11institutetext:  Jardel Vieira 22institutetext: Unidade Acadêmica Especial de
Matemática e Tecnologia - Universidade Federal de Goiás
Av. Dr. Lamartine Pinto de Avelar 1120, St. Universitário, 75704-020, Catalão,
Goiás, Brasil.
22email<EMAIL_ADDRESS>33institutetext: Eduardo Abreu 44institutetext:
Institute of Mathematics, Statistics and Scientific Computing - University of
Campinas
Rua Sérgio Buarque de Holanda, 651, Cidade Universitária ”Zeferino Vaz” -
Distr. Barão Geraldo, CEP 13083-859, Campinas, SP, Brasil
44email<EMAIL_ADDRESS>55institutetext: Joao B. Florindo 66institutetext:
Institute of Mathematics, Statistics and Scientific Computing - University of
Campinas
Rua Sérgio Buarque de Holanda, 651, Cidade Universitária ”Zeferino Vaz” -
Distr. Barão Geraldo, CEP 13083-859, Campinas, SP, Brasil
66email<EMAIL_ADDRESS>
# Texture image classification based on a pseudo-parabolic diffusion model
††thanks: J. B. Florindo gratefully acknowledges the financial support of São
Paulo Research Foundation (FAPESP) (Grant #2016/16060-0) and from National
Council for Scientific and Technological Development, Brazil (CNPq) (Grants
#301480/2016-8 and #423292/2018-8). E. Abreu gratefully acknowledges the
financial support of São Paulo Research Foundation (FAPESP) (Grant
#2019/20991-8), from National Council for Scientific and Technological
Development - Brazil (CNPq) (Grant #2 306385/2019-8) and PETROBRAS - Brazil
(Grant #2015/00398-0). E. Abreu and J. B. Florindo also gratefully acknowledge
the financial support of Red Iberoamericana de Investigadores en Matemáticas
Aplicadas a Datos (MathData).

Jardel Vieira  Eduardo Abreu  Joao B. Florindo

(Received: date / Accepted: date)

###### Abstract

This work proposes a novel method based on a pseudo-parabolic diffusion
process to be employed for texture recognition. The proposed operator is
applied over a range of time scales giving rise to a family of images
transformed by nonlinear filters. Therefore each of those images are encoded
by a local descriptor (we use local binary patterns for that purpose) and they
are summarized by a simple histogram, yielding in this way the image feature
vector. The proposed approach is tested on the classification of well
established benchmark texture databases and on a practical task of plant
species recognition. In both cases, it is compared with several state-of-the-
art methodologies employed for texture recognition. Our proposal outperforms
those methods in terms of classification accuracy, confirming its
competitiveness. The good performance can be justified to a large extent by
the ability of the pseudo-parabolic operator to smooth possibly noisy details
inside homogeneous regions of the image at the same time that it preserves
discontinuities that convey critical information for the object description.
Such results also confirm that model-based approaches like the proposed one
can still be competitive with the omnipresent learning-based approaches,
especially when the user does not have access to a powerful computational
structure and a large amount of labeled data for training.

###### Keywords:

Pseudo-parabolic equation Texture recognition Image classification Numerical
approximation methods for PDEs.

##  1 Introduction

The recognition and classification of texture images, either in isolated
conditions or “in the wild” is one of the most important tasks in computer
vision, with numerous applications in material sciences Naik and Khan (2019),
medicine Jasionowska and Przelaskowski (2019), remote sensing Tu et al.
(2019), agriculture Safdar et al. (2019), etc.

Despite the recent popularization of learning-based approaches like the deep
convolutional networks in this area Cimpoi et al. (2016), there still exists
space for the investigation of “hand-engineered” image descriptors, especially
in situations when there is a small amount of data for training, what is
typical, for instance, in medical applications, or when the introduction of
specific information from that particular problem (from the physical modeling
for example) can contribute with the performance of the computational
algorithm.

In this context, and inspired by the recent success of methods like the
nonlinear operator derived from a partial differential equation (PDE) in
Florindo and Bruno (2017); Florindo (2020) , we propose here new texture
descriptors based on the application of an operator that corresponds to
solutions of a pseudo-parabolic PDE (e.g., Abreu and Vieira (2017); Abreu et
al. (2020)) when the original image provides the initial condition of that
PDE. Pseudo-parabolic differential equations are characterized by having mixed
time and space derivatives appearing in the highest-order terms Karch (1997);
Ting (1969) and are applied to model many physical phenomena, for instance the
fluid dynamic in porous media Hassanizadeh and Gray (1993). Here our
particular interest on this type of PDE is explained by their ability to
regularize a function (image profile in our case) but at the same time that it
preserves discontinuities (here corresponding to edges and other relevant
local variations important for the image characterization).

To obtain the image representation (descriptors) we follow the PDE application
with an encoding process, which here is accomplished by local binary patterns
(LBP) Ojala et al. (2002). The descriptors are finally submitted to Karhunen-
Loève transform Pearson (1901) and classified by a linear discriminant
classifier Fisher (1936). The proposed method is assessed on state-of-the-art
benchmark databases (UIUC Lazebnik et al. (2005), UMD Xu et al. (2009) and
KTHTIPS-2b Hayman et al. (2004)) as well as in a practical problem of
Brazilian plant species identification Casanova et al. (2009). The results
confirm that the classification accuracy is competitive both with classical
texture descriptors as well as with modern approaches like convolutional
neural networks.

This text is divided into 7 sections, including this introduction. In the next
one, we present a brief review of other PDE-based methods for image analysis
and processing in the literature. In Section 3, the PDE model is presented
together with the numerical scheme used here to solve it. Section 4 shows the
methodology, including a short introduction to LBP theory and the proposed
method. Section 5 describes the setup for the texture classification
experiments. Section 6 presents the results of these experiments in comparison
with the state-of-the-art on this topic and corresponding discussion. Finally,
Section 7 concludes the work, summarizing the main points and raising some
general discussion on the impact of the proposed methodology.

##  2 Related works

Most methods for the classification of texture images can be divided into two
main categories: traditional approaches and deep learning methods Liu et al.
(2019). In the first group, we have the classical second order statistical
methods like Haralick descriptors Haralick et al. (1973) and Local Binary
Patterns (LBP) Ojala et al. (2002), more recently we have seen the Scale-
Invariant Feature Transform (SIFT) Lowe (2004), Filter Banks Varma and
Zisserman (2009), Fisher Vector Perronnin et al. (2010), Fractal Descriptors
Florindo and Bruno (2016), PCANet Chan et al. (2015), Locally Encoded
Transform Feature Histogram Song et al. (2018), and many others. On the second
category, deep learning methods, we have those based on Convolutional Neural
Networks (CNN), such as Deep Convolutional Activation Feature (DeCAF) Donahue
et al. (2014), Deep Filter Banks Cimpoi et al. (2016), Deep Texture Encoding
Network (DeepTEN) Zhang et al. (2017), Locally Transfered Fisher Vectors (LFV)
Song et al. (2017), Deep Texture Manifold (DEP) Xue et al. (2018), First and
Second Order Network (FASON) Dai et al. (2017), Multiple-Attribute-Perceived
(MAP-Net) Zhai et al. (2019), and others.

The disseminated use of PDEs to provide different viewpoints of a digital
image can be traced back to the space-scale theory of Witkin Witkin (1983),
formulated in terms of the canonical model of diffusion equation, i.e., the
heat equation. The well-known maximum principle of the elliptic equations was
employed in that occasion to ensure that spurious details not present in finer
scales would also not appear in coarser scales. This is a fundamental property
that must be satisfied by any robust multiscale theory Koenderink (1984).

However the smoothing effect intrinsic to diffusion equations can also
represent a problem in image analysis as at the same time that the control
over the smoothing parameter allows for fine tuning of the desired scale of
observation, it also smooths out image edges. Edges are known to be powerful
attributes in the discrimination of objects present in the image. To address
this issue, Perona and Malik proposed the anisotropic diffusion equation in
Perona and Malik (1990). This ensures the smoothing multiscale effect while
preserving edges by means of a less aggressive action of the operator in
regions with high-valued gradients.

Evolutions over the anisotropic model can be found in Catté et al. (1992),
where the image is regularized previous to the application of the PDE
operator. Cottet and Germain (1993) and Weickert (1996) employ diffusion
tensors to address the problem of “ghost” stair-case edges arising in some
situations after the application of the operator. A review of this family of
approaches can be found in Weickert (1997). Another strategy, focused
especially in speckle reduction and avoiding the direct use of gradients, is
presented in Yu and Acton (2002). Guidotti et al. (2013) and Guidotti (2009)
also propose an interesting alternative named Backward-Forward Regularized
Diffusion, again attenuating the “stair-case” effect and using two
regularizing parameters. More recently, a promising method combining the model
in Guidotti et al. (2013) with texture descriptors is also proposed in Barros
Neiva et al. (2018).

With regards to pseudo-parabolic equations, the early existence, uniqueness,
and regularity theory is well developed, for example, in Showalter (1969,
1975); Showalter and Ting (1970); Ting (1969). Such theory predicts that the
additional pseudo-parabolic term decreases the smoothing property
characteristic to parabolic problems. This character has consequences on the
behavior of the solution and it was also observed in subsequent works. For
instance, if the initial data has jump discontinuity at some point, then so
does the solution for every time Cuesta and Hulshof (2003); Cuesta and Pop
(2009). Here this is a particularly important feature as it allows a
controlled smoothing effect preserving relevant edges to some extent. It is
also worth to mention that, in general, there is no maximum principle for
pseudo-parabolic equations such as one expected of solutions to parabolic
equations Stecher and Rundell (1977), although the specific simplified model
adopted here still preserves that principle.

##  3 Proposed methodology

###  3.1 Partial differential equation and numerical modeling

The model of partial differential equation adopted here is that of a pseudo-
parabolic type. In its more general form, those equations can be represented
by

$\frac{\partial}{\partial t}\mathcal{A}(u)+\mathcal{B}(u)=0,$ (1)

where $\mathcal{A}$ and $\mathcal{B}$ are nonlinear elliptic operators. More
specifically, we build our model on the pseudo-parabolic equation:

$\frac{\partial u}{\partial
t}=\nabla\cdot\left[g(x,y,t)\,\nabla\left(u+\tau\,\frac{\partial u}{\partial
t}\right)\right].$ (2)

We also identify the pseudo-parabolic diffusive flux $\mathbf{w}$ as given by

$\mathbf{w}=g(x,y,t)\,\nabla\left(u+\tau\,\frac{\partial u}{\partial
t}\right).$ (3)

Let $\Omega\subset\mathbb{R}^{2}$ denote a rectangular domain and
${\displaystyle u(\,\cdot\,,\,\cdot\,,\,t):\Omega\rightarrow\mathbb{R}}$ be a
family of gray scale images that satisfies the pseudo-parabolic equation (2).
The original image corresponds to $t=0$, i.e., an initial value for the
differential problem. To completely define the differential problem, we impose
zero flux condition across the domain boundary $\partial\Omega$,

$\mathbf{w}\cdot\mathbf{n}=0,\qquad(x\,,\,y)\in\partial\Omega.$ (4)

In general, the coefficient $g(x\,,\,y\,,\,t)$ is a positive definite tensor
and, in the context of nonlinear diffusion models for image processing, it may
depend on $\nabla u$ Perona and Malik (1990); Catté et al. (1992); Weickert
(1997). In this work, we describe the numerical approach considering $g$ as a
scalar. However, the method may be directly extended for the others cases.

For the numerical modeling we employ a uniform rectangular mesh and cell-
centered finite differences. To construct this scheme, we use ideas learned
from the mixed finite element method in Abreu and Vieira (2017); Abreu et al.
(2020). The mixed finite element framework has two advantages Chavent and
Roberts (1991): it yields a very natural and physical discretization of the
boundary conditions; and it gives a consistent way of defining a gradient
flux, as in the mixed formulation.

Now, we will discuss a finite difference approach for the pseudo-parabolic
equation (2). Consider a uniform partition of $\Omega$ into rectangular
subdomains $\Omega_{i,j}$, for $i=1\,,\,\dots\,,\,m$ and
$j=1\,,\,\dots\,,\,l$, with dimensions $\Delta x\times\Delta y$. The center of
each subdomain $\Omega_{i,j}$ is denoted by $(x_{i}\,,\,y_{j})$. Given a final
time of simulation $T$, consider a uniform partition of the interval
$[0\,,\,T]$ into $N$ subintervals. The time step $\Delta t=T/N$ is usually
defined by a stability condition. We denote the time instants as
$t_{n}=n\Delta t$, for $n=0\,,\,\dots\,,\,N$.

Let $U_{i,j}^{n}$ be a finite difference approximation for
$u(x_{i}\,,\,y_{j}\,,\,t_{n})$. A discretization of (2) by the finite
difference method is given by

$\frac{U_{i,j}^{n+1}-U_{i,j}^{n}}{\Delta
t}=\frac{W_{i+\frac{1}{2},j}^{n+1}-W_{i-\frac{1}{2},j}^{n+1}}{\Delta
x}+\frac{W_{i,j+\frac{1}{2}}^{n+1}-W_{i,j-\frac{1}{2}}^{n+1}}{\Delta y},$ (5)

where the approximation of the diffusive flux is given by a centered
difference formula:

$W_{i+\frac{1}{2},j}^{n+1}=G_{i+\frac{1}{2},j}^{n}\left(\frac{U_{i+1,j}^{n+1}-U_{i,j}^{n+1}}{\Delta
x}\right)+\frac{\tau\,G_{i+\frac{1}{2},j}^{n}}{\Delta
t}\left(\frac{U_{i+1,j}^{n+1}-U_{i,j}^{n+1}}{\Delta
x}-\frac{U_{i+1,j}^{n}-U_{i,j}^{n}}{\Delta x}\right),$ (6a)
$W_{i,j+\frac{1}{2}}^{n+1}=G_{i,j+\frac{1}{2}}^{n}\left(\frac{U_{i,j+1}^{n+1}-U_{i,j}^{n+1}}{\Delta
y}\right)+\frac{\tau\,G_{i,j+\frac{1}{2}}^{n}}{\Delta
t}\left(\frac{U_{i,j+1}^{n+1}-U_{i,j}^{n+1}}{\Delta
y}-\frac{U_{i,j+1}^{n}-U_{i,j}^{n}}{\Delta y}\right).$ (6b)

The coefficients are chosen as the arithmetic mean at interfaces:

$G_{i+\frac{1}{2},j}^{n}=\frac{G_{i,j}^{n}+G_{i+1,j}^{n}}{2},\qquad
G_{i,j+\frac{1}{2}}=\frac{G_{i,j}^{n}+G_{i,j+1}^{n}}{2},$ (7)

where $G_{i,j}^{n}$ is an approximation for $g(x_{i}\,,\,y_{j}\,,\,t_{n})$.

We chose to use the coefficients at time $t_{n}$. This choice leads to an
algebraic linear system over the unknowns $U_{i,j}^{n+1}$. The algebraic
equations may be written as

$a_{i,j-\frac{1}{2}}\,U_{i,j-1}^{n+1}+a_{i-\frac{1}{2},j}\,U_{i-1,j}^{n+1}+a_{i,j}\,U_{i}^{n+1}+a_{i+\frac{1}{2},j}\,U_{i+1,j}^{n+1}+a_{i,j+\frac{1}{2}}\,U_{i,j+1}^{n+1}=b_{i,j},$
(8)

where,

$\begin{array}[]{ll}b_{i,j}=U_{i,j}^{n}-\tau&\left[\,\dfrac{G_{i-\frac{1}{2},j}^{n}\,U_{i-1,j}^{n}-(G_{i-\frac{1}{2},j}^{n}+G_{i+\frac{1}{2},j}^{n})U_{i,j}^{n}+G_{i+\frac{1}{2},j}^{n}\,U_{i+1,j}^{n}}{\Delta
x^{2}}\right.\\\
&\left.+\,\dfrac{G_{i,j-\frac{1}{2}}^{n}\,U_{i,j-1}^{n}-(G_{i,j-\frac{1}{2}}^{n}+G_{i,j+\frac{1}{2}}^{n})U_{i,j}^{n}+G_{i,j+\frac{1}{2}}^{n}\,U_{i,j+1}^{n}}{\Delta
y^{2}}\right],\end{array}$ (9)

and the coefficients are given by

$a_{i,j-\frac{1}{2}}=-\frac{(\Delta t+\tau)}{\Delta
y^{2}}\,G_{i,j-\frac{1}{2}}^{n},\qquad a_{i,j+\frac{1}{2}}=-\frac{(\Delta
t+\tau)}{\Delta y^{2}}\,G_{i,j+\frac{1}{2}}^{n},$ (10a)
$a_{i-\frac{1}{2},j}=-\frac{(\Delta t+\tau)}{\Delta
x^{2}}\,G_{i-\frac{1}{2},j}^{n},\qquad a_{i+\frac{1}{2},j}=-\frac{(\Delta
t+\tau)}{\Delta x^{2}}\,G_{i+\frac{1}{2},j}^{n},$ (10b)
$a_{i,j}=-(a_{i,j-\frac{1}{2}}+a_{i,j+\frac{1}{2}}+a_{i-\frac{1}{2},j}+a_{i+\frac{1}{2}},j)+1$
(10c)

Finally, to impose the boundary conditions, we have written the algebraic
equations for the subdomains that intersect the boundary.

For each time step, we have to solve a linear system
$\mathbf{A}^{n}\,\mathbf{U}^{n+1}=\mathbf{b}^{n}$, where $\mathbf{U}^{n+1}$ is
the vector of unknowns (numerical solution approximation at $t_{n+1}$) and
$\mathbf{b}^{n}$ is the vector of right-hand side terms (9). The matrix
$\mathbf{A}^{n}$ of linear system is symmetric positive definite and sparse,
thus efficient methods may be applied to solve the algebraic system. In this
work, we use the Preconditioned Conjugate Gradient Method (PCG).

In this work, we consider the $g(x\,,\ y\,,\,t)\equiv 1$, thus Equation (2)
becomes a linear pseudo-parabolic equation. For an image to be processed, each
subdomain $\Omega_{i,j}$ corresponds to a pixel, i.e., the mesh parameters are
$\Delta x=\Delta y=1$. We choose the time step $\Delta t=\Delta x$ and the
damping coefficient $\tau=5$.

Algorithms 1 and 2 shows the computational routines involved in the described
numerical scheme in a pseudo-code language. Algorithm 2 refers to the inner
subdomains. Those sudomains intercepting the boundary can be implemented
similarly with only small adaptations. More details can be found in Vieira and
Abreu (2018).

Algorithm 1 Algorithm.

1:$U^{0}$

2:$U^{n}$

3:for $n=1\textrm{ to }N$ do

4: $[A,b]\leftarrow\mathrm{LINSYST}(U^{n-1})$

5: $U^{n}\leftarrow\mathrm{PCG}(A,b,U^{n-1})$

6:end for

Algorithm 2 Linear system.

1:function LINSYST($G$,$U$)

2: $[m,l]\leftarrow\mathrm{size}(U)$

3: $A\leftarrow\mathrm{zeros}(m\cdot l\,,\,m\cdot l)$

4: $b\leftarrow\mathrm{zeros}(m\cdot l\,,\,1)$

5: for $j=2\textrm{ to }l-1$ do

6: for $i=2\textrm{ to }m-1$ do

7: $k=(j-1)\cdot m+i$

8: $A_{k,k}\leftarrow 4\,(1+\tau)+1$

9: $A_{k,k-m}\leftarrow-(1+\tau)$

10: $A_{k,k-1}\leftarrow-(1+\tau)$

11: $A_{k,k+1}\leftarrow-(1+\tau)$

12: $A_{k,k+m}\leftarrow-(1+\tau)$

13: $b_{k}\leftarrow
U_{i,j}-\tau\,[U_{i,j-1}+U_{i-1,j}-4\,U_{i,j}+U_{i+1,j}+U_{i,j+1}]$

14: end for

15: end for

16: return $[A,b]$

17:end function

Table 1: Variables used in Algorithm 1 and 2. $U^{0}$ | Matrix $m\times l$ containing the texture image in gray scales
---|---
$U^{n}$ |
$A$ | Matrix $m\cdot l\times m\cdot l$ for algebraic linear system
$b$ | Array $m\cdot l\times 1$ containing independent terms of linear system
Table 2: Pre-programmed routines supposed to be available by Algorithm 1 and 2. $\mathrm{size}(U)$ | Return the size (number of rows and columns) of the matrix $A$
---|---
$\mathrm{PCG}(A,b,U)$ | Return the solution of the linear system $Ax=b$ by the PCG method with
| initial approximation $U$

###  3.2 Local binary patterns

Local binary patterns (LBP) Ojala et al. (2002) are gray-level texture
descriptors, developed mainly for texture classification purposes. In its
original and most simple version, LBP assigns a code to each pixel (central
pixel) taking into consideration its gray value $g_{c}$ and the gray values
$g_{p}$ of $P$ neighbor pixels equally spaced on a circle centered at the
central pixel and with radius $R$. Such code is provided by

$LBP_{P,R}=\sum_{p=0}^{P-1}H(g_{p}-g_{c})2^{p},$ (11)

where $H$ is the step (Heaviside) function: $H(x)=1$ if $x\geq 0$ and
$H(x)=0$, otherwise, and the coordinates of $g_{p}$ are given by

$(x_{p},y_{p})=(x_{c}-R\sin(2\pi p/P),y_{c}+R\cos(2\pi p/P)),$ (12)

being $(x_{c},y_{c})$ the coordinates of the central pixel. If $(x_{p},y_{p})$
are not integer values, $g_{p}$ value is obtained by linear interpolation.

Here we use an improved version of (11), which ensures rotation invariance and
is more discriminative. It discards patterns that are very similar, the so-
called uniform patterns. The new descriptor is given by

$LBP_{P,R}^{riu2}=\left\\{\begin{array}[]{ll}\sum_{p=0}^{P-1}H(g_{p}-g_{c})2^{p}&\mbox{if
}\mathcal{U}(LBP_{P,R})\geq 2\\\ P+1&\mbox{otherwise},\end{array}\right.$ (13)

provided that

$\mathcal{U}(LBP_{P,R})=|H(g_{P-1}-g_{c})-H(g_{0}-g_{c})|+\sum_{p=1}^{P-1}|H(g_{p}-g_{c})-H(g_{p-1}-g_{c})|.$
(14)

###  3.3 Proposed descriptors

Recalling the notation in Section 3.1, the texture descriptors developed here
are obtained by evolving the pseudo-parabolic operator on the image
$u(\cdot,\cdot,t_{0})$ over a range of time $t_{1},t_{2},\cdots,t_{N}$. In
this way we have a family of transformed images $\\{u(\cdot,\cdot,t_{n})\\}$
for $n=1,\cdots,N$. Empirically we found out that $N=50$ provide a reasonable
compromise between computational cost and recognition accuracy.

In the next step, we apply the $LBP_{P,R}^{riu2}$ operator over each evolved
image. We adopted the combinations of $P,R$ values listed in Table 3.

Table 3: Parameters of $LBP_{P,R}^{riu2}$ codes used in this work. P | 1 | 2 | 3 | 4
---|---|---|---|---
R | 8 | 16 | 24 | 24

These combinations were chosen both based on recent publications on LBP, e.g.
Liu et al. (2017), which demonstrated their suitability, and on empirical
tests specific for the problem that we are addressing here.

Finally, as usual in the LBP pipeline, we compute the histogram $\mathrm{h}$
for all possible values of $LBP_{P,R}^{riu2}$ codes. We can formally summarize
the descriptors by

$\mathfrak{D}(u)=\bigcup_{\begin{subarray}{c}(P,R)=\\{(8,1),\\\
(16,2),(24,3),\\\
(24,4)\\}\end{subarray}}\bigcup_{n=0}^{N}\mathrm{h}(LBP_{P,R}^{riu2}(u(\cdot,\cdot,t_{n}))).$
(15)

We end up with a large feature vector for each image, with 4000 components in
each one. This requires some strategy to avoid issues like the dimensionality
curse. To cope with that we apply Karhunen-Loève transform Pearson (1901) and
select the number of uncorrelated features that provides sufficiently
discriminative capacity. Figure 1 shows a flow chart of the main steps
involved in the proposed method, whereas Figure 2 visually illustrates the
outcomes of those corresponding steps for an exemplar texture image. To
facilitate the visualization, we show only some of the parameter values
($t_{n}$, $P$ and $R$) used to provide the descriptors.

Figure 1: Flow chart of the proposed method. Figure 2: Outcomes of the most
important steps of the proposed descriptors: original texture image,
application of the pseudo-parabolic operator, LBP encoding and histogram.

###  3.4 Motivation

The numerical model employed here is inspired by works like van Duijn et al.
(2007), where the the Buckley-Leverett equation is extended to a two-phase
flow in porous media with dynamic capillary pressure, yielding the following
general equation:

$\frac{\partial\,}{\partial t}(\phi u)+\frac{\partial F(u)}{\partial
u}=\frac{\partial}{\partial x}\left(G(u)\frac{\partial}{\partial
x}\left(J(u)+\tau\frac{\partial\,}{\partial t}(\phi u)\right)\right).$ (16)

This model presents some important characteristics that are helpful for an
application like the one described here. First, it is sufficiently flexible to
represent a diversity of physical phenomena, partially due to the existence of
a third order mixed derivative term. It can be demonstrated, for example, that
the coefficient $\tau$ is critical for the type of solution profile with
respect to the regularized Buckley-Leverett problem. Such parameter acts as a
bifurcation threshold: if $\tau$ is within a certain range, the solution
profile is similar to the classical shock waves; on the other hand, for higher
values of $\tau$, the profile changes abruptly and new types of shock waves
take place in the solution. In some extreme cases, the solution may contain a
non-monotone profile or even exhibit damped oscillation.

Another interesting characteristic is that pseudo-parabolic equations like
that in van Duijn et al. (2007) can also be interpreted as a regularization of
the hyperbolic Buckley-Leverett equation Abreu and Vieira (2017); Abreu et al.
(2020). Nevertheless, unlike classical regularization schemes based, for
example, on standard diffusion processes, the pseudo-parabolic term attenuates
the smoothing effect typically associated to parabolic solutions. It is known,
for instance, that jump discontinuities present at some time persist in the
solution for the remaining of the time evolution. In practice, this implies
that the model employed here is strongly characterized by regularity
preservation. Whereas parabolic differential equations are known by their
effect of increasing regularity regardless the information conveyed by the
image, here the edges and other important elements of the image expressed by
discontinuities are not vanished by the operator with the same aggressiveness.
Preserving discontinuities is an important feature in space-scale approaches
to image analysis and well explored, for example, in the classical work of
Perona and Malik Perona and Malik (1990).

Generally speaking, the numerical modeling employed here consists of a fully
coupled space-time approach, capable of appropriately accounting for the
diffusive flux naturally appearing in the pseudo-parabolic equation. In terms
of its use as an image analysis tool, it implies in controlled conditions for
the regularization process. Being followed by the local encoding of binary
patterns makes the proposed descriptors robust in many aspects both to
geometrical variations addressed by LBP as well as to edge localization
preserved by the nonlinear differential operator.

##  4 Experiments

The accuracy of the proposed method in texture classification is assessed over
three benchmark data sets and compared with other approaches, some of them
classically used for this purpose, other ones from the state-of-the-art in the
literature.

UIUC Lazebnik et al. (2005) is a data set of gray-level texture images. Each
image has dimensions 256$\times$256 and we have 40 images per class and 25
classes, totalizing 1000 images. Those images are collected under uncontrolled
conditions with variations in scale, illumination, perspective and albedo. The
training/testing split follows the protocol classically used with half of the
images from each class randomly selected for training and the remaining ones
for testing. Such random division is repeated 10 times to compute the average
accuracy of the classification process.

UMD Xu et al. (2009) is another gray-scale texture database that shares some
similarities with UIUC, like the number of classes and images per class. The
images are also acquired under uncontrolled conditions. Nevertheless, UMD
images have high resolution, each sample has dimensions 1280$\times$960\. The
intra-class variation in scale and viewpoint is also more intense, which adds
difficulties to the class-wise discrimination. The training/testing split
follows the same protocol of UIUC.

Finally, KTHTIPS-2b Hayman et al. (2004) is a collection if color texture
images (here we convert the samples to gray-scales), comprising a total of
4752 images divided into 11 classes, each one corresponding to a particular
material, like wood, plastic, bread, etc. The most important motivation behind
the development of this database is the focus on the real material regardless
the conditions under which the object was photographed. Each class is divided
into 4 samples, and each sample represents a particular setting of
illumination, scale and pose. The training/testing split is the classically
used in the literature, where one sample is used for training and the
remaining ones are employed for testing.

The feature vectors resulting from the concatenation of different transform
parameters and LBP settings are large and a Karhunen-Loéve is used to reduce
dimensionality. Finally, we tested three classifiers to process the
descriptors, i.e., linear discriminant analysis (LDA) Fisher (1936), support
vector machines (SVM) Cortes and Vapnik (1995), and random forests (RF) Ho
(1995).

##  5 Results and Discussion

Figure 3 shows the classification accuracy when using LDA, SVM and RF. All
these classifiers use the data in its original form, without any
transformation. In this way they are expected to highlight the discriminative
power of the descriptors. LDA outperforms SVM and RF in this task with
relevant difference. The nature of each classifier can explain such
difference. LDA was originally developed to handle multi-class problems,
avoiding strategies like “one-against all” that need to be employed by SVM.
Especially in ideal situations when the classes are balanced, methods dealing
with all classes on an equal footing tend to work more appropriately.

Figure 3: Classification accuracy using different classifiers.

Figure 4 depicts the behavior of the classification accuracy when the
parameter $N$ (maximum iteration time of the PDE operator) is varied. This is
actually the most relevant free parameter that we have to tune in this method.
According to the plots, the results achieved by $N\geq 50$ are quite similar.
This is the critical point where the image is so smoothed that no more
relevant information can be captured by the image descriptors. Based on this
outcome, we employ $N=50$ for the remaining experiments.

|
---|---

---

Figure 4: Accuracy for different values of $N$.

Table 4 lists the accuracy (percentage of images assigned by the classifier to
the correct classes) for each database and compared with results recently
published in the literature on the same databases. Our approach outperforms
several “handcrafted” descriptors and even some learning-based methods like
those described in Cimpoi et al. (2016). Whereas in UIUC and UMD several
solutions have already been proposed that achieve accuracy close to 100%,
KTHTIPS-2b is much more challenging, especially due to its training protocol,
in which the algorithm should be able to recognize a material at different
viewpoints, illuminations, etc. based on features representing only the
material at an original condition. Even here, our proposal outperforms several
“hand-engineered” descriptors like SIFT and LBP and was also competitive with
the modern CNN-based approaches.

Table 4: Accuracy of the proposed descriptors compared with other texture
descriptors in the literature. A superscript 1 in KTHTIPS-2b means training on
three samples and testing on the remainder.

KTHTIPS-2b | UIUC
---|---
| Method | Acc. (%)
---|---
VZ-MR8 Varma and Zisserman (2005) | 46.3
LBP Ojala et al. (2002) | 50.5
VZ-Joint Varma and Zisserman (2009) | 53.3
BSIF Kannala and Rahtu (2012) | 54.3
LBP-FH Ahonen et al. (2009) | 54.6
CLBP Guo et al. (2010a) | 57.3
SIFT+LLC Cimpoi et al. (2016) | 57.6
ELBP Liu et al. (2012) | 58.1
SIFT + KCB Cimpoi et al. (2014) | 58.3
SIFT + BoVW Cimpoi et al. (2014) | 58.4
LBPriu2/VAR Ojala et al. (2002) | 58.511footnotemark: 1
PCANet (NNC) Chan et al. (2015) | 59.411footnotemark: 1
RandNet (NNC) Chan et al. (2015) | 60.711footnotemark: 1
ScatNet (NNC) Bruna and Mallat (2013) | 63.711footnotemark: 1
Proposed | 65.5
| Method | Acc. (%)
---|---
RandNet (NNC) Chan et al. (2015) | 56.6
PCANet (NNC) Chan et al. (2015) | 57.7
BSIF Kannala and Rahtu (2012) | 73.4
VZ-Joint Varma and Zisserman (2009) | 78.4
LBPriu2/VAR Ojala et al. (2002) | 84.4
LBP Ojala et al. (2002) | 88.4
ScatNet (NNC) Bruna and Mallat (2013) | 88.6
MRS4 Varma and Zisserman (2009) | 90.3
SIFT + KCB Cimpoi et al. (2014) | 91.4
MFS Xu et al. (2009) | 92.7
VZ-MR8 Varma and Zisserman (2005) | 92.8
DeCAF Cimpoi et al. (2014) | 94.2
FC-CNN VGGM Cimpoi et al. (2016) | 94.5
CLBP Guo et al. (2010a) | 95.7
Proposed | 96.1

UMD
---
| Method | Acc. (%)
---|---
PCANet (NNC) Chan et al. (2015) | 90.5
RandNet (NNC) Chan et al. (2015) | 90.9
ScatNet (NNC) Bruna and Mallat (2013) | 93.4
MFS Xu et al. (2009) | 93.9
LBPriu2/VAR Ojala et al. (2002) | 95.9
LBP Ojala et al. (2002) | 96.1
BSIF Kannala and Rahtu (2012) | 96.1
DeCAF Cimpoi et al. (2014) | 96.4
FC-CNN VGGM Cimpoi et al. (2016) | 97.2
SIFT + KCB Cimpoi et al. (2014) | 98.0
SIFT+BoVW Cimpoi et al. (2014) | 98.1
SIFT+LLC Cimpoi et al. (2016) | 98.4
CLBP Guo et al. (2010a) | 98.6
Proposed | 99.3

Another important test in texture recognition is on how the classifier
performance is affected by a reduced number of samples for training. In
particular, results on UIUC varying the number of training images are fairly
common in the literature, possibly because the usual protocol (20 images for
training) can be considered as not sufficiently challenging. Figure 5
illustrates how the accuracy of our proposal is affected by changing the
number of training samples in comparison with other methods whose
corresponding results are also published in the literature. More specifically,
we show the accuracy when using 10, 15 and 20 training samples and compare
with MRS4 Varma and Zisserman (2009), MR8 Varma and Zisserman (2009) and
affine spin Lazebnik et al. (2005). It can be noticed that the proposed
algorithm can be considered robust to the reduction of the training subset.
Its accuracy was only a little smaller than affine spin for 10 training images
but was larger than the compared approaches in other situations.

Figure 5: Accuracy when the number of training samples in UIUC database is set
to 5, 10, 15, and 20.

Figure 6 depicts the confusion matrices for the proposed method on the
benchmark datasets. In that representation, a gray map is employed to visually
express the number of samples assigned by the classifier to class $A$ and
actually pertaining to class $B$. An ideal matrix would present solid diagonal
and no gray point outside. The pictures confirm the accuracy achieved for each
database. UIUC and UMD for example has a nearly solid diagonal, as expected
from their success rate in the classification close to 100%. KTH-TIPS2b, on
the other hand, has a much more complex picture. It is interesting to observe
here the classes that were more difficult to classify. In this case, classes 3
(“corduroy”), 5 (“cotton”), 8 (“linen”), and 11 (“wool”) were the most usually
confused. All these materials are related, as they are types of fabric, which
makes such confusion somehow expected.

|  |
---|---
(a) | (b)
|
---
(c)
Figure 6: Confusion matrices. (a) KTHTIPS-2b. (b) UIUC. (c) UMD.

In summary, the proposed methodology achieved an accuracy in the
classification of benchmark texture databases that is competitive with the
state-of-the-art in this topic. And this was obtained by a method that does
not require neither so much computational power nor large amounts of data for
training. The use of pseudo-parabolic PDE-based descriptors was expected to
provide meaningful descriptors as it filters out unnecessary details and noise
while effectively preserving discontinuity regions. Here, this model confirmed
its capacity of providing an alternative description of the way that pixels
are locally related on the image. Here such locality is quantified by the
well-known LBP codes and such combination demonstrated its powerfulness in the
achieved results.

###  5.1 Application

We also apply the proposed descriptors to a practical problem, to know, the
identification of Brazilian plant species based on scanned images of leaf
surface. The samples are collected in vivo, washed and aligned with the
basal/apical axis. The analyzed data set is called 1200Tex Casanova et al.
(2009) and comprises a total of 1200 images, previously converted to
grayscales, corresponding to 20 species (classes). Each image has dimension
$128\times 128$.

Table 5 lists the accuracy for the proposed PDE descriptors on this task in
comparison with other results recently published in the literature for the
sabe database. It is particularly noticeable how deep learning approaches like
FC-CNN and FV-CNN are outperformed by the proposed method.

Table 5: State-of-the-art accuracies for 1200Tex. Method | Accuracy (%)
---|---
LBPV Guo et al. (2010b) | 70.8
Network diffusion Gonçalves et al. (2016) | 75.8
FC-CNN VGGM Cimpoi et al. (2016) | 78.0
FV-CNN VGGM Cimpoi et al. (2016) | 83.1
Gabor Casanova et al. (2009) | 84.0
FC-CNN VGGVD Cimpoi et al. (2016) | 84.2
Schroedinger Florindo and Bruno (2017) | 85.3
SIFT + BoVW Cimpoi et al. (2014) | 86.0
Proposed | 87.2

Figure 7 depicts the corresponding confusion matrix for the proposed
descriptors on 1200Tex. The matrix is nearly diagonal, as ideally expected,
and the most relevant confusion takes place at class 8 (confused with class
6). Figure 8 shows a few samples from those two groups illustrating the
complexity of separating the respective classes. It is known in Botany that
the most important elements to distinguish species when looking into leaf
surface are nervures, especially primary nervure. And in Figure 8 we observe
that those structures are pretty similar in both classes, such that the
limited performance was expected.

Figure 7: Confusion matrix for 1200Tex.
Figure 8: Illustrative samples of the most confused classes in 1200Tex.

In general terms, we see a good classification performance, competitive with
the most advanced methods in the state-of-the-art. This is quite interesting
considering that we have a model that is neither computationally expensive nor
depends on large amounts of data for training. It also confirms the potential
of the developed methodology in practice, especially in problems where visual
texture is a relevant attribute, a common situation for example in biological
applications and material sciences.

##  6 Conclusions

This work presented a methodology to recognize texture images based on the
application of a multiscale operator derived from the pseudo-parabolic
diffusion PDE model. Following the operator application, LBP codes are
extracted from the transformed images to compose the feature vector, which is
finally used as input to a classifier.

The method was tested on benchmark databases and in a practical application in
Botany. The classification accuracy was compared to classical and state-of-
the-art texture recognition approaches. The results showed that our proposal
was competitive even with the most recently published approaches on this
topic. This also confirmed how such “hand-crafted” descriptors can still be
useful, especially when the computational structure at hand is not
sufficiently powerful or when large amounts of data are not available, a
common situation in areas like medicine, for example.

It is also important to mention that the model presented here and its
effectiveness in a texture recognition task suggests the future investigation
of even more complex models, such as that in Abreu and Vieira (2017); Abreu et
al. (2020), for instance.

## Conflict of interest

The authors declare that they have no conflict of interest.

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