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- Eckhard Hitzer
- ArXiv
- 2013

We treat the quaternionic Fourier transform (QFT) applied to quaternion fields and investigate QFT properties useful for applications. Different forms of the QFT lead us to different Plancherel theorems. We relate the QFT computation for quaternion fields to the QFT of real signals. We research the general linear (GL) transformation behavior of the QFT withâ€¦ (More)

- Eckhard Hitzer
- ArXiv
- 2013

We explain the orthogonal planes split (OPS) of quaternions based on the arbitrary choice of one or two linearly independent pure unit quaternions f ,g. Next we systematically generalize the quaternionic Fourier transform (QFT) applied to quaternion fields to conform with the OPS determined by f ,g, or by only one pure unit quaternion f , comment on theirâ€¦ (More)

- Roxana Bujack, Ingrid Hotz, Gerik Scheuermann, Eckhard Hitzer
- IEEE Transactions on Visualization and Computerâ€¦
- 2015

The analysis of 2D flow data is often guided by the search for characteristic structures with semantic meaning. One way to approach this question is to identify structures of interest by a human observer, with the goal of finding similar structures in the same or other datasets. The major challenges related to this task are to specify the notion ofâ€¦ (More)

- Eckhard Hitzer
- 2005

This paper focuses on the symmetries of crystal cells and crystal space lattices. All two dimensional (2D) and three dimensional (3D) point groups of 2D and 3D crystal cells are exclusively described by vectors (two in 2D, three in 3D for one particular cell) taken from the physical cells. Geometric multiplication of these vectors completely generates allâ€¦ (More)

- Eckhard Hitzer
- ArXiv
- 2013

A split hypercomplex learning algorithm for the training of nonlinear finite impulse response adaptive filters for the processing of hypercomplex signals of any dimension is proposed. The derivation strictly takes into account the laws of hypercomplex algebra and hypercomplex calculus, some of which have been neglected in existing learning approaches (e.g.â€¦ (More)

- Roxana Bujack, Jens Kasten, Ingrid Hotz, Gerik Scheuermann, Eckhard Hitzer
- 2015 IEEE Pacific Visualization Symposiumâ€¦
- 2015

We generalize the framework of moments and introduce a definition of invariants for three-dimensional vector fields. To do so, we use the method of moment normalization that has been shown to be useful in the two dimensions. Using invariant moments, we show how to search for patterns in these fields independent from their position, orientation and scale.â€¦ (More)

- Robert Benjamin Easter, Eckhard Hitzer
- CGI
- 2016

We introduce Clifford geometric algebra based multivector modeling of quartic and general quadric surfaces, Darboux cyclides, Dupin cyclies, tori and pairs of standard CGA objects. A computer visualization is briefly described.

- Eckhard Hitzer
- ArXiv
- 2013

Geometric algebra is an optimal frame work for calculating with vectors. The geometric algebra of a space includes elements that represent all the its subspaces (lines, planes, volumes, ...). Conformal geometric algebra expands this approach to elementary representations of arbitrary points, point pairs, lines, circles, planes and spheres. Apart fromâ€¦ (More)

- Eckhard Hitzer
- 2014

Recently the general orthogonal planes split with respect to any two pure unit quaternions f, g âˆˆ H, f = g = âˆ’1, including the case f = g, has proved extremely useful for the construction and geometric interpretation of general classes of double-kernel quaternion Fourier transformations (QFT) [7]. Applications include color image processing, where theâ€¦ (More)

- Eckhard Hitzer
- ArXiv
- 2013

The Liouville theorem states that bounded holomorphic complex functions are necessarily constant. Holomorphic functions fulfill the socalled Cauchy-Riemann (CR) conditions. The CR conditions mean that a complex z-derivative is independent of the direction. Holomorphic functions are ideal for activation functions of complex neural networks, but the Liouvilleâ€¦ (More)