Efficient Hierarchical Multi-Object Segmentation in Layered Graphs

We propose a novel efficient seed-based method for the multi-object segmentation of images based on graphs, named Hierarchical Layered Oriented Image Foresting Transform (HLOIFT). It uses a tree of the relations between the image objects, with each node in the tree representing an object. Each tree node may contain different individual high-level priors of its corresponding object and defines a weighted digraph, named as layer. The layer graphs are then integrated into a hierarchical graph, considering the hierarchical relations of inclusion and exclusion. A single energy optimization is performed in the hierarchical layered weighted digraph leading to globally optimal results satisfying all the high-level priors. The experimental evaluations of HLOIFT, on medical, natural, and synthetic images, indicate promising results comparable to the related baseline methods that include structural information, but with lower computational complexity. Compared to the hierarchical segmentation by the min-cut/max-flow algorithm, our approach is less restrictive, leading to globally optimal results in more general scenarios, and has a better running time.

Enumeration of weighted paths on a digraph and block hook determinant

In this article, we evaluate determinants of “block hook” matrices, which are block matrices consist of hook matrices. In particular, we deduce that the determinant of a block hook matrix factorizes nicely. In addition we give a combinatorial interpretation of the aforesaid factorization property by counting weighted paths in a suitable weighted digraph.

Chains of three-dimensional evolution algebras: A description

In this paper we describe locally all the chains of three-dimensional evolution algebras (3-dimensional CEAs). These are families of evolution algebras with the property that their structure matrices with respect to a certain natural basis satisfy the Chapman-Kolmogorov equation. We do it by describing all 3-dimensional CEAs whose structure matrices have a fixed rank equal to 3, 2 and 1, respectively. We show that arbitrary CEAs are locally CEAs of fixed rank. Since every evolution algebra can be regarded as a weighted digraph, this allows us to understand and visualize time-dependent weighted digraphs with 3 nodes.

An Efficient Hierarchical Layered Graph Approach for Multi-Region Segmentation

We proposed a novel efficient seed-based method for the multiple region segmentation of images based on graphs, named Hierarchical Layered Oriented Image Foresting Transform (HLOIFT). It uses a tree of the relations between the image objects, represented by a node. Each tree node may contain different individual high-level priors and defines a weighted digraph, named as layer. The layer graphs are then integrated into a hierarchical graph, considering the hierarchical relations of inclusion and exclusion. A single energy optimization is performed in the hierarchical layered weighted digraph leading to globally optimal results satisfying all the high-level priors. The experimental evaluations of HLOIFT and its extensions, on medical, natural and synthetic images, indicate promising results comparable to the state-of-the-art methods, but with lower computational complexity. Compared to hierarchical segmentation by the min-cut/max-flow algorithm, our approach is less restrictive, leading to globally optimal results in more general scenarios, and has a better running time.

Computing extremal energy of a class of bicyclic weighted digraphs

The energy of a weighted digraph [Formula: see text] is the sum of absolute values of real part of its eigenvalues. Recently, the minimal and maximal energy of unicyclic weighted digraphs with cycle weight [Formula: see text] is studied. In this paper, we introduce a class [Formula: see text] of those bicyclic weighted digraphs of fixed order which contain vertex-disjoint weighted cycles of weights [Formula: see text] or [Formula: see text] and [Formula: see text] or [Formula: see text], where [Formula: see text]. We find digraphs in [Formula: see text] under certain conditions on [Formula: see text] and [Formula: see text] with extremal energy.