A linear algorithm for the grundy number of a tree

A coloring of a graph G = (V,E) is a partition {V1, V2, . . ., Vk} of V into independent sets or color classes. A vertex v Vi is a Grundy vertex if it is adjacent to at least one vertex in each color class Vj . A coloring is a Grundy coloring if every color class contains at least one Grundy vertex, and the Grundy number of a graph is the maximum number of colors in a Grundy coloring. We derive a natural upper bound on this parameter and show that graphs with sufficiently large girth achieve equality in the bound. In particular, this gives a linear time algorithm to determine the Grundy number of a tree

Deconvolution of Ultrasonic Signals in Porous Materials: Estimation of Acoustic Propagation Parameters andWave Separation.

Our study focuses on the development of assessment tools and nondestructive evaluation of porous materials from ultrasonic measurements. These materials are encountered in many industrial applications such as polyurethane foam used for insulation, aluminum foams used in aerospace or cancellous bone for biological applications. Acoustic propagation in these complex heterogeneous materials is governed by the Biot theory [1], involving the propagation of two types of waves: slow and fast wave, whose properties are respectively related to the fluid and solid phases constituting the material. During the propagation, these waves undergo deformations that can be characterized by related propagation models [2], defined by specific frequency-dependent attenuation and dispersion laws. Identification of these waves and of their related propagation parameters then provides a characterization of the material health. This may be a difficult problem in the case of porous materials of low thickness and/or with defects, since the different waves and their echoes may overlap, as shown in the example in Figure 1. Separation of these waveforms should however be possible, by taking into account reliable models describing the propagation of each wave. This paper presents a method for identifying such waves (arrival times and propagation parameters) from signals acquired in transmission or reflection, based on an optimization procedure that minimizes a nonlinear least-squares criterion, which is sufficiently constrained and properly initialized in order to produce robust results. The method is validated with numerical simulations and applied to a laboratory experiment with a porous ceramic plate. This work is partially supported by the French region “Pays de la Loire”, through the DECIMAP project

Wavelets and LPG-PCA for Image Denoising

In this chapter, a new image denoising approach is proposed. It combines two image denoising techniques. The first one is based on a wavelet transform (WT), and the second one is a two-stage image denoising by PCA (principal component analysis) with LPG (local pixel grouping). In this proposed approach, we first apply the first technique to the noisy image in order to obtain the first estimation version of the clean image. Then, we estimate the noise-level from the noisy image. This estimate is obtained by applying the third technique of noise estimation from noisy images. The third step of the proposed approach consists in using the first estimation of the clean image, the noisy image, and the estimate of the noise-level as inputs of the second image denoising system (LPG-PCA). A comparative study of the proposed technique and the two others denoising technique (one is based on WT and and the second is based on LPG-PCA), is performed. This comparative study used a number of noisy images, and the obtained results from PSNR (peak signal-to-noise ratio) and SSIM (structural similarity) computations show that the proposed approach outperforms the two other denoising approaches (the first one is based on WT and the second one is based on LPG-PCA)