A Fast, Memory-Efficient Alpha-Tree Algorithm using Flooding and Tree Size Estimation

The alpha-tree represents an image as hierarchical set of alpha-connected components. Computation of alpha-trees suffers from high computational and memory requirements compared with similar component tree algorithms such as max-tree. Here we introduce a novel alpha-tree algorithm using 1) a flooding algorithm for computational efficiency and 2) tree size estimation (TSE) for memory efficiency. In TSE, an exponential decay model was fitted to normalized tree sizes as a function of the normalized root mean squared deviation (NRMSD) of edge-dissimilarity distributions, and the model was used to estimate the optimum memory allocation size for alpha-tree construction. An experiment on 1256 images shows that our algorithm runs 2.27 times faster than Ouzounis and Soille's thanks to the flooding algorithm, and TSE reduced the average memory allocation of the proposed algorithm by 40.4%, eliminating unused allocated memory by 86.0% with a negligible computational cost

On the imaginary parts and infrared divergences of two-loop vector boson self-energies in thermal QCD

We calculate the imaginary part of the retarded two-loop self-energy of a static vector boson in a plasma of quarks and gluons of temperature T, using the imaginary time formalism. We recombine various cuts of the self-energy to generate physical processes. We demonstrate how cuts containing loops may be reinterpreted in terms of interference between Order $\alpha$ tree diagrams and the Born term along with spectators from the medium. We apply our results to the rate of dilepton production in the limit of dilepton invariant mass E>>T. We find that all infrared and collinear singularities cancel in the final result obtained in this limit.Comment: references added, typos corrected, slightly abridged, version accepted for publication in Phys. Rev.

A generalization of tree automata and its relation to matrix grammars

In this thesis we introduce alpha and beta tree acceptors, generalizations of tree automata. The alpha tree acceptors recognize a tree by final symbol and the beta tree acceptors by final state. We show that alpha and beta tree acceptors recognize the same sets of Gorn trees and demonstrate that there are sets of Gorn trees not recognizably by any alpha tree acceptor. The tree automata are equivalent to the one state alpha tree acceptors. We show that the languages of alpha tree acceptors are exactly those of matrix grammars and the derivation trees of matrix grammars are recognized by alpha tree acceptors. Finally we show that nondeterministic alpha tree acceptors are no more powerful than deterministic alpha tree acceptors

On morphological hierarchical representations for image processing and spatial data clustering

Hierarchical data representations in the context of classi cation and data clustering were put forward during the fties. Recently, hierarchical image representations have gained renewed interest for segmentation purposes. In this paper, we briefly survey fundamental results on hierarchical clustering and then detail recent paradigms developed for the hierarchical representation of images in the framework of mathematical morphology: constrained connectivity and ultrametric watersheds. Constrained connectivity can be viewed as a way to constrain an initial hierarchy in such a way that a set of desired constraints are satis ed. The framework of ultrametric watersheds provides a generic scheme for computing any hierarchical connected clustering, in particular when such a hierarchy is constrained. The suitability of this framework for solving practical problems is illustrated with applications in remote sensing

On the Graceful Cartesian Product of Alpha-Trees

A \emph{graceful labeling} of a graph $G$ of size $n$ is an injective assignment of integers from the set $\{0,1,\dots,n\}$ to the vertices of $G$ such that when each edge has assigned a \emph{weight}, given by the absolute value of the difference of the labels of its end vertices, all the weights are distinct. A graceful labeling is called an $\alpha$-labeling when the graph $G$ is bipartite, with stable sets $A$ and $B$, and the labels assigned to the vertices in $A$ are smaller than the labels assigned to the vertices in $B$. In this work we study graceful and $\alpha$-labelings of graphs. We prove that the Cartesian product of two $\alpha$-trees results in an $\alpha$-tree when both trees admit $\alpha$-labelings and their stable sets are balanced. In addition, we present a tree that has the property that when any number of pendant vertices are attached to the vertices of any subset of its smaller stable set, the resulting graph is an $\alpha$-tree. We also prove the existence of an $\alpha$-labeling of three types of graphs obtained by connecting, sequentially, any number of paths of equal size

A Fast Alpha-tree Algorithm for Extreme Dynamic Range Pixel Dissimilarities

The α-tree algorithm is a useful hierarchical representation technique which facilitates comprehension of imagessuch as remote sensing and medical images. Most α-tree algorithms make use of priority queues to process image edgesin a correct order, but because traditional priority queues areinefficient in α-tree algorithms using extreme-dynamic-rangepixel dissimilarities, they run slower compared with other relatedalgorithms such as component tree. In this paper, we proposea novel hierarchical heap priority queue algorithm that canprocess α-tree edges much more efficiently than other stateof-the-art priority queues. Experimental results using 48-bitSentinel-2A remotely sensed images and randomly generatedimages have shown that the proposed hierarchical heap priorityqueue improved the timings of the flooding α-tree algorithm byreplacing the heap priority queue with the proposed queue: 1.68times in 4-N and 2.41 times in 8-N on Sentinel-2A images, and2.56 times and 4.43 times on randomly generated images

Fast image and video segmentation based on alpha-tree multiscale representation

International audienceIn this paper, we propose to rely on a recent image representation model, namely the α-tree, to achieve efficient segmentation of images and videos. The α-tree is a multiscale representation of an image, based on its quasi-flat zones. An in-depth study of this tree reveals some interesting features of image pixels and regions. These features are then used in the design of both automatic and interactive segmentation algorithms. Interactivity is achieved thanks to a new and efficient implementation scheme. Experiments on the Berkeley Segmentation Dataset lead to very promising results