Handbook of Geometric Computing: Applications in Pattern Recognition, Computer Vision, Neuralcomputing, and Robotics PDF
Preview Handbook of Geometric Computing: Applications in Pattern Recognition, Computer Vision, Neuralcomputing, and Robotics
Handbook of Geometric Computing Eduardo Bayro Corrochano HHHHHaaaaannnnndddddbbbbbooooooooookkkkk ooooofffff GGGGGeeeeeooooommmmmeeeeetttttrrrrriiiiiccccc CCCCCooooommmmmpppppuuuuutttttiiiiinnnnnggggg Applications in Pattern Recognition, Computer Vision, Neuralcomputing, and Robotics With 277 Figures, 67 in color, and 38 Tables 123 Prof. Dr. Eduardo Bayro Corrochano Cinvestav Unidad Guadalajara Ciencias de la Computación P. O. Box 31-438 Plaza la Luna, Guadalajara Jalisco 44550 México [email protected] Library of Congress Control Number: 2004118329 ACM Computing Classification (1998): I.4, I.3, I.5, I.2, F. 2.2 ISBN-10 3-540-20595-0 Springer Berlin Heidelberg New York ISBN-13 978-3-540-20595-1 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com © Springer-Verlag Berlin Heidelberg 2005 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: KünkelLopka, Heidelberg Production: LE-TeX Jelonek, Schmidt & Vöckler GbR, Leipzig Typesetting: by the author Printed on acid-free paper 45/3142/YL - 5 4 3 2 1 0 Preface One importantgoalofhumancivilizationisto build intelligentmachines,not necessarily machines that can mimic our behavior perfectly, but rather ma- chines that can undertake heavy, tiresome, dangerous, and even inaccessible (forman)labortasks.Computersareagoodexampleofsuchmachines.With their ever-increasing speeds and higher storage capacities, it is reasonable to expect that in the future computers will be able to perform even more useful tasks for man and society than they do today, in areas such as health care, automated visual inspection or assembly, and in making possible intelligent man–machine interaction. Important progress has been made in the develop- mentofcomputerizedsensorsandmechanicaldevices.Forinstance,according toMoore’slaw,thenumberoftransistorsonachiproughlydoubleseverytwo years – as a result, microprocessors are becoming faster and more powerful and memory chips can store more data without growing in size. Developmentswith respectto concepts,unifiedtheory,andalgorithmsfor buildingintelligentmachineshavenotoccurredwiththesamekindoflightning speed.However,theyshouldnotbemeasuredwiththesameyardstick,because the qualitative aspects of knowledge development are far more complex and intricate. In 1999, in his work on building anthropomorphic motor systems, Rodney Brooks noted: “A paradigm shift has recently occurred – computer performance is no longer a limiting factor. We are limited by our knowledge of what to build.” On the other hand, at the turn of the twenty-firstcentury, it would seem we collectively know enough about the human brain and we have developed sufficiently advanced computing technology that it should be possible for us to find ways to construct real-time, high-resolution, verifiable models for significant aspects of human intelligence. Just as greatstrides in the dissemination of human knowledge were made possiblebytheinventionofthe printingpress,inthe samewaymodernscien- tificdevelopmentsareenhancedtoagreatextentbycomputertechnology.The Internet now plays an important role in furthering the exchange of informa- tion necessaryfor establishing cooperationbetweendifferent researchgroups. Unfortunately,thetheoryforbuildingintelligentmachinesorperception-and- VI Preface action systems is still in its infancy. We cannot blame a lack of commitment on the part of researchers or the absence of revolutionary concepts for this state of affairs. Remarkably useful ideas were proposed as early as the mid- nineteenth century, when Babbage was building his first calculating engines. Sincethen,usefulconceptshaveemergedinmathematics,physics,electronics, andmechanicalengineering–allbasicfieldsforthedevelopmentofintelligent machines. In its time, classical mechanics offered many of the necessary con- ceptual tools. In our own time, Lie group theory and Riemann differential geometry play a largerole in modern mathematics and physics.For instance, as a representationtool, symmetry, a visual primitive probably unattentively encoded, may provide an important avenue for helping us understand per- ceptual processes.Unfortunately, the application of these concepts in current work on image processing, neural computing, and robotics is still somewhat limited. Statistical physics and optimization theory have also proven to be useful in the fields of numerical analysis, nonlinear dynamics, and, recently, in neural computing. Other approaches for computing under conditions of uncertainty, like fuzzy logic and tensor voting, have been proposed in recent years. As we can see, since Turing’s pioneering 1950 work on determining whether machines areintelligent,the developmentof computersfor enhanced intelligence has undergone great progress. Thisnewhandbooktakesadecisivestepinbringingtogetherinonevolume varioustopicshighlighting the geometricaspectsnecessaryfor imageanalysis and processing, perception, reasoning, decision making, navigation, action, andautonomouslearning.Unfortunately,evenwithgrowingfinancialsupport for researchand the enhanced possibilities for communication brought about by the Internet, the various disciplines within the research community are still divorced from one another, still working in a disarticulated manner. Yet the effort to build perception–action systems requires flexible concepts and efficientalgorithms,hopefully developedinanintegratedandunifiedmanner. Itis ourhope that this handbookwillencourageresearchersto worktogether onproposalsandmethodologiessoastocreatethenecessarysynergyformore rapid progress in the building of intelligent machines. Structure and Key Contributions Thehandbookconsistsofninepartsorganizedbydiscipline,sothatthereader can form anunderstanding of how work among the variousdisciplines is con- tributing to progress in the area of geometric computing. Understanding in each individual field is a fundamental requirement for the development of perception-action systems. In this regard, a tentative list of relevant topics might include: • brain theory and neuroscience • learning • neurocomputing, fuzzy computing, and quantum computing Preface VII • image analysis and processing • geometric computing under uncertainty • computer vision • sensors • kinematics, dynamics, and elastic couplings • fuzzy and geometric reasoning • control engineering • robot manipulators, assembly, MEMS, mobile robots, and humanoids • path planning, navigation,reaching, and haptics • graphic engineering, visualization, and virtual reality • medical imagery and computer-aided surgery We have collected contributions from the leading experts in these diverse areas of study and have organized the chapters in each part to address low- level processingfirst before moving on to the more complex issues of decision making. In this way, the reader will be able to clearly identify the current state ofresearchfor eachtopic andits relevancefor the directionandcontent offutureresearch.Bygatheringthisworktogetherundertheumbrellaofbuil- dingperception–actionsystems,weareabletoseethateffortstowardthatgoal are flourishing in each of these disciplines and that they are becoming more interrelatedandareprofitingfromdevelopmentsintheotherfields.Hopefully, in the near future, we will see all of these fields interacting even more closely in the construction of efficient and cost-effective autonomous systems. Part I Neuroscience In Chapter 1 Haluk Öğmen reviews the fundamental properties of the pri- mate visual system, highlighting its maps and pathways as spatio-temporal informationencodingandprocessingstrategies.Heshowsthatretinotopicand spatial-frequencymapsrepresentthegeometryofthefusionbetweenstructure andfunctioninthenervoussystem,andthatmagnocellularandparvocellular pathways can resolve the trade-off between spatial and temporal deblurring. In Chapter 2 Hamid R. Eghbalnia, Amir Assadi, and Jim Townsend a- nalyze the important visual primitive of symmetry, probably unattentively encoded, which can have a central role in addressing perceptual processes. The authors argue that biological systems may be hardwired to handle fil- tering with extreme efficiency.They believe that it may be possible to appro- ximate this filtering, effectively preserving all the important temporal visual features, by using current computer technology. For learning, they favor the use ofbidirectionalassociativememories,usinglocalinformationin the spirit of a local-to-globalapproachto learning. VIII Preface Part II Neural Networks In Chapter 3 Hyeyoung Park, Tomoko Ozeki, and Shun-ichi Amari choose a geometric approach to provide intuitive insights on the essential properties of neural networks and their performance. Taking into account Riemann’s structureofthe manifoldofmultilayerperceptrons,they designgradientlear- ning techniques for avoidingalgebraicsingularitiesthat havea greatnegative influence on trajectories of learning. They discuss the singular structure of neuromanifolds and pose an interesting problem of statistical inference and learning in hierarchical models that include singularities. InChapter4GerhardRitterandLaurentiuIancupresentanewparadigm for neuralcomputing using the lattice algebraframework.They developmor- phological auto-associative memories and morphological feed-forward net- works based on dendritic computing. As opposed to traditional neural net- works,theirmodelsdonotneedhiddenlayersforsolvingnon-convexproblems, but rather they converge in one step and exhibit remarkable performance in both storage and recall. In Chapter 5 Tijl De Bie, Nello Cristianini, and Roman Rosipal de- scribe a large class of pattern-analysis methods based on the use of genera- lized eigenproblems and their modifications. These kinds of algorithms can be used for clustering, classification,regression,and correlationanalysis.The chapter presents all these algorithms in a unified framework and shows how they can all be coupled with kernels and with regularization techniques in order to produce a powerful class of methods that compare well with those of the support-vector type. This study provides a modern synthesis between several pattern-analysis techniques. Part III Image Processing In Chapter 6 Jan J. Koenderink sketches a framework for image processing that is coherent and almost entirely geometric in nature. He maintains that thetimeisripeforestablishingimageprocessingasasciencethatdepartsfrom fundamental principles, one that is developed logically and is free of hacks, unnecessaryapproximations,andmereshowpiecesonmathematicaldexterity. In Chapter 7 Alon Spira, Nir Sochen, and Ron Kimmel describe ima- ge enhancement using PDF-based geometric diffusion flows. They start with variationalprinciplesforexplainingtheoriginoftheflows,andthisgeometric approach results in some nice invariance properties. In the Beltrami frame- work,theimageisconsideredtobeanembeddedmanifoldinthespace-feature manifold, so that the requiredgeometric filters for the flows in gray-leveland colorimagesortexturewilltakeintoaccounttheinducedmetric.Thischapter presents numerical schemes and kernels for the flows that enable an efficient and robust implementation. In Chapter 8 Yaobin Mao and Guanrong Chen show that chaos theory is an excellent alternative for producing a fast, simple, and reliable image- encryption scheme that has a high degree of security. The chapter describes Preface IX a practical and efficient chaos-based stream-cipher scheme for still images. Fromanengineer’sperspective,thechaosimage-encryptiontechnologyisvery promisingforthereal-timeimagetransferandhandlingrequiredforintelligent discerning systems. Part IV Computer Vision In Chapter 9 Kalle Åström is concerned with the geometry and algebra of multiple one-dimensional projections in a 2D environment. This study is relevantfor 1Dcameras,for understanding the projectionoflines inordinary vision, and, on the application side, for understanding the ordinary vision of vehicles undergoing planar motion. The structure-of-motion problem for 1D cameras is studied at length, and all cases with non-missing data are solved. Cases with missing data are more difficult; nevertheless, a classification is introduced and some minimal cases are solved. InChapter 10 Anders Heydendescribes in-depth, n-viewgeometrywith allthe computationalaspects requiredfor achieving stratifiedreconstruction. He starts with camera modeling and a review of projective geometry. He de- scribesthemulti-viewtensorsandconstraintsandtheassociatedlinearrecon- struction algorithms. He continues with factorization and bundle adjustment methods and concludes with auto-calibration methods. InChapter 11AmnonShashuaandLiorWolfintroducea generalization of the classical collineation of Pn. The m-view tensors for Pn referred to as homographytensorsarestudiedindetailforthecasen=3,4inwhichtheindi- vidual points are allowed to move while the projective change of coordinates takes place. The authors show that without homographytensors a recovering of the alignment requires statistical methods of sampling, whereas with the tensor approach both stationary and moving points can be considered alike and part of a global transformation can be recovered analytically from some matchingpointsacrossmviews.Ingeneral,thehomographytensorsareuseful for recovering linear models under linear uncertainty. InChapter 12AbhijitOgale,CorneliaFermüllerandYiannis Aloimonos examinetheproblemofinstantaneousfindingofobjectsmovingindependently inavideoobtainedbyamovingcamerawitharestrictedfieldofview.Inthis problem,theimagemotioniscausedbythecombinedeffectofcameramotion, scene depth, and the independent motions of objects. The authors present a classification of moving objects and discuss detection methods; the first class isdetectedusingmotionclustering,theseconddependsonordinaldepthfrom occlusionsandthethirdusescardinalknowledgeofthedepth.Robustmethods for deducing ordinal depth from occlusions are also discussed. X Preface Part V Perception and Action In Chapter 13 Eduardo Bayro-Corrochano presents a framework of con- formal geometric algebra for perception and action. As opposed to standard projective geometry, in conformal geometric algebra, using the language of spheres,planes, lines, and points, one can deal simultaneously with incidence algebraoperations(meetandjoin)andconformaltransformationsrepresented effectively using bivectors. This mathematical system allows us to keep our intuitions and insights into the geometry of the problem at hand and it helps ustoreduceconsiderablythecomputationalburdenoftherelatedalgorithms. Conformalgeometric algebra,with its powerful geometric representationand rich algebraic capacity to provide a unifying geometric language, appears promising for dealing with kinematics, dynamics, and projective geometry problems without the need to abandon a mathematical system. In general, this can be a great advantage in applications that use stereo vision, range data, lasers, omnidirectionality, and odometry-based robotic systems. Part VI Uncertainty in Geometric Computations In Chapter 14 Kenichi Kanatani investigates the meaning of “statistical methods” for geometric inference on image points. He traces back the ori- ginoffeatureuncertaintytoimage-processingoperationsforcomputervision, andhediscussestheimplicationsofasymptoticanalysiswithreferenceto“ge- ometric fitting” and “geometric model selection.” The author analyzes recent progress in geometric fitting techniques for linear constraints and semipara- metric models in relation to geometric inference. In Chapter 15 Wolfgang Förstner presents an approach for geometric reasoning in computer vision performed under uncertainty. He shows that the great potential of projective geometry and statistics can be integrated easily for propagating uncertainty through reasoning chains. This helps to make decisions on uncertain spatial relations and on the optimal estimation of geometricentities andtransformations.The chapter discusses the essential link between statistics and projective geometry, and it summarizes the basic relations in 2D and 3D for single-view geometry. In Chapter 16 Gérard Medioni, Philippos Mordohai, and Mircea Nico- lescu presenta tensor voting frameworkfor computer visionthat can address awiderangeofmiddle-levelvisionproblemsinaunifiedway.Thisframework isbasedonadatarepresentationformalismthatusessecond-ordersymmetric tensors and an information propagationmechanism that uses a tensor voting scheme. The authors show that their approach is suitable for stereo and mo- tion analysis because it can detect perceptual structures based solely on the smoothness constraint without using any model. This property allows them to treat the arbitrary surfaces that are inherent in non-trivial scenes.
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