Line Enhancement and Completion via Linear Left Invariant Scale Spaces on SE(2)

Duits, Remco; Franken, Erik

doi:10.1007/978-3-642-02256-2_66

Cited by 19 publications

(22 citation statements)

References 11 publications

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“…To conclude this section, we recall a result from [2], see also [33], giving an explicit formula for the heat kernel on SE(2) via the non-commutative Fourier transform on SE (2). Via Mackey's machinery one obtains that unitary irreducible representations of SE(2) are parametrized by the disjoint union of the real half-line (0, +∞) with S 1 .…”

Section: Hypoelliptic Diffusions On Lie Groupsmentioning

confidence: 99%

A Semidiscrete Version of the Citti-Petitot-Sarti Model as a Plausible Model for Anthropomorphic Image Reconstruction and Pattern Recognition

Prandi

Gauthier

2018

SpringerBriefs in Mathematics

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PrefaceIn his beautiful book [66], Jean Petitot proposes a sub-Riemannian model for the primary visual cortex of mammals. This model is neurophysiologically justified. Further developments of this theory lead to efficient algorithms for image reconstruction, based upon the consideration of an associated hypoelliptic diffusion. The sub-Riemannian model of Petitot and Citti-Sarti (or certain of its improvements) is a left-invariant structure over the group SE(2) of rototranslations of the plane. Here, we propose a semi-discrete version of this theory, leading to a left-invariant structure over the group SE(2, N), restricting to a finite number of rotations. This apparently very simple group is in fact quite atypical: it is maximally almost periodic, which leads to much simpler harmonic analysis compared to SE(2). Based upon this semidiscrete model, we improve on previous image-reconstruction algorithms and we develop a pattern-recognition theory that leads also to very efficient algorithms in practice.

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Section: Hypoelliptic Diffusions On Lie Groupsmentioning

confidence: 99%

A Semidiscrete Version of the Citti-Petitot-Sarti Model as a Plausible Model for Anthropomorphic Image Reconstruction and Pattern Recognition

Prandi

Gauthier

2018

SpringerBriefs in Mathematics

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“…The invertibility property prevents information loss during the transformation, which is guaranteed by certain requirements of the kernel used for the transformation. The cake wavelets introduced by [42], [43] are proper wavelets, which satisfy these criteria. Similar to the Gabor wavelets, they are quadratic anisotropic filters, but unlike these, their summed Fourier transformations cover the entire frequency domain, making them spatially scale-independent.…”

Section: Lifting Retinal Images In the 5d Spacementioning

confidence: 99%

Curvature Integration in a 5D Kernel for Extracting Vessel Connections in Retinal Images

Abbasi-Sureshjani

Favali

Citti

et al. 2018

IEEE Trans. on Image Process.

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Tree-like structures, such as retinal images, are widely studied in computer-aided diagnosis systems for large-scale screening programs. Despite several segmentation and tracking methods proposed in the literature, there still exist several limitations specifically when two or more curvilinear structures cross or bifurcate, or in the presence of interrupted lines or highly curved blood vessels. In this paper, we propose a novel approach based on multi-orientation scores augmented with a contextual affinity matrix, which both are inspired by the geometry of the primary visual cortex (V1) and their contextual connections. The connectivity is described with a 5D kernel obtained as the fundamental solution of the Fokker-Planck equation modeling the cortical connectivity in the lifted space of positions, orientations, curvatures, and intensity. It is further used in a self-tuning spectral clustering step to identify the main perceptual units in the stimuli. The proposed method has been validated on several easy as well as challenging structures in a set of artificial images and actual retinal patches. Supported by quantitative and qualitative results, the method is capable of overcoming the limitations of current state-of-the-art techniques.

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“…where N (0, σ 2 ) is a normally distributed variable with zero mean and variance equal to σ 2 . The probability density of this process, denoted by v, was first used by Williams and Jacobs (Williams & Jacobs, 1997) to compute stochastic completion field, by August and Zucker (August & Zucker, 2000, 2003 to define the curve indicator random field, and more recently by R. Duits and Franken in (Duits & van Almsick, 2008;Duits & Franken, 2009) to perform contour completion, de-noising and contour enhancement. The kernel obtained integrating in time the density v 1…”

Section: A Model Of Cortical Connectivitymentioning

confidence: 99%

“…This method has been implemented in (Sanguinetti et al, 2008) and (Boscain et al, 2012). Exact solution of the Fokker-Planck equation has been provided by Duits and van Almsick (Duits & van Almsick, 2008), and their results have been applied by Duits and Franken (Duits & Franken, 2009) to image processing.…”

Section: Introductionmentioning

confidence: 99%

Local and Global Gestalt Laws: A Neurally Based Spectral Approach

2017

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A mathematical model of figure-ground articulation is presented, which takes into account both local and global gestalt laws and is compatible with the functional architecture of the primary visual cortex (V1). The local gestalt law of good continuation is described by means of suitable connectivity kernels, that are derived from Lie group theory, and quantitatively compared with long range connectivity in V1. Global gestalt constraints are then introduced in terms of spectral analysis of connectivity matrix derived from these kernels. This analysis performs grouping of local features and individuates perceptual units with the highest saliency. Numerical simulations are performed and results are obtained applying the technique to a number of stimuli.

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Line Enhancement and Completion via Linear Left Invariant Scale Spaces on SE(2)

Cited by 19 publications

References 11 publications

A Semidiscrete Version of the Citti-Petitot-Sarti Model as a Plausible Model for Anthropomorphic Image Reconstruction and Pattern Recognition

A Semidiscrete Version of the Citti-Petitot-Sarti Model as a Plausible Model for Anthropomorphic Image Reconstruction and Pattern Recognition

Curvature Integration in a 5D Kernel for Extracting Vessel Connections in Retinal Images

Local and Global Gestalt Laws: A Neurally Based Spectral Approach

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