Taking advantage of the topological and isotopic properties of binary digital images, we present here a new algorithm for connected component labeling (CLL). A local-to-global treatment of the topological information within the image, allows us to develop an inherent parallel approach. The time complexity order for an image of m × n pixels, under the assumption that a processing element exists for each pixel, is near O (log(m + n )) . Additionally, our method computes both the foreground and background CCL, and allows a straightforward computation of topological features like Adjacency Trees. Experiments show that our method obtains better performance metrics than other approaches. Our work aims at generating a new class of labeling algorithms: those centered in fully parallel approaches based on computational topology, thus allowing a perfect concurrent execution in multiple threads and preventing the use of critical sections and atomic instructions.

Given an nD digital image I based on cubical n-xel, to fully characterize the degree of internal topological dissimilarity existing in I when using different adjacency relations (mainly, comparing 2n or 2 n − 1 adjacency relations) is a relevant issue in current problems of digital image processing relative to shape detection or identification. In this paper, we design and implement a new self-dual representation for a binary 2D image I, called {4, 8}-region adjacency forest of I ({4, 8}-RAF , for short), that allows a thorough analysis of the differences between the topology of the 4-regions and that of the 8-regions of I. This model can be straightforwardly obtained from the classical region adjacency tree of I and its binary complement image I c , by a suitable region label identification. With these two labeled rooted trees, it is possible: (a) to compute Euler number of the set of foreground (resp. background) pixels with regard to 4-adjacency or 8-adjacency; (b) to identify new local and global measures and descriptors of topological dissimilarity not only for one image but also between two or more images. The parallelization of the algorithms to extract and manipulate these structures is complete, thus producing efficient and unsophisticated codes with a theoretical computing time near the logarithm of the width plus the height of an image. Some toy examples serve to explain the representation and some experiments with gray real images shows the influence of the topological dissimilarity when detecting feature regions, like those returned by the MSER (maximally stable extremal regions) method. Keywords: Hierarchical representation • Digital image • Topological dissimilarity • Parallelism • (4 • 8)-adjacency tree • {4,

A new algorithm for computing the α-tree hierarchical representation of a grey-scale digital image is presented here. The technique is based on an efficient simplified version of the Homological Spanning Forest (HSF ) for encoding homological and homotopy-based information of binary digital images. We create one Adjacency Tree (AdjT ) for each intensity contrast in a fully parallel manner. These trees, which define a Contrast Adjacency Forest (CAdjF ), are in turn transversely interconnected by another couple of trees: the classical α-tree, and a new one complementing it, called here the α * -tree. They convey the information of the contours and the flat regions of the original color image, plus the relations between them. Using both the α and α * -trees, this new topological representation prevents some classical drawbacks that appear when working with a single tree. An implementation in OCTAVE/MATLAB validates the correctness of our algorithm.

A novel, flexible (non-unique) and topologically consistent representation called CRIT (Contour-Region incidence Tree) for a color 2D digital image I is defined here. The CRIT is a tree containing all the inter and intra connectivity information of the constant-color regions. Considering I as an abstract cell complex (ACC), its topological information can be packed as a smaller (in terms of cells) ACC, whose 2-cells are the different constant-color regions of I. This modus operandi overcomes the classical connectivity paradoxes of digital images by working with lower-dimensional cells such as 0-cells, 1-cells, and 2-cells. The CRIT structure allows to describe this smaller ACC in a non-redundant way. The proposed technique is based on the previous construction of the Homological Spanning Forest (HSF) structures for encoding homological information of the ACCs canonically associated to I, in terms of rooted trees connecting digital object elements without redundancy.