Optimal Shape of a Blob
نویسندگان
چکیده
This paper presents the solution to the following optimization problem: What is the shape of the region that minimizes the average Lp distance between all pairs of points if the area of this region is held fixed? (The Lp-distance between two points x = (x1, x2) and y = (y1, y2) in < 2 is (|x1 − y1| p + |x2 − y2| p).) We use variational methods to determine the shape of the region that minimizes the average Lp distance when the area of this region is held fixed, for p ≥ 1. Specifically, we characterize the boundary curve h(x) of the optimal region and show that this boundary curve satisfies an integral equation. This integral equation is new and is the principal result of this paper. This paper generalizes two earlier studies. Karp, McKellar, and Wong [2] considered the two special extreme cases p = 1 and p = ∞. For these cases, they determined a differential equation that describes the boundary of the optimal region that minimizes the average distance between all points in the region. Bender et al. [1] studied the dimensionless average pairwise distance D[h] that characterizes the degree of optimality of the region for p = 1 and they used variational methods to minimize the value of this functional D[h]. This variational approach leads directly to the nonlinear differential equation that describes the boundary curve. Moreover, in [1] the numerical value of D[h] was computed. Here we extend the variational methods introduced in Ref. [1] to the general case p ≥ 1. For this situation the boundary curve satisfies an integral equation. We then examine three special cases: For p = 2 the integral equation reduces to a differential equation, whose solution, as one would expect, is a circle. For p = 1 and for p = ∞ the integral equation reduces to the second-order nonlinear differential equations that were studied previously in Refs. [1, 2]. In this talk we will also review the results of Ref. [1].
منابع مشابه
Reduced-Reference Image Quality Assessment based on saliency region extraction
In this paper, a novel saliency theory based RR-IQA metric is introduced. As the human visual system is sensitive to the salient region, evaluating the image quality based on the salient region could increase the accuracy of the algorithm. In order to extract the salient regions, we use blob decomposition (BD) tool as a texture component descriptor. A new method for blob decomposition is propos...
متن کاملConventional Voxel in Tomographic Reconstruction Based upon Plane-Integral Projections – Use It or Lose It?
Introduction: While the necessity of replacing voxels with blobs in conventional tomographic reconstruction based upon line-integrals is clear, it is not however well-investigated in plane- integral-based reconstruction. The problem is more challenging in convergent-plane projection reconstruction. In this work, we are aiming at utilizing blobs as alternative to voxels. <stron...
متن کاملUsing a Novel Concept of Potential Pixel Energy for Object Tracking
Abstract In this paper, we propose a new method for kernel based object tracking which tracks the complete non rigid object. Definition the union image blob and mapping it to a new representation which we named as potential pixels matrix are the main part of tracking algorithm. The union image blob is constructed by expanding the previous object region based on the histogram feature. The pote...
متن کاملAnalysis of the Characteristics, Physical Concepts and Entropy Generation in a Turbulent Channel Flow Using Vortex Blob Method
In this paper, using vortex blob method (VBM), turbulent flow in a channel is studied and physical concepts of turbulence are obtained and discussed. At first, time-averaged velocities, and , and then their fluctuations are calculated. To clarify turbulence structures, velocity fluctuations and are plotted. It is observed that turbulence structures occupy different positions and move with con...
متن کاملHigh Reynolds Viscous Flow Simulation Past the Elliptical Airfoil by Random Vortex Blob
In this paper, numerical simulation for a two-dimensional viscous and incompressible flow past the elliptical airfoil is presented by Random Vortex Blob (RVB). RVB is a numerical technique to solve the incompressible, two-dimensional and unsteady Navier-Stocks equations by converting them to rotational non-primitive formulations. In this method, the velocity vector at a certain point can be cal...
متن کامل