Catálogo de publicaciones - libros

Median filters for scalar-valued data are well-known tools for image denoising and analysis. They preserve discontinuities and are robust under noise. We generalise median filtering to matrix-valued data using a minimisation approach. Experiments on DT-MRI and fluid dynamics tensor data demonstrate that tensor-valued median filtering shares important properties of its scalar-valued counterpart, including the robustness as well as the existence of non-trivial steady states (root signals).

A straightforward extension of the definition allows the introduction of matrix-valued mid-range filters and, more general, M-smoothers. Mid-range filters can also serve as a building block in constructing further (e.g. supremum-based) tensor image filters.

The notions of maximum and minimum are the key to the powerful tools of greyscale morphology. Unfortunately these notions do not carry over directly to tensor-valued data. Based upon the Loewner ordering for symmetric matrices this chapter extends the maximum and minimum operation to the tensor-valued setting. This provides the ground to establish matrix-valued analogues of the basic morphological operations ranging from erosion/dilation to top hats. In contrast to former attempts to develop a morphological machinery for matrices, the novel definitions of maximal/minimal matrices depend continuously on the input data, a property crucial for the construction of morphological derivatives such as the Beucher gradient or a morphological Laplacian. These definitions are rotationally invariant and preserve positive semidefiniteness of matrix fields as they are encountered in DTMRI data. The morphological operations resulting from a component-wise maximum/minimum of the matrix channels disregarding their strong correlation fail to be rotational invariant. Experiments on DT-MRI images as well as on indefinite matrix data illustrate the properties and performance of our morphological operators.

Acquisition systems are not fully reliable since any real sensor will provide noisy and possibly incomplete and degraded data. Therefore, in tensor measurements, all problems dealt with in conventional multidimensional statistical signal processing are present with tensor signals. In this chapter we describe some noniterative approaches to tensor signal processing. Our schemes are achieved by the estimation of a local structure tensor, which is used as a key element in regularization. A stochastic point of view as well as a phase-invariant implementation are presented. This work also covers tensor extensions for common scalar operations such as anisotropic interpolation and filtering. An application of the structure tensor for regularization of deformation fields in tensor image registration is also shown. The techniques presented in this chapter suppose an alternative to variational and PDEs schemes, and another point of view of the tensor signal processing.

This chapter presents two techniques for regularization of tensor fields. We first present a nonlinear filtering technique based on normalized convolution, a general method for filtering missing and uncertain data. We describe how the signal certainty function can be constructed to depend on locally derived certainty information and further combined with a spatially dependent certainty field. This results in reduced mixing between regions of different signal characteristics, and increased robustness to outliers, compared to the standard approach of normalized convolution using only a spatial certainty field. We contrast this deterministic approach with a stochastic technique based on a multivariate Gaussian signal model in a Bayesian framework. This method uses a Markov random field approach with a 3D neighborhood system for modeling spatial interactions between the tensors locally. Experiments both on synthetic and real data are presented. The driving tensor application for this work throughout the chapter is the filtering of diffusion tensor MRI data.

Methods based on partial differential equations (PDEs) belong to those image processing techniques that can be extended in a particularly elegant way to tensor fields. In this survey chapter the most important PDEs for discontinuity-preserving denoising of tensor fields are reviewed such that the underlying design principles becomes evident. We consider isotropic and anisotropic diffusion filters and their corresponding variational methods, mean curvature motion, and selfsnakes. These filters preserve positive semidefiniteness of any positive semidefinite initial tensor field. Finally we discuss geodesic active contours for segmenting tensor fields. Experiments are presented that illustrate the behaviour of all these methods.