# Master Thesis Proposals

## Subject 1: Generalized Fourier transforms: discrete versus continuous

The classical Fourier transform (FT) can be related to a realization in terms of differential operators of the Lie algebra sl(2). This way of rewriting allows us to get far-reaching generalizations of the FT., by realizations of sl(2) (or even more complicated Lie (super) algebras) through new differential operators. During the last years, this is an active area of research, where both continuous and discrete transformations were introduced. In many cases, there is moreover an interesting quantum mechanical interpretation.

The purpose of this thesis is to study a number of such transformations in detail and make a comparing study of the discrete and continuous case.

**Supervisors**: Hendrik De Bie (S8, office 130.062) and Joris Van der Jeugt

## Subject 2: Study of the symplectic Dirac operator

The goal of this master thesis subject is to go through a recent book of Habermann and Habermann, which deals with the symplectic Dirac operator. This differential operator is, in comparison to the classical orthogonal invariant Dirac operator, invariant under the symplectic group. The consequence is that one has to use symplectic Clifford algebras, which act on the infinite dimensional space of symplectic spinors. Knowledge of differential geometry is necessary, since this operator has to be constructed very generally.

**Supervisor**: Hendrik De Bie (S8, office 130.062)

## Subject 3: Tensor product representations for Lie (super) algebras

The goal of this thesis is on the one hand a study of some known techniques from representation theory of Lie algebras, and on the other hand an application on open questions for Lie super algebras. First, the technique of Littlewood-Richardson and Young tableaus will be applied to tensor powers of the fundamental representations of the linear and orthogonal Lie algebra. Together with taking trace-representation, this offers a new framework to study the complete reducibility of representations of the orthosymplectic Lie super algebra.

**Supervisor**: Hendrik De Bie (S8, office 130.062)

## Subject 4: Hermitian Clifford analysis

Hermitian Clifford analysis is a refinement of orthogonal Clifford analysis, where the classical Dirac operator is rewritten as the sum of two complex differential operators. The consequence of this approach is that it leads to a much richer function theory, which consists of complex analysis in several vector variables.

The goal of this subject is to investigate a number of recent papers thoroughly. After a study of the basic frame work, there are a number of different possibilities, namely a study of the polynomial null solutions (the so-called h-monogenic system), fundamental solutions, Cauchy-Kowalevskaya extensions, etc.

**Supervisors**: Hendrik De Bie (S8, office 130.062) and Hennie De Schepper (S8, office 130.064)

## Subject 5: Special functions in Clifford analysis

The goal is to generalize a number of sets of orthogonal polynomials to the higher dimensional Clifford framework. We hereby think mainly of generalized Hermite polynomials. Also, special functions derived from the function exp(), arising in the Fourier-Laplace transform need to be investigated. These arise from the Fischer decomposition and the decomposition in spherical monogenics.

**Supervisor**: Frank Sommen (S8, office 130.063)

## Subject 6: Clifford analysis on varieties

Using the language of differential forms in combination with the Clifford algebra, it is possible to prove a Cauchy-Stokes formula for the tangential Dirac operator on a survace in the Euclidean space. The goal is to develop this theory and apply it to a number of examples such as the unit sphere, the Klein bottle, the real projective space, the Veronese embedding and the Grassmann varieties.

**Supervisor**: Frank Sommen (S8, office 130.063)

## Subject 7: Quaternion sparse representation model for color image processing

Many problems like image and video restoration, compression and coding, digital image inpainting and content analysis benefit from the sparse representation model. As these techniques are powerful and widely applicable, sparse representations of signals (including images and higher-dimensional data), attract the interest of researchers from different fields. The goal of such a sparse representation is to approximate well a signal using only few elements from a (typically redundant) dictionary (see Figures 1 and 2). One of the best known and widely used approaches for dictionary learning is the so-called K-SVD method. K-SVD is an extension of the K-means clustering method that allows efficient learning of the dictionary using the singular value decomposition (SVD). The common dictionary learning techniques, including the recent K-SVD methods, treat signals in a unified way irrespective of their dimensionality and the nature of different channels in the case of multicomponent data (such as color, multispectral or hyperspectral images). All the data within a 2-D window (in the case of a greyscale image) or a 3-D window (in the case of a multicomponent image) are simply stacked in an array, using a given scanning order and as such treated as a single 1-D vector.

A very recent trend in signal processing and machine learning attempts to build an improved sparse representation model of color images by introducing quaternions into dictionary construction. Quaternions are four-dimensional generalization of complex numbers (with three imaginary units instead of one). Due to their property to describe efficiently rotations in 3D, quaternions have many applications in theoretical and applied mathematics but also in different fields of engineering such as computer graphics and computer vision as well as in various applications including biomedical processing, remote sensing, hyperspectral image processing and many others. The quaternionic representation with three imaginary units is also ideally suited for representing three color channels, and therefore quaternions have already been used extensively in color image processing. A very recent method, so-called K-QSVD, which is a generalization of the K-SVD algorithm in the quaternionic framework, already showed remarkable results (see Figure 3). The potentials of quaternions in improving sparse representations of multicomponent images are yet to be explored, starting from the first encouraging results. The motivation is that the coefficient matrix preserves not only the correlation among the channels but also the orthogonal property. According to recent studies, this proves to be important in terms of computation complexity but also in terms of color fidelity in the reconstruction. However, many aspects of this approach are yet to be explored, both theoretically and in terms of the practical design. In this master thesis, the student will be guided by supervisors from the Image Processing and Interpretation research group and the Clifford research group.

This thesis should combine emerging and hugely popular technologies in image processing and computer vision with a solid mathematical theory to build a sound framework that will be validated in some concrete applications but even more widely applicable. The main goal of the thesis is to build a powerful method for encoding color images using quaternionic dictionaries starting from the literature and already developed algorithms, such as K-SVD and K-QSVD. A first task will be to explore the efficiency of K-QSVD dictionaries compared to the more traditional ones in terms of the approximation power (the goal is to compose the image as faithfully as possible by combining as few as possible elements at each local position). Secondly, the developed method will be applied in two concrete image processing applications - image denoising and digital inpainting (see an illustration in Figure 3). In these applications, the use of quaternionic dictionaries will be practically evaluated and compared to some of the current state-of-the-art methods that will be made available to the student.

Fig 1. - For input data Y, dictionary learning method aims to find dictionary D and a representation matrix X such that its columns Xi are sparse enough (https://research.csiro.au/data61/).

Fig 2. - An illustration of the color image dictionaries learned by K-SVD (on the left) and K-QSVD algorithm (on the right).

Fig 3. - An example showing the application of K-QSVD in image inpainting. Left: damaged image (70 % missing). Right: an image reconstructed using K-QSVD.

**Guidance**: Srđan Lazendić (S8, office 130.071), Tim Raeymaekers (S8, office 130.072)

**Supervisors**: Aleksandra Pižurica (TELIN), Hendrik De Bie (S8, office 130.062)

## Subject 8: Finsler geometry approach to Beltrami framework for image processing

Scientists from different fields of science and engineering are interested in image processing tasks due to their importance in medical sciences, geology and geography as well as in art investigation for restoration of old paintings. Quite often we have to deal with complicated scenarios and suitable representation and visualization of images is extremely important in order to be able to understand visual processing and to interpret image content properly. Geometric methods represent classical but at the same time modern tools for image processing and image analysis. The partial differential equations approaches such as scale-space methods and gradient descent methods represent powerful tools in image processing. Although differential geometry found its application in pattern recognition, shape reconstruction, edge detection, color-image enhancement and segmentation, new developments are yet to come. That this is indeed the case we can see in [1], which gives extremely nice survey of existing and widely applicable differential geometry methods and explains geometric framework in image processing in the combination with already mentioned PDE tools. Well known Beltrami flow, that represents one of the most important geometric frameworks in image processing, is also presented in [1] with all its potentials. In general, the geometric framework views images as manifolds (generalization of surfaces with locally Euclidean structure) embedded in a higher dimensional space-feature manifold. In Figure 1. we have a grayscale image represented as a surface (2D manifold), but also color and multispectral images as well as 3D medical images can be seen in this framework. In this framework many image processing problems can be formulated as the computation of minimal surfaces i.e. surfaces that locally minimize their area.

A special type of a manifold known as a Riemannian manifold is well suited for the representation of digital images due to the possibility to measure distances between the elements of a manifold. A very recent trend in geometrical image processing builds a generalized model of the Beltrami framework by the means of Finslerian geometry, which is a generalization of Riemannian geometry in the sense that objects in Finsler geometry typically depend not only on the position on the manifold, but also on the approaching direction between them (see [2]). In the generalized approach the images are still treated in the same manner as mentioned before, i.e. as manifolds embedded in a higher dimensional space and the goal is to find the minimal surface. For example, noise is represented as points of high curvature so the goal is to smooth the image surface by reducing the number of points of high curvature. The Beltrami flow is obtained by minimizing the area of the image manifold with respect to the intensity components: the surface moves according to the intensity component in the direction of the normal of the surface, which is the only direction which changes the shape of the image. In this way we are indeed changing the intensity components and removing the noise. Other image processing problems can be seen in this framework.

This thesis should combine already developed theoretical techniques in image processing and computer vision (introduced in [1,3]) with a solid mathematical theory of curves and surfaces in order to conduct experimental results and practical validation. The main goal of the thesis is to do the practical validation of the theoretical results presented in [3]. Since Beltrami flow already showed remarkable results, the new generalized Beltrami framework, so far only from the theoretical point of view, is also extremely promising method. As mentioned, practical validation is yet to be conducted and this should be the main task in this master thesis. The results should be compared with the traditional as well as with the state-of-the-art methods for image enhancing. All the existing code for the traditional Beltrami framework will be available to the student. The developed method will be applied in concrete image processing applications, which will be chosen in the agreement with the student and depending on his/her affinities.

Fig 1. - A Grayscale "Lena" image (left) can be considered as a 2D surface S=(x^{1},x^{2},I(x^{1},x^{2})) embedded in a 3D space (right); the surface point (x^{1},x^{2},I(x^{1},x^{2})) has the gray level I(x^{1},x^{2}).

[1] R. Kimmel, Numerical Geometry of Images, Springer-Verlag, 2004.

[2] M. Dahl, A Brief Introduction to Finsler Geometry, Lecture Notes, 2006. (Finsler.pdf)

[3] J. Stojanov, Anisotropic frameworks for dynamical systems and image processing, PhD Thesis, University of Novi Sad, 2015.

**Guidance**: Srđan Lazendić (S8, office 130.071), Astrid Massé (S8, office 130.072) **Supervisors**: Aleksandra Pižurica (TELIN), Hendrik De Bie (S8, office 130.062)

## Subject 9: Wavelets and shearlets using Clifford algebra techniques

Wavelets, and more recently also shearlets, are a very important tool in modern image processing, especially in the area of compression and reconstruction. In recent mathematical work, the wavelet transform was expanded to higher dimensions, by making use of Clifford analysis.

The goal of this subject is, on the one hand, to perform a theoretical study of these wavelets, but on the other hand also to investigate the practical applicability. Therefore, a comparison will be made between an existing algorithm, which uses so-called shearlets, with a new algorithm, which needs to be constructed, making use of Clifford wavelets.

Supervisors: Hendrik De Bie (S8, office 130.062) and Bart Goossens (TELIN)