Nonlinear Multilayered Representation of Graph-Signals

We propose a nonlinear multiscale decomposition of signals defined on the vertex set of a general weighted graph. This decomposition is inspired by the hierarchical multiscale (BV, L 2) decomposition of Tadmor, Nezzar, and Vese (Multiscale Model. Simul. 2(4):554–579, 2004). We find the decomposition by iterative regularization using a graph variant of the classical total variation regularization (Rudin et al, Physica D 60(1–4):259–268, 1992). Using tools from convex analysis, and in particular Moreau’s identity, we carry out the mathematical study of the proposed method, proving the convergence of the representation and providing an energy decomposition result. The choice of the sequence of scales is also addressed. Our study shows that the initial scale can be related to a discrete version of Meyer’s norm (Meyer, Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, 2001) which we introduce in the present paper. We propose to use the recent primal-dual algorithm of Chambolle and Pock (J. Math. Imaging Vis. 40:120–145, 2011) in order to compute both the minimizer of the graph total variation and the corresponding dual norm. By applying the graph model to digital images, we investigate the use of nonlocal methods to the multiscale decomposition task. Since the only assumption needed to apply our method is that the input data is living on a graph, we are also able to tackle the task of adaptive multi

Comparative evolution of some minerals, osteocalcin, 25(OH)D and 1,25(OH)2D in the female camel and its newborn in South Morocco

Postpartum kinetics of plasma minerals (Ca, P, Mg, Na and K), osteocalcin, 25(OH)D and 1,25(OH)2D were observed in the lactating camel (n = 5) and its newborn (n = 5) in South Morocco. Ca and P plasma levels (mg/l) were higher in camel calves than in dams from birth up to day 30 of life. At birth, Mg, Na and K plasma levels (mg/l) were lower in camel calves than in dams. OC plasma levels (ng/ml) were higher in camel calves than in their mothers from birth (3.4 ± 0.3 vs. 0.7 ± 0.3, P < 0.05) up to day 30 of life (5.2 ± 0.5 vs. 0.7 ± 0.3). These levels reached their maximum at day 3 after birth in camel calves (6.8 ± 0.5) and at day 4 postpartum in dams (2.3 ± 0.4). Plasma levels (pg/ml) for 25(OH)D and 1,25(OH)2D in newborn calves were lower at birth than those in dams (58.2 ± 12.4 vs. 480 ± 59.7, P < 0.05, and 834.8 ± 61.5 vs. 1301 ± 209, P < 0.05, respectively). These neonatal levels became significantly higher compared to those of the dams as early as day 15 after birth for 25(OH)D (176.4 ± 19.6 vs. 116.1 ± 30.4, P < 0.05), and day 7 after birth for 1,25(OH)2D (1215 ± 248 vs. 571 ± 64, P < 0.05). These results show that OC could be a reliable biomarker for neonatal bone growth in camel and that 25(OH)D and 1,25(OH)2D might play a major role in calcium homeostasis regulation in newborn calves and their lactating dams

Hierarchical Representation of Discrete Data on Graphs

International audienceWe propose a new hierarchical representation of discrete data sets living on graphs. The approach takes advantage of recent works on graph regularization. The role of the merging criterion that is common to hierarchical representations is greatly reduced due to the regularization step. The regularization is performed recursively with a decreasing fidelity parameter. This yields a robust representation of data sets. We show experiments on digital images and image databases