Optimal Paths for Variants of the 2D and 3D Reeds–Shepp Car with Applications in Image Analysis

Duits, Remco; Meesters, Spl Stephan; Mirebeau, Jean-Marie; Portegies, JM Jorg

doi:10.1007/s10851-018-0795-z

Cited by 61 publications

(152 citation statements)

References 58 publications

(203 reference statements)

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“…In our experiments, we aim to enhance contour and fiber trajectories in medical images and to remove noise. Lifting the image f : R d → R towards its orientation lift U : M → R defined on the space of positions and orientations M = R d S d−1 preserves crossings [21] and avoids leakage of wavefronts [18].…”

Section: Methodsmentioning

confidence: 99%

“…The OS is obtained by convolving the image with a set of rotated wavelets allowing for stable reconstruction [14,4,24]. 2nd row: Vessel-tracking in a 2D image via geodesic PDE-flows in OS that underly TVF: [4,18,8], with n = (cos θ, sin θ) T ∈ S 1 . 3rd row: CED-OS diffusion of a 3D image [24,17] visualized as a field of angular profiles.…”

Section: Figmentioning

confidence: 99%

“…Geodesic flows prior to steepest descent also rely on (related [32,5,8]) PDEs on M commuting with roto-translations. They can account for crossings/bifurcations/corners [4,18,8].…”

Section: Introductionmentioning

confidence: 99%

“…They employ (a)symmetric Finslerian geodesic models on M cf. [18]. As TVF falls short on invariance w.r.t.…”

Section: Introductionmentioning

confidence: 99%

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

Duits

St-Onge²,

Portegies

et al. 2019

Lecture Notes in Computer Science

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Total variation regularization and total variation flows (TVF) have been widely applied for image enhancement and denoising. To include a generic preservation of crossing curvilinear structures in TVF we lift images to the homogeneous space M = R d S d−1 of positions and orientations as a Lie group quotient in SE(d). For d = 2 this is called 'total roto-translation variation' by Chambolle & Pock. We extend this to d = 3, by a PDE-approach with a limiting procedure for which we prove convergence. We also include a Mean Curvature Flow (MCF) in our PDE model on M. This was first proposed for d = 2 by Citti et al. and we extend this to d = 3. Furthermore, for d = 2 we take advantage of locally optimal differential frames in invertible orientation scores (OS). We apply our TVF and MCF in the denoising/enhancement of crossing fiber bundles in DW-MRI. In comparison to data-driven diffusions, we see a better preservation of bundle boundaries and angular sharpness in fiber orientation densities at crossings. We support this by error comparisons on a noisy DW-MRI phantom. We also apply our TVF and MCF in enhancement of crossing elongated structures in 2D images via OS, and compare the results to nonlinear diffusions (CED-OS) via OS.

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Section: Methodsmentioning

confidence: 99%

Section: Figmentioning

confidence: 99%

“…Geodesic flows prior to steepest descent also rely on (related [32,5,8]) PDEs on M commuting with roto-translations. They can account for crossings/bifurcations/corners [4,18,8].…”

Section: Introductionmentioning

confidence: 99%

“…They employ (a)symmetric Finslerian geodesic models on M cf. [18]. As TVF falls short on invariance w.r.t.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Duits

St-Onge²,

Portegies

et al. 2019

Lecture Notes in Computer Science

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“…From the theoretical point of view, the lower-semicontinuity of these energies already is a challenge, which has been studied in many papers (in connexion to the applications to computer Riemannian and sub-Finslerian geodesic tracking in the roto-translation group [11,34].…”

Section: Introductionmentioning

confidence: 99%

Total roto-translational variation

Chambolle

Pock

2019

Numer. Math.

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We consider curvature depending variational models for image regularization, such as Euler's elastica. These models are known to provide strong priors for the continuity of edges and hence have important applications in shape-and image processing. We consider a lifted convex representation of these models in the roto-translation space: In this space, curvature depending variational energies are represented by means of a convex functional defined on divergence free vector fields. The line energies are then easily extended to any scalar function. It yields a natural generalization of the total variation to the roto-translation space. As our main result, we show that the proposed convex representation is tight for characteristic functions of smooth shapes. We also discuss cases where this representation fails. For numerical solution, we propose a staggered grid discretization based on an averaged Raviart-Thomas finite elements approximation. This discretization is consistent, up to minor details, with the underlying continuous model. The resulting non-smooth convex optimization problem is solved using a first-order primal-dual algorithm. We illustrate the results of our numerical algorithm on various problems from shape-and image processing. vision) since at least the 90s [12,13,16,14,17,15,30]. It is shown already in [12] that the boundary of many sets with cusps can be approximated by sets with smooth boundaries of bounded energy, showing that the relaxation of the Elastica for boundaries of sets is already far from trivial. The study of this lower semi-continuity of course enters the long history of the study of general curvature dependent energies of manifolds and in particular the Willmore energy [77].Quite early, it has been suggested to lift the manifold in a larger space where a variable represents its direction or orientation, by means in particular (for co-dimension one manifolds) of the Gauss map (x, ν(x)), ν(x) being the normal to the manifold at x [5,6]. Such approach allows to study very general curvature energies and has been successfully used for establishing lower-semicontinuity and existence results [6,7,32,31], and in particular to lines energies such as ours in higher codimension [2,1] or in dimension 2 [31]. We must mention also in this class an older approach based on "curvature varifolds" (which is very natural since varifolds are defined on the cross product of spatial and directional variables) which has allowed to show existence results since the 80s [45], see also [50].Interestingly, it was understood much earlier [44] that (cats') vision was functioning in a similar way (eg., using a sort of "Gauss map"), thanks to neurons sensitive to particular directions which were found to be stacked inside the visual cortex into ordered columns, making us sensitive to changes of orientations (and thus curvature intensity). These findings (which might explain some of Kanizsa's experiments) inspired some mathematical models quite early [48], however they were formalized into a consistent geometric inte...

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Probability of detection curve for the automatic visual inspection of steel bridges

Kompanets,

Leonetti,

Duits

et al. 2023

ce papers

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The damage tolerant design philosophy is based on periodical inspections and provides safe bridge operation preventing fatigue cracks from growing up to a critical size. It is possible to optimize the management costs of a bridge throughout its life by designing the inspection frequency, which depends on the capabilities of the inspection method. However, such optimization requires knowledge about the performance of inspections that are envisioned to take place during the use of the bridge. Regular visual inspections is the most frequently applied type of inspection of bridge structures. Recent advances in computer vision technologies provide a strong basis for the development of automatic damage detection systems that can support regular visual inspection, thus increasing the reliability of the inspection. Several automatic crack detection systems have been developed in the past years. However, the performances of such systems have not been evaluated in the way as for traditional non‐destructive inspection methods, i.e. in terms of probability of detection curves and detectability limits. This restricts the applicability of automatic visual inspections for inspection planning and damage tolerant design. This paper proposes an encoder‐decoder neural network for segmentation of cracks on images of steel bridges. A probability of detection curve is calculated for this neural network.

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Optimal Paths for Variants of the 2D and 3D Reeds–Shepp Car with Applications in Image Analysis

Cited by 61 publications

References 58 publications

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

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

Total roto-translational variation

Probability of detection curve for the automatic visual inspection of steel bridges

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