p-ADIC CELLULAR NEURAL NETWORKS APPLICATIONS TO IMAGE PROCESSING B. A. ZAMBRANO-LUNA AND W. A. Z UNIGA-GALINDO

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p-ADIC CELLULAR NEURAL NETWORKS: APPLICATIONS TO

IMAGE PROCESSING

B. A. ZAMBRANO-LUNA AND W. A. Z ´

U˜

NIGA-GALINDO

Abstract. The p-adic cellular neural networks (CNNs) are mathematical gen-

eralizations of the neural networks introduced by Chua and Yang in the 80s. In

this work we present two new types of CNNs that can perform computations

with real data, and whose dynamics can be understood almost completely. The

ﬁrst type of networks are edge detectors for grayscale images. The stationary

states of these networks are organized hierarchically in a lattice structure. The

dynamics of any of these networks consists of transitions toward some minimal

state in the lattice. The second type is a new class of reaction-diﬀusion net-

works. We investigate the stability of these networks and show that they can

be used as ﬁlters to reduce noise, preserving the edges, in grayscale images pol-

luted with additive Gaussian noise. The networks introduced here were found

experimentally. They are abstract evolution equations on spaces of real-valued

functions deﬁned in the p-adic unit ball for some prime number p. In practical

applications the prime pis determined by the size of image, and thus, only

small primes are used. We provide several numerical simulations showing how

these networks work.

1. Introduction

In the late 80s Chua and Yang introduced a new natural computing paradigm

called the cellular neural networks (or cellular nonlinear networks) CNN which

includes the cellular automata as a particular case [10], [11], [13]. This paradigm

has been extremely successful in various applications in vision, robotics and remote

sensing, see, e.g., [12], [32] and the references therein.

In [37] we introduce the p-adic cellular neural networks which are mathematical

generalizations of the classical CNNs. The new networks have inﬁnitely many cells

which are organized hierarchically in rooted trees, and also they have inﬁnitely many

hidden layers. Intuitively, the p-adic CNNs occur as limits of large hierarchical

discrete CNNs. A p-adic CNN is the dynamical system given by











∂X(x,t)

∂t =−X(x, t) + Z

A(x, y)Y(y, t)dy +Z

B(x, y)U(y)dy +Z(x)

Y(x, t) = f(X(x, t)),

(1.1)

where xis a p-adic number (x∈Qp), while tis a non-negative real number,

X(x, t)∈Ris the state of cell xat the time t,Y(x, t)∈Ris the output of

cell xat the time t,fis a sigmoidal nonlinearity, Uis the input of the CNN, and

Key words and phrases. Cellular neural networks, hierarchies, p-adic numbers, edge detectors,

denoising.

The second author was partially supported by the Lokenath Debnath Endowed Professorship.

arXiv:2210.14132v2 [nlin.CG] 16 Jan 2023

Zis the threshold of the CNN. In [37], we study the Cauchy problem associated to

(1.1) and also provide numerical methods for solving it.

The goal of this article is to show that p-adic CNNs can perform computations

using real data, and that the dynamics can be understood almost completely. We

present two new types of p-adic CNNs, one type for edge detection of grayscale

images, and the other, for denoising of grayscale images polluted with Gaussian

noise. It is important to emphasize that our goal is not to produce new techniques

for image processing, but to use these tasks to verify that p-adic CNNs can perform

relevant computations. On the other hand, classical CNNs have been implemented

in hardware for performing certain image processing tasks. We have used some of

the ideas introduced in [12], but our results go in a completely new direction.

We found experimentally that p-adic CNNs of the form







∂

∂t X(x, t) = −X(x, t) + aY (x, t)+(B∗U)(x) + Z(x), x ∈Zp, t ≥0;

Y(x, t) = f(X(x, t)),

(1.2)

can be used as edge detectors, here Zpis the p-adic unit ball, and Uis an image.

We develop numerical algorithms for solving the Cauchy problem attached to (1.2),

with initial datum X(x, 0) = 0. The simulations show that after a time suﬃciently

large the network outputs a black-and-white image approximating the edges of the

original image U(x). The performance of this edge detector is comparable to the

Canny detector, and other well-known detectors. But most importantly, we can

explain, reasonably well, how the network detects the edges of an image.

We determine all the stationary states of (1.2), i.e. the solutions of ∂

∂t X(x, t) = 0,

for any a∈R, see Lemma 1 and Theorem 1. We show that for a > 1, the set of all

possible stationary states Mof (1.2) has a hierarchical structure, more precisely,

(M,4) is a lattice, where 4is a partial order. Furthermore, we determine the

set of minimal elements of (M,4), see Theorem 2. The dynamics of the network

consists of transitions in a hierarchically organized landscape (M,4) toward some

minimal state. This is a reformulation of the classical paradigm asserting that the

dynamics of a large class of complex systems can be modeled as a random walk on

its energy landscape, see, e.g., [22], [23].

We found experimentally that p-adic CNNs of the form

∂X(x, t)

∂t =µX(x, t)+(λI −Dα

0)X(x, t) + Z

A(x−y)f(X(y, t))dy (1.3)

B(x−y)U(y)dy +Z(x)

can be used for denoising grayscale images polluted with Gaussian noise. In this

case, X(x, 0) is the input image, and X(x, t0) is the output image, for a suitable

(typically small) t0.

The CNN (1.3) is a reaction-diﬀusion network. The diﬀusion part corresponds

to ∂X(x, t)

∂t = (λI −Dα

0)X(x, t), x∈Zp, t ≥0,(1.4)

here Dα

0is the Vladimirov operator acting on functions supported in the unit ball,

α > 0. The equation (1.4) is a p-adic heat equation in the unit ball, this means that

there is a stochastic Markov process attached to it. The paths of this stochastic

process are discontinuous. p-Adic heat equations and the associated stochastic

processes have been studied intensively in the last thirty years in connection with

models of complex systems, see, e.g., [3]-[4], [14], [22], [25]-[23], [34]-[36], [38]-[39].

The reaction term in (1.3) gives an estimation of the edges of the image, while the

diﬀusion term produces a smoothed version of the image. Under suitable hypothe-

ses, see Theorem 3, we show that a solution of the initial value problem attached

to (1.3) is bounded at very time if µ≤0, otherwise, the solution is bounded by

Ceµt, where Cis a positive constant. Some numerical simulations show that our

ﬁlter eﬀectively reduces the noise while preserves the edges of the image, however,

its performance is inferior to the Perona-Malik ﬁlter, see, e.g., [31].

Finally, we want to mention that p-adic numbers have been used before in pro-

cessing image algorithms, see, e.g., [5]-[6], [26]. But these results are not directly

related with the ones presented here.

2. Basic facts on p-adic analysis

In this section we ﬁx the notation and collect some basic results about p-adic

analysis that we will use through the article. For a detailed exposition on p-adic

analysis the reader may consult [1], [33], [36]. For a quick review of p-adic analysis

the reader may consult [7], [27].

2.1. The ﬁeld of p-adic numbers. Throughout this article pwill denote a prime

number. The ﬁeld of p−adic numbers Qpis deﬁned as the completion of the ﬁeld

of rational numbers Qwith respect to the p−adic norm |·|p, which is deﬁned as

|x|p=(0 if x= 0

p−γif x=pγa

where aand bare integers coprime with p. The integer γ=ordp(x) with ordp(0) :=

+∞, is called the p−adic order of x. The metric space Qp,|·|pis a complete

ultrametric space. Ultrametric means that |x+y|p≤max n|x|p,|y|po. As a topo-

logical space Qpis homeomorphic to a Cantor-like subset of the real line, see, e.g.,

[1], [36].

Any p−adic number x6= 0 has a unique expansion of the form

x=pordp(x)

∞

j=0

xjpj,(2.1)

where xj∈ {0,1,2, . . . , p −1}and x06= 0. It follows from (2.1), that any x∈

Qpr{0}can be represented uniquely as x=pordp(x)u(x) and |x|p=p−ordp(x).

2.2. Topology of Qp.For r∈Z, denote by Br(a) = {x∈Qp;|x−a|p≤pr}the

ball of radius prwith center at a∈Qp, and take Br(0) := Br. The ball B0equals

the ring of p−adic integers Zp. We also denote by Sr(a) = {x∈Qp;|x−a|p=pr}

the sphere of radius prwith center at a∈Qp, and take Sr(0) := Sr. We notice

that S0=Z×

p(the group of units of Zp). The balls and spheres are both open

and closed subsets in Qp. In addition, two balls in Qpare either disjoint or one is

contained in the other.

As a topological space Qp,|·|pis totally disconnected, i.e. the only connected

subsets of Qpare the empty set and the points. A subset of Qpis compact if and

only if it is closed and bounded in Qp, see e.g. [36, Section 1.3], or [1, Section 1.8].

The balls and spheres are compact subsets. Thus Qp,|·|pis a locally compact

topological space.

Since (Qp,+) is a locally compact topological group, there exists a Haar measure

dx, which is invariant under translations, i.e. d(x+a) = dx. If we normalize this

measure by the condition RZpdx = 1, then dx is unique. For a quick review of

the integration in the p-adic framework the reader may consult [7], [27] and the

references therein.

Notation 1. We will use Ωp−r|x−a|pto denote the characteristic function of

the ball Br(a).

2.3. The Bruhat-Schwartz space. A real-valued function ϕdeﬁned on Qpis

called locally constant if for any x∈Qpthere exist an integer l(x)∈Zsuch that

ϕ(x+x0) = ϕ(x) for any x0∈Bl(x).(2.2)

A function ϕ:Qp→Cis called a Bruhat-Schwartz function (or a test function) if it

is locally constant with compact support. Any test function can be represented as

a linear combination, with real coeﬃcients, of characteristic functions of balls. The

R-vector space of Bruhat-Schwartz functions is denoted by D(Qp). For ϕ∈ D(Qp),

the largest number l=l(ϕ) satisfying (2.2) is called the exponent of local constancy

(or the parameter of constancy) of ϕ. Let Ube an open subset of Qp, we denote

by D(U) the R-vector space of all test functions with support in U. For instance

D(Zp) is the R-vector space of all test functions with supported in the unit ball

Zp. A function ϕin D(Zp) can be written as

ϕ(x) =

j=1

ϕ(exj) Ω prj|x−exj|p,

where the exj,j= 1, . . . , M, are points in Zp, the rj,j= 1, . . . , M, are integers,

and Ω prj|x−exj|pdenotes the characteristic function of the ball B−rj(exj) =

exj+prjZp.

2.4. Some function spaces. Given ρ∈[1,∞), we denote by Lρ:= Lρ(Zp) :=

Lρ(Zp, dx),the R−vector space of all functions g:Zp→Rsatisfying

kgkρ="R

|g(x)|ρdx#1

<∞.

We denote by C(Zp) the R-vector space of continuous functions f:Zp→Rsatis-

fying

kfk∞:= max

x∈Zp

|f(x)|<∞. (2.3)

3. p-Adic continuous CNNs

3.1. A type p-adic continuous CNNs. In this section we present new edge

detectors based on p-adic CNNs for grayscale images. We take B∈L1(Zp) and

U, Z ∈ C(Zp), a,b∈R, and ﬁx the sigmoidal function f(s) = 1

2(|s+ 1| − |s−1|)

for s∈R. In this section we consider the following p-adic CNN:







∂

∂t X(x, t) = −X(x, t) + aY (x, t)+(B∗U)(x) + Z(x), x ∈Zp, t ≥0;

Y(x, t) = f(X(x, t)).

(3.1)

We denote this p-adic CNN as CNN (a, B, U, Z), where a, B, U, Z are the param-

eters of the network. In applications to edge detection, we take U(x) to be a

grayscale image, and take the initial datum as X(x, 0) = 0.

3.2. Stationary states. We say that Xstat(x) is a stationary state of the network

CNN(a, B, U, Z), if







Xstat(x) = aYstat(x)+(B∗U)(x) + Z(x), x ∈Zp;

Ystat(x) = f(Xstat(x)).

(3.2)

Remark 1. Let ˜

X(x)be any solution of (3.2). Then

X(x) = (a+ (B∗U)(x) + Z(x)if e

X(x)>1

−a+ (B∗U)(x) + Z(x)if e

X(x)<−1,(3.3)

and

(1 −a)e

X(x) = (B∗U)(x) + Z(x)if e

X(x)≤1.(3.4)

Lemma 1. (i) If a < 1, then the network CNN(a, B, U, Z)has a unique stationary

state Xstat(x)∈ C(Zp)given by

Xstat(x) = 





a+ (B∗U)(x) + Z(x)if (B∗U)(x) + Z(x)>1−a

−a+ (B∗U)(x) + Z(x)if (B∗U)(x) + Z(x)<−1 + a

(B∗U)(x)+Z(x)

1−aif |(B∗U)(x) + Z(x)| ≤ 1−a.

(3.5)

(ii) If a= 1 , then the network CNN(a, B, U, Z)has a unique stationary state

Xstat(x)∈L1(Zp)given by

Xstat(x) = 





1+(B∗U)(x) + Z(x)if (B∗U)(x) + Z(x)>0

−1+(B∗U)(x) + Z(x)if (B∗U)(x) + Z(x)<0

0if (B∗U)(x) + Z(x) = 0.

(3.6)

Proof. If a < 1, it follows from (3.3)-(3.4) that (3.5) is a continuous stationary

state since by the dominated convergence theorem (B∗U)(x) is continuous. To

establish the uniqueness of the solution, let X(x)∈ C(Zp) be another stationary

state. Consider a point x0∈Zpsuch that X(x0)>1. Then by (3.3), X(x0) =

a+ (B∗U)(x0) + Z(x0)>1 consequently (B∗U)(x0) + Z(x0)>1−aand therefore

X(x0) = a+ (B∗U)(x0) + Z(x0) := Xstat(x0).

The cases X(x0)<−1 and X|(x0)|<1 are treated in a similar way.

The case a= 1 follows from (3.4), in this case we have that Xstat(x)∈L1(Zp)

since Xstat(x) is bounded. The continuity of Xstat(x) requires further hypotheses

on B, U, Z.

Deﬁnition 1. Assume that a > 1. Given

I+⊆ {x∈Zp; 1 −a < (B∗U)(x) + Z(x)},

I−⊆ {x∈Zp; (B∗U)(x) + Z(x)< a −1},

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p-ADICCELLULARNEURALNETWORKS:APPLICATIONSTOIMAGEPROCESSINGB.A.ZAMBRANO-LUNAANDW.A.ZU~NIGA-GALINDOAbstract.Thep-adiccellularneuralnetworks(CNNs)aremathematicalgen-eralizationsoftheneuralnetworksintroducedbyChuaandYanginthe80s.InthisworkwepresenttwonewtypesofCNNsthatcanperformcomputationswithrealdata...

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