A Hybrid High-Order method for Leray-Lions elliptic equations on general meshes

In this work, we develop and analyze a Hybrid High-Order (HHO) method for steady non-linear Leray-Lions problems. The proposed method has several assets, including the support for arbitrary approximation orders and general polytopal meshes. This is achieved by combining two key ingredients devised at the local level: a gradient reconstruction and a high-order stabilization term that generalizes the one originally introduced in the linear case. The convergence analysis is carried out using a compactness technique. Extending this technique to HHO methods has prompted us to develop a set of discrete functional analysis tools whose interest goes beyond the specific problem and method addressed in this work: (direct and) reverse Lebesgue and Sobolev embeddings for local polynomial spaces, $L^{p}$-stability and $W^{s,p}$-approximation properties for $L^{2}$-projectors on such spaces, and Sobolev embeddings for hybrid polynomial spaces. Numerical tests are presented to validate the theoretical results for the original method and variants thereof

Visual Computing Tools for Studying Micro-scale Diffusion

In this dissertation, we present novel visual computing tools and techniques to facilitate the exploration, simulation, and visualization of micro-scale diffusion. Our research builds upon the latest advances in visualization, high-performance computing, medical imaging, and human perception. We validate our research using the driving applications of nano-assembly and diffusion kurtosis imaging (DKI). In both of these applications, diffusion plays a central role. In the former it mediates the process of transporting micron-sized particles through moving lasers, and in the latter it conveys brain micro-geometry. Nanocomponent-based devices, such as bio-sensors, electronic components, photonic devices, solar cells, and batteries, are expected to revolutionize health care, energy, communications, and the computing industry. However, in order to build such useful devices, nanoscale components need to be properly assembled together. We have developed a hybrid CPU/GPU-based computing tool to understand complex interactions between lasers, optical beads, and the suspension medium. We demonstrate how a high-performance visual computing tool can be used to accelerate an optical tweezers simulation to compute the force applied by a laser onto micro particles and study shadowing (refraction) behavior. This represents the first steps toward building a real-time nano-assembly planning system. A challenge in building such a system, however, is that optical tweezers systems typically lack stereo depth cues. We have developed a visual tool to provide an enhanced perception of a scene's 3D structure using the kinetic depth effect. The design of our tool has been informed by user studies of stereo perception using the kinetic-depth effect on monocular displays. Diffusion kurtosis imaging is gaining rapid adoption in the medical imaging community due to its ability to measure the non-Gaussian property of water diffusion in biological tissues. Compared with the traditional diffusion tensor imaging (DTI), DKI can provide additional details about the underlying microstructural characteristics of neural tissues. It has shown promising results in studies on changes in gray matter and mild traumatic brain injuries, where DTI is often found to be inadequate. However, the highly detailed spatio-angular fields in DKI datasets present a special challenge for visualization. Traditional techniques that use glyphs are often inadequate for expressing subtle changes in the DKI fields. In this dissertation, we outline a systematic way to manage, analyze, and visualize spatio-angular fields using spherical harmonics lighting functions to facilitate insights into the micro-structural properties of the brain

Parametric temporal alignment for the detection of facial action temporal segments

In this paper we propose the very first weakly supervised approach for detecting facial action unit temporal segments. This is achieved by means of behaviour similarity matching, where no training of dedicated classifiers is needed and the input facial behaviour episode is compared to a template. The inferred temporal segment boundaries of the test sequence are those transferred from the template sequence. To this end, a parametric temporal alignment algorithm is proposed to align a single exemplar sequence to the test sequence. The proposed strategy can accommodate flexible time warp functions, does not need to exhaustively align all frames in both sequences, and the optimal warp parameters can be found by an efficient Gauss-Newton gradient descent search. We show that our approach produces the best results to date for the problem at hand, and provides a promising opportunity to studying facial actions from a new perspective

Pozostałości pików oddechowych w widmie zmienności rytmu serca dorosłego człowieka z szybkim rytmem oddechowym i wolnym rytmem serca

Wstęp: Analiza zmienności rytmu serca (HRV) w dziedzinie częstotliwości jest użytecznym narzędziem do badania fluktuacji rytmu zatokowego powodowanych przez autonomiczny układ nerwowy. Jednakże, nie jest jasne, jaka jest górna granica częstotliwości, która może być analizowana w widmie HRV. Materiał i metody: Badamy widma HRV dorosłego ochotnika z szybkim rytmem oddechowym, gdy jego rytm serca jest zwalniany bądź przyspieszany. Stosując szybką transformatę Fouriera (FFT), analizujemy serie odstępów między uderzeniami serca w dziedzinie częstotliwości. Przeprowadzamy również komputerowe symulacje mierzonych sygnałów i porównujemy otrzymane widma z widmami ochotnika. Wyniki: Jeżeli częstość oddechów przekracza połowę średniej częstości uderzeń serca, w widmie HRV są obecne dwa piki reprezentujące ten sam sygnał oddechowy. Jeden z pików jest wynikiem zjawiska aliasingu, natomiast drugi jest pozostałością prawdziwego piku oddechowego, jako efekt właściwości symetrii FFT. Żaden z nich nie przynosi rzetelnej, ilościowej informacji o fali oddechowej w widmie HRV. Wnioski: Górna granica częstotliwości w widmie HRV jest zdeterminowana przez połowęśredniej częstotliwości serca. Przed obliczaniem widma należy więc określić górny możliwy do analizy limit częstotliwości, by uniknąć sytuacji, w której analizowane zakresy lokalizują się poza tą granicą. Sytuacja taka jest wielce prawdopodobna w przypadku, gdy pacjent wykazuje wolną akcję serca