Hypoelliptic Diffusion and Human Vision: A Semidiscrete New Twist

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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“…We solve this by using approximate reconstruction Eq. (13). Alternatively, one could replace (5), with small.…”

Section: Construction Of Line and Edge Detectorsmentioning

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 starting point of our work is the sub-Riemannian model of the primary visual cortex V1 [14,6], and our recent contributions [3,1,2,4]. This model has also been deeply studied in [8,11].…”

Section: The Semi-discrete Modelmentioning

confidence: 99%

“…EvolveDiffusion: Given a function Lh on SE(2, N ) evolves it according to (1). An efficient way to compute this diffusion is presented in [1], and recalled in Algorithm 2. 4.…”

Section: Image Inpaintingmentioning

confidence: 99%

“…According to this model, the primary visual cortex V1 lifts grey-scale images, given as functions f : R 2 → [0, 1], to functions Lf defined on the projectivized tangent bundle of the plane P T R 2 = R 2 × P 1 . Recently, in [1], the authors presented a promising semidiscrete variant of this model where the Euclidean group of rototranslations SE (2), which is the double covering of P T R 2 , is replaced by SE(2, N ), the group of translations and discrete rotations. In particular, in [15], an implementation of this model allowed for state-of-the-art image inpaintings.…”

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confidence: 99%

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Image Processing in the Semidiscrete Group of Rototranslations

Prandi

Boscain

Gauthier

2015

Lecture Notes in Computer Science

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“…This is the definition used in [10], where Hopf extends the classical Poincaré-Hopf Theorem to the case of line fields. Line fields appear often in nature as for instance in fingerprints [13,15,22], liquid crystals [5,8,17] and in the pinwheel structure of the visual cortex V1 of mammals [3,4,6,11,16]. Contrarily to what happens for vector fields, where the topology of the manifold forces the vector fields to have zeros, the topology of the manifold forces line fields to have singularities (i.e., points where a line field is not defined).…”

Section: Introductionmentioning

confidence: 99%

Generic singularities of line fields on 2D manifolds

Boscain

Sacchelli

Sigalotti

2016

Differential Geometry and its Applications

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Generic singularities of line fields have been studied for lines of principal curvature of embedded surfaces. In this paper we propose an approach to classify generic singularities of general line fields on 2D manifolds. The idea is to identify line fields as bisectors of pairs of vector fields on the manifold, with respect to a given conformal structure. The singularities correspond to the zeros of the vector fields and the genericity is considered with respect to a natural topology in the space of pairs of vector fields. Line fields at generic singularities turn out to be topologically equivalent to the Lemon, Star and Monstar singularities that one finds at umbilical points.

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Hypoelliptic Diffusion and Human Vision: A Semidiscrete New Twist

Cited by 60 publications

References 42 publications

Design and Processing of Invertible Orientation Scores of 3D Images

Design and Processing of Invertible Orientation Scores of 3D Images

Image Processing in the Semidiscrete Group of Rototranslations

Generic singularities of line fields on 2D manifolds

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