Active Contours

1 Introduction.- 1.1 Organisation of the book.- 1.2 Applications.- 2 Active shape models.- 2.1 Snakes.- 2.2 Deformable templates.- 2.3 Dynamic contours.- I Geometrical Fundamentals.- 3 Spline curves.- 3.1 B-spline functions.- 3.2 Finite bases.- 3.3 Multiple knots.- 3.4 Norm and inner product for spline functions.- 3.5 B-spline parametric curves.- 3.6 Curves with vertices.- 3.7 Control vector.- 3.8 Norm for curves.- 3.9 Areas and moments.- 4 Shape-space models.- 4.1 Representing transformations in shape-space.- 4.2 The space of Euclidean similarities.- 4.3 Planar affine shape-space.- 4.4 Norms and moments in a shape-space.- 4.5 Perspective and weak perspective.- 4.6 Three-dimensional affine shape-space.- 4.7 Key-frames.- 4.8 Articulated motion.- 5 Image processing techniques for feature location.- 5.1 Linear scanning.- 5.2 Image filtering.- 5.3 Using colour.- 5.4 Correlation matching.- 5.5 Background subtraction.- 6 Fitting spline templates.- 6.1 Regularised matching.- 6.2 Normal displacement in curve fitting.- 6.3 Recursive solution of curve-fitting problems.- 6.4 Examples.- 7 Pose recovery.- 7.1 Calculating the pose of a planar object.- 7.2 Pose recovery for three-dimensional objects.- 7.3 Separation of rigid and non-rigid motion.- II Probabilistic Modelling.- 8 Probabilistic models of shape.- 8.1 Probability distributions over curves.- 8.2 Posterior distribution.- 8.3 Probabilistic modelling of image features.- 8.4 Validation gate.- 8.5 Learning the prior.- 8.6 Principal Components Analysis (PCA).- 9 Dynamical models.- 9.1 Some simple dynamical prior distributions.- 9.2 First-order Auto-regressive processes.- 9.3 Limitations of first-order dynamical models.- 9.4 Second-order dynamical models.- 9.5 Second-order AR processes in shape-space.- 9.6 Setting dynamical parameters.- 10 Dynamic contour tracking.- 10.1 Temporal fusion by Kaiman filter.- 10.2 Tracking performance.- 10.3 Choosing dynamical parameters.- 10.4 Case study.- 11 Learning motion.- 11.1 Learning one-dimensional dynamics.- 11.2 Learning AR process dynamics in shape-space.- 11.3 Dynamical modes.- 11.4 Performance of trained trackers.- 12 Non-Gaussian models and random sampling algorithms.- 12.1 Factored sampling.- 12.2 The CONDENSATION algorithm.- 12.3 An observation model.- 12.4 Applications of the CONDENSATION algorithm.- A Mathematical background.- A.1 Vectors and matrices.- A.2 B-spline basis functions.- A.3 Probability.- B Stochastic dynamical systems.- B.1 Continuous-time first-order dynamics.- B.2 Second-order dynamics in continuous time.- B.3 Accuracy of learning.- C Farther shape-space models.- C.1 Recursive synthesis of shape-spaces.- Glossary of notation.- Author Index.