In this chapter we focus on formulating and implementing on abstract domains such as arbitrary graphs popular methods and techniques developed for image analysis, in particular multiscale morphology and active contours. To this goal we extend existing work on graph morphology to multiscale dilation and erosion and implement them recursively using level sets of functions defined on the graph’s n...

Heterogeneous multiscale methods (HMM) have been introduced by E and Engquist [Commun. Math. Sci. 1 (2003), pp. 87-132] as a general methodology for the numerical computation of problems with multiple scales. In this paper we discuss finite element methods based on the HMM for multiscale partial differential equations (PDEs). We give numerous examples of such multiscale problems, including elli...

A novel multiscale framework is introduced for the representation of a clam of fractal point processes. Using this framework, efficient algorithms are developed for the Rynthesis of fractal point processes from a mixture of Poisson processes. Multiscale analysis algorithms are also developed within this framework for computing Maximum-Likelihood fractal dimension estimates of such processes fro...

Abstract: Multiscale entropy (MSE) was introduced in the 2000s to quantify systems’ complexity. MSE relies on (i) a coarse-graining procedure to derive a set of time series representing the system dynamics on different time scales; (ii) the computation of the sample entropy for each coarse-grained time series. A refined composite MSE (rcMSE)—based on the same steps as MSE—also exists. Compared ...

We describe a framework for multiscale image analysis in which line segments play a role analogous to the role played by points in wavelet analysis. The framework has five key components. The beamlet dictionary is a dyadicallyorganized collection of line segments, occupying a range of dyadic locations and scales, and occurring at a range of orientations. The beamlet transform of an image f(x, y...

Pattern formation in developing tissues involves dynamic spatio-temporal changes in cellular organization and subsequent evolution of functional adult structures. Branching morphogenesis is a developmental mechanism by which patterns are generated in many developing organs, which is controlled by underlying molecular pathways. Understanding the relationship between molecular signaling, cellular...

Textures often show multiscale properties and hence multiscale techniques are considered useful for texture analysis. Scale-space theory as a biologically motivated approach may be used to construct multiscale textures. In this paper various ways are studied to combine features on different scales for texture classification of small image patches. We use the N-jet of derivatives up to the secon...

Based on the statistical property of SAR image speckle noise and the property that the multiscale geometric analysis can capture the intrinsic geometrical structure of image, combining curvelet transform with BivaShrink denoising model, a method of SAR image denoising based on curvelet domain is presented in this paper. According to calculation of variance homogeneous measurement and curvelet c...

The multiscale behavior of a recently reported model for stock markets is presented. It has been shown that indexes of real-world markets display absolute returns with memory properties on a long-time range, a phenomenon known as cluster volatility. The multiscale characteristics of an index are studied by analyzing the power-law scaling of the volatility correlations which display nonunique sc...