Interpretation of Visual Motion

1 Introduction 1.1 Problem Description 1.2 Motivations for the Study 1.3 The Approach 1.4 Strategy of Formulation and Analysis 1.5 Summary of Results 1.6 Conclusions and New Questions 1.7 Organization of the Book2 Background 2.1 Introduction 2.2 Visual Ambiguity and Assumptions 2.3 Discrete Approach 2.4 Continuous Approach 2.4.1 Computation of Image Flow from Intensity Derivatives 2.4.2 Computation of Image Flow from Features of the Intensity Distribution 2.5 Interpretation of Image Flow 2.6 Some Comments on the Approaches 2.7 What is New?3 Formulation 3.1 Computational Theory and Approach 3.2 Image Formation 3.3 Formulation: Instantaneous Image Flow Due to Rigid Motion 3.4 The Nature of the Image Flow Equations4 Solving the Image Flow Equations 4.1 Solution for Motion And Slopes in Terms of Θ and r 4.2 Planar Surfaces 4.2.1 The Planar Surface Equations 4.2.2 Solving for Θ and r 4.2.3 Bounds on the Velocity of Approach from First Order Flow Parameters 4.2.4 The Two-Fold Ambiguity 4.2.5 Resolving the Two-Fold Ambiguity of Interpretation Using Spatial or Temporal Consistency 4.3 Curved Surfaces 4.3.1 Solving the Image Flow Equations 4.3.2 Conditions for the Presence of Multiple Interpretations 4.3.3 Resolving Ambiguity of Interpretation 4.4 Overall Summary of the Nature of the Solutions 4.5 Fully Reduced Flow Parameters for Image Flow Analysis 4.5.1 Motion Parameters 4.5.2 Structure Parameters 4.5.3 Image Flow Parameters 4.5.4 Image Flow Equations 4.5.5 Solving the Image Flow Equations5 Using Temporal Information: Rigid Motion 5.1 Formulation and Solution 5.2 Relating the First Order Image Flow Parameters and Scene Parameters 5.3 Rigid and Uniform Motion 5.3.1 The Case When V is Uniform with Respect to the Camera 5.4 The Case When the Direction of V Changes 5.4.1 The Case When the Camera Tracks a Point 5.5 Accelerated Motions 5.5.1 An Example of Non-Uniform Motion6 The General Formulation 6.1 Representation and Formulation 6.1.1 An Example of Non-Rigid Motion 6.2 Arbitrarily Time-Varying 3D Scenes: Non-Rigid and Nonuniform Motion of General Surfaces 6.2.1 Time and Space-Time Derivatives of Image Flow 6.2.2 The Nature of the Problem7 Error Sensitivity and Numerical Examples 7.1 Estimation of Maximum Absolute Error 7.2 Numerical Examples of Multiple Solution Cases8 ConclusionsAppendicesA Expressing a Surface in Terms of Image CoordinatesB Some Degenerate Cases and Conditions 1 Some Degenerate Cases 2 Condition for the Moving Surface to be Planar 3 Condition for the Moving Surface to be Curved 4 Solution for Orientation and Motion in Terms of r and ΘC Solving the Image Flow Equations: Planar Surfaces 1 Bounds from First Order Flow Parameters 2 A Method for Determining the Structure and Motion Parameters 3 Solving for Θ and r 4 Summary of the Solution Method 5 Interpretation of the Bounds on Vz and ΩzD Nature of Solutions for Planar Surfaces 1 Relation Between the Two InterpretationsE Resolving the Ambiguity of Interpretation: Planar Surfaces 1 Evolution of Planar Surface Solutions 2 Uniqueness of Interpretation: Two Flow FieldsF Solving the Image Flow Equations: Curved Surfaces 1 Solution for Curvatures 2 Constraints on r and Θ 113 3 Summary of Computational AlgorithmG Conditions for Multiple Interpretations: Curved Surfaces 1 Solutions for Θ 2 Multiple Interpretations Due o Multiple Solutions for Θ 2.1 A Necessary Condition for Multiple Solutions to Θ 2.2 Triple Solution Theorem 2.3 Double Solution Theorem 1 2.4 The Four Solution Theorem 3 Multiple Interpretations Due to Multiple Solutions for r 3.1 Double Solution Theorem 2 4 Condition for Uniqueness of InterpretationH Solving for Θ When the Direction of V ChangesI Solving for r and Θ When the Camera Tracks a PointJ Surface Deformation Parameters 1 Interpretation of the Velocity Gradient Tensor 2 Interpretation of the Motion and Deformation Parameters aijBibliographyIndex