scholarly journals MULTISCALE IMAGE ANALYSIS BASED ON ROBUST AND ADAPTIVE MORPHOLOGICAL SCALE-SPACES

2014 ◽  
Vol 33 (2) ◽  
pp. 39 ◽  
Author(s):  
El Hadji Samba Diop ◽  
Jesus Angulo
Keyword(s):  
Image Analysis ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Mathematical Morphology ◽  
Image Features ◽  
Cauchy Problems ◽  
Numerical Schemes ◽  
Morphological Operators ◽  
Scale Spaces ◽  
Multiscale Image Analysis

Mathematical morphology is a powerful tool for image analysis; however, classical morphological operators suffer from lacks of robustness against noise, and also intrinsic image features are not accounted at all in the process. We propose in this work a new and different way to overcome such limits, by introducing both robustness and locally adaptability in morphological operators, which are now defined in a manner such that intrinsic image features are accounted. Dealing with partial differential equations (PDEs) for generalized Cauchy problems, we show that proposed PDEs are equivalent to impose robustness and adaptability of corresponding sup-inf operators, to structuring functions. Accurate numerical schemes are also provided to solve proposed PDEs, and experiments conducted for both synthetic and real images, show the efficiency and robustness of our approach.

scholarly journals Review of Methods Inspired by Algebraic-Multigrid for Data and Image Analysis Applications

2015 ◽  
Vol 8 (2) ◽  
pp. 283-312 ◽  
Author(s):  
Meirav Galun ◽  
Ronen Basri ◽  
Irad Yavneh
Keyword(s):  
Image Analysis ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Algebraic Multigrid ◽  
Partial Differential ◽  
Structured Grids ◽  
Common Principle ◽  
Flow Of Information ◽  
Coarse To Fine

AbstractAlgebraic Multigrid (AMG) methods were developed originally for numerically solving Partial Differential Equations (PDE), not necessarily on structured grids. In the last two decades solvers inspired by the AMG approach, were developed for non PDE problems, including data and image analysis problems, such as clustering, segmentation, quantization and others. These solvers share a common principle in that there is a crosstalk between fine and coarse representations of the problems, with flow of information in both directions, fine-to-coarse and coarse-to-fine. This paper surveys some of these problems and the AMG-inspired algorithms for their solution.

scholarly journals STOCHASTIC SOLUTIONS OF A CLASS OF HIGHER ORDER CAUCHY PROBLEMS IN ℝd

2010 ◽  
Vol 10 (03) ◽  
pp. 341-366 ◽  
Author(s):  
ERKAN NANE
Keyword(s):  
Partial Differential Equations ◽  
Markov Process ◽  
Differential Equations ◽  
Stable Process ◽  
Higher Order ◽  
Cauchy Problems ◽  
Partial Differential ◽  
Iterated Brownian Motion ◽  
Special Cases ◽  
First Time

We study solutions of a class of higher order partial differential equations in bounded domains. These partial differential equations appeared first time in the papers of Allouba and Zheng [4], Baeumer, Meerschaert and Nane [10], Meerschaert, Nane and Vellaisamy [37], and Nane [42]. We express the solutions by subordinating a killed Markov process by a hitting time of a stable subordinator of index 0 < β < 1, or by the absolute value of a symmetric α-stable process with 0 < α ≤ 2, independent of the Markov process. In some special cases we represent the solutions by running composition of k independent Brownian motions, called k-iterated Brownian motion for an integer k ≥ 2. We make use of a connection between fractional-time diffusions and higher order partial differential equations established first by Allouba and Zheng [4] and later extended in several directions by Baeumer, Meerschaert and Nane [10].

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