A Tensor Framework for Multidimensional Signal Processing

C.-F. Westin
PhD Dissertation
Linköping University, Sweden
1994

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Abstract

This thesis deals with filtering of multidimensional signals. A large part of the thesis is devoted to a novel filtering method termed ''Normalized convolution''. The method performs local expansion of a signal in a chosen filter basis which not necessarily has to be orthonormal. A key feature of the method is that it can deal with uncertain data when additional certainty statements are available for the data and/or the filters. It is shown how false operator responses due to missing or uncertain data can be significantly reduced or eliminated using this technique. Perhaps the most well-known of such effects are the various edge effects which invariably occur at the edges of the input data set. The method is an example of the signal/certainty - philosophy, i.e. the separation of both data and operator into a signal part and a certainty part. An estimate of the certainty must accompany the data. Missing data are simply handled by setting the certainty to zero. Localization or windowing of operators is done using an applicability function, the operator equivalent to certainty, not by changing the actual operator coefficients. Spatially or temporally limited operators are handled by setting the applicability function to zero outside the window.

The use of tensors in estimation of local structure and orientation using spatio-temporal quadrature filters is reviewed and related to dual tensor bases. The tensor representation conveys the degree and type of local anisotropy. For image sequences, the shape of the tensors describe the local structure of the spatio-temporal neighbourhood and provides information about local velocity. The tensor representation also conveys information for deciding if true flow or only normal flow is present. It is shown how normal flow estimates can be combined into a true flow using averaging of this tensor field description.

Important aspects of representation and techniques for grouping local orientation estimates into global line information are discussed. The uniformity of some standard parameter spaces for line segmentation is investigated. The analysis shows that, to avoid discontinuities, great care should be taken when choosing the parameter space for a particular problem. A new parameter mapping well suited for line extraction, the Mobius strip parameterization, is defined. The method has similarities to the Hough Transform.

Estimation of local frequency and bandwidth is also discussed. Local frequency is an important concept which provides an indication of the appropriate range of scales for subsequent analysis. One-dimensional and two-dimensional examples of local frequency estimation are given. The local bandwidth estimate is used for defining a certainty measure. The certainty measure enables the use of a normalized averaging process increasing robustness and accuracy of the frequency statements.

The thesis cover figure shows a visualization of a symmetric tensor in three dimensions. The object in the gure is the sum of a spear, a plate and a sphere. The spear describes the principal direction of the tensor, where the length is proportional to the largest eigenvalue. The plate describes the plane spanned by the eigenvec- tors corresponding to the two largest eigenvalues. The sphere, with a radius proportional to the smallest eigenvalue, shows how isotropic the tensor is. The visualization is done using AVS.

Reference

Westin CF. A Tensor Framework for Multidimensional Signal Processing. Ph.D. thesis, Linköping University, Sweden, S-581 83 Linköping, Sweden, 1994. Dissertation No 348, ISBN 91-7871-421-4.

Bibtex entry

@PhdThesis{westin94,
  author         = {C.-F. Westin},                                             
  title          = {A Tensor Framework for Multidimensional Signal Processing},
  school         = {Link\"oping University, Sweden},                           
  year           = 1994,                                                       
  address        = {S-581 83 Link\"oping, Sweden},                             
  note           = {Dissertation No 348, ISBN 91-7871-421-4}
}

Research areas

Tensor, Normalized Convolution