Klette R., Kozera R., Noakes L., Weickert J. (eds.) Geometric Properties for Incomplete Data

Springer, 2006. — 390 p.Computer vision and image analysis requires interdisciplinary collaboration between mathematics and engineering. This book addresses the area of high-accuracy measurements of length, curvature, motion parameters and other geometrical quantities from acquired image data. It is a common problem that these measurements are incomplete or noisy, such that considerable efforts are necessary to regularise the data, to fill in missing information, and to judge the accuracy and reliability of these results. This monograph brings together contributions from researchers in computer vision, engineering and mathematics who are working in this area. A number of the chapters are expanded from invited lectures presented at the international computer science center at Dagstuhl/Germany which is funded by the German federal and two state governments. The editors gratefully acknowledge this support.
The invited authors are well-known experts in their area and have been encouraged to stress the survey character of their contributions. These are fundamental work, directed to applications in computer vision and engineering. Each paper was reviewed by at least three independent referees, and we thank all referees (see list below) for their efforts. Their comments were very valuable in order to improve the quality of this research monograph. The help of Jovisa Zunic in finalizing the Latex files is very much appreciated.
By its nature, it was difficult to categorize the chapters in this book. We decided for a division into three parts, which should be of help to demonstrate the themes running through this volume. Although geometry underlies most contributions in the present volume, it is possible to draw distinctions. Contributions in Part I are mainly concerned with continuous geometry, including algebraic tools designed for geometry, such as Clifford algebras in the chapters by Sommer and Perwass, and quaternions in the contribution of Farouki. The chapters by Noakes and Kozera concern geometrical aspects of approximation and interpolation. A somewhat more specific application of geometry to computer vision is given in the chapter by Robles-Kelly.
Discrete geometry in computer vision (Part II), including the important emerging area of digitization, has a different flavour. Even more than with other contributions to this volume, incomplete information is ever-present. The two chapters by Eckhardt, and by Doerksen-Reiter and Debled-Rennesson discuss segmentations of borders in a digital image into convex and concave parts. Linh, Imiya and Torii consider polygonalization and polyhedrization algorithms for approximating borders in 2D and 3D images.
Binary tomography (i.e., reconstruction of binary images from projections) is the subject of the chapter by Weber, Schn.orr, Schule and Hornegger. Linear discriminant analysis for features derived from digital images is applied by Skarbek, Kucharski and Bober for face recognition. Huxley, Klette and ˇZuni.c study the accuracy of approximating real moments based on data available in digital images. 3D shape recovery based on digital images is the subject of the following two chapters; Tankus, Sochen and Yeshurun use shading models, and Imiya considers shadows.
Part III is concerned with approximation and regularisation methods that can be interpreted in a statistical or deterministic way. Typical applications include robust denoising of signals and images, the reliable estimation of model parameters, and motion estimation in image sequences. This area is characterised by transparent mathematical models, optimality results and performance evaluation. It includes two contributions on motion analysis: The chapter by Bruhn and Weickert evaluates a novel confidence measure for variational optic flow methods, while Kanatani and Sugaya review their contributions on feature point tracking in video sequences. The subsequent chapters deal with approximation methods: Fenn and Steidl establish connections between robust local estimation methods in image processing and approximation theoretical techniques for scattered data. The contribution by Mr.azek et al. presents a unified framework for edge-preserving denoising and interpolation, while M.uhlich’s work deals with data fitting to geometric manifolds. The contribution by Obereder et al. concludes this category by presenting an analysis of higher order bounded variation regularisation in terms of generalised G norms.
Finally, it has to be said that the majority of contributions transcend the categories we have (sometimes arguably) assigned. This reflects the interdisciplinary nature of the work. We hope that the reader will enjoy an exciting journey.Continuous Geometry
The Twist Representation of Free-form Objects
Uncertain Geometry with Circles, Spheres and Conics
Algorithms for Spatial Pythagoreanhodograph Curves
Cumulative Chords, Piecewise- Quadratics and Piecewise-Cubics
Spherical Splines
Graph-Spectral Methods for Surface Height Recovery from Gauss Maps
Discrete Geometry
Segmentation of Boundaries into Convex and Concave Parts
Convex and Concave Parts of Digital Curves
Polygonalisation and Polyhedralisation by Optimisation
Binary Tomography by Iterating Linear Programs
Cascade of Dual LDA Operators for Face Recognition
Precision of Geometric Moments in Picture Analysis
Shape-from-Shading by Iterative Fast Marching for Vertical and Oblique Light Sources
Shape from Shadows
Approximation and Regularization
A Confidence Measure for Variational Optic Flow Methods
Video Image Sequence Analysis: Estimating Missing Data and Segmenting Multiple Motions
Robust Local Approximation of Scattered Data
On Robust Estimation and Smoothing with Spatial and Tonal Kernels
Subspace Estimation with Uncertain and Correlated Data
On the Use of Dual Norms in Bounded Variation Type Regularization