodology in accurately delineating spine tumors.

Figure 8: IoU Analysis: Visualizing Intersection Over Union Scores for Test
Images.

###  IV-E Ablation Study

Our conducted ablation study delved into the approach of employing fuzzy
c-means clustering followed by random forest for spine tumor segmentation,
comparing it against solely utilizing random forest. The findings revealed a
substantial enhancement in accuracy when employing the combined approach. In
particular, while direct employment of random forest yielded a commendable 94%
accuracy, integrating fuzzy c-means clustering before random forest gave
better performance, resulting in an approximate 99% accuracy rate. These
results underscore the effectiveness of incorporating fuzzy c-means clustering
as a prior step, enhancing the robustness and precision of tumor segmentation
in spinal imaging. Thus, our study advocates for the adoption of this combined
methodology as a superior approach in medical image analysis for spine tumor
segmentation.

###  IV-F Comparative Study for Tumour Classification

Our model demonstrates proficiency in classifying three types of tumors:
Intramedullary, Intradural-Extramedullary, and Extra Dural Tumors. Through
rigorous testing against established model architectures such as VGG16 [32],
VGG19 [32], MobileNetV2 [33], ResNet50 [34], AlexNet [35], and ZFNet [36], our
proposed CNN architecture has consistently outperformed them all. This
superiority is evident in the model accuracy results depicted in Figure 9.

Figure 9: Accuracy Comparison among various CNN Architectures.

During our experimentation and testing phase, where we evaluated different
configurations of CNN layers, we observed a significant trend in the results.
As depicted in Figure 10, a CNN architecture consisting of six layers yielded
the most promising outcomes among all the variations we explored.

This finding underscores the critical role of model architecture in achieving
optimal performance in our tumor classification task. By leveraging a six-
layer CNN configuration, our model demonstrates enhanced capabilities in
effectively discerning between different types of tumors with higher accuracy
and reliability. This insight highlights the importance of fine-tuning the
architectural aspects of neural networks to achieve the best possible outcomes
in medical image analysis tasks like tumor classification.

Figure 10: Analyzing the Impact of Varying CNN Layer Depth on Training
Accuracy.

As shown in Figure 11, we observe a gradual and consistent decrease in loss
values across 40 training epochs of training and validation. This downward
trend signifies a crucial aspect of the efficacy and robustness of our model.
The progressive reduction in loss values over successive epochs indicates that
our model is effectively learning from the training data and generalizing well
to unseen validation data. This phenomenon is pivotal in machine learning,
particularly in tasks like tumor classification, where accurate predictions
are paramount for clinical diagnosis and decision-making.

Figure 11: Loss Curve During Training & Validation.

The fact that both training and validation loss curves exhibit a decreasing
trend suggests that our model is not only capable of capturing intricate
patterns within the training data but also avoids overfitting by maintaining
good performance on unseen validation data. This balance between learning from
the training set and generalizing to new data indicates a well-designed and
well-trained model. We have generated a confusion matrix to visually represent
the classification performance of our model, as illustrated in Figure 12.

Figure 12: Visualizing Tumor Type Identification Performance: Confusion Matrix
Analysis of Our CNN Model.

##  V Discussion & Conclusion

Significant progress has been achieved in the detection and segmentation of
spinal cord tumors, leveraging advancements in deep learning and sophisticated
imaging techniques. Nevertheless, persistent challenges stem from the limited
availability and quality of spine tumor data, leading to biases and
compromised accuracy. Notably, algorithms such as Zhuo et al.’s deep learning
pipeline framework [24] have demonstrated commendable accuracy, albeit
struggling with distinguishing tumors from back muscles during training,
resulting in a 96% accuracy rate. This underscores the complexity of handling
typical tumor characteristics and tissue overlap, particularly in the face of
limited data. Moreover, while multi-model fusion frameworks hold promise, they
often overlook crucial clinical parameters beyond age and MRI images,
necessitating thorough validation and integration into standard practice. What
sets our model apart is the exceptional quality of its training data,
meticulously curated through real-time data integration and extensive
consultations with medical professionals across diverse clinical scenarios.
Our rigorous training program ensures superior performance by effectively
addressing challenges such as tumor and tissue feature overlaps, resulting in
a remarkable 99% accuracy rate. Furthermore, our model excels in enhancing
diagnostic precision by precisely identifying the tumor’s origin and impacted
vertebrae, surpassing previous methods and offering a comprehensive solution
for precise spinal cord tumor delineation.

Our research enhances medical imaging by automating 3D spine tumor data
generation. We use a combination of random forest and fuzzy c-means algorithms
for 99% accurate tumor segmentation in T2 MRI lumbar spine images. A 6-layer
CNN achieves 98% accuracy in tumor type classification. Our novel data
augmentation method combats scarcity, improving both segmentation and
classification. We also extend our approach to multi-class vertebrae labeling,
achieving a 99.5% tumor localization accuracy. Integrating our method with
robotic surgical tools could revolutionize surgical planning for more precise
procedures, benefiting patient outcomes. Ultimately, our work aims to empower
medical professionals with advanced tools for improved diagnosis and treatment
planning, which leads to benefiting patient outcomes.

## Acknowledgment

We extend heartfelt gratitude to Dr. Alok Pandit from the Bangur Institute of
Neurosciences, IPGMER Kolkata; Dr. Nihar Ranjan Sarkar, a distinguished
radiology specialist at IPGMER Hospital, and Dr. Soumyadip Roy, MBBS in
WBPDCL, for their invaluable contributions to our data analysis. Their
expertise, insights, and dedication have greatly enriched our understanding.

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