We present a discrete stochastic model which represents many of the salient features of the biological process of wound healing. The model describes fronts of cells invading a wound. We have numerical results in one and two dimensions. In one dimension we can give analytic results for the front speed as a power series expansion in a parameter, p, that gives the relative size of proliferation and diffusion processes for the invading cells. In two dimensions the model becomes the Eden model for p ≈ 1. In both one and two dimensions for small p, front propagation for this model should approach that of the Fisher-Kolmogorov equation. However, as in other cases, this discrete model approaches Fisher-Kolmogorov behavior slowly.

whether to change its preference to the opponent as biased by the game outcome, preferring but not absolutely certain to go with the winner, repeating this process indefinitely. In the simplest implementation of this process, the probability p of choosing the winner is kept constant across voters and games played, with p > 1/2 because on average the winner should be recognized as the better team and p < 1 to allow a given voter to argue that the losing team is still the better team (moreover, the p = 1 limit can be mathematically more complicated in certain scenarios, as discussed in section 3). The voting automatons are nothing more than independent, biased random walkers on a graph connecting the teams (vertices) by their head-to-head games (edges). These "voters" thereby obey idealized behavioral rules dictated by one of the most natural arguments relating the relative ranking of two teams: "my team beat your team." Indeed, the statistics of such biased random walkers can be presented as nothing more than the logical extension of this argument repeated ad infinitum.This algorithm is easy to explain in terms of the "microscopic" behavior of individual walkers who randomly change their opinion about which team is best (biased by the outcomes of individual games). Of course, this behavior is grossly simplistic compared with real-world poll voters. In fact, under the specified range of p, a single walker will never reach a definitive conclusion about which team is the best; rather, it will forever change its allegiance from one team to another, ultimately traversing the entire graph. However, the "macroscopic" total number of votes cast for each team by an aggregate of random-walking voters quickly reaches a statistically-steady ranking.The advantage of the algorithm discussed here is that it can be easily understood in terms of single-voter behavior. Additionally, it has a single explicit, precisely-defined parameter with a meaningful interpretation at the single-voter level. We do not claim that this ranking is superior to other algorithms, nor do we review the vast number of ranking systems available, as numerous reviews are already available (see, for example, [11], [20], [7], [14] for reviews of different ranking methodologies and the list of algorithms and "Bibliography on College Football Ranking Systems" maintained by [22]). We do not even claim that this ranking algorithm is wholly novel; indeed, the resulting linear algebra problem is in the class of "direct methods" discussed by Keener [11] and has many similarities to the linear algebra problem solved by Colley [6]. Rather, we propose this random-walker ranking on the strength of its simple interpretation: our intent is to show that this simplydefined ranking yields reasonable results.THOMAS CALLAGHAN is a graduate student in the Institute for Computational and Mathematical Engineering at Stanford University. He is a 2005 graduate of the Georgia Institute of Technology, where he started this work as an REU project at a time when all three authors were...

We consider the problem of synthetic aperture radar (SAR) imaging and motion estimation of complex scenes. By complex we mean scenes with multiple targets, stationary and in motion. We use the usual setup with one moving antenna emitting and receiving signals. We address two challenges: (1) the detection of moving targets in the complex scene and (2) the separation of the echoes from the stationary targets and those from the moving targets. Such separation allows high resolution imaging of the stationary scene and motion estimation with the echoes from the moving targets alone. We show that the robust principal component analysis (PCA) method which decomposes a matrix in two parts, one low rank and one sparse, can be used for motion detection and data separation. The matrix that is decomposed is the pulse and range compressed SAR data indexed by two discrete time variables: the slow time, which parametrizes the location of the antenna, and the fast time, which parametrizes the echoes received between successive emissions from the antenna. We present an analysis of the rank of the data matrix to motivate the use of the robust PCA method. We also show with numerical simulations that successful data separation with robust PCA requires proper data windowing. Results of motion estimation and imaging with the separated data are presented, as well.

We introduce from first principles a synthetic aperture radar (SAR) imaging and target motion estimation method that is combined with compensation for radar platform trajectory perturbations. The main steps of the method are (a) segmentation of the data into properly calibrated small apertures, (b) motion or platform trajectory perturbation estimation using the Wigner transform and the ambiguity function of the data, in a complementary way, (c) combination of small aperture estimates and construction of high resolution images over wide apertures. The analysis provides quantitative criteria for implementing the aperture segmentation and the parameter estimation process. X-band persistent surveillance SAR is a specific application that is covered by our analysis. Detailed numerical simulations illustrate the robust applicability of the theory and validate the theoretical resolution analysis.

No abstract