Mathematical morphology (MM) is a powerful non-linear theory that can be used for signal and image processing and analysis. Although MM can be very well defined on complete lattices, which are partially ordered sets with well defined extrema operations, there is no natural ordering for multivalued images such as hyper-spectral and color images. Thus, a great deal of effort has been devoted to ordering schemes for multivalued MM. In a reduced ordering, in particular, elements are ranked according to the so-called ordering mapping. Despite successful applications, morphological operators based on reduced orderings are usually too reliant on the ordering mapping. In many practical situations, however, the ordering mapping may be subject to uncertainties such as measurement errors or the arbitrariness in the choice of the mapping. In view of this remark, in this paper we present two approaches to multivalued MM based on an uncertain reduced ordering. The new operators are formulated as the solution of an optimization problem which, apart from the uncertainty, can circumvent the false value problem and deal with irregularity issues.
Symmetry is present in many tasks in computer vision, where the same class of objects can appear transformed, e.g. rotated due to different camera orientations, or scaled due to perspective. The knowledge of such symmetries in data coupled with equivariance of neural networks can improve their generalization to new samples. Differential invariants are equivariant operators computed from the partial derivatives of a function. In this paper we use differential invariants to define equivariant operators that form the layers of an equivariant neural network. Specifically, we derive invariants of the Special Euclidean Group SE(2), composed of rotations and translations, and apply them to construct a SE(2)-equivariant network, called SE(2) Differential Invariants Network (SE2DINNet). The network is subsequently tested in classification tasks which require a degree of equivariance or invariance to rotations. The results compare positively with the state-of-the-art, even though the proposed SE2DINNet has far less parameters than the compared models.
Equivariance of neural networks to transformations helps to improve their performance and reduce generalization error in computer vision tasks, as they apply to datasets presenting symmetries (e.g. scalings, rotations, translations). The method of moving frames is classical for deriving operators invariant to the action of a Lie group in a manifold. Recently, a rotation and translation equivariant neural network for image data was proposed based on the moving frames approach. In this paper we significantly improve that approach by reducing the computation of moving frames to only one, at the input stage, instead of repeated computations at each layer. The equivariance of the resulting architecture is proved theoretically and we build a rotation and translation equivariant neural network to process volumes, i.e. signals on the 3D space. Our trained model overperforms the benchmarks in the medical volume classification of most of the tested datasets from MedMNIST3D.
Morfologia Matemática foi concebida como uma ferramenta para a análise e processamento de imagens binárias e foi subsequentemente generalizada para o uso em imagens em tons de cinza e imagens multivaloradas. Reticulados completos, que são conjuntos parcialmente ordenados em que todo subconjunto tem extremos bem definidos, servem como a base matemática para uma definição geral de morfologia matemática. Em contraste a imagens em tons de cinza, imagens multivaloradas não possuem uma ordem não-ambígua. Essa dissertação trata das chamadas ordens reduzidas para imagens multivaloradas. Ordens reduzidas são definidas por meio de uma relação binária que ordena os elementos de acordo com uma função h do conjunto de valores em um reticulado completo. Ordens reduzidas podem ser classificadas em ordens não-supervisionadas e ordens supervisionadas. Numa ordem supervisionada, o função de ordenação h depende de conjuntos de treinamento de valores de foreground e de background. Nesta dissertação, estudamos ordens supervisionadas da literatura. Também propomos uma ordem supervisionada baseada em valores fuzzy.Valores fuzzy generalizam cores fuzzy -conjuntos fuzzy que modelam o modo que humanos percebem as cores -para imagens multivaloradas. Em particular, revemos como construir o mapa de ordenação baseado em conjuntos fuzzy para o foreground e para o background. Também introduzimos uma função de pertinência baseada numa estrutura neuro-fuzzy e generalizamos a função de pertinência baseada no diagrama de Voronoi. Por fim, as ordens supervisionadas são avaliadas num experimento de segmentação de imagens hiperespectrais baseado num perfil morfológico modificado.
The fuzzy transform is a mathematical tool that unifies the traditional concept of function transforms with fuzzy rulebased systems. In this project, we studied applications of the fuzzy transform for edge detection and compression of grayscale images. To this end, definition and mathematical properties of the fuzzy transform were studied. Finally, computational experiments were performed to quantitatively evaluate the studied approaches.
Processamento de sinais é uma área do conhecimento com muitas aplicações, incluindo, entre outras, o processamento de áudio e vídeo, telecomunicação, sistemas de radar e medicina. Nesse trabalho de iniciação científica, estudamos o problema de reconstrução de um fragmento de um sinal de áudio discreto no tempo. A principal ferramenta utilizada para a reconstrução do fragmento foi o modelo linear preditivo, combinado com o método dos quadrados mínimos e técnicas de otimização não-linear. Para avaliar a eficiência dos métodos desenvolvidos, efetuamos alguns experimentos computacionais utilizando dados sintéticos e reais.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.