New Developments in the Visualization and Processing of by Hans Knutsson, Carl-Fredrik Westin, Mats Andersson (auth.),

By Hans Knutsson, Carl-Fredrik Westin, Mats Andersson (auth.), David H. Laidlaw, Anna Vilanova (eds.)

Bringing jointly key researchers in disciplines starting from visualization and photo processing to functions in structural mechanics, fluid dynamics, elastography, and numerical arithmetic, the workshop that generated this edited quantity was once the 3rd within the profitable Dagstuhl sequence. Its target, mirrored within the caliber and relevance of the papers awarded, was once to foster collaboration and clean traces of inquiry within the research and visualization of tensor fields, which provide a concise version for various actual phenomena. regardless of their application, there is still a dearth of equipment for learning all however the easiest ones, a scarcity the workshops goal to address.

Documenting the newest development and open examine questions in tensor box research, the chapters replicate the thrill and suggestion generated by way of this most modern Dagstuhl workshop, held in July 2009. the subjects they handle diversity from purposes of the research of tensor fields to purer study into their mathematical and analytical houses. They exhibit how cooperation and the sharing of rules and knowledge among these engaged in natural and utilized learn can open new vistas within the research of tensor fields.

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8 Sum of Monomial Tensors Even more careful weighting of spherical harmonic filter products will give positive semi-definite tensors, in this case the result will be equivalent to a sum of tensor estimates over different order, n 0 (even and odd), monomial filter matrices, Qn . X TSM D wn Qn QTn (54) n 7 Table of Structure Tensor Related Algorithms The table below is a chronological list of a number of image processing algorithms directly or indirectly related to local structure tensor estimation.

1 . Inserting this into Eq. (16) yields: ln jC1 D ln j C 2˛ Ä Rj . ln2 Â ˛ ln2 C ln2 exp. 1 / exp 4˛ ln jC1 jC1 Ä ˛ ln2 . j C1 ˛ ln j ˛ j C1 / C 1 / 4˛ 2 1 4 ˛2 Ã 2 ln ln 2 ln ln jC1 Ã jC1 (17) Structure Tensor Estimation: Introducing Monomial Quadrature Filter Sets 11 Fig. u/. Multiplying the lognormal filter (blue) with u (dotted black) results in a new lognormal filter function (red) with the same relative bandwidth at the next scale. Note that the new filter is odd. e. odd and even D 1 / RjC1 .

Figure 5 (and 4) illustrates this relation. 12 H. Knutsson et al. Fig. 5 Figure showing the fundamental linking of scale, order and the gradient operator that is provided by lognormal monomial filter sets. The gradient operator increases the order by one and shifts the center frequency towards higher frequency by a constant factor r 4 Monomial Filter Response Matrices The next step is to apply a monomial filter matrix to a signal, thus obtaining a monomial filter response matrix. This can be done by convolving the signal with each of the filters in the filter matrix and storing the results in the corresponding positions.

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