In 1979, the Nobel Prize for Medicine and Physiology was awarded jointly to Allan McLeod Cormack and Godfrey Newbold Houns eld, the two pioneering scienti- engineers primarily responsible for the development, in the 1960s and early 1970s, of computerized axial tomography, popularly known as the CAT or CT scan. In his papers [13], Cormack, then a Professor at Tufts University, in Massachusetts, dev- oped certain mathematical algorithms that, he envisioned, could be used to create an image from X-ray data. Working completely independently of Cormack and at about the same time, Houns eld, a research scientist at EMI Central Research Laboratories in the United Kingdom, designed the rst operational CT scanner as well as the rst commercially available model. (See [22] and [23]. ) Since 1980, the number of CT scans performed each year in the United States has risen from about 3 million to over 67 million. What few people who have had CT scans probably realize is that the fundamental problem behind this procedure is essentially mathematical: If we know the values of the integral of a two- or three-dimensional fu- tion along all possible cross-sections, then how can we reconstruct the function itself? This particular example of what is known as an inverse problem was studied by Johann Radon, an Austrian mathematician, in the early part of the twentieth century.

This book constitutes selected papers from the First International Symposium on Geometry and Vision, ISGV 2021, held in Auckland, New Zealand, in January 2021. Due to the COVID-19 pandemic the conference was held in partially virtual format.
The 29 papers were thoroughly reviewed and selected from 50 submissions. They cover topics in areas of digital geometry, graphics, image and video technologies, computer vision, and multimedia technologies.

One of the ultimate goals in Robotics is to create autonomous robots. Such robots will accept high-level descriptions of tasks and will execute them without further human intervention. The input descriptions will specify what the user wants done rather than how to do it. The robots will be any kind of versatile mechanical device equipped with actuators and sensors under the control of a computing system. Making progress toward autonomous robots is of major practical inter­ est in a wide variety of application domains including manufacturing, construction, waste management, space exploration, undersea work, as­ sistance for the disabled, and medical surgery. It is also of great technical interest, especially for Computer Science, because it raises challenging and rich computational issues from which new concepts of broad useful­ ness are likely to emerge. Developing the technologies necessary for autonomous robots is a formidable undertaking with deep interweaved ramifications in auto­ mated reasoning, perception and control. It raises many important prob­ lems. One of them - motion planning - is the central theme of this book. It can be loosely stated as follows: How can a robot decide what motions to perform in order to achieve goal arrangements of physical objects? This capability is eminently necessary since, by definition, a robot accomplishes tasks by moving in the real world. The minimum one would expect from an autonomous robot is the ability to plan its x Preface own motions.

This book contains the proceedings of the 4th International Conference on Data Analysis and Processing held in Cefalu' (Palermo, ITALY) on September 23-25 1987. The aim of this Conference, now at its fourth edition, was to give a general view of the actual research in the area of methods and systems for achieving artificial vision as well as to have an up-dated information of the current activity in Europe. A number of invited speakers presented overviews of statistical classification problems and methods, non conventional archi­ tectures, mathematical morphology, robotic vision, analysis of range images in vision systems, pattern matching algorithms and astronomical data processing. Finally a survey of the discussion on the contribution of AI to Image Analysis is given. The papers presented at the Conference have been subdivided in four sections: knowledge based approaches, basic pattern recognition tools, multi features system based solutions, image analysis-applications. We must thank the IBM-Italia and the Digital Equipment Corpo­ ration for sponsoring this Conference. We feel that the days spent at Cefalu' were an important step toward the mutual exchange of scientific information within the image processing community. v. Cantoni Pavia University V. Di Gesu' Palermo University S. Levialdi Rome University v CONTENTS INVITED LECTURES . • • • • . • • • 3 Morphological Optics.

This book presents theories and techniques for perception of textures by computer. Texture is a homogeneous visual pattern that we perceive in surfaces of objects such as textiles, tree barks or stones. Texture analysis is one of the first important steps in computer vision since texture provides important cues to recognize real-world objects. A major part of the book is devoted to two-dimensional analysis of texture patterns by extracting statistical and structural features. It also deals with the shape-from-texture problem which addresses recovery of the three-dimensional surface shapes based on the geometry of projection of the surface texture to the image plane. Perception is still largely mysterious. Realizing a computer vision system that can work in the real world requires more research and ex­ periment. Capability of textural perception is a key component. We hope this book will contribute to the advancement of computer vision toward robust, useful systems. vVe would like to express our appreciation to Professor Takeo Kanade at Carnegie Mellon University for his encouragement and help in writing this book; to the members of Computer Vision Section at Electrotechni­ cal Laboratory for providing an excellent research environment; and to Carl W. Harris at Kluwer Academic Publishers for his help in preparing the manuscript.

If you have ever hiked up a steep hill to reach a viewpoint, you will know that sensing can involve the expenditure of effort. More generally, the choice of which movement an intelligent system chooses to make is usually based on information gleaned from sensors. But the information required to make the motion decision may not be immediately to hand, so the system . first has to plan a motion whose purpose is to acquire the needed sensor information. Again, this conforms to our everyday experience: I am in the woods and don't know which direction to go, so I climb up to the ridge to get my bearings; I am lost in a new town, so I plan to drive to the next junction where there is sure to be a roadsign, failing that I will ask someone who seems to be from the locality. Why, if experiences such as these are so familiar, has the problem only recently been recognised and studied in Robotics? One reason is that until quite recently Robotics research was dominated by work on robot arms with limited reach and fixed in a workcell.

In recent years there has been an increasing interest in Second Generation Image and Video Coding Techniques. These techniques introduce new concepts from image analysis that greatly improve the performance of the coding schemes for very high compression. This interest has been further emphasized by the future MPEG 4 standard.

The book provides an introduction to Differential Geometry of Curves and Surfaces. The theory of curves starts with a discussion of possible definitions of the concept of curve, proving in particular the classification of 1-dimensional manifolds. We then present the classical local theory of parametrized plane and space curves (curves in n-dimensional space are discussed in the complementary material): curvature, torsion, Frenet’s formulas and the fundamental theorem of the local theory of curves. Then, after a self-contained presentation of degree theory for continuous self-maps of the circumference, we study the global theory of plane curves, introducing winding and rotation numbers, and proving the Jordan curve theorem for curves of class C2, and Hopf theorem on the rotation number of closed simple curves. The local theory of surfaces begins with a comparison of the concept of parametrized (i.e., immersed) surface with the concept of regular (i.e., embedded) surface. We then develop the basic differential geometry of surfaces in R3: definitions, examples, differentiable maps and functions, tangent vectors (presented both as vectors tangent to curves in the surface and as derivations on germs of differentiable functions; we shall consistently use both approaches in the whole book) and orientation. Next we study the several notions of curvature on a surface, stressing both the geometrical meaning of the objects introduced and the algebraic/analytical methods needed to study them via the Gauss map, up to the proof of Gauss’ Teorema Egregium. Then we introduce vector fields on a surface (flow, first integrals, integral curves) and geodesics (definition, basic properties, geodesic curvature, and, in the complementary material, a full proof of minimizing properties of geodesics and of the Hopf-Rinow theorem for surfaces). Then we shall present a proof of the celebrated Gauss-Bonnet theorem, both in its local and in its global form, using basic properties (fullyproved in the complementary material) of triangulations of surfaces. As an application, we shall prove the Poincaré-Hopf theorem on zeroes of vector fields. Finally, the last chapter will be devoted to several important results on the global theory of surfaces, like for instance the characterization of surfaces with constant Gaussian curvature, and the orientability of compact surfaces in R3.

The author has maintained two open-source MATLAB Toolboxes for more than 10 years: one for robotics and one for vision. The key strength of the Toolboxes provide a set of tools that allow the user to work with real problems, not trivial examples. For the student the book makes the algorithms accessible, the Toolbox code can be read to gain understanding, and the examples illustrate how it can be used —instant gratification in just a couple of lines of MATLAB code. The code can also be the starting point for new work, for researchers or students, by writing programs based on Toolbox functions, or modifying the Toolbox code itself.

Spatial reasoning and planning is a core constituent in robotics, graphics, computer-aided design, and geographic information systems.

The concept of ridges has appeared numerous times in the image processing liter­ ature. Sometimes the term is used in an intuitive sense. Other times a concrete definition is provided. In almost all cases the concept is used for very specific ap­ plications. When analyzing images or data sets, it is very natural for a scientist to measure critical behavior by considering maxima or minima of the data. These critical points are relatively easy to compute. Numerical packages always provide support for root finding or optimization, whether it be through bisection, Newton's method, conjugate gradient method, or other standard methods. It has not been natural for scientists to consider critical behavior in a higher-order sense. The con­ cept of ridge as a manifold of critical points is a natural extension of the concept of local maximum as an isolated critical point. However, almost no attention has been given to formalizing the concept. There is a need for a formal development. There is a need for understanding the computation issues that arise in the imple­ mentations. The purpose of this book is to address both needs by providing a formal mathematical foundation and a computational framework for ridges. The intended audience for this book includes anyone interested in exploring the use­ fulness of ridges in data analysis.

Appendix 164 3. A 3. A. 1 Approximate Estimation of Fundamental Matrix from General Matrix 164 3. A. 2 Estimation of Affine Transformation 165 4 RECOVERY OF EPIPOLAR GEOMETRY FROM LINE SEGMENTS OR LINES 167 Line Segments or Straight Lines 168 4. 1 4. 2 Solving Motion Using Line Segments Between Two Views 173 4. 2. 1 Overlap of Two Corresponding Line Segments 173 Estimating Motion by Maximizing Overlap 175 4. 2. 2 Implementation Details 4. 2. 3 176 Reconstructing 3D Line Segments 4. 2. 4 179 4. 2. 5 Experimental Results 180 4. 2. 6 Discussions 192 4. 3 Determining Epipolar Geometry of Three Views 194 4. 3. 1 Trifocal Constraints for Point Matches 194 4. 3. 2 Trifocal Constraints for Line Correspondences 199 4. 3. 3 Linear Estimation of K, L, and M Using Points and Lines 200 4. 3. 4 Determining Camera Projection Matrices 201 4. 3. 5 Image Transfer 203 4. 4 Summary 204 5 REDEFINING STEREO, MOTION AND OBJECT RECOGNITION VIA EPIPOLAR GEOMETRY 205 5. 1 Conventional Approaches to Stereo, Motion and Object Recognition 205 5. 1. 1 Stereo 205 5. 1. 2 Motion 206 5. 1. 3 Object Recognition 207 5. 2 Correspondence in Stereo, Motion and Object Recognition as 1D Search 209 5. 2. 1 Stereo Matching 209 xi Contents 5. 2. 2 Motion Correspondence and Segmentation 209 5. 2. 3 3D Object Recognition and Localization 210 Disparity and Spatial Disparity Space 210 5.

Scale is a concept the antiquity of which can hardly be traced. Certainly the familiar phenomena that accompany sc ale changes in optical patterns are mentioned in the earliest written records. The most obvious topological changes such as the creation or annihilation of details have been a topic to philosophers, artists and later scientists. This appears to of fascination be the case for all cultures from which extensive written records exist. For th instance, chinese 17 c artist manuals remark that "distant faces have no eyes" . The merging of details is also obvious to many authors, e. g. , Lucretius mentions the fact that distant islands look like a single one. The one topo­ logical event that is (to the best of my knowledge) mentioned only late (by th John Ruskin in his "Elements of drawing" of the mid 19 c) is the splitting of a blob on blurring. The change of images on a gradual increase of resolu­ tion has been a recurring theme in the arts (e. g. , the poetic description of the distant armada in Calderon's The Constant Prince) and this "mystery" (as Ruskin calls it) is constantly exploited by painters.