Shape Calculus for Shape Energies in Image Processing

Many image processing problems are naturally expressed as energy minimization or shape optimization problems, in which the free variable is a shape, such as a curve in 2d or a surface in 3d. Examples are image segmentation, multiview stereo reconstruction, geometric interpolation from data point clouds. To obtain the solution of such a problem, one usually resorts to an iterative approach, a gradient descent algorithm, which updates a candidate shape gradually deforming it into the optimal shape. Computing the gradient descent updates requires the knowledge of the first variation of the shape energy, or rather the first shape derivative. In addition to the first shape derivative, one can also utilize the second shape derivative and develop a Newton-type method with faster convergence. Unfortunately, the knowledge of shape derivatives for shape energies in image processing is patchy. The second shape derivatives are known for only two of the energies in the image processing literature and many results for the first shape derivative are limiting, in the sense that they are either for curves on planes, or developed for a specific representation of the shape or for a very specific functional form in the shape energy. In this work, these limitations are overcome and the first and second shape derivatives are computed for large classes of shape energies that are representative of the energies found in image processing. Many of the formulas we obtain are new and some generalize previous existing results. These results are valid for general surfaces in any number of dimensions. This work is intended to serve as a cookbook for researchers who deal with shape energies for various applications in image processing and need to develop algorithms to compute the shapes minimizing these energies

Energy efficient engine high-pressure turbine detailed design report

The energy efficient engine high-pressure turbine is a single stage system based on technology advancements in the areas of aerodynamics, structures and materials to achieve high performance, low operating economics and durability commensurate with commercial service requirements. Low loss performance features combined with a low through-flow velocity approach results in a predicted efficiency of 88.8 for a flight propulsion system. Turbine airfoil durability goals are achieved through the use of advanced high-strength and high-temperature capability single crystal materials and effective cooling management. Overall, this design reflects a considerable extension in turbine technology that is applicable to future, energy efficient gas-turbine engines

Laser velocimeter measurements of the flowfield generated by an advanced counterrotating propeller

Results are presented of an investigation to measure the flowfield generated by an advanced counterrotating pusher propeller model similar to the full-scale Unducted Fan demonstrator engine. A laser Doppler velocimeter was used to measure the velocity field in several planes normal to the centerline of the model at axial stations upstream and downstream of each rotor. During this investigation, blades of the F4/A4 type were installed on the model which was operating in a freestream Mach 0.72 regime, with the advance ratio of each rotor set at 2.80. The measured data indicate only a slight influence of the potential field of each front rotor blade on the flowfield upstream of the rotor. The data measured downstream of the front rotor characterize the tip vortices, vortex sheets and potential field nonuniformities generated by the front rotor. The unsteadiness of the flow in the rotating frame of reference of the aft rotor is also illustrated

Lattice Boltzmann study of chemically-driven self-propelled droplets

We numerically study the behavior of self-propelled liquid droplets whose motion is triggered by a Marangoni-like flow. This latter is generated by variations of surfactant concentration which affect the droplet surface tension promoting its motion. In the present paper a model for droplets with a third amphiphilic component is adopted. The dynamics is described by Navier-Stokes and convection-diffusion equations, solved by lattice Boltzmann method coupled with finite-difference schemes. We focus on two cases. First the study of self-propulsion of an isolated droplet is carried on and, then, the interaction of two self-propelled droplets is investigated. In both cases, when the surfactant migrates towards the interface, a quadrupolar vortex of the velocity field forms inside the droplet and causes the motion. A weaker dipolar field emerges instead when the surfactant is mainly diluted in the bulk. The dynamics of two interacting droplets is more complex and strongly depends on their reciprocal distance. If, in a head-on collision, droplets are close enough, the velocity field initially attracts them until a motionless steady state is achieved. If the droplets are vertically shifted, the hydrodynamic field leads to an initial reciprocal attraction followed by a scattering along opposite directions. This hydrodynamic interaction acts on a separation of some droplet radii otherwise it becomes negligible and droplets motion is only driven by Marangoni effect. Finally, if one of the droplets is passive, this latter is generally advected by the fluid flow generated by the active one.Comment: 14 pages, 9 figures. In press on EPJ