Image-A : Applicable Mathematics in Image Engineering - 2010 - Vol. I, Nº 2

URI permanente para esta colecciónhttps://hdl.handle.net/11441/2591

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Obtaining cell complexes associated to four dimensional digital objects
(Universidad de Sevilla, 2010) Pacheco Martínez, Ana María; Mari, Jean-Luc; Real Jurado, Pedro
In this paper, we determine a cell complex representation of a 80–adjacent doxelbased 4-dimensional object. The homological information of this polyhedral cell complex can be employed to specify topological features and characteristics of a digital object. This homological information (for example, Euler characteristic, homological classification of cycles, homology generators, relations among them...) of a discrete object can be extracted from some specific boundary operators for each cell of an object (see [3]). The different (up to isometry) polyhedral cells are 400 configurations and their local boundary information can be suitably glued for determining the global boundary of an object and consequently, its corresponding homological information. This fact allows us to implement this technique using a look-up table for the different basic configurations and its corresponding boundary operators.
On the Recognition of Tori Embedded in R3
(Universidad de Sevilla, 2010) Arnaud, Hélène
Many known algorithms allow us to topologically recognize surfaces in 3D images. However, none of them permits us to distinguish different types of embeddings of surfaces. In this paper, we restrict our study to the case of embedded tori, and focus on their recognition up to isotopy. We recall the mathematical definition of isotopy, then we define the two key elements which will enable us to classify an embedded torus in R3 up to isotopy : its state and the knot associated with it. At the end of the paper, we bring up two algorithms which aim at finding the isotopy type of an embedded torus, by determining its state and computing its associated knot.
Strong separating (k, k)−surfaces on Z3
(Universidad de Sevilla, 2010) Ciria Cosculluela, José; Domínguez Murillo, Eladio; Francés Román, Ángel Ramón; Quintero Toscano, Antonio Rafael
For each adjacency pair (k, k) != (6, 6), k, k ∈ {6, 18, 26}, we introduce a new family Skk of surfaces in the discrete space Z3 that strictly contains several families of surfaces previously defined, and other objects considered as surfaces, in the literature. Actually, Skk characterizes the strongly k−separating objects of the family of digital surfaces, defined by means of continuous analogues, of the universal (k, k)−spaces introduced in [6].
Improved Locally Adaptive Sampling Criterion for Topology Preserving Reconstruction of Multiple Regions
(Universidad de Sevilla, 2010) Tcherniavski, Leonid; Hnisch, Christian; Meine, Hans
Volume based digitization processes often deal with non-manifold shapes. Even though many reconstruction algorithms have been proposed for non-manifold surfaces, they usually don’t preserve topological properties. Only recently, methods were presented which—given a finite set of surface sample points—result in a mesh representation of the original boundary preserving all or certain neighbourhood relations, even if the sampling is sparse and highly noise corrupted. We show that the required sampling conditions of the algorithm called “refinement reduction” limit the guaranteed correctness of the outcome to a small class of shapes. We define new locally adaptive sampling conditions that depend on our new pruned medial axis and finally prove without any restriction on shapes that under these new conditions, the result of “refinement reduction” corresponds to a superset of a topologically equivalent mesh.
Topology-preserving perceptual segmentation using the Combinatorial Pyramid
(Universidad de Sevilla, 2010) Antúnez Ortiz, Esther; Marfil Robles, Rebeca; Bandera Rubio, Antonio
Scene understanding and other high-level visual tasks usually rely on segmenting the captured images for dealing with a more efficient mid-level representation. Although this segmentation stage will consider topological constraints for the set of obtained regions (e.g., their internal connectivity), it is typical that the importance of preserving the topological relationships among regions will be not taken into account. Contrary to other similar approaches, this paper presents a bottom-up approach for perceptual segmentation of natural images which preserves the topology of the image. The segmentation algorithm consists of two consecutive stages: firstly, the input image is partitioned into a set of blobs of uniform colour (pre-segmentation stage) and then, using a more complex distance which integrates edge and region descriptors, these blobs are hierarchically merged (perceptual grouping). Both stages are addressed using the Combinatorial Pyramid, a hierarchical structure which can correctly encode relationships among image regions at upper levels. The performance of the proposed approach has been initially evaluated with respect to groundtruth segmentation data using the Berkeley Segmentation Dataset and Benchmark. Although additional descriptors must be added to deal with small regions and textured surfaces, experimental results reveal that the proposed perceptual grouping provides satisfactory scores.
Combining regular decimation and dual graph contraction for hierarchical image segmentation
(Universidad de Sevilla, 2010) Torres García, Fuensanta; Marfil Robles, Rebeca; Haxhimusa, Yll; Bandera Rubio, Antonio
The Bounded Irregular Pyramid (BIP) is a hierarchical structure for image representation whose aim is to combine concepts from regular and irregular pyramids. The data structure is a combination of the simplest regular and irregular structures: the 2 × 2/4 regular one and the simple graph representation. However, simple graphs only take into account adjacency relationships, being unable to correctly encode the topology of the image. This paper proposes a new version of the BIP, where the regular decimation process is now merged with a stochastic graph decimation strategy. Experiments demonstrate that this new irregular pyramid is able to provide qualitative good segmentation results and to preserve the topology of the input image at higher levels of its hierarchy.
Invariant Spectral Hashing of Image Saliency Graph
(Universidad de Sevilla, 2010) Taquet, Maxime; Jacques, Laurent; Vleeschouwer, Christophe de
Image hashing is the process of associating a short vector of bits to an image. The resulting summaries are useful in many applications including image indexing, image authentication and pattern recognition. These hashes need to be invariant under transformations of the image that result in similar visual content, but should drastically differ for conceptually distinct contents. This paper proposes an image hashing method that is invariant under rotation, scaling and translation of the image. The gist of our approach relies on the geometric characterization of salient point distribution in the image. This is achieved by the definition of a saliency graph connecting these points jointly with an image intensity function on the graph nodes. An invariant hash is then obtained by considering the spectrum of this function in the eigenvector basis of the graph Laplacian, that is, its graph Fourier transform. Interestingly, this spectrum is invariant under any relabeling of the graph nodes. The graph reveals geometric information of the image, making the hash robust to image transformation, yet distinct for different visual content. The efficiency of the proposed method is assessed on a set of MRI 2-D slices and on a database of faces.

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