In this paper we develop a new filtering framework for tensor signal processing using the theory of vector spaces. From this point of view, signals are regarded as elements of vector spaces and operators as mappings from the input space to the output space. Hence, it is possible to generalize the principle of superposition to any operator defined on the signal spaces. Systems that obey that generalization of the principle of superposition are referred to as homomorphic and they can be decomposed in a cascade of three homomorphic subsystems: the first one operates on the input signal space, the second one is a linear system in the usual sense and the third one operates on the output signal space. Thus, suitable input and output subsystems can be chosen to deal with input signals, which defines a whole family of homomorphic filters. To apply this idea for DT-MRI signals, which consist of positive semi-definite matrices, we identify input and output signal spaces as the set of those real symmetric positive semi-definite matrices. Our homomorphic filtering framework not only guarantees a positive-semidefinite output tensor field whatever linear filter is used to regularize the noisy input, but also reduces the swelling effect produced by a faster regularization of diffusivities rather than orientations, as demonstrate the encouraging results that have been obtained.
We present an anisotropic filtering scheme which uses a nonlinear version of the local structure tensor to dynamically adapt the shape of the neighborhood used to perform the estimation. In this way, only the samples along the orthogonal direction to that of maximum signal variation are chosen to estimate the value at the current position, which helps to better preserve boundaries and structure information. This idea sets the basis of an anisotropic filtering framework which can be applied for different kinds of linear filters, such as Wiener
or LMMSE, among others. In this paper, we describe the underlying idea using anisotropic gaussian filtering which allows us, at the same time, to study the influence of nonlinear structure tensors in filtering schemes, as we compare the performance to that obtained with classical definitions of the structure tensor.