Bezier and splines in image processing and machine vision

Digital image processing and machine vision have grown considerably during the last few decades. Of the various techniques, developed so far, splines play a significant role in many of them. This book deals with various image processing and machine vision problems efficiently with splines and includes: the significance of Bernstein Polynomial in splines, detailed coverage of Beta-splines applications which are relatively new, Splines in motion tracking, various deformative models and their uses.

Finally the book covers wavelet splines which are efficient and effective in different image applications.

Author Notes

Dr Sambhunath Biswas is a system analyst at the Indian Statistical Institute, Calcutta where he teaches Machine Vision in M Tech (Computer Science)
Professor Brian Lovell is a Research Leader in National ICT Australia and Research Director of the Intelligent Real-Time Imaging and Sensing Research Group at the University of Queensland

Part I Early Background
1 Bernstein Polynomial and Bezier-Bernstein Spline	p. 3
1.1 Introduction	p. 3
1.2 Significance of Bernstein Polynomial in Splines	p. 3
1.3 Bernstein Polynomial	p. 5
1.3.1 Determination of the Order of the Polynomial	p. 6
1.3.2 Bezier-Bernstein Polynomial	p. 8
1.4 Use in Computer Graphics and Image Data Approximation	p. 9
1.4.1 Bezier-Bernstein Curves	p. 10
1.4.2 Bezier-Bernstein Surfaces	p. 13
1.4.3 Curve and Surface Design	p. 13
1.4.4 Approximation of Binary Images	p. 14
1.5 Key Pixels and Contour Approximation	p. 15
1.5.1 Key Pixels	p. 15
1.5.2 Detection of Inflection Points	p. 21
1.6 Regeneration Technique	p. 23
1.6.1 Method 1	p. 23
1.6.2 Method 2	p. 24
1.6.3 Recursive Computation Algorithm	p. 25
1.6.4 Implementation Strategies	p. 26
1.7 Approximation Capability and Effectiveness	p. 28
1.8 Concluding Remarks	p. 31
2 Image Segmentation	p. 33
2.1 Introduction	p. 33
2.2 Two Different Concepts of Segmentation	p. 33
2.2.1 Contour-based Segmentation	p. 34
2.2.2 Region-based Segmentation	p. 35
2.3 Segmentation for Compression	p. 35
2.4 Extraction of Compact Homogeneous Regions	p. 36
2.4.1 Partition/Decomposition Principle for Gray Images	p. 41
2.4.2 Approximation Problem	p. 43
2.4.3 Polynomial Order Determination	p. 44
2.4.4 Algorithms	p. 46
2.4.5 Merging of Small Regions	p. 47
2.5 Evaluation of Segmentation	p. 48
2.6 Comparison with Multilevel Thresholding Algorithms	p. 50
2.6.1 Results and Discussion	p. 51
2.7 Some Justifications for Image Data Compression	p. 52
2.8 Concluding Remarks	p. 55
3 1-d B-B Spline Polynomial and Hilbert Scan for Graylevel Image Coding	p. 57
3.1 Introduction	p. 57
3.2 Hilbert Scanned Image	p. 58
3.2.1 Construction of Hilbert Curve	p. 58
3.3 Shortcomings of Bernstein Polynomial and Error of Approximation	p. 63
3.4 Approximation Technique	p. 64
3.4.1 Bezier-Bernstein (B-B) Polynomial	p. 64
3.4.2 Algorithm 1: Approximation Criteria of f(t)	p. 65
3.4.3 Implementation Strategy	p. 67
3.4.4 Algorithm 2	p. 69
3.5 Image Data Compression	p. 70
3.5.1 Discriminating Features of the Algorithms	p. 71
3.6 Regeneration	p. 72
3.7 Results and Discussion	p. 73
3.8 Concluding Remarks	p. 81
4 Image Compression	p. 83
4.1 Introduction	p. 83
4.2 SLIC: Subimage-based Lossy Image Compression	p. 84
4.2.1 Approximation and Choice of Weights	p. 88
4.2.2 Texture Coding	p. 90
4.2.3 Contour Coding	p. 91
4.3 Quantitative Assessment for Reconstructed Images	p. 95
4.4 Results and Discussion	p. 98
4.4.1 Results of SLIC Algorithm for 64 X 64 Images	p. 99
4.4.2 Results of SLIC Algorithm for 256 X 256 Images	p. 101
4.4.3 Effect of the Increase of Spatial Resolution on Compression and Quality	p. 103
4.5 Concluding Remarks	p. 106
Part II Intermediate Steps
5 B-Splines and Its Applications	p. 109
5.1 Introduction	p. 109
5.2 B-Spline Function	p. 110
5.2.1 B-Spline Knot Structure for Uniform, Open Uniform, and Nonuniform Basis	p. 110
5.3 Computation of B-Spline Basis Functions	p. 112
5.3.1 Computation of Uniform Periodic B-spline Basis	p. 113
5.4 B-Spline Curves on Unit Interval	p. 114
5.4.1 Properties of B-Spline Curves	p. 117
5.4.2 Effect of Multiplicity	p. 117
5.4.3 End Condition	p. 117
5.5 Rational B-Spline Curve	p. 118
5.5.1 Homogeneous Coordinates	p. 118
5.5.2 Essentials of Rational B-Spline Curves	p. 120
5.6 B-Spline Surface	p. 121
5.7 Application	p. 121
5.7.1 Differential Invariants of Image Velocity Fields	p. 121
5.7.2 3D Shape and Viewer Ego-motion	p. 123
5.7.3 Geometric Significance	p. 124
5.7.4 Constraints	p. 125
5.7.5 Extraction of Differential Invariants	p. 127
5.8 Recovery of Time to Contact and Surface Orientation	p. 129
5.8.1 Braking and Object Manipulation	p. 129
5.9 Concluding Remarks	p. 130
6 Beta-Splines: A Flexible Model	p. 133
6.1 Introduction	p. 133
6.2 Beta-Spline Curve	p. 133
6.3 Design Criteria for a Curve	p. 136
6.3.1 Shape Parameters	p. 138
6.3.2 End Conditions of Beta Spline Curves	p. 138
6.4 Beta-Spline Surface	p. 141
6.5 Possible Applications in Vision	p. 142
6.6 Concluding Remarks	p. 142
Part III Advanced Methodologies
7 Discrete Splines and Vision	p. 145
7.1 Introduction	p. 145
7.2 Discrete Splines	p. 145
7.2.1 Relation Between [alpha subscript i,k] and B[subscript i,k], k > 2	p. 148
7.2.2 Some Properties of [alpha subscript i,k](j)	p. 151
7.2.3 Algorithms	p. 152
7.3 Subdivision of Control Polygon	p. 154
7.4 Smoothing Discrete Splines and Vision	p. 155
7.5 Occluding Boundaries and Shape from Shading	p. 155
7.5.1 Image Irradiance Equation	p. 156
7.5.2 Method Based on Regularization	p. 157
7.5.3 Discrete Smoothing Splines	p. 157
7.5.4 Necessary Condition and the System of Equations	p. 158
7.5.5 Some Important Points About DSS	p. 159
7.6 A Provably Convergent Iterative Algorithm	p. 159
7.6.1 Convergence	p. 160
7.7 Concluding Remarks	p. 161
8 Spline Wavelets: Construction, Implication, and Uses	p. 163
8.1 Introduction	p. 163
8.2 Cardinal Splines	p. 164
8.2.1 Cardinal B-Spline Basis and Riesz Basis	p. 167
8.2.2 Scaling and Cardinal B-Spline Functions	p. 170
8.3 Wavelets	p. 172
8.3.1 Continuous Wavelet Transform	p. 172
8.3.2 Properties of Continuous Wavelet Transform	p. 173
8.4 A Glimpse of Continuous Wavelets	p. 174
8.4.1 Basic Wavelets	p. 174
8.5 Multiresolution Analysis and Wavelet Bases	p. 176
8.6 Spline Approximations	p. 179
8.6.1 Battle-Lemarie Wavelets	p. 181
8.7 Biorthogonal Spline Wavelets	p. 182
8.8 Concluding Remarks	p. 184
9 Snakes and Active Contours	p. 187
9.1 Introduction	p. 187
9.1.1 Splines and Energy Minimization Techniques	p. 187
9.2 Classical Snakes	p. 189
9.3 Energy Functional	p. 190
9.4 Minimizing the Snake Energy Using the Calculus of Variations	p. 194
9.5 Minimizing the Snake Energy Using Dynamic Programming	p. 196
9.6 Problems and Pitfalls	p. 207
9.7 Connected Snakes for Advanced Segmentation	p. 207
9.8 Conclusions	p. 211
10 Globally Optimal Energy Minimization Techniques	p. 213
10.1 Introduction and Timeline	p. 213
10.2 Cell Image Segmentation Using Dynamic Programming	p. 214
10.3 Globally Optimal Geodesic Active Contours (GOGAC)	p. 219
10.3.1 Fast Marching Algorithm	p. 221
10.4 Globally Minimal Surfaces (GMS)	p. 224
10.4.1 Minimum Cuts and Maximum Flows	p. 225
10.4.2 Development of the GMS Algorithm	p. 227
10.4.3 Applications of the GMS Algorithm	p. 229
10.5 Conclusions	p. 233
References	p. 235
Index	p. 245

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Author Notes

Table of Contents