Mikola Lysenko

Group morphology with convolution algebras

Group morphology is an extension of mathematical morphology with classical Minkowski sum and difference operations generalized respectively to Minkowski product and quotient operations over arbitrary groups. We show that group morphology is a proper setting for unifying, formulating and solving a number of important problems, including translational and rotational configuration space problems, mechanism workspace computation, and symmetry detection. The proposed computational approach is based on group convolution algebras, which extend classical convolutions and the Fourier transform to non-commutative groups. In particular, we show that all Minkowski product and quotient operations may be represented implicitly as sublevel sets of the same real-valued convolution function.

Path-Based Mathematical Morphology on Tensor Fields

Traditional path-based morphology allows finding long, approximately straight, paths in images. Although originally applied only to scalar images, we show how this can be a very good fit for tensor fields. We do this by constructing directed graphs representing such data, and then modifying the traditional path opening algorithm to work on these graphs. Cycles are dealt with by finding strongly connected components in the graph. Some examples of potential applications are given, including path openings that are not limited to a specific set of orientations.

Non-commutative morphology: Shapes, filters, and convolutions

Group morphology is a generalization of mathematical morphology which makes an explicit distinction between shapes and filters. Shapes are modeled as point sets, for example binary images or 3D solid objects, while filters are collections of transformations (such as translations, rotations or scalings). The action of a filter on a shape generalizes the basic morphological operations of dilation and erosion. This shift in perspective allows us to compose filters independent of shapes, and leads to a non-commutative generalization of the Minkowski sum and difference which we call the Minkowski product and quotient respectively. We show that these operators are useful for unifying, formulating and solving a number of important problems, including translational and rotational configuration space problems, mechanism workspace computation, and symmetry detection. To compute these new operators, we propose the use of group convolution algebras, which extend classical convolution and the Fourier transform to non-commutative groups. In particular, we show that all Minkowski product and quotient operations may be represented implicitly as sublevel sets of the same real-valued convolution function.

Rapid Mapping and Exploration of Configuration Space

We describe a GPU-based computational platform for six-dimensional configuration mapping, which is the description of the configuration space of rigid motions in terms of collision and contact constraints. The platform supports a wide range of computations in design and manufacturing, including three and six dimensional configuration space obstacle computations, Minkowski sums and differences, packaging problems, and sweep computations. We demonstrate dramatic performance improvements in the special case of configuration space operations that determine interference-free or containment-preserving configurations between moving solids. Our approach treats such operations as convolutions in the six dimensional configuration space that are efficiently computed using the Fast Fourier Transform (FFT). The inherent parallelism of FFT algorithms facilitates a straightforward implementation of convolution on GPUs with existing and freely available libraries, making all such six dimensional configuration space computations practical, and often interactive.Copyright

Fourier collision detection

We investigate a new approach to narrowphase collision detection for rigid objects based on a Fourier series expansion. This new collision test scales with respect to accuracy (in the Hausdorff sense), which we show rigorously in the case of translational motions. Because our new form of the collision test is also a smooth inequality, it can be used as a holonomic unilateral constraint in many applications, such as path planning, rigid body dynamics, nesting or tool placement, replacing the need for more ad-hoc normal/contact-based constraint solvers. Moreover, we also show how this constraint can be directly differentiated via Fourier multipliers with only a constant factor overhead, which leads to a simple method for constructing a Jacobian for both normal forces and rotational torques.

Real-Time Machinability Analysis of Free Form Surfaces on the GPU

In this paper a new hardware accelerated method is presented to evaluate the machinability of free-form surfaces. This method works on tessellated models that are commonly used by CAD systems to render three-dimensional shaded images of solid models. Modern Graphics Processing Units (GPUs) can be programmed in hardware to accelerate specialized rendering techniques. In this research, we have developed new algorithms that utilize the programmability of GPUs to evaluate machinability of free-form surfaces. The method runs in real time on fairly inexpensive hardware (<

Effective contact measures

600), and performs well regardless of the surface type. The complexity of the method is dictated by the size of the projected view of the model. The proposed method can be used as a plug-in in a CAD system to evaluate manufacturability of a part at early design stages. The efficiency and the speed of the proposed method are demonstrated on some complex objects.Copyright

Configuration Workspaces of Series-Parallel Mechanisms

Contact area is an important geometric measurement in many physical systems. It is also difficult to compute due to its extreme sensitivity to infinitesimal perturbations. In this paper, we propose a new concept called an effective contact measure, which acts as a smooth version of contact area. Effective contact measures incorporate a notion of scale into the definition of contact area, allowing one to consider the degree of contact at different sizes. We show how effective contact measures can yield useful statistics for a number of applications, including analysis of multiphase materials and docking/alignment problems. Introduces a new concept of effective contact measure as an approximation of surface contact area.Proposes 3 new concepts of effective contact measures.Discusses application to alignment problems.

A Framework for Megascale Agent Based Model Simulations on Graphics Processing Units

The workspace of a mechanism is the set of positions and orientations that is reachable by its end effector. Workspaces have numerous applications, including motion planning, mechanism design, and manufacturing process planning, but their representation and computation is challenging due to high dimensionality and geometric/topological complexity. We propose a new formulation of the workspace computation problem for a large class of mechanisms represented by series-parallel constraint graphs. A wide variety of allowable constraints include all lower pair, some higher pair, and non-collision constraints. We show that the workspace of such mechanisms may be computed by a constraint propagation algorithm. After the space of all rigid body motions is discretized, these operations can be efficiently implemented using the Fast Fourier Transform and a depth first search. In contrast to algebraic formulations, the proposed method assures that all configurations in the computed workspace not only satisfy pairwise constraints but can be reached without breaking and reassembling the mechanism.© 2012 ASME