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Review of the book of Vladimir Kovalevsky The title of the book may create an impression that it is for geometers and topologists. That is right, but I think that the main purpose of the book is toconstruct a broad road for specialists in digital geometry, image recognition,and other branches of computer and applied mathematics. Where to? Into theworld of computational and algorithmic topology. Topological notions such assurface, manifold, connectedness, boundary, orientation, dimension unavoid-ably appear when you start to think about basic theoretical problems in theabove-mentioned branches of science. However, existing topological books aredevoted to the purely theoretical interior problems of topology and are very farfrom practical applications. The theory of locally finite spaces presented in the330-page monograph of Professor Vladimir Kovalevsky fills this gap. The the-oretic part of the book contains numerous new definitions and theorems whichexpress in a form understandable for non-specialists applications of those no-tions of set topology which are needed for computer imagery. The part devotedto applications presents some algorithms for investigating topological and geo-metrical properties of digitized two- and three-dimensional images. The bookassumes a certain level of preparation, but some elementary knowledge in topol-ogy and in image processing would be sufficient. One of the important featuresof this book is that it provides a means which gives the possibility to overcomethe existing discrepancy between theory and applications: The traditional wayof research consists in making theory in Euclidean space with real coordinateswhile applications deal only with finite discrete sets and rational numbers. Thereason is that even the smallest part of the Euclidean space cannot be explic-itly represented in a computer and computations with irrational numbers areimpossible since there exists no arithmetic of irrational numbers. The authordemonstrates that locally finite spaces are explicitly representable in a com-puter, only rational coordinates are used and the theory of those spaces is in accordance with classical topology. The book consists of 14 chapters: an intro-ductory section followed by thirteen main sections. The introductory sectionpresents a short retrospect to the origin of the book followed by an overview ofthe contents and of the aims of the monograph. Section 2 presents a new setof axioms and a proof that classical axioms of a topological space follow fromthe new axioms as theorems. This means that a locally finite space satisfyingthe new axioms (ALF spaces) is a particular case of a classical space: it isa T0 Alexandroff space which is not a T1 space. I would like to stress thatthe axioms are actually designed to applications. In particular, they are veryconvenient while working with computer presentations of geometric objects.
Section 3 is devoted to the theory of the spaces under consideration. It con-tains numerous theorems about the properties of ALF spaces and definitionsof balls, spheres and of the dimension of the space elements. These definitionsare important for describing combinatorial homeomorphisms between spaces.
This section also contains a new generalization of the orientation of simplicialcomplexes to the general case of abstract complexes and a generalization ofthe classical notion of a boundary. Section 4 considers mappings among locallyfinite spaces. The author has demonstrated that classical homeomorphismsbased on continuous maps being applied to locally finite spaces degrade to iso-morphisms. He suggests to replace them by a much more convenient notion ofconnectedness preserving correspondences (CPMs), which can map one spaceelement to many. He also has demonstrated that a combinatorial homeomor-phism based on elementary subdivisions of space elements uniquely defines acontinuous CPM whose inverse is also continuous. Sections 6 to 9 are devotedto a new concept of digital geometry which reflects Euclidean geometry nu-merically. There is among others a new and complete theory of digital straightsegments being considered as one-dimensional complexes rather than sequencesof pixels. A digital straight segment is considered here as a subset of the bound-ary of a digital half-plane rather than as digitization of a Euclidean straightline. This theory leads to interesting efficient applications to image analysis(Section 11.3). Sections 10 to 13 are devoted to applications. They contain de-scriptions of numerous algorithms based on the theory of locally finite spaces.
There are among them algorithms for tracing and encoding boundaries in two-and three-dimensional digital images, exactly reconstructing images from theirboundary codes, labeling connected components, computing skeletons, con-structing convex hulls and others. Some sections of the book contain problemsto be solved which will stimulate further research. Particularly interesting isSection 14 ”Topics for Discussion”. The author discusses here the possibilityto avoid irrational numbers and to use finite differences instead of derivatives.
He demonstrates the possibilities of his approach while presenting an inferenceof the Taylor formula based on finite differences. The book is unique: it isthe first one in its own way presenting a self-contained theory of locally finitespaces that is independent of the Hausdorff topology. It also contains a conceptof digital geometry that is independent of Euclidean geometry. I do not know any similar books. The book is carefully and clearly written and contains nu-merous well-made excellent illustrations. It is understandable for those whoseinterests are close to digital geometry / topology as well as for specialists intopology craving for applications. This book would be very useful and guidingfor students and researchers, or for anyone interested in digital topology, dig-ital geometry and computer imagery. It also may be of interest for physicistsworking on quantum gravity since it presents a well founded theory of spacesthat could serve as the base of this branch of physics. My main conclusion isthat the book should certainly find its way into university libraries and ontomany private book shelves.
Sergei Matveev (matveev@csu.ru)Corresponding member of RASDepartment of Mathematics, Chelyabinsk State University, Kashirin Brothersstreet 129, Chelyabinsk 454021, Russia.

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