On Self‐Affine and Self‐Similar Graphs of Fractal Interpolation Functions Generated from Iterated Function Systems

This chapter provides a brief and coarse discussion on the theory of fractal interpolation functions and their recent developments including some of the research made by the authors. It focuses on fractal interpolation as well as on recurrent fractal interpolation in one and two dimensions. The resulting self‐affine or self‐similar graphs, which usually have non‐integral dimension, were generated through a family of (discrete) dynamic systems, the iterated function system, by using affine transformations. Specifically, the fractal interpolation surfaces presented here were constructed over triangular as well as over polygonal lattices with triangular subdomains. A further purpose of this chapter is the exploration of the existent breakthroughs and their application to a flexible and integrated software that constructs and visualises the above‐mentioned models. We intent to supply both a panoramic view of interpolating functions and a useful source of links to assist a novice as well as an expert in fractals. The ideas or findings contained in this paper are not claimed to be exhaustive, but are intended to be read before, or in parallel with, technical papers available in the literature on this subject

Image analysis in medical imaging: recent advances in selected examples

Medical imaging has developed into one of the most important fields within scientific imaging due to the rapid and continuing progress in computerised medical image visualisation and advances in analysis methods and computer-aided diagnosis. Several research applications are selected to illustrate the advances in image analysis algorithms and visualisation. Recent results, including previously unpublished data, are presented to illustrate the challenges and ongoing developments

Statistical shape analysis for bio-structures : local shape modelling, techniques and applications

A Statistical Shape Model (SSM) is a statistical representation of a shape obtained from data to study variation in shapes. Work on shape modelling is constrained by many unsolved problems, for instance, difficulties in modelling local versus global variation. SSM have been successfully applied in medical image applications such as the analysis of brain anatomy. Since brain structure is so complex and varies across subjects, methods to identify morphological variability can be useful for diagnosis and treatment. The main objective of this research is to generate and develop a statistical shape model to analyse local variation in shapes. Within this particular context, this work addresses the question of what are the local elements that need to be identified for effective shape analysis. Here, the proposed method is based on a Point Distribution Model and uses a combination of other well known techniques: Fractal analysis; Markov Chain Monte Carlo methods; and the Curvature Scale Space representation for the problem of contour localisation. Similarly, Diffusion Maps are employed as a spectral shape clustering tool to identify sets of local partitions useful in the shape analysis. Additionally, a novel Hierarchical Shape Analysis method based on the Gaussian and Laplacian pyramids is explained and used to compare the featured Local Shape Model. Experimental results on a number of real contours such as animal, leaf and brain white matter outlines have been shown to demonstrate the effectiveness of the proposed model. These results show that local shape models are efficient in modelling the statistical variation of shape of biological structures. Particularly, the development of this model provides an approach to the analysis of brain images and brain morphometrics. Likewise, the model can be adapted to the problem of content based image retrieval, where global and local shape similarity needs to be measured

Computational Modeling for Abnormal Brain Tissue Segmentation, Brain Tumor Tracking, and Grading

This dissertation proposes novel texture feature-based computational models for quantitative analysis of abnormal tissues in two neurological disorders: brain tumor and stroke. Brain tumors are the cells with uncontrolled growth in the brain tissues and one of the major causes of death due to cancer. On the other hand, brain strokes occur due to the sudden interruption of the blood supply which damages the normal brain tissues and frequently causes death or persistent disability. Clinical management of these brain tumors and stroke lesions critically depends on robust quantitative analysis using different imaging modalities including Magnetic Resonance (MR) and Digital Pathology (DP) images. Due to uncontrolled growth and infiltration into the surrounding tissues, the tumor regions appear with a significant texture variation in the static MRI volume and also in the longitudinal imaging study. Consequently, this study developed computational models using novel texture features to segment abnormal brain tissues (tumor, and stroke lesions), tracking the change of tumor volume in longitudinal images, and tumor grading in MR images. Manual delineation and analysis of these abnormal tissues in large scale is tedious, error-prone, and often suffers from inter-observer variability. Therefore, efficient computational models for robust segmentation of different abnormal tissues is required to support the diagnosis and analysis processes. In this study, brain tissues are characterized with novel computational modeling of multi-fractal texture features for multi-class brain tumor tissue segmentation (BTS) and extend the method for ischemic stroke lesions in MRI. The robustness of the proposed segmentation methods is evaluated using a huge amount of private and public domain clinical data that offers competitive performance when compared with that of the state-of-the-art methods. Further, I analyze the dynamic texture behavior of tumor volume in longitudinal imaging and develop post-processing frame-work using three-dimensional (3D) texture features. These post-processing methods are shown to reduce the false positives in the BTS results and improve the overall segmentation result in longitudinal imaging. Furthermore, using this improved segmentation results the change of tumor volume has been quantified in three types such as stable, progress, and shrinkage as observed by the volumetric changes of different tumor tissues in longitudinal images. This study also investigates a novel non-invasive glioma grading, for the first time in literature, that uses structural MRI only. Such non-invasive glioma grading may be useful before an invasive biopsy is recommended. This study further developed an automatic glioma grading scheme using the invasive cell nuclei morphology in DP images for cross-validation with the same patients. In summary, the texture-based computational models proposed in this study are expected to facilitate the clinical management of patients with the brain tumors and strokes by automating large scale imaging data analysis, reducing human error, inter-observer variability, and producing repeatable brain tumor quantitation and grading

Unitary Representations of Wavelet Groups and Encoding of Iterated Function Systems in Solenoids

For points in $d$ real dimensions, we introduce a geometry for general digit sets. We introduce a positional number system where the basis for our representation is a fixed $d$ by $d$ matrix over \bz. Our starting point is a given pair $(A, \mathcal D)$ with the matrix $A$ assumed expansive, and $\mathcal D$ a chosen complete digit set, i.e., in bijective correspondence with the points in \bz^d/A^T\bz^d. We give an explicit geometric representation and encoding with infinite words in letters from $\mathcal D$. We show that the attractor $X(A^T,\mathcal D)$ for an affine Iterated Function System (IFS) based on $(A,\mathcal D)$ is a set of fractions for our digital representation of points in \br^d. Moreover our positional "number representation" is spelled out in the form of an explicit IFS-encoding of a compact solenoid \sa associated with the pair $(A,\mathcal D)$. The intricate part (Theorem \ref{thenccycl}) is played by the cycles in \bz^d for the initial $(A,\mathcal D)$-IFS. Using these cycles we are able to write down formulas for the two maps which do the encoding as well as the decoding in our positional $\mathcal D$-representation. We show how some wavelet representations can be realized on the solenoid, and on symbolic spaces

A holistic approach to structure from motion

This dissertation investigates the general structure from motion problem. That is, how to compute in an unconstrained environment 3D scene structure, camera motion and moving objects from video sequences. We present a framework which uses concatenated feed-back loops to overcome the main difficulty in the structure from motion problem: the chicken-and-egg dilemma between scene segmentation and structure recovery. The idea is that we compute structure and motion in stages by gradually computing 3D scene information of increasing complexity and using processes which operate on increasingly large spatial image areas. Within this framework, we developed three modules. First, we introduce a new constraint for the estimation of shape using image features from multiple views. We analyze this constraint and show that noise leads to unavoidable mis-estimation of the shape, which also predicts the erroneous shape perception in human. This insight provides a clear argument for the need for feed-back loops. Second, a novel constraint on shape is developed which allows us to connect multiple frames in the estimation of camera motion by matching only small image patches. Third, we present a texture descriptor for matching areas of extended sizes. The advantage of this texture descriptor, which is based on fractal geometry, lies in its invariance to any smooth mapping (Bi-Lipschitz transform) including changes of viewpoint, illumination and surface distortion. Finally, we apply our framework to the problem of super-resolution imaging. We use the 3D motion estimation together with a novel wavelet-based reconstruction scheme to reconstruct a high-resolution image from a sequence of low-resolution images

Iterative geometric design for architecture

This work investigates on computer aided integrated architectural design and production. The aim is to provide integral solutions for the design and the production of geometrically complex free-form architecture. Investigations on computer aided geometric design and integrated manufacturing are carried out with equal importance. This research is considering an integral and interdisciplinary approach, including computer science, mathematics and architecture. Inspired by fractal geometry, the IFS formalism is studied with regards to discrete architectural geometric design. The geometric design method studied provides new shape control possibilities unifying two separate design paradigms of rough and smooth objects. Capable to design fractal geometric figures, the method also covers the generation of classical objects such as conics and NURBS-curves. Close attention has been paid to the design of iterative free-form surfaces, which are composed entirely out of planar elements. A surface method based on projected vector sums is proposed. The resulting geometric figures are expressed in a discrete form and can be easily translated into a coherent set of constructional elements. The studies for translation of the geometrical elements into constructional elements consider integrated manufacturing. Addressing and numbering of the elements by iterative geometric design are investigated and compared to lexicographically ordered addressing systems, in order to provide an adequate data structure for the design, production and assembly of the constructional elements. For the generation of the data describing constructional elements, problems related to thickening and offset meshes are discussed. Once the global geometry of the constructional part has been computed, parameters are defined for generic automated detailing. Hereby the entire description of the constructional elements is completed. These elements are mapped and packed with regards to the coordinate system of a CNC-machine and the properties and the dimensions of the raw material, providing the complete set of workshop plans needed for integrated manufacturing. For automated generation of machine instructions (G-code), machining strategies – depending on the type of machine used, tool and material properties – are elaborated. Finally, the integrated digital design methods studied within the scope of this thesis are tested and verified by the realization of different reduced scale prototypes. The studied applications range from bearing vault structures to fractal and smooth timber panel shell structures. The developed methods have shown to be efficient for the design and the realization of geometrically complex architectural objects. The required planning effort to handle and manipulate the design and the production data has been greatly reduced. Some of the proposed methods have proved to be robust and general enough to be applied on real world applications. Iterative geometric design provides high degree of design possibilities offering an efficient tool for the creation of smooth and rough free form objects. The possibility to incorporate successive folds in free-form objects allows structural applications

Quantification of tumour heterogenity in MRI

Cancer is the leading cause of death that touches us all, either directly or indirectly. It is estimated that the number of newly diagnosed cases in the Netherlands will increase to 123,000 by the year 2020. General Dutch statistics are similar to those in the UK, i.e. over the last ten years, the age-standardised incidence rate1 has stabilised at around 355 females and 415 males per 100,000. Figure 1 shows the cancer incidence per gender. In the UK, the rise in lifetime risk of cancer is more than one in three and depends on many factors, including age, lifestyle and genetic makeup

Geometry–aware finite element framework for multi–physics simulations: an algorithmic and software-centric perspective

In finite element simulations, the handling of geometrical objects and their discrete representation is a critical aspect in both serial and parallel scientific software environments. The development of codes targeting such envinronments is subject to great development effort and man-hours invested. In this thesis we approach these issues from three fronts. First, stable and efficient techniques for the transfer of discrete fields between non matching volume or surface meshes are an essential ingredient for the discretization and numerical solution of coupled multi-physics and multi-scale problems. In particular L2-projections allows for the transfer of discrete fields between unstructured meshes, both in the volume and on the surface. We present an algorithm for parallelizing the assembly of the L2-transfer operator for unstructured meshes which are arbitrarily distributed among different processes. The algorithm requires no a priori information on the geometrical relationship between the different meshes. Second, the geometric representation is often a limiting factor which imposes a trade-off between how accurately the shape is described, and what methods can be employed for solving a system of differential equations. Parametric finite-elements and bijective mappings between polygons or polyhedra allow us to flexibly construct finite element discretizations with arbitrary resolutions without sacrificing the accuracy of the shape description. Such flexibility allows employing state-of-the-art techniques, such as geometric multigrid methods, on meshes with almost any shape.t, the way numerical techniques are represented in software libraries and approached from a development perspective, affect both usability and maintainability of such libraries. Completely separating the intent of high-level routines from the actual implementation and technologies allows for portable and maintainable performance. We provide an overview on current trends in the development of scientific software and showcase our open-source library utopia