Conf. univ. dr. Eugen ZAHARESCU- Statement of Research Interest

Statement of Research Interest

Conf. univ. dr. Eugen ZAHARESCU

1. Introduction

My research interest ranges from mathematical morphology, used in image analysis and processing, to large scale image ontology generation.

I have started with the extension of mathematical morphology theory to multivariate images and I have continued with the exploration of adapting the whole set of morphological operators for the images represented in a logarithmic environment.

Finally, I have developed algorithmic and mathematical concepts and tools used as data analysis and modeling for machine learning, computer vision and image understanding.

2. Mathematical Morphology Theory. Basic Concepts

The Mathematical Morphology (MM) theory represents a powerful image processing framework based on order and set theory and Euclidian geometry. The main advantage of morphological operators lies in their inherent ability to exploit the spatial relationships of pixels. Moreover, they can also adapt to vectorial strategies in order to process multivariate data. Based upon the MM theory, the precise detection of the object’s pixels along with pertinent classification features, proved to be more computationally efficient then other approaches (i.e. statistical methods concentrate on individual pixel values and therefore they are more computation-intensive).

For recalling the preliminary definitions of some basic morphological operators and the theory of MM, the reader should refer to [7] and [9].

Let f : D → T be a grayscale image, with D the discrete coordinate grid while T represents the set of possible grey values. In short, MM studies the transformations of an image, when it interacts through operators with a matching pattern E, called structuring element (SE). E is considered as a subset of D. At the very basis of the MM operators pyramid are the morphological operators, dilation (δE ( f )) and erosion (εE ( f)).

Subsequently, a multitude of operators is derived from dilation and erosion such as opening γE(f)=δĚ(εE(f)), closing φE(f)=εĚ(δE(f)) and morphological gradient grad(f)=δE(f)-εE(f).

The skeleton of the set X represents the union of all intersections of the differences between the erosion and its opening through variable sized discs E (i.e. White Top-Hat transforms) [7]:

SK ( X ) = 0 0λ µ≥ ≥UI { ε λE (X ) \ γ µE ( ε λE (X )) }

The Hit-or-Miss transform, locates either pixels from erosion X-E1 (i.e. “hit") and from erosion Xc-E2

c, (i.e. “miss") [9]:

Conf. univ. dr. Eugen ZAHARESCU- Statement of Research Interest

HMTE(X ) = εE1 ( X ) I εE2 ( X c )

Using these basic concepts I have developed a morphological algorithm for automated color objects detection and classification presented in [11].

Figure 1. Experiments for automated color objects classification

3. New Morphological Operators in Logarithmic Image Processing Theory The Logarithmic Image Processing (LIP) theory is a mathematical framework that provides a set of specific algebraic and functional operations and structures that are well adapted to the representation and processing of non-linear images, and more generally of non-linear signals, valued in a bounded intensity range.

This very well structured theory determined using the logarithmic image representation in my approach for defining a new set of mathematical morphology operators based upon structuring elements with a variable geometrical shape or adaptable structuring elements.

Therefore, I have defined and analyzed new multiplicative logarithmic morphological operators used in medical image enhancement. Finally, the experimental results reveal that this method has wide potential areas of impact which may include: Digital X-ray, Digital Mammography, Computer Tomography Scans, Nuclear Magnetic Resonance Imagery and Telemedicine Applications.

Conf. univ. dr. Eugen ZAHARESCU- Statement of Research Interest

In the paper [12] I have presented several possibilities of defining structuring elements with a variable geometrical shape within the logarithmic image processing theory, i.e. using only vector-oriented operations with images represented as gray-tone functions operands.

In fact, we can demonstrate that all the classical mathematical morphology operators are solely a particular case of the logarithmic mathematical morphology operators, based upon the definition of the structuring elements with a variable geometrical shape or adaptable structuring elements.

At least three categories of mathematical morphology operators can be defined within the context of a logarithmic image representation:

1. Multiplicative logarithmic morphological operators;

2. Additive logarithmic morphological operators;

3. Additive-multiplicative logarithmic morphological operators;

As an example, here are the definitions of multiplicative logarithmic morphological operators from the basic levels of their pyramid following the classical morphological paradigm. 3.1 Multiplicative Logarithmic Morphological Erosion Definition 1: Multiplicative logarithmic morphological erosion for the image f by structuring element g, represents the gray-tone function defined as follows:

(f ML ğ)( x ) = inf{ k ⊗ (f ( y) - g( y-x )) | y∈ R2 } . (1)

where ğ is defined as: ∀x∈Rn, ğ(x)=g(-x) and ⊗ represents LIP product of a gray-tone function

with a real scalar. Observation 1: Scalar k can be constant for the entire image or can be, as well, another gray-tone function (an adaptable scalar multiplication) defined like this:

∀ x ∈ R2, k (x) = ( )f xM

. (2)

Within LIP model we will obtain the following definition:

( f ML ğ)( x ) =inf{ ( ( ( ) - ( - ))) - ( - ( ( ) - ( - )))

( ( ( ) - ( - ))) ( - ( ( ) - ( - )))

+⋅

+ +

k k

k kM f y g y x M f y g y x

MM f y g y x M f y g y x

| y∈R2} . (3)

3.2 Multiplicative Logarithmic Morphological Dilation

Definition 2: Multiplicative logarithmic morphological dilation for the image f by structuring element g, represents the gray-tone function defined as follows:

(f ⊕ML ğ)(x) = sup{k⊗(f (y) + g( y-x ))| y∈R2} . (4)

where ğ is defined as: ∀x∈Rn, ğ(x)=g(-x) and ⊗ represents LIP product of a gray-tone function

with a real scalar. Within LIP model we will obtain the following definition:

( f ⊕ML ğ)( x )= =sup{ ( ( ( ) ( - ))) - ( - ( ( ) ( - )))

( ( ( ) ( - ))) ( - ( ( ) ( - )))

+ + +⋅

+ + + +

k k

k kM f y g y x M f y g y x

MM f y g y x M f y g y x

| y∈R2} (5)

Conf. univ. dr. Eugen ZAHARESCU- Statement of Research Interest

a) Original color image of a mouth infection

b) Logarithmic morphological enhanced contrast and sharpness

c) Original color image of eye retina d) Contrast and sharpness enhancement using logarithmic

morphological transforms

e) Original color image “Blood Vessels on Eye Retina” f) Contrast and sharpness enhancement using logarithmic

morphological transforms

Figure 2. Experiments for logarithmic morphological enhancement in medical imagery 4. Future research In the future I would like to work on the challenging problems of multivariate image morphogical segmentation. Also I would investigate how a multiscale decomposition could lead to a better

Conf. univ. dr. Eugen ZAHARESCU- Statement of Research Interest

segmentation approach. I would like to combine the ideas of hierarchical decomposition and variational techniques with those of other branches of mathematics and engineering to come up with more effective way of approaching the problems in image processing. I am very much interested in applying mathematical image processing techniques to satellite imaging technology.

Today, our world has evolved to a point where the problems facing mankind have grown extremely interdisciplinary. As a researcher in the domain of applied mathematics, I am more than willing to work with experts from various disciplines and solve problems which can not be categorized anymore.

References

[1] J. Angulo, J. Serra, "Color segmentation by ordered mergings", Proc. of IEEE International Conference on Image Processing (ICIP'03), IEEE, Vol. 2, Barcelona, Spain, Sept. 2003, p. 125-128.

[2] A. Hanbury, J.Serra, "Color Image Analysis in 3D-polar coordinates", Dans Proc. of DAGM symposium, Vienna, April 2003.

[3] A. Hanbury and J. Serra, Morphological Operators on the Unit Circle, IEEE Transactions on Image Processing, 10(12) (2001) 1842-1850.

[4] M. Jourlin, J.C. Pinoli, “Logarithmic Image Processing. The mathematical and physical framework for the representation and processing of transmitted images”, Advances in Imaging and Electron Physics, vol. 115:130-196, 2001.

[5] V. Pătraşcu, V. Buzuloiu, “A Mathematical Model for Logaritmic Image Processing”, The 5-th World Multi-Conference on Systemics, Cybernetics and Informatics, SCI2001, July 22-25, 2001, Orlando, USA

[6] P. Salambier, "Morphological Multi-scale Segmentation for Image Coding", Signal Processing, Vol. 38, pp. 359-386, 1994.

[7] J. Serra, “Image Analysis and Mathematical Morphology” Academic Press, London, 1982. [8] J. Serra, Morphological Segmentation of Color Images by Merging of Partitions, in Proc. of International

Symposium on Mathematical Morphology (ISMM '05), Kluwer, 2005,151-176. [9] P. Soille, “Morphological Image Analysis”, Springer-Verlag, Berlin, Heidelberg, New York, 1999. [10] C. Vertan, M. Zamfir, E.Zaharescu, V.Buzuloiu, C. Fernandez-Maloigne, “Nonlinear Color Image Filtering by

Color to Planar Shape Matching”, ICIP 2003, Barcelona, Spain, 2003. [11] E. Zaharescu, "Morphological Algorithm for Color Objects Classification", IEEE Computer Society Proceedings of

the Fourth Balkan Conference in Informatics, 17-19 September 2009, Thessaloniki, Greece, pag.152-156, Library of Congress No. 2009934322, (ISBN 978-0-7695-3783-2)

[12] E. Zaharescu, “Morphological Enhancement of Medical Images in a Logarithmic Image Environment”, IEEE Proceedings of International Symposium on Signals Circuits and Systems-ISSCS 2007, Iaşi, Romania, 2007, pag.352-356 (ISSN 1-4244-0968-1) ISI-Inspec® http://ieeexplore.ieee.org/xpl/

[13] E. Zaharescu, M. Zamfir, C.Vertan: “Color Morphology-like Operators based on Color Geometric Shape Characteristics”, Proc. of International Symposium on Signals Circuits and Systems SCS 2003, Iaşi, Romania, 2003.