The Hessian of Axially Symmetric Functions on SE(3) and Application in 3D Image Analysis

Janssen, Mark; Haije, T.C.J. Dela; Martin, Frank C.; Bekkers, Erik J.; Duits, Remco

doi:10.1007/978-3-319-58771-4_51

Cited by 5 publications

(2 citation statements)

References 11 publications

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“…(processed) orientation scores only on 8 orientations. On M 3 we compute regularized orientation lifts on a grid with 162 orientations, where we rely on efficient numerical schemes for PDEs on M 3 relying on the low-order PDE discretization schemes explained in [17,37], instead of higher-order schemes via spherical harmonics [34,Ch. 3.4], in order to reduce computation time.…”

Section: Choice Of Geometrymentioning

confidence: 99%

Total Variation and Mean Curvature PDEs on the Homogeneous Space of Positions and Orientations

Smets

Portegies

St-Onge³

et al. 2020

J Math Imaging Vis

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Two key ideas have greatly improved techniques for image enhancement and denoising: the lifting of image data to multi-orientation distributions and the application of nonlinear PDEs such as total variation flow (TVF) and mean curvature flow (MCF). These two ideas were recently combined by Chambolle and Pock (for TVF) and Citti et al. (for MCF) for two-dimensional images. In this work, we extend their approach to enhance and denoise images of arbitrary dimension, creating a unified geometric and algorithmic PDE framework, relying on (sub-)Riemannian geometry. In particular, we follow a different numerical approach, for which we prove convergence in the case of TVF by an application of Brezis–Komura gradient flow theory. Our framework also allows for additional data adaptation through the use of locally adaptive frames and coherence enhancement techniques. We apply TVF and MCF to the enhancement and denoising of elongated structures in 2D images via orientation scores and compare the results to Perona–Malik diffusion and BM3D. We also demonstrate our techniques in 3D in the denoising and enhancement of crossing fiber bundles in DW-MRI. In comparison with data-driven diffusions, we see a better preservation of bundle boundaries and angular sharpness in fiber orientation densities at crossings.

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Section: Choice Of Geometrymentioning

confidence: 99%

Total Variation and Mean Curvature PDEs on the Homogeneous Space of Positions and Orientations

Smets

Portegies

St-Onge³

et al. 2020

J Math Imaging Vis

Self Cite

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“…The diffusion system in Eq. (110) has been set up in previous work [45] with constant D 11 and D 33 .…”

Section: Cedosmentioning

confidence: 99%

Design and Processing of Invertible Orientation Scores of 3D Images

Janssen

Bekkers

et al. 2018

J Math Imaging Vis

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The enhancement and detection of elongated structures in noisy image data are relevant for many biomedical imaging applications. To handle complex crossing structures in 2D images, 2D orientation scores U : R 2 × S 1 → C were introduced, which already showed their use in a variety of applications. Here we extend this work to 3D orientation scores U : R 3 ×S 2 → C. First, we construct the orientation score from a given dataset, which is achieved by an invertible coherent state type of transform. For this transformation we introduce 3D versions of the 2D cake wavelets, which are complex wavelets that can simultaneously detect oriented structures and oriented edges. Here we introduce two types of cake wavelets: the first uses a discrete Fourier transform, and the second is designed in the 3D generalized Zernike basis, allowing us to calculate analytical expressions for the spatial filters. Second, we propose a nonlinear diffusion flow on the 3D roto-translation group: crossing-preserving coherence-enhancing diffusion via orientation scores (CEDOS). Finally, we show two applications of the orientation score transformation. In the first application we apply our CEDOS algorithm to real medical image data. In the second one we develop a new tubularity measure using 3D orientation scores and apply the tubularity measure to both artificial and real medical data.

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Recent Geometric Flows in Multi-orientation Image Processing via a Cartan Connection

Duits¹,

Smets²,

Wemmenhove³

et al. 2021

Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging

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The Hessian of Axially Symmetric Functions on SE(3) and Application in 3D Image Analysis

Cited by 5 publications

References 11 publications

Total Variation and Mean Curvature PDEs on the Homogeneous Space of Positions and Orientations

Total Variation and Mean Curvature PDEs on the Homogeneous Space of Positions and Orientations

Design and Processing of Invertible Orientation Scores of 3D Images

Recent Geometric Flows in Multi-orientation Image Processing via a Cartan Connection

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