A NEW PARALLELIZABLE DEFORMATION METHOD

Automatic Comparision between Foot and Last

J. M. Buades, M. Gonzalez-Hidalgo, Francisco J. Perales, S. Ramis-Guarinos,

A. Oliver and V. Blanch

Computer Graphics Vision and Artificial Intelligence Group, Department of Math. and Comp. Science,

University of Balearic Island, Anselm Turmeda Building, Crta. Valldemossa Km 7.5, E-07122 Palma de Mallorca, Spain

Keywords: Biometrics, Shoes, Foot, Footwear, Deformation, Anatomy.

Abstract: This document shows the research project developed by the Computer Graphics and Vision Group in

collaboration with Inescop.

The main objective of this project is to define a new methodology to design

shoes exactly adapted to the shape foot. The system proposed is based on a biomechanical anatomical

structure of the foot and using a deformable shape. Using with several joints adapted the different foots to

different shoe last evaluating the error between the surfaces. The system is automatic driven in the selection

of significant foot points. We consider several anthropometrical parts of the foot in order to apply the

deformations of the shape with different axis. The proposed system is implemented in software. An

associated Application Programming Interface (API) is also developed for commercial use of the system.

Also the results presented evaluate the error between deformations.

1 INTRODUCTION

The footwear manufacturers from Europe, and

particularly from Spain, need to go towards highly

technified and added value products, since they

cannot compete in price against the low cost

producing countries invading the markets.

Personalisation is an ideal strategy: It offers a high

added value and links the client to the enterprise.

Once the scanned feet of the client are registered, the

client could demand his personalised shoes from the

catalogue, even through the Internet, knowing the

shoes will fit. The system presented in this paper

project completes and follows one of the research

and development lines which is being carried out at

the INESCOP for several years, and is a clear

example of collaboration between industry and

university research group. Until now this project

(European ERGOSHOE project: http://www.ergo

shoe.inescop.es; also CEC-made shoe project: www.

cec-made-shoe.com) led to the development of a low

cost 3D foot scanner, a high precision 3D last

scanner and a software tool (Forma-3D) for last

design which also allows to superimpose and

compare different volumes (e.g. foot and last).

However there are still unsolved issues. With the

general aim in mind of improve the automatic

process of personalising footwear, in this paper we

address the issue of virtually deform the foot in

order to predict his geometry once it is placed upon

an elevated sole profile (e.g. high heels), basing on

the scan performed at feet flat on the ground and the

profile of the last sole. It is necessary to be able to

utilise the foot scan of the flat foot for multiple lasts,

and most of the lasts have a certain amount of heel

rise. The elevated sole profile is determined by the

last.

The idea of "measuring" the shoe fit to the foot is

not new. Jacob Lowe in 1927 obtained a patent for a

fluoroscope intended to check the fit of the shoes

visually (Lowe, 1927). However, and having the

right technology, have not made great progress since

then. The process of digitizing a foot and change the

last shape from its numerical representation is

something that is already being done at the

theoretical and practical level (Luximon et al., 2005;

Luximon et al., 2003; Mochimaru et al., 2000).

There have been some advances in foot volume

parameterization from 3D coordinates of discrete

points (Luximon et al., 2003; Luximon et al., 2004;

Luximon et al., 2003b), and numerous mathematical

models have appeared for describing the foot motion

(Arampatzis et al., 2002; Carson et al., 2001),

however, offer no information on the volume of the

foot. There are also models that describe the shape

363

M. Buades J., Gonzalez-Hidalgo M., J. Perales F., Ramis-Guarinos S., Oliver A. and Blanch V..

A NEW PARALLELIZABLE DEFORMATION METHOD - Automatic Comparision between Foot and Last.

DOI: 10.5220/0003860803630369

In Proceedings of the 1st International Conference on Pattern Recognition Applications and Methods (SADM-2012), pages 363-369

ISBN: 978-989-8425-98-0

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

Figure 1: Description of the deformation virtual process.

of the foot, but these tend to be static (Luximon et

al., 2004), some of then described the deformation of

the foot depending on the load (Houston et al., 2006)

and there are also attempts to record real-time full

volume during walking (CEC made shoe project).

But, in our concern, we have not found articles

performing deformation models of the foot for

applications with time restrictions.

Foot deformation into a high level position, as

we can see in Fig. 4, is not so simple. Techniques

such as Free Form Deformation (Mochimaru et al.,

2000) or warping to the silhouette of the shoe last

will not produce satisfactory results for such cases.

Using an analytical model of foot bones that will

lead to surface deformation could produce useful

results (Carson et al., 2001), but has a high

consuming time. Leon Kos et al. (2002) propose that

the scanned foot in the flat position should be

matched with a similar foot from the database to

obtain land mark similarities for the fitness analysis.

J. Leng et al. (2005) propose to use a distance map

to indicate how well the selected shoe last fits the

specific foot shape and to guide the deformation, in

this case, the idea is to deform existing last model to

fit a consumer’s foot shape with minimally affecting

the original last style, this process is achieved by

minimizing an equation. Only results using men’s

shoes with low profile are shown. As we can see a

typical idea is to select from an existing shoe lasts

database or deform an existing shoe last model into

one that fits the scanned foot data (Luximon et al.,

2005; Li et al., 2004; Mochimaru et al., 2000; Kim

et al., 2002; Cheng et al., 1999).

In this paper we propose a system to predict and

study the deformation of a consumer's foot when it

adapts to different lasts, for in a subsequent process,

design the best last that fits its foot.

The proposed system uses as input, a scanned

three-dimensional model of the foot, obtained

through the INESCOP 3D scanner (see (Telfer et al,

2010) for other 3D surface scanning), and also a

model of the last sole. The foot is scanned over a flat

plane and its orientation is irrelevant, since the

system is designed to automatically reposition the

foot to agree with the last sole orientation. The

process described in this paper is shown in Figure 1.

Moreover, the computations for each vertex of the

3D foot model are independently, therefore the

whole process is easily parallelizable. This fact is

important because it can have to compare the last

and foot deformation obtained with several last sole,

then the computation time is extremely important.

The paper is organized as follows. Section 2 is

devoted to the description of the deformation

process of the foot. The analysis of the results is

carried out in Section 3. Finally, the paper ends with

the conclusions and future work

2 DEFORMATION

In this section, we describe the method used for foot

deformation.

Through a 3D capture system for feet, we scan

three fundamentals elements: the foot, the platform

shoe and both together. We can process these three

elements by means of our algorithm. We will

describe the five steps of our algorithm: obtaining

significant foot points, division of foot areas,

calculation of the deformed sole, calculation of the

foot’s top and fusion of both parts.

2.1 Significant Foot Points

To obtain an accurate deformation we need to

calculate several significant foot points. The most

important points are IH, IF, MT, MF, HA and HB

(see Figure 2). To obtain these points we have that

the length of the foot is the x-coordinate, the width is

the y-coordinate and the height is the z-coordinate.

HF is the point with lowest x-coordinate and IF is

the point with the lowest distance between the point

HF and all the points of the top’s foot border line.

Now we divide the length of the foot in two

parts: forefoot (from the middle of the foot to the

toes) and hindfoot (from the middle of the foot to the

heel). In the forefoot we find the points MF and MT.

MF is the point with the lowest y-coordinate and MT

is the point with the highest y-coordinate. Doing the

same in the hindfoot we have that HA is the point

with the lowest y-coordinate and HB is the point

with the highest y-coordinate.

With the points MT and MF we create the plane

Deformed

model

Deformation

of the top

Comparison

with the

real foot

Scanned 3D

coordinates

(Foot, upswing,

foot raised)

Obtaining

significant

points

Deformation

of the sole

from a last shoe

ICPRAM 2012 - International Conference on Pattern Recognition Applications and Methods

364

Figure 2: Foot dimensions.

that divides the toes from the instep (Figure 3, left)

and with the points IF and HF we create the plane

that divide the instep from the ankle (Figure 3, left).

2.2 Division of the Foot Areas

The planes delimit the different parts of the foot. We

divide the foot into three areas as you can see on the

left of the Figure 3.

Figure 3: Left: planes of the foot. Right: areas of the foot.

We separate the foot into two principal parts:

(see Figure 3, right): the sole (purple colour) and the

top (red, blue and green areas). Each part will suffer

a different type of deformation according to their

characteristics.

2.2.1 Cut Section of the Sole

The sole of the shoe last is a fundamental part to

create a custom shoe because its shape will depend

on the user’s weight distribution on footwear. This

fact influences in the comfort of the shoe.

From the scanned and the significant points we

determine the cut of the sole, which is malleable to

the deformation line (see the red line of the Fig. 4).

The calculation of this cut is realized with the MT

point (Figure 2). All points that are below this point

are the sole.

2.2.2 Cut Section of the Top

The top requires a different type of deformation.

Must be generate a virtual skeletal from the 3D foot

model scanned and the joint axes allowing a more

accurate deformation. The foot model has two

principal joints: the metatarsal joint and the ankle

joint. Then the top is considered as the part that it

doesn’t belong to the sole.

2.2.2.1

Metatarsal Joint

The metatarsal bones are those responsible for the

toes movement. This joint is essential in the virtual

deformation, because it allows the correct rotation of

the forefoot. In this paper, we consider that this joint

is the most important for a good deformation,

because it is the most influential in the final shape of

the foot.

To find this joint, we get the most prominent

points of the forefoot (MT and MF). The axis of

rotation passes through these two points (see Fig. 2).

2.2.2.2 Ankle Joint

The ankle joint is obtained from the most prominent

points of the hindfoot (HA and HB). The axis of

rotation passes through these two points (see Figure

2) allowing joint movement.

2.3 Deformation of the Sole

The deformation of the sole is the first step to obtain

the final deformed foot. This part must be adapted

with precision to the deformation line. So, it needs a

different deformation to the top of the foot.

Once obtained the cut section that delimits the

sole, we need to obtain the deformation line. This

line is obtained from the platform. We cut the

platform by a plane that crosses longitudinally.

Then, we obtain the normal to every point of the

line. This is necessary to correctly locate points on

the deformed sole. The distance between them varies

according to deformation line. Thus we get simulate

the deformation suffered by the foot. In the Figure 4,

we show an example of the final deformation where

you can see the final deformation of the sole.

Figure 4: Deformed foot and the final deformation of the

sole in purple.

2.4 Deformation of the Top

On the right of Figure 3 we divide the foot into four

areas, but as we only want the top, we have three

parts: the toes area, the instep area and the ankle

area.

Each area is separated by a joint and it has a

different rotation angle. To join all parts, we must

A NEW PARALLELIZABLE DEFORMATION METHOD - Automatic Comparision between Foot and Last

365

make a smoothing. The result meets the C1

continuity. Therefore, we apply an interpolation

between the different areas (see Figure 4).

2.4.1 Rigid Areas

The toes, the instep and the ankle are considered

rigid parts of the foot. When the foot is deformed,

we apply it a composition of transformations:

)(

1−

⋅⋅= TRTM

(1)

It is basically moves the object to the origin (T),

rotates (R) and returns it to its original point (T

-1

Then,

1000

100

010

001

⎟

⎠

⎞

⎜

⎝

⎛

1000

100

010

001

⎟

⎠

⎞

⎜

⎝

⎛

−

(2)

1000

0cos0sin

0010

0sin0cos

⎟

⎠

⎞

⎜

⎝

⎛

−

αα

(3)

The axis of rotation is the y-axis both for the

metatarsal joint and the ankle joint, as we explained

above. Then,

1000

cossincos0sin

0010

sincossin0cos

⎟

⎠

⎞

⎜

⎝

⎛

+−−

zzx

xzx

αααα

(4)

To determine the rotation angles, we consider

three areas:

Figure 5: Left: rotation angle of the toes. Right: rotation

angle of the instep.

 Toes area. It calculates the most prominent

point of the foot, with the foot without

deforming (point A) and with the deformed

foot (point B). Then you need the furthest

point in the toes area. It is located at the height

of the plane that divides the toes area and the

instep area. We find this point with the foot

without deforming (point C) and with the

deformed foot (point D). The angle is formed

by the vectors C-A, D-B (see Figure 5, left).

 Instep area. You take the same point C and D

calculated for the toes area. We estimated the

farthest point that it is in the heel: the point

with the foot without deforming (point E) and

with the deformed foot (point F). The angle,

as shown in Figure 5 on the right, is the vector

formed by the E-C, D-F.

 Ankle area. The angle used to rotate the ankle

is the same that is used for the instep area,

except that now it will be negative. With this

we get that the ankle turn back to be straight

because first the instep is rotated and this

affects the ankle too (Figure 6). So, it was

subsequently applied the same rotation but

with a negative angle (Figure 7 and 8).

Figure 6: Deformed foot without rotated ankle.

It should be noted that the IF and HF points (Fig.

2) create the first plane that delimit the ankle area.

Figure 7: Left: unrotated ankle (one plane). Middle:

rotated ankle (one plane). Right: interpolation with one

plane.

As shown in Figure 7, with one plane; the

vertices get into the sole. To avoid this problem in

the first stage is realized an interpolation in this area

(see the right side of Figure 7), but results are not

entirely satisfactory. This is because the cut of the

sole has a different deformation. Thus, it was

considered convenient to use also a second plane

(Figure 8) to avoid the problem. Later we will

realize an interpolation that it is described below.

When we apply the interpolation we are obtained

more satisfactory results and more tight according to

the line of deformation. We can compare the results

obtained by the interpolation using one plane (Figure

7, right) and two planes (Figure 8, right),

respectively.

ICPRAM 2012 - International Conference on Pattern Recognition Applications and Methods

366

Figure 8: Left: unrotated ankle (two planes). Middle:

rotated ankle (two planes). Right: interpolation with two

planes.

2.4.2 Interpolation Areas

Figure 9 shows the interpolation areas, which are

bounded by the planes where there are the colour

transitions. The interpolation process is similar to

that used in the paper (Kavan, 2003).

Figure 9: Interpolation areas.

This zone is delimited by t

min

and t

max

. Where t

min

is the lower plane and t

max

is the higher plane. The t’

is the distance between the point and the lower

plane. It is positive for points in the n direction.

Negative for points behind the plane (in the –n

direction) and zero for points incident to the plane.

The interpolation parameter is:

minmax

min

−

(5)

Therefore, we apply an interpolation of the angle

(β) when we rotate all vertex of this zone. Where β =

α·t, α is the angle calculated for the rigid zones. That

is,

1000

cossincos0sin

0010

sincossin0cos

⎟

⎠

⎞

⎜

⎝

⎛

+−−

zzx

xzx

ββαββ

ββββ

(6)

This way, we obtain a simple implementation of

the deformation, which allows for an acceptable

computational cost. This allows obtaining tight

results to real deformation that suffers the foot.

2.5 Fusion of Parts

Once we have both the top and bottom of the

deformed model, it is necessary to merge both into a

single three-dimensional model that represents the

client's foot placed on the platform. This 3D model

can be compared a posteriori with different shoe

lasts and so, you can check the comfort of them.

The fusion is a simple process, since it is based

on linking two three-dimensional models into a

single one, considering the shared points. These

points will form a union that can present certain

anomalies product of the deformation process, so it

is smoothed using the Laplacian smoothing

technique. This process is also known as diffusion.

Fusion equation takes the form,

),(XL

∂

(7)

where X is the vertex of the mesh, is the Laplacian,

and λ is a scalar which controls the rate of diffusion.

Assuming that the Laplacian operator is linear, the

smoothing equation can be written as the following

difference equation forward.

).()()1( nXdtLInX

(8)

Doing this we do not achieve changing the

connectivity of the mesh. Each step changes the

position of the vertices, but the mesh topology

remains unchanged. The relaxation of a given vertex

only requires information about its immediate

neighbours. In figures 11 and 12 you can see the

deformation obtained at the end of the fusion

process.

3 RESULTS

To check the results, we must conduct a

comprehensive study comparing the error between

the deformed foot and foot on platform real. In the

first phase of the project we scanned sixteen

women's feet with Europe size 37, considered a

common size.

Figure 10: Example of different feet scanned. From left to

right: foot 1, foot 2.

After obtaining the data (Figure 10), they are

filtered and processed with the deformation

algorithm. The following figures show the results of

these deformations (Figures 11 and 12) with

A NEW PARALLELIZABLE DEFORMATION METHOD - Automatic Comparision between Foot and Last

367

platform of 4, 25, 35, and 75 mm, used to lift the

foot.

Platform 4 Platform 25

Platform 35 Platform 75

Figura 11: Deformation and inclined platform adjustment

of the foot 1 in Figure 10. The deformation line is the red

line.

Platform 4 Platform 25

Platform 35 Platform 75

Figure 12: Deformation and inclined platform adjustment

of the foot 2 in Figure 10. The deformation line is the red

line.

To estimate the error, the algorithm calculates

the intersection of the normal of each polygon of the

deformed model with each polygon of the real

model, so that it generates a colour map. In Figure

13 you can see the map. Green means it is above the

actual model, the red that it is below the actual

model and finally the gray coincides with the model.

We realize a comparison of the deformed foot

with the real foot on the same rise. We obtained the

average error for each area (toes, instep and heel).

The data obtained are shown in Table 1.

The error numbers can seem highs because of the

process of scanning the foot on the platform. Some

Figure 13: Colour map of the deformed foot.

Table 1: Error obtained.

Maximum average error 5.45 mm.

Minimum average error 0,93 mm.

Total average error 2,45 mm.

people put the foot obliquely on the platform,

because the foot wasn’t fixed to the platform. This

causes that the toes protrude from the model on the

outer part and the heel on the inside part.

Footwear experts did a visual supervision of the

results. They concluded that the result is more

precise than initially estimated.

4 CONCLUSIONS AND FUTURE

WORK

We have achieved virtual deformation of a

consumer’s foot and adapt it to a previously selected

platform. In this way we will be able to determine

the shoe that best fits to this foot.

With the final goal in mind of a global process of

adaptation of the shoes to the feet of the customer,

the speed of the process is essential. For this reason,

we adopt in this simulation a deformation model

based on geometry rather than based on physics. We

sacrifice realism for speed and efficiency. A client

can not wait long time to find out what type of shoe

is best suited to his/her feet.

In this paper we present a system with low

computational cost and a low error rate. The error

rate has been supervised by experts in the design of

footwear and has considered it acceptable. The

control points needed to perform the deformation are

detected automatically. It has been created a pleasant

user interface easy to use. So this paper can be seen

as an important basis for future improvements of a

global process, in order to obtain a better fit of the

footwear to the foot of the customer.

As we have seen the tests have been satisfactory,

as well as we have seen the system is easily

parallelizable. Therefore, an immediate future work

will be to parallelize on GPU the deformation

process, so it could be implanted in the footwear

ICPRAM 2012 - International Conference on Pattern Recognition Applications and Methods

368

industry. Moreover will be performed a more

comprehensive study of the error and is expected to

evolve the deformation to make it more realistic in

the critical points.

ACKNOWLEDGEMENTS

We thank the reviewers for their constructive

feedback. This work is subsidized by the national

project DPI2009-14738-C02-01 of the MICIIN

Spanish Government coordinated by INESCOP

(Asoc. investigación industrias del calzado y

conexas) and developed in collaboration with UIB.

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