### Synopses & Reviews

Digital image processing and machine vision have grown considerably during the last few decades. Of the various techniques, developed so far splines play a positive and significant role in many of them. Strong mathematical theory and ease of implementations is one of the keys of their success in many research issues. This book deals with various image processing and machine vision problems efficiently with splines and includes: • the significance of Bernstein Polynomial in splines • effectiveness of Hilbert scan for digital images • detailed coverage of Beta-splines, which are relatively new, for possible future applications • discrete smoothing splines and their strength in application • snakes and active contour models and their uses • the significance of globally optimal contours and surfaces Finally the book covers wavelet splines which are efficient and effective in different image applications. Dr Biswas is a system analyst at the Indian Statistical Institute, Calcutta where he teaches Machine Vision in M Tech (Computer Science). His research interests include image processing, computer vision, computer graphics, pattern recognition, neural networks and wavelet image-data analysis. Professor 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. His research interests are currently focussed on optimal image segmentation, real-time video analysis and face recognition.

#### Synopsis

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.

#### Synopsis

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.

### Table of Contents

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.-