Perimeter | EPFL Graph Search

A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference. Calculating the perimeter has several practical applications. A calculated perimeter is the length of fence required to surround a yard or garden. The perimeter of a wheel/circle (its circumference) describes how far it will roll in one revolution. Similarly, the amount of string wound around a spool is related to the spool's perimeter; if the length of the string was exact, it would equal the perimeter. Formulas The perimeter is the distance around a shape. Perimeters for more general shapes can be calculated, as any path, with \int_0^L \mathrm{d}s, where L is the length of the path and ds is an infinitesimal line element. Both of these must be replaced by algebraic forms in order to be practically calculated. If the

Propriétés magnétiques de nanostructures de cobalt adsorbées

Nicolas Weiss

This thesis reports results on magnetic properties of supported cobalt nanostructures. The nanostructures were grown on single crystal metal surface by Molecular Beam Epitaxy in an Ultra High Vacuum chamber. Our experimental setup is based on two in situ measurement techniques. The first is the Scanning Tunneling Microscope (STM) allowing to investigate the nanostructure morphology. The second, called Surface Magneto-Optical Kerr Effect (SMOKE), probes the magnetism of the nanostructures. By combining the data obtained with both techniques we were able to investigate the magnetism of 2D nanoparticles down to the atomic level. The first aim of the thesis consisted in disclosing the role played by the differently coordinated atoms in determining the Magnetic Energy Anisotropy in 2D nanostructures. To this purpose, we performed experiments on the Co/Pt(111) system, chosen as a model system since the Pt surface is well known to increase the out-of-plane uniaxial anisotropy as well as the magnetic moment of the Co islands. A preliminary work has been devoted to the growth of the Co islands as a function of the deposition and annealing temperatures. As a result, we learned to grow three different island's shapes : ramified, compact and double-layer islands. We then concentrated on the magnetism of these islands. Due to the nonlinear relationship between perimeter length and surface area, we were able to distinguish the different contributions to the anisotropy energy of surface and perimeter atoms. We found that the magnetic anisotropy in 2D Co nanostructures is predominantly due to the low-coordinated edge atoms, for which the anisotropy energy can be as large as 20 times the bulk value. This finding opens new possibilities to separately tune the anisotropy and moment of nanostructures. To exemplify this, and to illustrate once more the role of perimeter atoms, we produced Co island with a non-magnetic Pt core. As expected, these bimetallic islands have identical anisotropy and lower magnetic moment to their equally shaped pure Co counterpart. The second aim we focused on was the investigation of the magnetic properties of ultra-high density arrays of nanostructures created by self-assembly. We chose Co/Au(788) as a model system since the nanoscopic Co dots were found to have a very narrow size distribution and to be placed onto a lattice which is phase coherent on a macroscopic length scale. Moreover, the array has a density of 26 Tdot/in2 which allows the investigation of inter-particle interaction in ultra high density arrays. We showed that the narrow size distribution of the dots corresponds to an unprecedented narrow Magnetic Anisotropy Energy distribution with 35% width at half of the maximum. Moreover, we demonstrated the weak magnetic coupling between the Co dots despite their high density. In parallel to these fundamental researches, we developed a new experimental setup that will replace the current one. We designed a UHV chamber equipped with a homemade variable temperature STM combined with a SMOKE experiment. A quadrupole, focused on the sample, generates a 0.35 T magnetic field. This field is one order of magnitude larger than the present one and can be uniformly rotated in the plane of the quadrupole. Furthermore, the design of the new chamber allows STM imaging in this magnetic field. In this report, we present the adopted solutions as well as the first tests of the components (STM, SMOKE, cryostat).

EPFL2004

Shapes from Pixels

Mitra Fatemi

In today's digital world, sampling is at the heart of any signal acquisition device. Imaging devices are ubiquitous examples that capture two-dimensional visual signals and store them as the pixels of discrete images. The main concern is whether and how the pixels provide an exact or at least a fair representation of the original visual signal in the continuous domain. This motivates the design of exact reconstruction or approximation techniques for a target class of images. Such techniques benefit different imaging tasks such as super-resolution, deblurring and compression. This thesis focuses on the reconstruction of visual signals representing a shape over a background, from their samples. Shape images have only two intensity values. However, the filtering effect caused by the sampling kernel of imaging devices smooths out the sharp transitions in the image and results in samples with varied intensity levels. To trace back the shape boundaries, we need strategies to reconstruct the original bilevel image. But, abrupt intensity changes along the shape boundaries as well as diverse shape geometries make reconstruction of this class of signals very challenging. Curvelets and contourlets have been proved as efficient multiresolution representations for the class of shape images. This motivates the approximation of shape images in the aforementioned domains. In the first part of this thesis, we study generalized sampling and infinite-dimensional compressed sensing to approximate a signal in a domain that is known to provide a sparse or efficient representation for the signal, given its samples in a different domain. We show that the generalized sampling, due to its linearity, is incapable of generating good approximation of shape images from a limited number of samples. The infinite-dimensional compressed sensing is a more promising approach. However, the concept of random sampling in this scheme does not apply to the shape reconstruction problem. Next, we propose a sampling scheme for shape images with finite rate of innovation (FRI). More specifically, we model the shape boundaries as a subset of an algebraic curve with an implicit bivariate polynomial. We show that the image parameters are solutions of a set of linear equations with the coefficients being the image moments. We then replace conventional moments with more stable generalized moments that are adjusted to the given sampling kernel. This leads to successful reconstruction of shapes with moderate complexities from samples generated with realistic sampling kernels and in the presence of moderate noise levels. Our next contribution is a scheme for recovering shapes with smooth boundaries from a set of samples. The reconstructed image is constrained to regenerate the same samples (consistency) as well as forming a bilevel image. We initially formulate the problem by minimizing the shape perimeter over the set of consistent shapes. Next, we relax the non-convex shape constraint to transform the problem into minimizing the total variation over consistent non-negative-valued images. We introduce a requirement -called reducibility- that guarantees equivalence between the two problems. We illustrate that the reducibility effectively sets a requirement on the minimum sampling density. Finally, we study a relevant problem in the Boolean algebra: the Boolean compressed sensing. The problem is about recovering a sparse Boolean vector from a few collective binary tests. We study a formulation of this problem as a binary linear program, which is NP hard. To overcome the computational burden, we can relax the binary constraint on the variables and apply a rounding to the solution. We replace the rounding procedure with a randomized algorithm. We show that the proposed algorithm considerably improves the success rate with only a slight increase in the computational cost.

EPFL2016

Upper bounds for the perimeter of plane convex bodies

Filip Moric

We show that the maximum total perimeter of k plane convex bodies with disjoint interiors lying inside a given convex body C is equal to , in the case when C is a square or an arbitrary triangle. A weaker bound is obtained for general plane convex bodies. As a consequence, we establish a bound on the perimeter of a polygon with at most k reflex angles lying inside a given plane convex body.

Springer2014