Foot Pathologies Classiﬁcation From Pressure

Distribution Over The Foot Plant

⋆

Jos

e Garc

ıa-Hern

andez

, Roberto Paredes

, David Garrido

and Carlos Soler

Instituto Tecnol

ogico de Inform

atica

Instituto de Biomec

anica de Valencia

Universidad Polit

ecnica de Valencia

Camino de Vera s/n, 46071 Valencia (Spain)

Abstract. Nowadays, approximately 35% of the population suffers problems in

the feet. In many cases, some foot pathologies are treated by means of especial

insoles. These insoles require customisation for the success of their treatment.

However, there is a great lack of knowledge in this sector. As an initial step it

is essential to develop a new objective tool capable to classify among the dif-

ferent foot pathologies. In this paper we present a system to classify different

foot pathologies which uses the Nearest Neighbor (NN) classiﬁcation rule based

on weighted metrics and prototypes selection. The system uses as input data the

pressure distribution over the foot plant.

1 Introduction

In the developed countries, approximately 35% of the population suffers problems in the

feet enough serious to need medical attention. The illnesses of the feet are recognised

as a great social problem in most of these countries, being most of the people affected

women. A high percentage of the population of the European Union over 40 years old

has some type of problem in their feet (above 90% in Spain, United Kingdom, Germany

or Holland).

In many cases, some foot pathologies are treated by means of especial insoles whose

principal aim is to protect the feet against external aggressions (overpressures, shear

forces, etc.) and to maintain the articulations in a suitable anatomical position. These

insoles require customisation for the success of their treatment. However, there is a great

lack of knowledge in this sector. On the one hand, there is not information regarding

the best orthetic solution depending on the pathology. On the other hand, in the health

area the ﬁnal adaptation of the insole to the patient highly depends on the specialist

experience and on the capability of the patient to transmit the problems to the orthotics

use. This lack of knowledge, together with a low technology in the design process,

causes that the orthotics customisation process is performed at present without objective

criteria and in a traditional way which implies a process with low efﬁciency.

⋆

Work supported by the “Agencia Valenciana de Ciencia y Tecnolog

ıa (AVCiT)” under grant

GRUPOS03/031 and the Spanish Project DPI2004-08279-C02-02

Gar

cıa-Hernández J., Paredes R., Garrido D. and Soler C. (2005).

Foot Pathologies Classiﬁcation From Pressure Distribution Over The Foot Plant.

In Proceedings of the 1st International Workshop on Biosignal Processing and Classiﬁcation, pages 87-94

DOI: 10.5220/0001195200870094

 SciTePress

Therefore, as an initial step and taking into account the information presented above,

it is essential to develop a new objective tool capable to classify among the different

foot pathologies, disorders or problems in order to optimise and focus the strategies of

design to improve treatments and footwear functionality.

In this paper we present a system to classify different foot pathologies from pressure

distribution over the foot plant during walking. For this purpose, we compare the k-

Nearest Neighbor (kNN) classiﬁcation rule using the well known Euclidean Distance

with the 1-NN classiﬁcation rule improved by the Learning Prototypes and Distance

(LPD) method [1, 2]. While at section 2 we present the LPD method, at section 3 is

shown the database used in the experiments presented at section 4.

2 Methodology

The Nearest neighbor (NN) classiﬁcation rule has successfully been used in many pat-

tern recognition applications. The good behaviour of this rule with unbounded number

of prototypes is well known. However, in many practical classiﬁcation problems only a

small number of prototypes is usually available.

The NN performance can be improved by using appropriately trained distance mea-

sures of metrics. Improvements can be particularly signiﬁcant for small prototype sets.

Trained metrics can be global (the same for all the prototypes [3–5], class-dependent

(shared by all the prototypes of the same class) [6–8] and/or locally-dependent (the

distance measure depends on the particular position of the prototypes) in the feature

space [9–12].

More concretely, in this work we have used (comparing with the well known Euclid-

ean Distance) the approach called Learning Prototypes and distances (LPD), proposed

in [1,2]. It stars with an initial selection of a small number of randomly selected proto-

types from the training set. Then it iteratively adjusts both the position (features) of the

prototypes themselves and the corresponding local-metric weights, so that the resulting

combination of prototypes and metric minimises a suitable estimation of the probabil-

ity of classiﬁcation error. The adjustment rules are derived by solving the minimisation

problem through gradient descent.

2.1 Approach

Let T = {x

, . . . , x

} be a training set; i.e., a collection of training vectors or class-

labelled points x

∈ E, 1 ≤ i ≤ N in a suitable representation space. In our case, each

training vector (prototype) x

represents each subject. The components of this vector

are the pressure values of the subject’s plant foot acquired from a speciﬁc sensor. Being

M the number of available plant foot sensors, the training vector x

∈ R

, that is the

representation space E = R

. Each prototype belongs to a particular class, among C

different classes, being each class a different foot pathology.

The goal of the LPD algorithm is to use T to obtain a reduced set of prototypes,

P = {y

, . . . , y

} ⊂ E, n ≪ N, and a suitable weighted distance d : E × P → R

associated to P , which optimise the Nearest Neighbor (NN) classiﬁcation performance.

Initially P is a subset of T randomly selected.

The weighted distance from an arbritrary vector x ∈ E to a prototype y

∈ P is

deﬁned as:

(x, y

) =

j=1

− y

)

(1)

where w

is a weight associated with the feature j of the prototype y

. That is, in our

problem, w

is the weight associated with the plant foot sensor j. These weights can

be represented as an N × M weight matrix W = {w

, 1≤i ≤ N, 1≤j ≤ M }. In what

follows, whenever the weight matrix W can be understood from the context, d

(x, y

)

will be simply denoted as d(x, y

Note that this deﬁnition assigns separate weights to the different dimensions or fea-

tures of the representation space. Note also that it is asymmetric and local in that it

depends on the particular position of each y

Learning the Prototypes and their Weights In order to ﬁnd both a matrix W and a

suitable reduced set of prototypes P that results in a low error rate of the NN classiﬁer,

we propose minimise a criterion index which is an approximation to the NN classiﬁca-

tion error of T using P and d(·, ·). This NN error estimate can be written as:

J(P, W ) =

x∈T

step

d(x, y

)

d(x, y

)

(2)

where the step function is deﬁned as

step(z) =

(

0 if z ≤ 1

1 if z > 1

and y

, y

∈ P are, respectively, the same-class and different-class NNs of x. Note

that each term of the above sum (2) involves two different prototypes y

, y

, and their

associated weights.

As in previous work [13, 14, 8, 15], a gradient descent procedure is used to minimise

this index. This requires J to be differentiable with respect to all the parameters to be

optimised; i.e.: y

and w

, 1 ≤ i ≤ k, 1 ≤ j ≤ M. Therefore some approximations

are needed. First, the step function will be approximated by using a sigmoid function,

deﬁned as:

(z) =

1 + e

β(1−z)

(3)

With this approximation, the proposed index becomes:

J(P, W ) ≈

x∈T

(r(x)) (4)

where r(x) =

d(x, y

)

d(x, y

)

(5)

Clearly, if β is large S

(z) ≈ step(z), ∀z ∈ R, and this approximation is very accurate.

On the other hand, if β is not so large, the contribution of each NN classiﬁcation error

(or success) to the index J depends upon the corresponding quotient of the distances

responsible for the error (or the success). As will be discussed later, in some cases this

can be a desirable property which may make the sigmoid approximation preferable to

the exact step function. The derivative of S

(·) will be needed throughout the paper:

′

(z) =

d S

(z)

d z

βe

β(1−z)

(1 + e

β(1−z)

)

(6)

′

(z) is a “windowing” function which has its maximum for z = 1 and vanishes for

|z − 1| >> 0. Note that if β is large S

′

(z) approaches the Dirac delta function, while

if it is small S

′

(z) is approximately constant for a wide range of values of z.

To obtain the partial derivatives from (4-5), required for gradient descent, it should

be noted that J depends on P and W through the distances d(·, ·) in two different ways.

First, it depends directly through the prototypes and weights involved in the deﬁnition

of d(·, ·) (1). The second, more subtle dependence is due to the fact that, for some

x ∈ T , y

and y

may be different as prototype positions and their associated weights

are varied.

While the derivatives due to the ﬁrst, direct dependence can be developed from (4-5),

the secondary dependence is non-linear and is thus more problematic. Therefore a sim-

ple approximation will be followed here by assuming that the secondary dependence

is not signiﬁcant compared to the direct one. In other words, we will assume that, for

sufﬁciently small variations of the positions and weights, the prototype neighborhoods

remain unchanged. Correspondingly, we can derive from (1) and (4-5):

∂J

∂y

≈

∀x∈T :

index(y

) = i

′

(r(x)) r(x)

− x

)

(x, y

)

−

∀x∈T :

index(y

) = i

′

(r(x)) r(x)

− x

)

(x, y

)

(7)

∂J

∂w

≈

∀x∈T :

index(y

) = i

′

(r(x)) r(x)

− x

)

(x, y

)

−

∀x∈T :

index(y

) = i

′

(r(x)) r(x)

− x

)

(x, y

)

(8)

where r(x) and S

′

(·) are as in (5) and (6), respectively. Using these derivatives leads

to the corresponding gradient descent update equations. A simple manner to implement

these equations is by visiting each prototype x in T and updating the positions and the

weights associated with the same-class and different-class NNs of x.

The effects of the update equations in the LPD algorithm are intuitively clear. For

each training vector x, its same-class NN, y

= y

, is moved towards x, while its

different-class NN, y

= y

, is moved away from x. Similarly, the feature-dependent

weights associated with y

are modiﬁed so as to make it appear closer to x in a feature-

dependent manner, while those of y

are modiﬁed so that it will similarly appear farther

from x. Since these update steps are weighted by the distance ratio, r(x), their impor-

tance depends upon the relative proximity of x to y

or y

. This is further divided by

the corresponding squared distance, thereby reducing the update importance for large

distances. Finally, the resulting steps are windowed by the derivative of the sigmoid

function applied to the distance ratio, r(x). This way, only those prototypes (and their

weights) which are sufﬁciently close to the decision boundaries are actually updated.

3 The Database Description

For the classiﬁcation of the foot problems, pressure distribution over the foot plant

during walking was used (ﬁgure 1). To acquire the plantar pressure distribution, the

Biofoot

IBV pressure insoles were used (ﬁgure 1). The structure of the pressure ﬁles

(plain text) exported by the application Biofoot

IBV is based on a matrix with as many

rows as the sample frequency and the duration of the test deﬁne. The number of columns

will depend on the number of sensor of the instrumented insole, in our case 64. So, for

each subject (each training vector) n and for each foot, we have a matrix in the form:







. . . x

. . .

. . . x







being x

the registered pressure by the sensor j in the time t

. ¿From the matrixes

we compute the average pressure for each sensor from each foot, that is:

i=1

As there is a matrix for each foot, we built each training vector as:

subject



. . . x

128



Being x

from right foot when 1 ≤ i ≤ 64 and from the left one when 65 ≤ i ≤ 128.

Note that E = R

128

. Note also that x

subject

is the training vector acquired from the

subject A and the database can be represented as: T = {x

subject

, . . . , x

subject

As it is said above, the database is acquired by the Biofoot

IBV. It is a tool that

has been developed by the Institute of Biomechanics of Valencia and it consists of a

ﬂexible insole with up to 64 piezoelectric ceramics (ﬁgure 1) distributed according to

foot physiology in such a way that a greater density of sensors is placed under bony

areas where the pressure uses to reach the highest values, mainly on the heel and the

metatarsal heads.

Fig.1. Example of pressures (left) and Biofoot

IBV Insole (center and right).

Biofoot

IBV allows the study of the foot pressures during the normal conditions,

walking and wearing shoes. Besides, by means of a telemetry system the signal is trans-

mitted wireless to a PC, where it is saved (ﬁgure 2). On the other hand, and in order to

avoid the effect in pressure values of the walking speed, two photocell barriers placed

at a known distance were used. This procedure permits the researchers to remove the

trials out of the interval [1,1.5] m/s, which deﬁnes the normal speed gait.

Fig.2. Biofoot

IBV Use.

In order to obtain a representative sample of the foot problems from a statistically

point of view, 60 subject of heterogeneous characteristics carried out the trials with

their usual footwear. A specialist in foot diseases classiﬁed them in different groups of

pathologies (a class label) after an exhaustive evaluation. Working in this way, for each

subject we obtain a labelled training vector. The pathologies diagnosed were:

– Rheumatoid arthritis.

– Pain in the heel or in other foot parts different of plant.

– Plantar foot pain.

– Hallus Valgus (bunion).

– Hemiplegia.

– Neuropathic foot.

– Degenerative problems of the Central Nervous System.

– Diabetic foot.

– Consequences of fractures.

Due to the high incidence of Pain in the heel or in other foot parts different of plant

and Plantar foot pain, and the fact that the 85% of the subjects were classiﬁed inside of

these groups, the study was only focused on these pathologies. Besides, in the database

the other pathologies has not a number of subject as great as representative for this

work. So, the used database is composed by 49 subjects, that is, 49 training vectors,

that are included in the two pathologies mentioned above (17 labelled subjects as “Pain

in the heel or in other foot parts different of plant” and 32 as “Plantar foot pain”).

4 Experiments

The experiments were carried out with the Biofoot

IBV database. Only two classes

were taken into account, as is mentioned above. The classiﬁcation error of the proposed

method (LPD) is compared with the classiﬁcation error of the well known k-NN rule

based on the Euclidean Distance. This error is estimated using the well known Leaving

One Out (LOO) estimation procedure due to the reduced number of available training

vectors. In LOO, given N labelled training vectors, two prototypes sets are built. In the

ﬁrst one, called training set, we include N − 1 vectors and the other vector is included

(only it) in other set called test set. This process is repeated N times, each one including

a different vector in the test set and the others N − 1 in the training set. So, we do N

different experiments; each one with different training and test sets. In each experiment,

we train a classiﬁer with the training set and test its performance with the test set. The

idea is to work as we did not know the real label of the test vector (that is; the real

pathology of the subject that corresponds with this vector) and see if the classiﬁer is

able to ﬁnd the class. The ﬁnal result is the average over the N tests.

Results are shown in table 1. As can be seen, LPD results with reduced training

set and local weighted distances, are better than Euclidean Distance with the whole

training set results. The best result with LPD is obtained reducing the training set to

3 prototypes per class (14.3 % error rate), this result is a improvement of 20 % over

the best results obtained using k-NN with the Euclidean distance and all the training

prototypes (18.4 %). The LPD with a reduced set of the vectors is able to improve the

classiﬁcation error in front of the original training set.

Table 1. k-NN error rate using Euclidean distance with different k values (left) and 1-NN LPD

error rate for different prototypes per class (right).

k Error Rate

1 18.4 %

2 18.4 %

3 28.6 %

4 24.5 %

5 40.8 %

Prototypes per class Error Rate

1 16.3 %

2 16.3 %

3 14.3 %

4 16.3 %

These results are still worse than those given by an expert, which obtains about 5%

error rate. This relative bad result of the automatic methods is due to the low num-

ber of samples available to learn adequately the class distributions. Also, the proposed

preprocessing step takes the average of all the pressures along the time loosing some

important temporal relations that are present in the plant during walking. We hope to

improve the error rate in future works with more training samples and the use of other

classiﬁcation methods, as Hidden Markov Models (HMM), which allows us to treat the

foot plant pressures as a sequence of pressure vectors.

5 Conclusions and Future Work

A system to classify different foot pathologies, from pressure distribution over the foot

plant, has been presented. For this purpose, the Nearest Neighbor (NN) classiﬁcation

rule using the Euclidean Distance and the NN classiﬁcation rule improved by the LPD

method have been compared. The LPD obtained the best classiﬁcation results with a

(14.3 % error rate), an 20% of improvement over the NN rule using the original train-

ing set and the Euclidean distance. Future work will focus on obtaining more samples

and using appropriate classiﬁcation models (Hidden Markov Models) that allows us to

process the pressure of the foot plant as a temporal sequence of pressures.

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