Although algebraic or so-called “implicit polynomial” curves have been studied rather extensively for several decades, to the best of our knowledge, a dynamic formulation of them, similar to active contours, has not been done yet. This paper develops a dynamic formulation for implicit polynomial curves based on level set formalism. In particular, it is shown that utilization of an implicit poly...

Although algebraic or so called “implicit polynomial” curves/surfaces have been studied rather extensively for several decades, to the best of our knowledge, a dynamic formulation of them, similar to active contours, has not been done yet. This paper proposes to use implicit polynomial functions instead of signed distance functions in classical level set formulation and shows potential of such ...

We give a deterministic polynomial-time algorithm that computes a nontrivial rational point on an elliptic curve over a finite field, given a Weierstrass equation for the curve. For this, we reduce the problem to the task of finding a rational point on a curve of genus zero.

For any a := (a1, a2, . . . , an) ∈ (R) , define ∆Pa(x, t) := (x+ a1t)(x+ a2t) · · · (x + ant) − x and Sa(x, y) := a1x + a2xy + · · · + any. The two homogeneous polynomials ∆Pa(x, t) and tSa(x, y) are comparable in the positive octant x, y, t ∈ R. Recently the authors [2] studied the inequality ∆Pa(x, t) > tSa(x, y) and its reverse and noted that the boundary between the corresponding regions i...

Multimedia is going more and more popular day by day and the major reason is the rapid advancement in technology and continuous reduction in price. Boundless amount of data in all media formats are now available locally in our own computers and online over the internet, most of these are however unorganized and scattered. There is an emerging need of classification, summarization and indexing o...

What is tropical geometry about? Back in 2005 an influential paper by Richter-Gebert, Sturmfels and Theobald [11] answered that question in the following way: “Tropical algebraic geometry is the geometry of the tropical semiring (R,min,+). Its objects are polyhedral cell complexes which behave like complex algebraic varieties.” Let us look at plane algebraic curves and their tropicalizations to...

Consider a plane curve given parametrically by a generalized trigonometric polynomial, that is, x + iy = P n k=1 a k e ik. In this paper, we obtain an impliciti-zation of the curve, that is, an equation in x and y which captures all the points on the curve and, if any, only nitely many more points.

We propose a new method for registering a cloud of points in 2D onto a planar curve. This method does not require the knowledge of an initial guess for the position of the point cloud and proceeds without having to order, smooth out or otherwise process the points of the query point cloud in any way. The method consists in representing the planar curve by an algebraic curve, and in fitting the ...

When fitting parametric polynomial curves to sequences of points or derivatives we have to choose suitable parameter values at the interpolation points. This paper investigates the effect of the parameterization on the approximation order of the interpolation. We show that chord length parameter values yield full approximation order when the polynomial degree is at most three. We obtain full ap...

We deal with the problem of fitting a discrete polynomial curve to 2D data in the presence of outliers. Finding a maximal inlier set from given data that describes a discrete polynomial curve is equivalent with finding the feasible region corresponding to the set in the parameter space. When iteratively adding a data point to the current inlier set, how to update its feasible region is a crucia...