Topics in geometry, analysis and inverse problems

The thesis consists of three independent parts. Part I: Polynomial amoebas We study the amoeba of a polynomial, as de ned by Gelfand, Kapranov and Zelevinsky. A central role in the treatment is played by a certain convex function which is linear in each complement component of the amoeba, which we call the Ronkin function. This function is used in two di erent ways. First, we use it to construct a polyhedral complex, which we call a spine, approximating the amoeba. Second, the Monge-Ampere measure of the Ronkin function has interesting properties which we explore. This measure can be used to derive an upper bound on the area of an amoeba in two dimensions. We also obtain results on the number of complement components of an amoeba, and consider possible extensions of the theory to varieties of codimension higher than 1. Part II: Differential equations in the complex plane We consider polynomials in one complex variable arising as eigenfunctions of certain differential operators, and obtain results on the distribution of their zeros. We show that in the limit when the degree of the polynomial approaches innity, its zeros are distributed according to a certain probability measure. This measure has its support on the union of nitely many curve segments, and can be characterized by a simple condition on its Cauchy transform. Part III: Radon transforms and tomography This part is concerned with different weighted Radon transforms in two dimensions, in particular the problem of inverting such transforms. We obtain stability results of this inverse problem for rather general classes of weights, including weights of attenuation type with data acquisition limited to a 180 degrees range of angles. We also derive an inversion formula for the exponential Radon transform, with the same restriction on the angle.

A variational approach to camera motion smoothing

[EN] In this paper we study a variational problem derived from a computer vision application: video camera calibration with smoothing constraint. By video camera calibration we meanto estimate the location, orientation and lens zoom-setting of the camera for each video frame taking into account image visible features. To simplify the problem we assume that the camera is mounted on a tripod, in such case, for each frame captured at time t , the calibration is provided by 3 parameters : (1) P(t) (PAN) which represents the tripod vertical axis rotation, (2) T(t) (TILT) which represents the tripod horizontal axis rotation and (3) Z(t) (CAMERA ZOOM) the camera lens zoom setting. The calibration function t -> u(t) = (P(t),T(t),Z(t)) is obtained as the minima of an energy function I[u] . In thIs paper we study the existence of minima of such energy function as well as the solutions of the associated Euler-Lagrange equations.