1 Bounding Constraint Propagation for Optical Flow Estimation.- 1.1 Introduction.- 1.2 The Gradient Constraint Equation.- 1.3 Gradient-Based Algorithms.- 1.4 Coping with Smoothness Violations.- 1.4.1 Thresholding for Smoothness.- 1.4.2 Continuous Adaptation to Errors.- 1.5 Results.- 1.6 Discussion.- 2 Image Flow: Fundamentals and Algorithms.- 2.1 Introduction.- 2.1.1 Background.- 2.1.2 Applications for Image Flow.- 2.1.3 Summary.- 2.2 Simple Image Flows.- 2.2.1 Image Flow Equation for Simple Flows.- 2.2.2 Algorithms for Simple Image Flows.- 2.2.3 Summary of Simple Image Flows.- 2.3 Discontinuous Image Flow.- 2.3.1 Surfaces and Projections.- 2.3.2 Image Irradiance Discontinuities.- 2.3.3 Velocity Field Discontinuities.- 2.3.4 Validity of the Image Flow Equation.- 2.3.5 Related Work.- 2.4 Analysis of Discontinuous Image Flows.- 2.4.1 Discontinuities in Continuous Image Functions.- 2.4.2 Sampling of Discontinuous Image Flows.- 2.4.3 Directional Selectivity.- 2.4.4 Summary of Discontinuous Image Flows.- 2.5 Algorithms for Discontinuous Image Flows.- 2.5.1 Background.- 2.5.2 Problem Statement.- 2.5.3 Constraint Line Clustering.- 2.5.4 Summary.- 2.6 Smoothing Discontinuous Image Flows.- 2.6.1 Motion Boundary Detection.- 2.6.2 Velocity Field Smoothing.- 2.6.3 Interleaved Detection and Smoothing.- 2.7 Summary and Conclusions.- 3 A Computational Approach to the Fusion of Stereopsis and Kineopsis.- 3.1 Introduction.- 3.2 Integrating Optical Flow to Stereopsis for Motion.- 3.3 Perception of Rigid Objects in Motion.- 3.4 Examples.- 3.5 Summary.- 4 The Empirical Study of Structure from Motion.- 4.1 Introduction.- 4.2 Viewer-Centered vs. Object-Centered Depth.- 4.2.1 Orthographic Projections of Rotation in Depth.- 4.2.2 Recovery of Structure from Velocity Gradients.- 4.3 The Correspondence Problem.- 4.3.1 Point Configurations.- 4.3.2 Contour Deformation.- 4.3.3 Texture Deformation.- 4.4 Rigidity.- 4.5 Perception of Self Motion.- 4.6 A Theory of Observers.- 4.7 An Empirical Test of Constraints.- 4.8 Summary and Conclusions.- 5 Motion Estimation Using More Than Two Images.- 5.1 Introduction.- 5.2 General Description of the Method.- 5.2.1 Establishing the Equations.- 5.2.2 Simplifying the Equations.- 5.2.3 Solving the Equations.- 5.2.4 Calculating the Motion Parameters.- 5.2.5 Advantages of this Approach.- 5.2.6 Limitations of Our Approach.- 5.3 Results.- 5.3.1 Synthetic Test Data.- 5.3.2 Real Test Data.- 5.4 Comparison with Other Methods.- 5.4.1 Error Analysis.- 5.5 Conclusions.- 6 An Experimental Investigation of Estimation Approaches for Optical Flow Fields.- 6.1 Introduction.- 6.2 Feature Based Estimation.- 6.2.1 The Monotonicity Operator.- 6.2.2 From Feature Positions to Optical Flow Vectors.- 6.2.3 Test Sequence.- 6.2.4 Moving Object Detection.- 6.2.5 Performance Analysis of the Monotonicity Operator.- 6.2.6 Robustness of the Monotonicity Operator Against Parameter Changes.- 6.2.7 Reduction to Two Classes.- 6.3 Analytical Approach for the Estimation of Optical Flow Vector Fields.- 6.3.1 The "Oriented Smoothness" Constraint.- 6.3.2 Evaluation at Local Extrema of the Picture Function.- 6.4 Discussion.- 7 The Incremental Rigidity Scheme and Long-Range Motion Correspondence.- 7.1 The Rigidity-Based Recovery of Structure from Motion.- 7.1.1 The Perception of Structure from Motion by Human Observers.- 7.1.2 Computational Studies of the Recovery of Structure from Motion.- 7.1.3 Additional Requirements for the Recovery of Structure from Motion.- 7.1.4 A Hypothesis: Maximizing Rigidity Relative to the Current Internal Model.- 7.2 The Incremental Rigidity Scheme.- 7.2.1 The Basic Scheme.- 7.2.2 Possible Modifications.- 7.2.3 Implementation.- 7.3 Experimental Results.- 7.3.1 Rigid Motion.- 7.3.2 Non-Rigid Motion.- 7.4 Additional Properties of the Incremental Rigidity Scheme.- 7.4.1 Orthographic and Perspective Projections.- 7.4.2 The Effect of the Number of Points.- 7.4.3 On Multiple Objects.- 7.4.4 Convergence to the Local Minimum.- 7.5 Possib
The physical processes which initiate and maintain motion have been a major concern of serious investigation throughout the evolution of scientific thought. As early as the fifth century B. C. questions regarding motion were presented as touchstones for the most fundamental concepts about existence. Such wide ranging philosophical issues are beyond the scope of this book, however, consider the paradox of the flying arrow attri buted to Zeno of Elea: An arrow is shot from point A to point B requiring a sequence of time instants to traverse the distance. Now, for any time instant, T, of the sequence the arrow is at a position, Pi' and at Ti+! the i arrow is at Pi+i> with Pi ::I-P+. Clearly, each Ti must be a singular time i 1 unit at which the arrow is at rest at Pi because if the arrow were moving during Ti there would be a further sequence, Til' of time instants required for the arrow to traverse the smaller distance. Now, regardless of the level to which this recursive argument is applied, one is left with the flight of the arrow comprising a sequence of positions at which the arrow is at rest. The original intent of presenting this paradox has been interpreted to be as an argument against the possibility of individuated objects moving in space.
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