. Every day a huge amount of digital data is generated. Processing such big data encourages eﬃcient data structure and parallelized operations. In this regard, this paper proposes a graph-based method reducing the memory requirement of the data storage. Graphs as a versa-tile representative tool in intelligent systems and pattern recognition may consist of many nonessential edges accumulating memory. This paper deﬁnes the structure of such redundant edges in the neighborhood graph of a 2D binary image. We introduce a novel approach for contracting the edges that simultaneously assists in determining the structurally redundant edges. In addition, ﬁnding a set of independent edges, the redundant edges are removed in parallel with the complexity O (1). Theoretically, we prove that the maximum number of redundant edges is bounded by half of all edges. Practical results show the memory requirement decreases signiﬁcantly depending on the input data in diﬀerent categories of binary image data sets. Using the combinatorial map as the data structure, ﬁrst the topological structure of the graph is preserved. Second, the method can be extended to higher dimensions (nD).

Removing Redundancies in Binary Images

Parallel 풪(log(n)) Computation of the Adjacency of Connected Components

: Irregular Pyramids are powerful hierarchical structures in pattern recognition and image processing. They have high potential of parallel processing that makes them useful in processing of a huge amount of digital data generated every day. This paper presents a fast method for constructing an irregular pyramid over a binary image where the size of the images is more than 2000 in each of 2/3 dimensions. Selecting the contraction kernels (CKs) as the main task in constructing the pyramid is investigated. It is shown that the proposed fast labeled spanning tree (FLST) computes the equivalent contraction kernels (ECKs) in only two steps. To this purpose, first, edges of the corresponding neighborhood graph of the binary input image are classified. Second, by using a total order an efficient function is defined to select the CKs. By defining the redundant edges, further edge classification is performed to partition all the edges in each level of the pyramid. Finally, two important applications are presented : connected component labeling (CCL) and distance transform (DT) with lower parallel complexity 𝒪( 𝑙𝑜𝑔 ( 𝛿 )) where the 𝛿 is the diameter of the largest connected component in the image.

https://www.jenrs.com/?download_id=3074&smd_process_download=1

Fast Labeled Spanning Tree in Binary Irregular Graph Pyramids

: Nowadays a huge amount of digital data are generated every moment in a broad spectrum of application domains such as biomedical imaging, document processing, geosciences, remote sensing, video surveillance, etc. Processing such big data requires an efficient data structure, encouraging the algorithms with lower complexity and parallel operations. In this paper, first, a new method for computing the distance transform (DT) as the fundamental operation in binary images is presented. The method computes the DT with the parallel logarithmic complexity O ( log ( n )) where n is the maximum diameter of the largest foreground region in the 2D binary image. Second, we define the DT in the combinatorial map (CM) structure. In the CM, by replacing each edge with two darts a smoother DT with the double resolution is derived. Moreover, we compute n different distances for the nD-map. Both methods use the hierarchical irregular pyramid structure and have the advantage of preserving topological information between regions. The operations of the proposed algorithms are totally local and lead to parallel implementations. The GPU implementation of the algorithm has high performance while the bottleneck is the bandwidth of the memory or equivalently the number of available independent processing elements. Finally, the logarithmic complexity of the algorithm speeds up the execution and suits it, particularly for large images.

Distance Transform in Parallel Logarithmic Complexity

This paper presents a new logarithmic-time algorithm which simultaneously assigns labels to all connected components of a binary image in parallel. The irregular graph pyramid of an input binary image is constructed based on the optimized combinatorial structure. The novel small built pyramid has only three levels and is created at worst case with $O(log(N^{2}))$ complexity for a $N\times N$ -sized (2D) binary image. To assign a label to each connected component, instead of the common linear-time raster scan techniques (with complexity $O(N^{2}))$ , only two important movements, namely bottom-up and top-down traversing, are needed. First, in the bottom-up traversing, the redundant connections are removed and the contraction kernels are contracted. This results in a simpler graph at top of the pyramid each of its vertices has a unique label identifying corresponding connected component. Such reduced graph preserves all connecting relations including inclusions. Second, in the top-down traversing, each unique label propagates down into each individual corresponding pixel at the base level. The complexity of the labeling propagation procedure in worst cases is $O$ (log(image – size)). The GPU implementation of the algorithm has high performance and the bottleneck is the bandwidth of the memory or equivalently the number of available independent processing elements. Finally, the experimental results show the proposed algorithm outperforms the other state-of-the-art methods.

Majid Banaeyan

Papers

Pyramidal Connected Component Labeling by Irregular Graph Pyramid