Fabio Dias

Morphological filtering on graphs

We study some basic morphological operators acting on the lattice of all subgraphs of an arbitrary (unweighted) graph G. To this end, we consider two dual adjunctions between the edge set and the vertex set of G. This allows us (i) to recover the classical notion of a dilation/erosion of a subset of the vertices of G and (ii) to extend it to subgraphs of G. Afterward, we propose several new openings, closings, granulometries and alternate filters acting (i) on the subsets of the edge and vertex set of G and (ii) on the subgraphs of G. The proposed framework is then extended to functions that weight the vertices and edges of a graph. We illustrate with applications to binary and grayscale image denoising, for which, on the provided images, the proposed approach outperforms the usual one based on structuring elements.

Comparison of different camera calibration approaches for underwater applications.

The purpose of this study was to compare three camera calibration approaches applied to underwater applications: (1) static control points with nonlinear DLT; (2) moving wand with nonlinear camera model and bundle adjustment; (3) moving plate with nonlinear camera model. The DVideo kinematic analysis system was used for underwater data acquisition. The system consisted of two gen-locked Basler cameras working at 100 Hz, with wide angle lenses that were enclosed in housings. The accuracy of the methods was compared in a dynamic rigid bar test (acquisition volume-4.5×1×1.5 m(3)). The mean absolute errors were 6.19 mm for the nonlinear DLT, 1.16 mm for the wand calibration, 1.20 mm for the 2D plate calibration using 8 control points and 0.73 mm for the 2D plane calibration using 16 control points. The results of the wand and 2D plate camera calibration methods were less associated to the rigid body position in the working volume and provided better accuracy than the nonlinear DLT. Wand and 2D plate camera calibration methods presented similar and highly accurate results, being alternatives for underwater 3D motion analysis.

Some morphological operators on simplicial complex spaces

In this work, we propose a framework that allows to build morphological operators for processing and filtering objects defined on (abstract) simplicial complex spaces. We illustrate with applications to mesh and image processing, for which, on the provided examples, the proposed approach outperforms the classical one.

Dimensional operators for mathematical morphology on simplicial complexes

In this work we study the framework of mathematical morphology on simplicial complex spaces. Simplicial complexes are widely used to represent multidimensional data, such as meshes, that are two dimensional complexes, or graphs, that can be interpreted as one dimensional complexes. Mathematical morphology is one of the most powerful frameworks for image processing, including the processing of digital structures, and is heavily used for many applications. However, mathematical morphology operators on simplicial complex spaces is not a concept fully developed in the literature. Specifically, we explore properties of the dimensional operators, small, versatile operators that can be used to define new operators on simplicial complexes, while maintaining properties from mathematical morphology. These operators can also be used to recover many morphological operators from the literature. Matlab code and additional material, including the proofs of the original properties, are freely available at http://code.google.com/p/math-morpho-simplicial-complexes.

Improved accuracy in 3D analysis using DLT after lens distortion correction

This study aimed at assessing the applicability of a robust method to determine and correct lens distortion before using the direct linear transformation (DLT) algorithm in three-dimensional motion analysis. The known length of a rigid bar was reconstructed under different conditions of working volume (interpolation or extrapolation), number of cameras (2 or 4), position of the cameras (wide or narrow angle between optical axes), camera focal distance (4 or 8 mm) and number of control points (CPs; 8, 12, 18 or 162), through four different camera set-ups. The accuracy (percent root mean square error) of Set-up 2 (non-extrapolated working volume; two cameras; 4 mm focal distance; narrow optical axes angle) decreased with less CPs (162: 0.73%; 8: 2.78%). Set-up 1 (non-extrapolated working volume; two cameras; 8 mm focal distance; wide optical axes angle), Set-up 3 (Set-ups 1 and 2 used simultaneously) and Set-up 4 (extrapolated working volume; two cameras; 4 mm focal distance; wide optical axes angle) showed minor differences in accuracy across groups of CPs, with maximum values of 0.84%, 1.20% and 1.71%, respectively. Random errors were the main source of decreased accuracy of Set-ups 2 and 4.The proposed procedure enables accurate results with no modification in the DLT-based analysis system, even with smaller calibration frames, less CPs and wide field-of-view cameras.

Wavelet-Based Visual Analysis of Dynamic Networks

Dynamic networks naturally appear in a multitude of applications from different fields. Analyzing and exploring dynamic networks in order to understand and detect patterns and phenomena is challenging, fostering the development of new methodologies, particularly in the field of visual analytics. In this work, we propose a novel visual analytics methodology for dynamic networks, which relies on the spectral graph wavelet theory. We enable the automatic analysis of a signal defined on the nodes of the network, making viable the robust detection of network properties. Specifically, we use a fast approximation of a graph wavelet transform to derive a set of wavelet coefficients, which are then used to identify activity patterns on large networks, including their temporal recurrence. The coefficients naturally encode the spatial and temporal variations of the signal, leading to an efficient and meaningful representation. This methodology allows for the exploration of the structural evolution of the network and their patterns over time. The effectiveness of our approach is demonstrated using usage scenarios and comparisons involving real dynamic networks.

Watersheds on Hypergraphs for Data Clustering

We present a novel extension of watershed cuts to hypergraphs, allowing the clustering of data represented as an hypergraph, in the context of data sciences. Contrarily to the methods in the literature, instances of data are not represented as nodes, but as edges of the hypergraph. The properties associated with each instance are used to define nodes and feature vectors associated to the edges. This rich representation is unexplored and leads to a data clustering algorithm that considers the induced topology and data similarity concomitantly. We illustrate the capabilities of our method considering a dataset of movies, demonstrating that knowledge from mathematical morphology can be used beyond image processing, for the visual analytics of network data. More results, the data, and the source code used in this work are available at https://github.com/015988/hypershed.

Wavelet-Based Visual Analysis for Data Exploration

The conventional wavelet transform is widely used in image and signal processing, where a signal is decomposed into a combination of known signals. By analyzing the individual contributions, the behavior of the original signal can be inferred. In this article, the authors present an introductory overview of the extension of this theory into graphs domains. They review the graph Fourier transform and graph wavelet transforms that are based on dictionaries of graph spectral filters, namely, spectral graph wavelet transforms. Then, the main features of the graph wavelet transforms are presented using real and synthetic data.