The Hidden Geometry of Total Variation

We want to investigate geometrical characteristics and eigenfunctions of (anisotropic-, vectorial- and second order) total-variation minimization problems. Minimizing scalar total variation can be reformulated as a set-optimization problem, where for each levelset we obtain an optimization problem involving perimeter and area. In the scalar case eigenfuctions are characteristics of sets whose curvature is smaller than the ratio of perimeter over area (geometrical condition). Explicit construction of such minimizers allows calculating the duals of characteristic functions and give them a geometrical interpretation. Dual total variation functionals seem to be connected to a limiting stability parameter in mechanical problems. Analog results for total variation for vector fields or higher order total variation have not been investigated yet, since the methods used in the scalar case (co-area-formula, morphological operators) are not directly applicable to these cases. The applications to be studied here, demonstrate, that these theoretical considerations have an immense practical significance: 1.) Parameter identification in fluid mechanics: We consider the problem of identifying the limiting buoyancy parameter at which a solid in a fluid is not moving. This problem reduces to an eigenvalue problem where we want to obtain analytical solutions in several geometries by calculating dual total variation functionals for characteristic functions/vector fields. In the simplest case the problem reduces to a scalar problem (scalar total variation). The difficulty of the problem increases dramatically when reducing the symmetry of the setup. The major goal is to study these complex vectorial case. 2.) Parameter identification in hybrid imaging: Hybrid imaging will be one of the primary cancer-diagnostic tools in the future. Ist techniques utilize couplings of physical modalities. Recently there has been an enormous research interest in this area because these methods promise very high resolution without losing contrast. One of these promising methods is Current Density Impedance Imaging, a new imaging technique that can non-invasively measure the conductivity distribution inside a medium. The mathematical methods behind these techniques are challenging and require to solve parameter identification problems (inverse problems). In many cases the devices can only measure data inside the test sample but not on ist boundary. In many applications knowledge of the structure of the parameter is sufficient information (i.e., crack in a material, boundary of a tumour). Hence we propose a method based on vector-total variation, that uses available interior data in order to reconstruct this structural information. We consider fundamental problems that can lead to better understanding and should yield the basis for completely new approaches in TV-type optimization problems.

Solving Partial Differential equations (PDEs) numerically is quite standard nowadays in order to solve mechanical, electrical or any type of problems in industry. Can we trust the numerical solution? Explicit solutions are only known for some special kinds of PDEs, but knowing properties of solutions, can also help to estimate how well a numerical solution can approximate the real solution? Why are we interested in explicit solutions? Having explicit solutions, for basic problems helps to get an idea of the behaviour of solutions and gives us a tool to test/develop numerical solutions for that special kind of PDE. The type of PDEs that I looked at, included a term with fraction that can be 1,-1 in regions where the solution is not constant, and would be 0/0 whenever the solution is constant. A division through zero is always problematic, because it leads to things that are either not really defined or not unique. The trick to overcome this difficulty was to reformulate the PDE into a set of optimization problems that can be solved using morphological image operation, that are normally used in image processing. To "close" a geometric object (like for instance a triangle) in the morphological sense, one needs to move a circle with a fixed radius inside his geometric object. It is like drawing a triangle with chalk on a blackboard, then using a sponge with circular cross-section, cleaning the black-board inside the triangle, trying not to touch the boarder of the triangle. The cleaned area inside the triangle is called the closingof the triangle. Now where is the connection to the partial differential equation? We could show that it is possible to solve a special type of partial differential equations (the involved domains are convex objects) can be transformed to an optimization problem involving area and perimeter of this domain and its morphological closed versions (the sponge cleaned versions of it). As an application, we worked on a fluid-dynamics problem: Imagine putting a really long cylinder inside the ocean. Depending on the material of this cylinder and the viscosity of the ocean, this cylinder might sink to the ground, swim on top or just float (meaning neither sinking nor swimming). It does no only depend on the material of the cylinder but also on the radius of it. Instead of solving a complex PDE, one can just solve an equation that includes parameters such as area and perimeter of the cross-section of the cylinder (and its morphological closed versions). The main goal was to find new methods to calculate explicit solutions for higher order PDEs or higher-dimensional PDEs.