Curvature Estimation for Enhancement of Crossing Curves

Franken, Erik; Duits, Remco; Romeny, Bart ter Haar

doi:10.1109/iccv.2007.4409145

Cited by 5 publications

(3 citation statements)

References 4 publications

(4 reference statements)

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“…The idea of a gauge frame in image processing (Florack et al 1993) is to define, at each spatial position independently, a data-dependent orthogonal coordinate frame that spans the tangent space, such that the basis vectors are in alignment with some local feature of interest in the image. This paper extends earlier work presented at conferences (Franken et al 2007a(Franken et al , 2007b with new results, particularly the gauge frame, stability bounds of the numerical schemes, and the concept of deviation from horizontality. In this paper we focus on the applied image processing and algorithmic part of our work.…”

Section: Introductionsupporting

confidence: 80%

Crossing-Preserving Coherence-Enhancing Diffusion on Invertible Orientation Scores

2009

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Many image processing problems require the enhancement of crossing elongated structures. These problems cannot easily be solved by commonly used coherenceenhancing diffusion methods. Therefore, we propose a method for coherence-enhancing diffusion on the invertible orientation score of a 2D image. In an orientation score, the local orientation is represented by an additional third dimension, ensuring that crossing elongated structures are separated from each other. We consider orientation scores as functions on the Euclidean motion group, and use the group structure to apply left-invariant diffusion equations on orientation scores. We describe how we can calculate regularized left-invariant derivatives, and use the Hessian to estimate three descriptive local features: curvature, deviation from horizontality, and orientation confidence. These local features are used to adapt a nonlinear coherence-enhancing, crossing-preserving, diffusion equation on the orientation score. We propose two explicit finite-difference schemes to apply the nonlinear diffusion in the orientation score and provide a stability analysis. Experiments on both artificial and medical images show that preservation of crossings is the main advantage compared to standard coherenceenhancing diffusion. The use of curvature leads to improved enhancement of curves with high curvature. Furthermore,

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Section: Introductionsupporting

confidence: 80%

Crossing-Preserving Coherence-Enhancing Diffusion on Invertible Orientation Scores

2009

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“…it indicates how much the tangent vector of the optimally fitting exponential curve (tangent spiral) points out of the horizontal plane, [20], [19]; see Figure 11.…”

Section: )mentioning

confidence: 99%

“…The curvature estimation is now given by For numerical experiments on the proposed curvature estimation (comparing the two methods above and the approach by van Ginkel [37]) on orientation scores of noisy example images, see [20], [19], [16].…”

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Left-invariant parabolic evolutions on $SE(2)$ and contour enhancement via invertible orientation scores Part II: Nonlinear left-invariant diffusions on invertible orientation scores

2010

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Abstract. By means of a special type of wavelet unitary transform we construct an orientation score from a grey-value image. This orientation score is a complex-valued function on the 2D Euclidean motion group SE(2) and gives us explicit information on the presence of local orientations in an image. As the transform between image and orientation score is unitary we can relate operators on images to operators on orientation scores in a robust manner. Here we consider nonlinear adaptive diffusion equations on these invertible orientation scores. These nonlinear diffusion equations lead to clear improvements of the celebrated standard "coherence enhancing diffusion" equations on images as they can enhance images with crossing contours. Here we employ differential geometry on SE(2) to align the diffusion with optimized local coordinate systems attached to an orientation score, allowing us to include local features such as adaptive curvature in our diffusions.

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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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Curvature Estimation for Enhancement of Crossing Curves

Abstract: In this paper we describe a method for estimating curvature

Cited by 5 publications

References 4 publications

Crossing-Preserving Coherence-Enhancing Diffusion on Invertible Orientation Scores

Crossing-Preserving Coherence-Enhancing Diffusion on Invertible Orientation Scores

Left-invariant parabolic evolutions on $SE(2)$ and contour enhancement via invertible orientation scores Part II: Nonlinear left-invariant diffusions on invertible orientation scores

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

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