Numerical simulations in 3-dimensions of reaction–diffusion models for brain tumour growth

We work with a well-known model of reaction–diffusion type for brain tumour growth and accomplish full 3-dimensional (3d) simulations of the tumour in time on two types of imaging data, the 3d Shepp–Logan head phantom image and an MRI T1-weighted brain scan from the Internet Brain Segmentation Repository. The source term is such that we have logistic growth. These simulations are obtained using standard finite difference approximations with novel calculations to increase speed and accuracy. Moreover, biological background to the model, its well-posedness together with a variational formulation are given. The variational formulation enable the feasibility of different derivations and modifications of the model

Variational Tensor-Based Models for Image Diffusion in Non-Linear Domains

This dissertation addresses the problem of adaptive image filtering. Although the topic has a long history in the image processing community, researchers continuously present novel methods to obtain ever better image restoration results. With an expanding market for individuals who wish to share their everyday life on social media, imaging techniques such as compact cameras and smart phones are important factors. Naturally, every producer of imaging equipment desires to exploit cheap camera components while supplying high quality images. One step in this pipeline is to use sophisticated imaging software including, e.g., noise reduction to reduce manufacturing costs, while maintaining image quality. This thesis is based on traditional formulations such as isotropic and tensor-based anisotropic diffusion for image denoising. The difference from main-stream denoising methods is that this thesis explores the effects of introducing contextual information as prior knowledge for image denoising into the filtering schemes. To achieve this, the adaptive filtering theory is formulated from an energy minimization standpoint. The core contributions of this work is the introduction of a novel tensor-based functional which unifies and generalises standard diffusion methods. Additionally, the explicit Euler-Lagrange equation is derived which, if solved, yield the stationary point for the minimization problem. Several aspects of the functional are presented in detail which include, but are not limited to, tensor symmetry constraints and convexity. Also, the classical problem of finding a variational formulation to a given tensor-based partial differential equation is studied. The presented framework is applied in problem formulation that includes non-linear domain transformation, e.g., visualization of medical images. Additionally, the framework is also used to exploit locally estimated probability density functions or the channel representation to drive the filtering process. Furthermore, one of the first truly tensor-based formulations of total variation is presented. The key to the formulation is the gradient energy tensor, which does not require spatial regularization of its tensor components. It is shown empirically in several computer vision applications, such as corner detection and optical flow, that the gradient energy tensor is a viable replacement for the commonly used structure tensor. Moreover, the gradient energy tensor is used in the traditional tensor-based anisotropic diffusion scheme. This approach results in significant improvements in computational speed when the scheme is implemented on a graphical processing unit compared to using the commonly used structure tensor.VIDINACIPGARNICSEMC^

A Variational Approach to Image Diffusion in Non-Linear Domains

Image filtering methods are designed to enhance noisy images captured in situations that are problematic for the camera sensor. Such noisy images originate from unfavourable illumination conditions, camera motion, or the desire to use only a low dose of ionising radiation in medical imaging. Therefore, in this thesis work I have investigated the theory of partial differential equations (PDE) to design filtering methods that attempt to remove noise from images. This is achieved by modeling and deriving energy functionals which in turn are minimized to attain a state of minimum energy. This state is obtained by solving the so called Euler-Lagrange equation. An important theoretical contribution of this work is that conditions are put forward determining when a PDE has a corresponding energy functional. This is in particular described in the case of the structure tensor, a commonly used tensor in computer vision.A primary component of this thesis work is to model adaptive image filtering such that any modification of the image is structure preserving, but yet is noise suppressing. In color image filtering this is a particular challenge since artifacts may be introduced at color discontinuities. For this purpose a non-Euclidian color opponent transformation has been analysed and used to separate the standard RGB color space into uncorrelated components.A common approach to achieve adaptive image filtering is to select an edge stopping function from a set of functions that have proven to work well in the past. The purpose of the edge stopping function is to inhibit smoothing of image features that are desired to be retained, such as lines, edges or other application dependent characteristics. Thus, a step from ad-hoc filtering based on experience towards an application-driven filtering is taken, such that only desired image features are processed. This improves what is characterised as visually relevant features, a topic which this thesis covers, in particular for medical imaging.The notion of what are relevant features is a subjective measure may be different from a layman's opinion compared to a professional's. Therefore, we advocate that any image filtering method should yield an improvement not only in numerical measures but also a visual improvement should be experienced by the respective end-userNACIP, VIDI, GARNIC

A Variational Approach to Image Diffusion in Non-Linear Domains

Image filtering methods are designed to enhance noisy images captured in situations that are problematic for the camera sensor. Such noisy images originate from unfavourable illumination conditions, camera motion, or the desire to use only a low dose of ionising radiation in medical imaging. Therefore, in this thesis work I have investigated the theory of partial differential equations (PDE) to design filtering methods that attempt to remove noise from images. This is achieved by modeling and deriving energy functionals which in turn are minimized to attain a state of minimum energy. This state is obtained by solving the so called Euler-Lagrange equation. An important theoretical contribution of this work is that conditions are put forward determining when a PDE has a corresponding energy functional. This is in particular described in the case of the structure tensor, a commonly used tensor in computer vision.A primary component of this thesis work is to model adaptive image filtering such that any modification of the image is structure preserving, but yet is noise suppressing. In color image filtering this is a particular challenge since artifacts may be introduced at color discontinuities. For this purpose a non-Euclidian color opponent transformation has been analysed and used to separate the standard RGB color space into uncorrelated components.A common approach to achieve adaptive image filtering is to select an edge stopping function from a set of functions that have proven to work well in the past. The purpose of the edge stopping function is to inhibit smoothing of image features that are desired to be retained, such as lines, edges or other application dependent characteristics. Thus, a step from ad-hoc filtering based on experience towards an application-driven filtering is taken, such that only desired image features are processed. This improves what is characterised as visually relevant features, a topic which this thesis covers, in particular for medical imaging.The notion of what are relevant features is a subjective measure may be different from a layman's opinion compared to a professional's. Therefore, we advocate that any image filtering method should yield an improvement not only in numerical measures but also a visual improvement should be experienced by the respective end-userNACIP, VIDI, GARNIC

A Variational Approach to Image Diffusion in Non-Linear Domains

Image filtering methods are designed to enhance noisy images captured in situations that are problematic for the camera sensor. Such noisy images originate from unfavourable illumination conditions, camera motion, or the desire to use only a low dose of ionising radiation in medical imaging. Therefore, in this thesis work I have investigated the theory of partial differential equations (PDE) to design filtering methods that attempt to remove noise from images. This is achieved by modeling and deriving energy functionals which in turn are minimized to attain a state of minimum energy. This state is obtained by solving the so called Euler-Lagrange equation. An important theoretical contribution of this work is that conditions are put forward determining when a PDE has a corresponding energy functional. This is in particular described in the case of the structure tensor, a commonly used tensor in computer vision.A primary component of this thesis work is to model adaptive image filtering such that any modification of the image is structure preserving, but yet is noise suppressing. In color image filtering this is a particular challenge since artifacts may be introduced at color discontinuities. For this purpose a non-Euclidian color opponent transformation has been analysed and used to separate the standard RGB color space into uncorrelated components.A common approach to achieve adaptive image filtering is to select an edge stopping function from a set of functions that have proven to work well in the past. The purpose of the edge stopping function is to inhibit smoothing of image features that are desired to be retained, such as lines, edges or other application dependent characteristics. Thus, a step from ad-hoc filtering based on experience towards an application-driven filtering is taken, such that only desired image features are processed. This improves what is characterised as visually relevant features, a topic which this thesis covers, in particular for medical imaging.The notion of what are relevant features is a subjective measure may be different from a layman's opinion compared to a professional's. Therefore, we advocate that any image filtering method should yield an improvement not only in numerical measures but also a visual improvement should be experienced by the respective end-userNACIP, VIDI, GARNIC

On the Choice of Tensor Estimation for Corner Detection, Optical Flow and Denoising

Many image processing methods such as corner detection,optical flow and iterative enhancement make use of image tensors. Generally, these tensors are estimated using the structure tensor. In this work we show that the gradient energy tensor can be used as an alternativeto the structure tensor in several cases. We apply the gradient energy tensor to common image problem applications such as corner detection, optical flow and image enhancement. Our experimental results suggest that the gradient energy tensor enables real-time tensor-based image enhancement using the graphical processing unit (GPU) and we obtain 40% increase of frame rate without loss of image quality.VID

On Tensor-Based PDEs and their Corresponding Variational Formulations with Application to Color Image Denoising

The case when a partial differential equation (PDE) can be considered as an Euler-Lagrange (E-L) equation of an energy functional, consisting of a data term and a smoothness term is investigated. We show the necessary conditions for a PDE to be the E-L equation for a corresponding functional. This energy functional is applied to a color image denoising problem and it is shown that the method compares favorably to current state-of-the-art color image denoising techniques.NACIPGARNICSELLII