Bézier and Splines in Image Processing and Machine Vision (eBook)

Part I Early Background.- 1 Bernstein Polynomial and Bézier-Bernstein Spline.- 1.1 Introduction.- 1.2 Significance of Bernstein Polynomial in Splines.- 1.3 Bernstein Polynomial.- 1.3.1 Determination of the Order of the Polynomial.- 1.4 Use in Computer Graphics and Image Data Approximation.- 1.4.1 Bézier-Bernstein Curves.- 1.4.2 Bézier-Bernstein Surfaces.- 1.4.3 Curve and Surface Design.- 1.4.4 Approximation of Binary Images.- 1.5 Key Pixels and Contour Approximation.- 1.5.1 Key Pixels.- 1.5.2 Detection of Inflexion Points.- 1.6 Regeneration Technique.- 1.6.1 Method 1.- 1.6.2 Method 2.- 1.6. 3 Recursive Computation Algorithm.- 1.6.4 Implementation Strategies.- 1.7 Approximation Capability and Effectiveness.- 1.8 Concluding Remarks.- 2 Image Segmentation.- 2.1 Introduction.- 2.2 Two Different Concepts of Segmentation.- 2.2.1 Contour Based Segmentation.- 2.2.2 Region Based Segmentation.- 2.3 Segmentation for Compression.- 2.4 Extraction of Compact Homogeneous Regions.- 2.4.1 Partition/ Decomposition Principle for Gray Images.- 2.4.2 Approximation Problem.- 2.4.3 Polynomial Order Determination.- 2.4.4 Algorithms.- 2.4.5 Merging of Small Regions.- 2. 5 Evaluation of Segmentation.- 2.6 Comparison with Multilevel Thresholding Algorithms.- 2.6.1 Results and Discussion.- 2.7 Some Justifications for Image Data Compression.- 2.8 Concluding Remarks.- 3 1-d B-B Spline Polynomial and Hilbert Scan for Graylevel Image Coding.- 3.1 Introduction.- 3.2 Hilbert Scanned Image.- 3.2.1 Construction of Hilbert Curve.- 3.3 Shortcomings of Bernstein Polynomial and Error of Approximation.- 3.4 Approximation Technique.- 3.4.1 Bézier-Bernstein (B-B) Polynomial.- 3.4.2 Algorithm 1: Approximation Criteria of f(t).- 3.4.3 Implementation Strategy.- 3.4. 4 Algorithm 2.- 3.5 Image Data Compression.- 3.5.1 Discrimination Features of the Algorithms.- 3.6 Regeneration.- 3. 7 Results and Discussion.- 3. 8 Concluding Remarks.- 4 Image Compression.- 4.1 Introduction.- 4.2 SLIC: Sub-image Based Lossy Image Compression.- 4.2.1 Approximation and Choice of Weights.-4.2.2 Texture Coding.- 4.2.3 Contour Coding.- 4.3 Quantitative Assessment for Reconstructed Images.- 4.4 Results and Discussion.- 4.4.1 Result of SLIC Algorithm for 64 x 64 Images.- 4.4.2 Results of SLIC Algorithm for 256 x 256 Images.- 4.4.3 Effects of the Increase of Spatial Resolution on Compression and Quality.- 4.5 Concluding Remarks.- Part II Intermediate Steps.- 5 B-Splines and its Applications.- 5.1 Introduction.- 5.2 B-Spline Function.-5.2.1 B-spline Knot for Uniform, Open Uniform and Nonuniform basis.- 5.3 Computation of B-Spline Basis Functions.- 5.3.1 Computaion of Uniform Periodic B-spline Basis.- 5.4 B-Spline Curves on Unit Interval.- 5.4.1 Properties of B-spline Curves.- 5.4.2 Effect of Multiplicity.- 5.4.3 End Condition.-5.5 Rational B-Spline Curve.- 5.5.1 Homogeneous Co-ordinates.-5.5.2 Essentials of Rational B-spline Curves.- 5.6 B-Spline Surface.- 5.7 Application.- 5.7.1 Differential invariants of Image Velocity Fields.-5.7.2 3D Shape and Viewer Ego-motion.- 5.7.3 Geometric Significance.- 5.7.4 Constraints.-5.7.5 Extraction of Differential Invariants.-5.8 Recovery of Time to Contact and Surface Orientation.- 5.8.1 Braking and Object Manipulation.- 5.9 Concluding Remarks.- 6 Beta-Splines: A Flexible Model.- 6.1 Introduction.- 6.2 Beta-Spline Curve.- 6.3 Design Criteria for a Curve.-6.3.1 Shape Parameters.-6.3.2 End Conditions of Beta spline Curves.- 6.4 Beta-Spline Surface.-6.5 Possible Applications in Vision.- 6.6 Concluding Remarks.-Part III Advanced Methodologies.- 7 Discrete Spline and Vision.- 7.1 Introduction.- 7.2 Discrete Splines.- 7.2.1 Relation between ai,k and Bi, ,k>2.- 7.2.2 Some Properties of ai,k (j).- 7.2.3 Algorithms.- 7.3 Subdivision of Control Polygon.- 7.4 Smoothing Discrete Spline and Vision.- 7.5. Occluding Boundaries and Shape from Shading.- 7.5.1 Image Irradiance Equation.- 7.5.2 Method Based on Regularization.- 7.5.3 Discrete Smoothing Splines.- 7.5.4 Necessary Condition and the System of Equations.-7.5.5 Some Important Points about Algorithm.-7.6 A Provably Convergent Iterative Algorithm.- 7.6.1 Convergence.-7.7 Concluding Remarks.-8 Spline Wavelets: Construction, Implication and Uses.- 8.1 Introduction.-8.2 Cardinal Splines.-8.2.1 Cardinal B-spline Basis and Riesz Basis.- 8.2.2 Scaling and Cardinal B-spline Functions.- 8.3 Wavelets.- 8.3.1 Continuous Wavelet Transform.-8.4 A Glimpse of Continuous Wavelets.- 8.4.1 Basic Wavelets.-8.5 Multiresolution Analysis and Wavelet bases.- 8.6 Spline Approximations.-8.6.1 Battle-Lemarie wavelets.-8.7 Biorthogonal Spline Wavelets.-8.8 Concluding Remarks.-9 Snakes and Active Contours.- 9.1 Introduction.- 9.1.1 Splines and Energy Minimisation Techniques.-9.2 Classical Snakes.-9.3 Energy Functional.-9.4 Minimizing the Snake Energy Using the Calculus of Variations.-9.5 Minimising the Snake Energy Using Dynamic Programming.-9.6 Problems and Pitfalls.-9.8 Concluding Remarks.- 10 Globally Optimal Energy Minimisation Techinques.-10.1 Introduction and Time-Line.-10.2 Cell Image Segmentation using Dynamic Programming.-10.3 Globally Optimal Geodesic Active Contours (GOGAC).- 10.3.1 Fast Marching Algorithm.- 10.4 Globally Minimal Surfaces (GMS).- 10.4.1 Minimum Cuts and Maximum Flows.-10.4.2 Development of the GMS Algorithm.- 10.4.3 Applications of the GMS Algorithm.- 11 Acknowledgements.- References.- Index.-