| Early Background | |
| Bernstein Polynomial and Bezier-Bernstein Spline | p. 3 |
| Introduction | p. 3 |
| Significance of Bernstein Polynomial in Splines | p. 3 |
| Bernstein Polynomial | p. 5 |
| Determination of the Order of the Polynomial | p. 6 |
| Bezier-Bernstein Polynomial | p. 8 |
| Use in Computer Graphics and Image Data Approximation | p. 9 |
| Bezier-Bernstein Curves | p. 10 |
| Bezier-Bernstein Surfaces | p. 13 |
| Curve and Surface Design | p. 13 |
| Approximation of Binary Images | p. 14 |
| Key Pixels and Contour Approximation | p. 15 |
| Key Pixels | p. 15 |
| Detection of Inflection Points | p. 21 |
| Regeneration Technique | p. 23 |
| Method 1 | p. 23 |
| Method 2 | p. 24 |
| Recursive Computation Algorithm | p. 25 |
| Implementation Strategies | p. 26 |
| Approximation Capability and Effectiveness | p. 28 |
| Concluding Remarks | p. 31 |
| Image Segmentation | p. 33 |
| Introduction | p. 33 |
| Two Different Concepts of Segmentation | p. 33 |
| Contour-based Segmentation | p. 34 |
| Region-based Segmentation | p. 35 |
| Segmentation for Compression | p. 35 |
| Extraction of Compact Homogeneous Regions | p. 36 |
| Partition/Decomposition Principle for Gray Images | p. 41 |
| Approximation Problem | p. 43 |
| Polynomial Order Determination | p. 44 |
| Algorithms | p. 46 |
| Merging of Small Regions | p. 47 |
| Evaluation of Segmentation | p. 48 |
| Comparison with Multilevel Thresholding Algorithms | p. 50 |
| Results and Discussion | p. 51 |
| Some Justifications for Image Data Compression | p. 52 |
| Concluding Remarks | p. 55 |
| 1-d B-B Spline Polynomial and Hilbert Scan for Graylevel Image Coding | p. 57 |
| Introduction | p. 57 |
| Hilbert Scanned Image | p. 58 |
| Construction of Hilbert Curve | p. 58 |
| Shortcomings of Bernstein Polynomial and Error of Approximation | p. 63 |
| Approximation Technique | p. 64 |
| Bezier-Bernstein (B-B) Polynomial | p. 64 |
| Algorithm 1: Approximation Criteria of f(t) | p. 65 |
| Implementation Strategy | p. 67 |
| Algorithm 2 | p. 69 |
| Image Data Compression | p. 70 |
| Discriminating Features of the Algorithms | p. 71 |
| Regeneration | p. 72 |
| Results and Discussion | p. 73 |
| Concluding Remarks | p. 81 |
| Image Compression | p. 83 |
| Introduction | p. 83 |
| SLIC: Subimage-based Lossy Image Compression | p. 84 |
| Approximation and Choice of Weights | p. 88 |
| Texture Coding | p. 90 |
| Contour Coding | p. 91 |
| Quantitative Assessment for Reconstructed Images | p. 95 |
| Results and Discussion | p. 98 |
| Results of SLIC Algorithm for 64 X 64 Images | p. 99 |
| Results of SLIC Algorithm for 256 X 256 Images | p. 101 |
| Effect of the Increase of Spatial Resolution on Compression and Quality | p. 103 |
| Concluding Remarks | p. 106 |
| Intermediate Steps | |
| B-Splines and Its Applications | p. 109 |
| Introduction | p. 109 |
| B-Spline Function | p. 110 |
| B-Spline Knot Structure for Uniform, Open Uniform, and Nonuniform Basis | p. 110 |
| Computation of B-Spline Basis Functions | p. 112 |
| Computation of Uniform Periodic B-spline Basis | p. 113 |
| B-Spline Curves on Unit Interval | p. 114 |
| Properties of B-Spline Curves | p. 117 |
| Effect of Multiplicity | p. 117 |
| End Condition | p. 117 |
| Rational B-Spline Curve | p. 118 |
| Homogeneous Coordinates | p. 118 |
| Essentials of Rational B-Spline Curves | p. 120 |
| B-Spline Surface | p. 121 |
| Application | p. 121 |
| Differential Invariants of Image Velocity Fields | p. 121 |
| 3D Shape and Viewer Ego-motion | p. 123 |
| Geometric Significance | p. 124 |
| Constraints | p. 125 |
| Extraction of Differential Invariants | p. 127 |
| Recovery of Time to Contact and Surface Orientation | p. 129 |
| Braking and Object Manipulation | p. 129 |
| Concluding Remarks | p. 130 |
| Beta-Splines: A Flexible Model | p. 133 |
| Introduction | p. 133 |
| Beta-Spline Curve | p. 133 |
| Design Criteria for a Curve | p. 136 |
| Shape Parameters | p. 138 |
| End Conditions of Beta Spline Curves | p. 138 |
| Beta-Spline Surface | p. 141 |
| Possible Applications in Vision | p. 142 |
| Concluding Remarks | p. 142 |
| Advanced Methodologies | |
| Discrete Splines and Vision | p. 145 |
| Introduction | p. 145 |
| Discrete Splines | p. 145 |
| Relation Between [alpha subscript i,k] and B[subscript i,k], k > 2 | p. 148 |
| Some Properties of [alpha subscript i,k](j) | p. 151 |
| Algorithms | p. 152 |
| Subdivision of Control Polygon | p. 154 |
| Smoothing Discrete Splines and Vision | p. 155 |
| Occluding Boundaries and Shape from Shading | p. 155 |
| Image Irradiance Equation | p. 156 |
| Method Based on Regularization | p. 157 |
| Discrete Smoothing Splines | p. 157 |
| Necessary Condition and the System of Equations | p. 158 |
| Some Important Points About DSS | p. 159 |
| A Provably Convergent Iterative Algorithm | p. 159 |
| Convergence | p. 160 |
| Concluding Remarks | p. 161 |
| Spline Wavelets: Construction, Implication, and Uses | p. 163 |
| Introduction | p. 163 |
| Cardinal Splines | p. 164 |
| Cardinal B-Spline Basis and Riesz Basis | p. 167 |
| Scaling and Cardinal B-Spline Functions | p. 170 |
| Wavelets | p. 172 |
| Continuous Wavelet Transform | p. 172 |
| Properties of Continuous Wavelet Transform | p. 173 |
| A Glimpse of Continuous Wavelets | p. 174 |
| Basic Wavelets | p. 174 |
| Multiresolution Analysis and Wavelet Bases | p. 176 |
| Spline Approximations | p. 179 |
| Battle-Lemarie Wavelets | p. 181 |
| Biorthogonal Spline Wavelets | p. 182 |
| Concluding Remarks | p. 184 |
| Snakes and Active Contours | p. 187 |
| Introduction | p. 187 |
| Splines and Energy Minimization Techniques | p. 187 |
| Classical Snakes | p. 189 |
| Energy Functional | p. 190 |
| Minimizing the Snake Energy Using the Calculus of Variations | p. 194 |
| Minimizing the Snake Energy Using Dynamic Programming | p. 196 |
| Problems and Pitfalls | p. 207 |
| Connected Snakes for Advanced Segmentation | p. 207 |
| Conclusions | p. 211 |
| Globally Optimal Energy Minimization Techniques | p. 213 |
| Introduction and Timeline | p. 213 |
| Cell Image Segmentation Using Dynamic Programming | p. 214 |
| Globally Optimal Geodesic Active Contours (GOGAC) | p. 219 |
| Fast Marching Algorithm | p. 221 |
| Globally Minimal Surfaces (GMS) | p. 224 |
| Minimum Cuts and Maximum Flows | p. 225 |
| Development of the GMS Algorithm | p. 227 |
| Applications of the GMS Algorithm | p. 229 |
| Conclusions | p. 233 |
| References | p. 235 |
| Index | p. 245 |
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