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2014 | Book

Introduction Read chapter Read first chapter

Geometrical Multiresolution Adaptive Transforms

Theory and Applications

Author: Agnieszka Lisowska

Publisher: Springer International Publishing

Book Series : Studies in Computational Intelligence

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

Modern image processing techniques are based on multiresolution geometrical methods of image representation. These methods are efficient in sparse approximation of digital images. There is a wide family of functions called simply ‘X-lets’, and these methods can be divided into two groups: the adaptive and the nonadaptive. This book is devoted to the adaptive methods of image approximation, especially to multismoothlets.

Besides multismoothlets, several other new ideas are also covered. Current literature considers the black and white images with smooth horizon function as the model for sparse approximation but here, the class of blurred multihorizon is introduced, which is then used in the approximation of images with multiedges. Additionally, the semi-anisotropic model of multiedge representation, the introduction of the shift invariant multismoothlet transform and sliding multismoothlets are also covered.

Geometrical Multiresolution Adaptive Transforms should be accessible to both mathematicians and computer scientists. It is suitable as a professional reference for students, researchers and engineers, containing many open problems and will be an excellent starting point for those who are beginning new research in the area or who want to use geometrical multiresolution adaptive methods in image processing, analysis or compression.

Table of Contents

Frontmatter
Chapter 1. Introduction
Abstract
In this chapter, the motivation of this book was presented based on the human visual system. Then, the state-of-the-art review was given of the geometrical multiresolution methods of image approximation together with the contribution of this book. The chapter ends with the outline of this book.
Agnieszka Lisowska

Multismoothlet Transform

Frontmatter
Chapter 2. Smoothlets
Abstract
In this chapter the family of functions, called smoothlets, was presented. A smoothlet is defined as a generalization of a wedgelet and a second order wedgelet. It is based on any curve beamlet, named as a curvilinear beamlet. Smoothlets, unlike the other adaptive functions, are continuous functions. Thanks to that they can adapt to edges of different blur. In more details, the smoothlet can adapt to location, scale, orientation, curvature and blur. Additionally, a sliding smoothlet was introduced. It is the smoothlet with location and size defined freely within an image. The Rate-Distortion dependency and the \(\mathcal {M}\)-term approximation of smoothlets were also discussed.
Agnieszka Lisowska
Chapter 3. Multismoothlets
Abstract
In this chapter, the theory of multismoothlets was introduced. A multismoothlet is defined as a vector of smoothlets. Such a vector can adapt efficiently to multiple edges. So, the multismoothlet can adapt to edges of different multiplicity, location, scale, orientation, curvature and blur. Additionally, a notion of sliding multismoothlet was introduced. It is the multismoothlet with location and size defined freely within an image. Based on that, the shift invariant multismoothlet transform was proposed as well. The Rate-Distortion dependency and the \(\mathcal {M}\)-term approximation of multismoothlets were also discussed.
Agnieszka Lisowska
Chapter 4. Moments-Based Multismoothlet Transform
Abstract
In this chapter, the moments-based multismoothlet transform was proposed. It is based on Custom-Built moments used to compute multiedge parameters. The transform was presented in the consecutive steps, starting from a linear beamlet computation. Further, smoothlet parameters are computed. And finally, multismoothlet parameters are determined. At the end of this chapter, the computational complexity of the presented transform was discussed followed by some numerical results.
Agnieszka Lisowska

Applications

Frontmatter
Chapter 5. Image Compression
Abstract
In this chapter, the compression methods of binary and grayscale still images were presented. They are based on curvilinear beamlets and smoothlets, respectively. Both methods are based on quadtree decomposition of images. Each description of the compression method was followed by the results of numerical experiments. These results were further compared to the known state-of-the-art methods.
Agnieszka Lisowska
Chapter 6. Image Denoising
Abstract
In this chapter, the image denoising algorithm based on the multismoothlet transform was presented. The algorithm works in the way that image representations are computed for different values of the penalization factor and the optimal approximation is taken as the result. The algorithm description was followed by the results of numerical experiments. These results were further compared to the known state-of-the-art methods. The proposed algorithm assures the best denoising results in the most cases.
Agnieszka Lisowska
Chapter 7. Edge Detection
Abstract
In this chapter, the edge detection methods based on multismoothlets were proposed. The first one is based on the multismoothlet transform. The second one is based on sliding multismoothlets. Both methods were compared to the state-of-the-art methods. As follows from the performed experiments, the method based on sliding multismoothlets leads to the best results of edge detection.
Agnieszka Lisowska
Chapter 8. Summary
Abstract
In this chapter, the concluding remarks and the further directions in the area of multismoothlets, which were introduced in this book, were presented. In this book, the theory of multismoothlets was presented. This theory is the generalization of the concepts of geometrical multiresolution adaptive methods of image approximation proposed so far. Such generalization leads to new possibilities and applications to image processing and analysis. There is still plenty of work that can be done in this field. Some open problems are described in this section.
Agnieszka Lisowska
Backmatter
Metadata
Title
Geometrical Multiresolution Adaptive Transforms
Author
Agnieszka Lisowska
Copyright Year
2014
Electronic ISBN
978-3-319-05011-9
Print ISBN
978-3-319-05010-2
DOI
https://doi.org/10.1007/978-3-319-05011-9

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