SPATIAL RECONSTRUCTION OF LOCALLY SYMMETRIC

OBJECTS BASED ON STEREO MATE IMAGES

Leonid Mestetskiy

Department of Mathematical Methods of Forecasting, Moscow State University, Moscow, Russia

Archil Tsiskaridze

Control/Management and Applied Mathematics, Moscow Institute of Physics and Technology, Moscow, Russia

Keywords: Silhouette, Stereo mate, Middle axes, Continuous skeleton, Camera calibration, Recognition of gestures,

Palm, Human body.

Abstract: Restoration of spatial objects characteristics with locally symmetric elements is proposed in this paper. An

approach based on the model of a spatial flexible object defined as a family of spheres with the centres on a

graph with a tree-like structure is proposed. A method of real time identification of such objects using the

stereo mate images of their silhouettes is introduced. Image processing comprises construction of

continuous skeletons of silhouettes. Application to real time gesture recognition is considered.

1 INTRODUCTION

Reconstruction of spatial objects using several two-

dimensional images is a well-known problem and

has many real life applications. The essence of our

approach is that two-dimensional images are

considered to be a binary image and represent only

silhouettes of a spatial object. Such statement of the

problem, in particular, arises in recognition of

gestures by means of standard inexpensive

equipment. We suppose the initial data are low

resolution (480×640) images received from standard

WEB-cameras. In 3D pose estimation of object, such

as human hand or body, texture is less important and

much of the information can be extracted from the

silhouettes alone. Recognition of a gesture requires

reconstruction of the spatial form of such a complex

and variable object as a human palm or body. The

relevance of such statement of a problem is caused

by the fact that the range of potential users of

gesture recognition systems includes a great number

of people (disabled, hard of hearing, etc.) not

capable of obtaining expensive equipment but still

deeply needing real-time gesture understanding

systems. There are works devoted to creation of deaf

alphabet understanding software (Burger and

Caplier, 2007) as well as to developing gesture-

driven computer systems (Keshkin, 2005).

The lack of texture details makes it impossible to

analyse images at texture level and apply well

known object reconstruction methods based on

automatic identification of matching points on stereo

mate images. Obviously, the boundary points are the

only points that may be reliably identified on the

silhouette images. The problem is that the boundary

points of a silhouette on one of the stereo mate

images have, as a rule, no matching points on the

boundary of the silhouette on the other image. Thus,

it is impossible to directly identify the matching

points on the stereo mate silhouette images.

One can still try to identify the matching points

making some assumptions on the nature of the

original object. As far as gesture recognition deals

with images of a human palm or body we propose to

approximately represent these objects as a union of

several "cylindrical" elements having local axial

symmetry and solve the problem using a well known

notion of a planar image skeleton. Such objects are

also called “generalised cylinders” or “tubular

objects”. To be more precise a cylindrical element is

a spatial body formed by a family of spheres with

the centres on some curve. Such objects are called

spatial fat curves. We are interested in objects that

can be represented as a union of several fat curves.

Such locally symmetric objects can be used as

models for the description of a human palm or body.

443

Mestetskiy L. and Tsiskaridze A. (2009).

SPATIAL RECONSTRUCTION OF LOCALLY SYMMETRIC OBJECTS BASED ON STEREO MATE IMAGES.

In Proceedings of the Fourth International Conference on Computer Vision Theory and Applications, pages 443-448

DOI: 10.5220/0001769104430448

Copyright

c

SciTePress

It is natural that the accuracy of such description

of a human body or palm that uses generalized

cylinders is very low. Therefore, the proposed

approach cannot be used for the high-precision

reconstruction of shape and surface of 3D objects,

as, for instance, in (German Cheung Baker, 2003).

But for the recognition of gestures or poses the high

accuracy of the description of shapes and surfaces is

not required. It is sufficient to recognise only

substantial changes in the shape of these objects,

which characterise gestures. This approach makes it

possible to obtain solution of the problem with the

use of simple and inexpensive equipment under the

normal conditions.

2 THE PROPOSED METHOD

The proposed approach is based on the revealing the

of symmetry axes of the locally symmetric objects.

Although, these axes are invisible on the stereo mate

images, they still can be calculated for each image

by processing a silhouette presented on it.

We assume that the observed object does not

have occlusion. This means that all elements of the

object, for example, the fingers of a palm are visible

in the silhouette image. For objects with occlusions

it is intended to use a sequential segmentation of

initial grey scaled image to reveal overlapped parts.

Considering the silhouettes of stereo mate

images as projections of the spatial fat curves onto

the corresponding planes, we can expect that the

projections of the axes of the fat curves coincide

with the middle axes of the silhouettes.

In reality, the silhouette of a sphere is an ellipsis.

For the simplified case, when a radius of a sphere is

constant, there is a precise method of restoration

based on one silhouette image analysis

(Caglioti, 2006). For the images which we deal with,

the difference between this ellipsis and a circle is so

small, that it can be neglected.

We shall consider some (invisible) points which

are not the boundary points of the silhouettes as the

common matching points of stereo mate images.

Such reference points are provided by middle axes

of the silhouettes constituting its skeleton.

Implementation of the proposed approach poses

several problems. First we need to build the

skeletons of the silhouettes in a way that allows

identification of the points of different skeletons.

Then we have to restore the spatial form of the

whole object using the results of the identification of

the pair of skeletons. It is worth mentioning that all

calculations should be performed in the framework

of the computer vision system in real time which

requires processing of several stereo mate images

per second. This demands developing highly

efficient computational algorithms.

The notion of a flat flexible object is introduced

in (Mestetskiy, 2007) and an effective method of

comparing flexible objects on the basis of a

boundary-skeletal model is proposed. In the present

paper, we propose a generalisation of the notion of a

flat flexible object to the spatial case.

We define spatial flexible object as a set of

spheres of various sizes with centres on a spatial

tree. Stereo mate image processing allows

reconstructing the spatial structure of the object.

Reconstructing spatial characteristics of the

object allows monitoring the displacement dynamics

of the elements constituting the object, as well as the

changes in the object's shape. Applied to the human

palm or body this allows tracking their gestures or

movements.

Implementation of the proposed approach

includes solving of several subtasks.

2.1 Silhouette Acquisition

It is assumed that there is a pair of video cameras

which allows receiving synchronized images of an

object. An example of such stereo mate images is

presented on fig.1. In our experiments the standard

web-cameras connected to the desktop computer

were used. Each image is separately segmented, then

a silhouette is extracted and represented as a binary

raster image. There are different ways of

segmentation. All of them depend on specific

applications. One can note that in gestures

recognition the requirements to the quality of

silhouette images are not very demanding. Figure 2

shows the result of palm segmentation obtained

using the background subtraction method. In this

example a simple method of background subtraction

and thresholding was used.

Figure 1: The palm stereo mate.

2.2 Continuous Skeleton

Construction of the silhouette skeletons (fig. 3) is

VISAPP 2009 - International Conference on Computer Vision Theory and Applications

444

carried out by a method described in (Mestetskiy,

2008). A skeleton represents a geometric locus of

the centres of the circles inscribed in a silhouette.

The main advantage of the skeleton method used is

that the skeleton is represented as a graph with edges

as continuous lines. As we show later, this feature

allows successfully resolving the problem of

identification of skeleton points on different images.

Additionally, the method has the advantage of high

computing efficiency which allows solving the

problem in real time in the framework of computer

vision system.

Figure 2: The silhouettes received from two cameras.

Figure 3: Skeletons of obtained silhouettes.

2.3 Camera Calibration

Each point in the space is characterised by the

coordinates in a fixed orthogonal system which we

call a laboratory system. At the same time each

camera has its own orthogonal coordinate system,

with the centre located in the centre of the camera, z

axis is directed along the optical axis of the camera

and the two others are parallel to the coordinate axes

of the image. This camera model is called central

projection and despite its simplicity it often

constitutes an acceptable approximation to the

process of image acquisition. Camera calibration

process implies the problem of determining the

camera location in a certain laboratory coordinate

system and adjusting its internal parameters.

Another calibration method is based directly on

processing the stereo mate images, which requires

the identification of 5-8 points depending on the

method chosen (Brückner, 2008). The most

complicated part of this approach is allocation and

identification of the distinguishable points on the

images. Solving this problem with traditional

methods require a large amount of computations and

is inevitably accompanied by plenty of errors. Thus,

such approach is unacceptable when the problem

needs to be solved in real time and with the use of

web-cameras. The quality of obtained images, due to

their low resolution, does not allow reliably

detecting and identifying required number of points

on the stereo mate images. However, for locally

symmetric objects the use of skeletons makes this

problem essentially simpler. The skeleton nodes can

be used as the reference points. Thus, the problem is

reduced to identification of the nodes of two stereo

mate image skeletons.

2.4 Identification of the Reference

Points on the Skeleton

We assume that the projection of the axes of a

locally symmetric object approximately coincides

with a skeleton of the silhouette and this allows to

calculate these axes. Let

C be a point on one of the

stereo mate images. There is a straight line in the

space which is projected in this point. The image of

this straight line on the other picture is a so called

epipolar line of the point

C . For a given point on a

skeleton its stereo mate coincides with the point of

intersection of the other skeleton and the epipolar

line of this point. (fig. 4).

Figure 4: Stereo mate points found on skeletons.

2.5 Spatial Object Reconstruction

Having constructed axes on the basis of

identification of stereo mate points, it is possible to

calculate a spatial structure of a skeleton of the

object. Then, using the information on the width of

the object, with respect to the middle axes, we

restore a surface of the spatial object.

3 SKELETON POINTS

IDENTIFICATION

We describe our method of model construction for a

human palm example. We will consider stereo mate

images of a human palm (fig. 1) and the

corresponding axial graphs (fig. 4). Obviously,

fingers are locally symmetric objects. We assume

O

A

B

D

α

β

A`

B`

D`

C`

α

β

Epipolar

O`

C

SPATIAL RECONSTRUCTION OF LOCALLY SYMMETRIC OBJECTS BASED ON STEREO MATE IMAGES

445

that the projection of a spatial axis of a finger

coincides with a skeleton of a finger silhouette (fig.

4, curves

AB and

B

A

′′

).

Experiments show that the centres of the big

circles on both silhouettes (points

O and O

′

) are

the stereo mates with sufficient accuracy. Hence, it

can be assumed that the set of stereo mate points on

the sub-tree

OA

of the axial graph of the silhouette

coincides with the sub-tree

AO

′′

of the other

silhouette's axial graph. This allows constructing a

curve in the space.

If we consider the curve

OA as a continuous

mapping

2

]1,0[: Rf →

, and the curve AO

′

′

as

2

]1,0[: Rg →

, the problem reduces to finding a

mapping

]1,0[]1,0[: →w which maps each point

)(tf into its stereo mate ))(( twg . (fig 5). Obviously,

there are restrictions imposed on

w : the mapping

should be monotonous and continuous.

Figure 5: Dependence w(t).

Let

C

be a point on one of the stereo mate

images. For a given point

)(tfC = on the curve

OA , its stereo mate

))(( twgC

=

′

is located at the

intersection of the curve

AO

′′

and the epipolar of

C . Thus, using the epipolar lines, it is possible to

identify the stereo mate points on axial graphs and

determine the spatial arrangement of the axes of the

fat lines.

However, the difficulty arises when the

intersection angel

)(t

θ

of the curve

AO

′

′

and the

epipolar line is small, and therefore

C

′

is defined

with great inaccuracy. We can avoid this, by

imposing the following restriction

mi

n

)(

θθ

>t . The

value of

w(t) can be calculated only if

min

θ>θ(t) . In order to determine w(t) when

min

θθ(t) ≤ we interpolate using already obtained

values of

w . The application of the linear

interpolation is quite comprehensible to our

problem.

In figure 5, restriction

mi

n

θθ(t) > is violated on

curve

OB

, thus this curve is a segment.

Based on identification of the skeleton points we

obtain complete spatial configuration of the axes of

the given locally symmetric object. In many cases,

this representation is enough to handle the problems

of gesture recognition. However, the method of

skeletal representation contains not only the

information on the mid-axes of an object, but also

the information on the width of an object, since the

radii of inscribed circles with the centres on the mid-

axes are known. This information on the width of

the object makes it possible to visualise a

constructed spatial model.

4 VISUALIZATION

Having constructed spatial axes and calculated the

sizes of spheres with the centres on these axes, we

can reconstruct a spatial image of the object. For

each point

γ

of the spatial axial graph we define a

corresponding sphere in the following way: let

Q

be a point of the axial graph of one of the silhouettes

which is the image of

γ

. There exists a

corresponding maximal sized circle S with the centre

in

Q which is inscribed in the silhouette. The given

circle is the image of a sphere

r

S with the centre in

γ

and radius

r

. Let's choose an arbitrary point

),(, vuPSP

=

∈

. It determines a ray l which

starts in the centre of the first camera and is tangent

to the sphere

r

S . Then

r

is a distance between the

point

γ

and the ray l .

Thus, the sphere radius can be calculated. The

model of the object is a surface enveloping the set of

these spheres. An example of a human palm model

visualization obtained from the stereo mate images

is presented on fig. 6.

One can see from this example that the

visualization is not quite realistic, since in the model

not only the fingers are described as the fat curves,

but also a palm part between the fingers and a wrist

is considered as a fat curve. This fault of

visualization can be easily eliminated because the

spatial position of the fingers makes it possible to

calculate the plane where this palm part is located

and slightly flatten the sphere towards this plane.

The result of such an improvement is presented on

fig. (7).

(1, 1)

t

(0, 0)

A

C

B

D

w(t)

VISAPP 2009 - International Conference on Computer Vision Theory and Applications

446

Figure 6: Spatial model visualization.

Figure 7: Corrected spatial model.

5 EXPERIMENTAL RESULTS

Experiments with the reconstruction of the spatial

model of a human body were conducted with dolls

of a size of 30 cm. The only purpose of using the

dolls was to simplify the process of taking photos in

the laboratory conditions. The results can easily be

extrapolated for a case of a real human body. Figure

7 shows the initial stereo mate images, their

silhouettes and the resulting spatial objects.

The experimental estimation of the accuracy of

restoration of human body shape can be obtained

with the use of "Kung-Fu Girl" data, presented by

Graphics-Optics-Vision group at the Max-Planck

Institute for Informatics (MPII). These data consist

of synthetic scenes (size 240×320) obtained from the

virtual cameras, for which the precise values of the

calibration parameters are known. The results of the

experiments are shown on figure 9. Visual analysis

shows that the accuracy of restoration of body shape

and surface is very poor. But, this accuracy looks

like completely sufficient for restoring poses and

gestures. We intend to investigate the formal

quantitative criterion of accuracy and the methods

for its calculation.

The performance of the algorithm implemented

on Intel Pentium IV, Core 2 Duo, 2800 Mhz, 1Gb

RAM computer is more than 5 frames per second.

This makes it possible to use the proposed method as

a real time tool in the framework of the computer

vision systems.

Figure 8: Stereo mate of initial images, their silhouettes

and the received spatial objects.

SPATIAL RECONSTRUCTION OF LOCALLY SYMMETRIC OBJECTS BASED ON STEREO MATE IMAGES

447

Figure 9: Stereo mate of initial images and received spatial

objects for “Kungfu girl”.

ACKNOWLEDGEMENTS

Authors are grateful to the Russian Foundation of

Basic Researches, which has supported this work

(grant 05-01-00542).

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