Automatic Identification and Elastic Properties of Deformed Objects Using their Microscopic Images

Automatic Identification and Elastic Properties of Deformed Objects Using their Microscopic Images

C. Papaodysseus (National Technical University of Athens, Greece), P. Rousopoulos (National Technical University of Athens, Greece), D. Arabadjis (National Technical University of Athens, Greece), M. Panagopoulos (National Technical University of Athens, Greece) and P. Loumou (National Technical University of Athens, Greece)
DOI: 10.4018/978-1-60566-314-2.ch023
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Abstract

In this chapter the state of the art is presented in the domain of automatic identification and classification of bodies on the basis of their deformed images obtained via microscope. The approach is illustrated by means of the case of automatic recognition of third-stage larvae from microscopic images of them in high deformation instances. The introduced methodology incorporates elements of elasticity theory, image processing, curve fitting and clustering methods; a concise presentation of the state of the art in these fields is given. Combining proper elements of these disciplines, we first evaluate the undeformed shape of a parasite given a digital image of a random parasite deformation instance. It is demonstrated that different orientations and deformations of the same parasite give rise to practically the same undeformed shape when the methodology is applied to the corresponding images, thus confirming the consistency of the approach. Next, a pattern recognition method is introduced to classify the unwrapped parasites into four families, with a high success rate. In addition, the methodology presented here is a powerful tool for the exact evaluation of the mechano-elastic properties of bodies from images of their deformation instances.
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Background

Some Necessary Notions and Relations from Elasticity Theory

We now proceed to state some notions that are commonly used in most approaches of Elasticity theory (Chandrasekharaiah and Debnath, 1994). These will later be used in the analysis on which the unwrapping of the parasite is based.

Definition of the Stress and the Strain Tensor

Consider a one-to-one correspondence between all points of the deformed and the undeformed parasite states. Thus, let us consider two arbitrary points, say and of an undeformed parasite element and let and be their unique images in the deformed parasite body. In other words, due to the deformation, point moves to and point to . We consider the lengths of and : and

Now one defines the relative elongations along the x- and y-axes:

Along the x-axis, we set , in which case, and above become and , respectively. Thus, the relative elongation is defined to be

(1)

Similarly, along the y-axis we set and thus the relative elongation εyy is defined to be:

(2)

If the relative elongations are small, after expanding the square roots of (1) and (2) in Taylor series, we obtain

, (3).

Moreover, using the same assumption, the tangent of the angle of deformation of the x-axis, , and the y-axis,, become and respectively (Figure1). Adopting the hypothesis of small relative elongations, it results that , implying that the initially right angle is deformed by

(4) .
Figure 1.

Element differential deformation

Therefore, after using the shear stress definition , we adopt the standard strain tensor definition: .

Next, in order to study the elastic forces’ distribution throughout the parasite’s body, we proceed by considering an arbitrary differential element in the parasite body, starting at point with vertices, , , (Figure2). Let the force per unit area/length acting on the side be , where the first subscript denotes the axis to which the side is vertical, while the second subscript denotes the vector component axis. Similarly, . It is evident that the four functions , , , , suffice to determine the stress condition of the differential element under consideration. Hence, we define the standard stress tensor .

Figure 2.

Differential element strain forces

Hypothesis on the Parasite Constitutive Equation

The parasite constitutive equation relates the stress tensor with the strain tensor . These two tensors can be related through any functional form, i.e. . However, in many practical circumstances, this functional form can be considered to be linear, namely , where is a constant matrix (generalized Hooke ’s law).

Key Terms in this Chapter

Image Operations: Actions performed on an image that change the colour content of its pixels usually to detect or bring out some image characteristics.

Elastic Deformation Invariants: Quantities, shapes or characteristics of a body, e.g. a parasite, which remain invariant during its elastic deformation.

Automatic Curve Classification: A process which automatically classifies curves into different groups according to their similarity.

Parasite Image Segmentation: The automated procedure that isolates parasite body in its microscopic image and perhaps locates the various parasite body regions.

Curve Fitting Methods: Techniques that optimally fit a curve of desired functional form into a set of pixels or data points.

Parasite Mechano-Elastic Properties: The quantities and properties that characterize the body of a parasite, from the point of view of Mechanics and Elasticity Theory.

Pattern Classification Techniques: A set of methods that classify the members of a data set in different groups according to a number of group-characteristic patterns.

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