p-ADIC CELLULAR NEURAL NETWORKS APPLICATIONS TO IMAGE PROCESSING B. A. ZAMBRANO-LUNA AND W. A. Z UNIGA-GALINDO
2025-05-06
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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
Qp
A(x, y)Y(y, t)dy +Z
Qp
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.
1
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
Zp
A(x−y)f(X(y, t))dy (1.3)
+Z
Zp
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
2
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
b,
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,|·|pis 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)
∞
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,|·|pis totally disconnected, i.e. the only connected
subsets of Qpare the empty set and the points. A subset of Qpis compact if and
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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,|·|pis 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 Ωp−r|x−a|pto 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) =
M
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|pdenotes 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
Zp
|g(x)|ρdx#1
ρ
<∞.
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
4
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
e
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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