João Paixão

Parameterized Complexity of Discrete Morse Theory

Optimal Morse matchings reveal essential structures of cell complexes that lead to powerful tools to study discrete geometrical objects, in particular, discrete 3-manifolds. However, such matchings are known to be NP-hard to compute on 3-manifolds through a reduction to the erasability problem. Here, we refine the study of the complexity of problems related to discrete Morse theory in terms of parameterized complexity. On the one hand, we prove that the erasability problem is W[P]-complete on the natural parameter. On the other hand, we propose an algorithm for computing optimal Morse matchings on triangulations of 3-manifolds, which is fixed-parameter tractable in the treewidth of the bipartite graph representing the adjacency of the 1- and 2-simplices. This algorithm also shows fixed-parameter tractability for problems such as erasability and maximum alternating cycle-free matching. We further show that these results are also true when the treewidth of the dual graph of the triangulated 3-manifold is bounded. Finally, we discuss the topological significance of the chosen parameters and investigate the respective treewidths of simplicial and generalized triangulations of 3-manifolds.

Tuning Manifold Harmonics Filters

There are several techniques for automatic music visualization, which are included with virtually any media player. The basic ingredient of those techniques is spectral analysis of the sound, used to automatically generate parameters for procedural image generation. However, only a few music visualizations rely on 3d models. This paper proposes to use spectral mesh processing techniques, namely manifold harmonics, to produce 3d music visualization. The images are generated from 3d models by deforming an initial shape, mapping the sound frequencies to the mesh harmonics. A concise representation of such frequency mapping is proposed to permit for an animated gallery interface with genetic reproduction. Such galleries allow the user to quickly navigate between visual effects. Rendering such animated galleries in real-time is a challenging task, since it requires computing and rendering the deformed shapes at a very high rate. This paper introduces a direct GPU implementation of manifold harmonics filters, which allows to display animated gallery.

Topology Aware Vector Field Denoising

Recent developments in data acquisition technology enable to directly capture real vector fields, helping for a better understanding of physical phenomena. However measured data is corrupted by noise, puzzling the understanding of the phenomena. This turns the task of removing noise, i.e. denoising, an essential preprocessing step for a better analysis of the data. Nonetheless a careful use of denoising is required since usual algorithms not only remove the noise but can also eliminate information, in particular the vector field singularities, which are fundamental features in the analysis. This paper proposes a semi-automatic vector field denoising methodology, where the user visually controls the topological changes caused by classical vector field filtering in scale-spaces.

Stereo music visualization through manifold harmonics

Music visualizations are nowadays included with virtually any media player. They usually rely on harmonic analysis of each sound channel, which automatically generate parameters for procedural image generation. However, only few music visualizations make use of 3d shapes. This paper proposes to use spectral mesh processing techniques, here manifold harmonics, to produce 3d stereo music visualization. The images are generated from 3d models by deforming an initial shape, mapping the sound frequencies to the mesh harmonics. A symmetry criterion is introduced to enhance the stereo effects on the deformed shape. A concise representation of the frequency mapping is proposed to allow for an animated gallery interface with genetic reproduction. Such galleries let the user quickly navigate between visual effects. Rendering such animated galleries in real time is a challenging task, since it requires computing and rendering the deformed shapes at a very high rate. This paper introduces a direct GPU implementation of manifold harmonics filters, which allows the displaying of the animated galleries.

Random Walks for Vector Field Denoising

In recent years, several devices allow to directly measure real vector fields, leading to a better understanding of fundamental phenomena such as fluid simulation or brainwater movement. This turns vector field visualization and analysis important tools for many applications in engineering and in medicine. However, real data is generally corrupted by noise, puzzling the understanding provided by those tools.Those tools thus need a denoising step as preprocessing, although usual denoising removes discontinuities, which are fundamental for vector field analysis. This paper proposes a novel method for vector field denoising based on random walks which preserve those discontinuities. It works in a meshless setting; it is fast, simple to implement, and shows a better performance than the traditional gaussian denoising technique.