Spherical Tensor Calculus for Local Adaptive Filtering

Reisert, Marco; Burkhardt, Hans

doi:10.1007/978-1-84882-299-3_7

Cited by 13 publications

(19 citation statements)

References 16 publications

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“…Here we give an introduction about some important conceptions for working on the 2-sphere(S 2 ), in the scope of this paper. For more details, we refer to [3,17,20].…”

Section: Preliminaries For Working On the 2-spherementioning

confidence: 99%

3D Rotation-Invariant Description from Tensor Operation on Spherical HOG Field

Liu¹,

Skibbe²,

Schmidt³

et al. 2011

Procedings of the British Machine Vision Conference 2011

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Rotation-invariant descriptions are required in many 3D volumetric image analysis tasks. The histogram-of-oriented-gradient (HOG) is widely used in 2D images and proves to be a very robust local description. This paper concentrates on how to use the HOG feature in 3D volumetric images when rotation-invariance is concerned. This is challenging because of the complexity of 3D rotations. We present a decent solution based on the spherical harmonics theory which is an effective tool for analysing 3D rotations, together with the spherical tensor operations which explore high order tensor information in spherical coordinates. The design is quite general and could be used for different applications. It achieves high scores on Princeton Shape Benchmark and SHREC 2009 Generic Shape Benchmark, and also produces promising results when applying on biological microscopy images.

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“…Here we give an introduction about some important conceptions for working on the 2-sphere(S 2 ), in the scope of this paper. For more details, we refer to [3,17,20].…”

Section: Preliminaries For Working On the 2-spherementioning

confidence: 99%

3D Rotation-Invariant Description from Tensor Operation on Spherical HOG Field

Liu¹,

Skibbe²,

Schmidt³

et al. 2011

Procedings of the British Machine Vision Conference 2011

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show abstract

“…In the following we give a short introduction on spherical tensor algebra based on the definitions and notation used in [5]. The spherical tensor algebra is necessary for expanding higher order tensor fields (e.g.…”

Section: Spherical Tensor Fieldsmentioning

confidence: 99%

“…Similar to Cartesian tensor fields, where Kronecker products connect tensor fields of different rank, there exist spherical products [5,2] that connect spherical tensor fields of different rank. In fact, for two given spherical tensor fields v ∈ T 1 and w ∈ T 2 , there exists a whole set of different products • to build new spherical tensor fields.…”

Section: Spherical Tensor Fieldsmentioning

confidence: 99%

“…In order to obtain a representation of spherical tensor fields with tensor valued Fourier basis functions having the same convenient rotation properties known from the spherical harmonics, we perform a spherical tensor field expansion based on tensorial harmonics [5]. In addition to the tensorial harmonic expansion given in [5] for representing spherical tensor fields on the 2-sphere, we use the spherical Bessel function j (r) for representing the radial part (see [10] for definition).…”

Section: Tensorial Bessel Harmonicsmentioning

confidence: 99%

“…In addition to the tensorial harmonic expansion given in [5] for representing spherical tensor fields on the 2-sphere, we use the spherical Bessel function j (r) for representing the radial part (see [10] for definition). This directly extends the spherical Fourier basis B k (r) := Y (r)j (kr) presented in [9] to higher order tensor fields, where k ∈ R >0 represents the frequency in radial direction.…”

Section: Tensorial Bessel Harmonicsmentioning

confidence: 99%

See 2 more Smart Citations

3D Object Detection Using a Fast Voxel-Wise Local Spherical Fourier Tensor Transformation

Skibbe

Reisert

Schmidt

et al. 2010

Lecture Notes in Computer Science

Self Cite

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Abstract. In this paper we present a novel approach for expanding spherical 3D-tensor fields of arbitrary order in terms of a tensor valued local Fourier basis. For an efficient implementation, a two step approach is suggested combined with the use of spherical derivatives. Based on this new transformation we conduct two experiments utilizing the spherical tensor algebra for computing and using rotation invariant features for object detection and classification. The first experiment covers the successful detection of non-spherical root cap cells of Arabidopsis root tips presented in volumetric microscopical recordings. The second experiment shows how to use these features for successfully detecting α−helices in cryo-EM density maps of secondary protein structures, leading to very promising results.

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Harmonic Filters for Generic Feature Detection in 3D

Reisert¹,

Burkhardt

2009

Lecture Notes in Computer Science

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Spherical Tensor Calculus for Local Adaptive Filtering

Cited by 13 publications

References 16 publications

3D Rotation-Invariant Description from Tensor Operation on Spherical HOG Field

3D Rotation-Invariant Description from Tensor Operation on Spherical HOG Field

3D Object Detection Using a Fast Voxel-Wise Local Spherical Fourier Tensor Transformation

Harmonic Filters for Generic Feature Detection in 3D

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