Journal of Mathematical Imaging and Vision (2018) 60:763–765https://doi.org/10.1007/s10851-018-0824-y

Differential Geometry and Orientation Analysis in Image Processing

Remco Duits1 · Giovanna Citti2 · Andrea Fuster1 · Thomas Schultz3

Published online: 26 May 2018© Springer Science+Business Media, LLC, part of Springer Nature 2018

The synergy between the mathematical fields of partial dif-ferential equations, geometric control, Lie group analysis,harmonic analysis, variational methods and the applied fieldsof image analysis, numerical analysis, neurogeometry, andneuroimaging is increasing rapidly and has attracted manyresearchers.

An exciting playground for interaction between thesefields is multi-orientation image analysis. Here, differentialgeometrical concepts play a fundamental role in PDE- andODE-based techniques for pattern recognition and imageanalysis. Image processing applications are then providedwith effective differential geometrical tools to deal generi-cally with complex structures in images.

This JMIV special issue aims to further stimulate this syn-ergy and to provide a showcase for recent state-of-the-artmethods combining differential geometry and image pro-cessing. The 11 articles are high-quality works in accordancewith the high standards of JMIV. The submissions can be cat-egorized into:

1. Geometric models for symmetric and asymmetric geo-desic front propagation above the base manifold ofpositions and orientations. Such geodesically equidistantfront propagations yield (data-driven) sub-Finslerian,sub-Riemannian spheres, or value functions for elas-tica models. They allow for globally optimal geodesictracking and segmentation in the combined space of posi-tions and orientations. The main advantage over existinggeodesic tracking methods acting in the image domain(position space) is that such methods deal with complexstructures (crossings and bifurcations), as geodesic frontsdo not leak at such structures.

2. Geometric perceptual organization models of brightnessand orientation. This area aims for perceptional organi-

B Remco Duitsr.duits@tue.nl

1 TU/e, Eindhoven, The Netherlands

2 UNIBO, Bologna, Italy

3 University of Bonn, Bonn, Germany

zation methods for orientation and brightness perceptionthat live on higher dimensional manifolds including ori-entation, color, and scale. The advantage over existinggeodesic tracking methods acting directly in the imagedomain is that the geometry in higher dimensional mani-folds allows for better results in correction and denoisingof color images and in perceptional grouping assignments(e.g., of blood vessel patches). It also puts establishedpractical imaging techniques and cortical line perceptionmodels into mathematical context.

3. Geometric models for efficient shape and multi-orientation analysis. This research area aims for anal-ysis of shapes and includes shape registration modelsand shape deformation models. Here, sparse encod-ing of the data leads to efficient differential geometricmulti-orientation methods for local shape analysis, pathfollowing and visual tracking.

Next, we address each of these items, where we categorizethe submissions in our special issue and where we highlightper category both the novel image analysis techniques andthe novel mathematical results.

The first category covers various new image processingtechniques and mathematical results, for example on:

– effective image segmentation via antisymmetric frontpropagation in position and orientation space in the work“Fast Asymmetric Fronts Propagation for Image Seg-mentation” by Da Chen and Laurent Cohen. This workcontains new (analysis of) data-driven Randers metricsfor effective image segmentation, with clear advantagesfor image segmentation over (anisotropic) Riemannianapproaches.

– efficient and new numerical implementations of frontpropagation in position and orientation space in the work“Fast Marching methods for Curvature Penalized Short-est Paths” by Jean-Marie Mirebeau. This work containsnew theoretic results onmonotonicity, causality, and con-vergence of new discretization schemes for anisotropicfast-marching methods to compute distance maps and

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geodesics in Finslerian models on the combined mani-fold of positions and orientations.

– tracking of complex vasculature and neural fibers via newmodels and solutions of the Reed-Shepp car in “OptimalPaths for Variants of the 2D and 3D Reeds-Shepp Carwith Applications in Image Analysis” by Remco Duits,StephanMeesters, Jean-MarieMirebeau, and Jorg Porte-gies. This work improves geodesic vessel/fiber trackingin lifts of image data on the manifold M of positions andorientations. It contains new theorems on:

– strong convergence of Finslerian minimizing geo-desics to sub-Finslerian minimizing geodesics on M.

– the presence/absence of topological interest points onFinslerian geodesics, and a tangible solution to theproblem of cusps in sub-Riemannian image analysismodels.

– the back-tracking of optimal asymmetric Finslergeodesics via manifold splitting into symmetric man-ifolds (explaining automatic placement of in-placerotations at corners and bifurcations).

The second category covers various new image processingtechniques and mathematical results, for example, on:

– differential geometrical models for brightness percep-tion that integrate covariant derivatives to model colorappearance phenomena for improved correction tasksof color images, as explained in “A Geometric Modelof Brightness Perception and its Application to ColorImages Correction” by Thomas Batard and MarceloBartelmío. Theoretical results are obtained by inclusionof Weber’s/Weber-Fechner’s laws in covariant deriva-tives in associated vector bundles that are solutions ofa variational problem parametrized by a scalar and a Liegroup. Here, conditional vanishing of the curvature of theunderlying connection one forms has been shown.

– improved perceptional grouping of local orientations.Local fragments of blood vessels are grouped both effi-ciently and correctly via nilpotent approximations ofsub-Riemannian distances in the roto-translation groupsSE(2) and SE(3) in “Nilpotent Approximations of Sub-Riemannian Distances for Fast Perceptual Grouping ofBlood Vessels in 2D and 3D” by Erik Bekkers, Da Chen,and Jorg Portegies. Here, simple local analytic approx-imations of sub-Riemannian distances on position andorientation space with improved anisotropies and con-stants in existing estimations are used for better and fasterperceptual grouping algorithms.

– improved modeling of the generation of orientation pref-erence maps in the primary visual cortex, consideringboth orientation and scale features in “A GeometricalModel of Multiscale Orientation Preference Maps via

Gabor Functions” by Emre Baspinar, Giovanna Citti,and Alessandro Sarti. Here, the functional architectureof the orientation preference maps in the primary visualcortex is considered as a principal fiber bundle over the2D retinal plane. The visual stimulus is then lifted in afour-dimensional space with coordinate variables, posi-tion, orientation, and scale, through a linear filtering withGabor functions. Simulation results are compared to rel-evant neurophysiological findings in the literature.

The third category covers various new techniques andmathematical results:

– shape registration in the work “Fast Blended Transfor-mations for Partial Shape Registration” by Alon Shtern,Matan Sela, and Ron Kimmel. Here, over-completedictionary representations are integrated in shape defor-mation problems. The approach relies on blend skinningmodels on the space of positions and orientations andextends the subspace of deformations and controls thesmoothness of the reconstructed mesh. It allows for acomputation of a newpose froma small number of knownfeature points. Analysis, splitting, and sparsification ofvariational problems for as-rigid-as-possible models forshape registration are achieved.

– efficient algorithms for channel representations, whichcan be used in computer vision applications such as pathfollowing and visual tracking, are presented in the work"Approximative Coding Methods for Channel Represen-tations" by Kristoffer Öfjäll and Michael Felsberg. Inparticular, they present new approaches for continuousdensity reconstruction, for nonlinear gridding, and forestimating Copulas.

– a method for representation of 3Dmeshes using the spec-tral geometry of the Hamiltonian operator is presented inthe work “Sparse Approximation of 3D Meshes Usingthe Spectral Geometry of the Hamiltonian Operator” byYoni Choukroun, Gautam Pai, and Ron Kimmel. Com-pression performance is improved by sparsely encodingthe shape geometry using a data-dependent basis, com-pared to previous results using the standard Laplacianbasis.

– the stochastic version of the Beg algorithm is derived andapplied to shape and image matching in the paper “StringMethods for Stochastic Image and Shape Matching” byAlexis Arnaudon, Darry Holm, and Stefan Sommer. Thesimilarity between the presented algorithm and the finitetemperature string method used in rare event sampling isalso shown.

– a linear algebraic and tensorial framework for the shape-from-shading problem is presented in the work “Tensors,Differential Geometry and Statistical Shading Analysis”by Daniel Niels Holtmann-Rice, Benjamin S. Kuns-

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berg, and Steven W. Zucker. Using this framework, theyinvestigated when image derivatives exhibit invariance tochanging illumination and build a boot-strapping algo-rithm to find the surface solutions.

Altogether, this special issue provides a comprehensiveoverview of state-of-the-art differential geometrical methodsin image processing, with an emphasis on image processingthat takes place on the base manifold of positions and orien-tations. To avoid any conflicts of interest, manuscripts that

have been co-authored by one guest editor have been handledby another guest editor.

This special issue was preceded by an international work-shop on “Geometry, PDE’s and Lie Groups in Image Analy-sis”, URL:http://bmia.bmt.tue.nl/people/RDuits/workshop/index.html, held at Eindhoven University of Technology(August 24–26, 2016, Eindhoven, the Netherlands).

The guest editors, Giovanna Citti, Remco Duits, AndreaFuster, Thomas Schultz.

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