Advances in Imaging and Electron Physics (eBook)

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2011 | 1. Auflage
288 Seiten
Elsevier Science (Verlag)
978-0-08-056912-3 (ISBN)

Infos

Advances in Imaging and Electron Physics merges two long-running serials-Advances in Electronics and Electron Physics and Advances in Optical and Electron Microscopy. This series features extended articles on the physics of electron devices (especially semiconductor devices), particle optics at high and low energies, microlithography, image science and digital image processing, electromagnetic wave propagation, electron microscopy, and the computing methods used in all these domains.

Publication of this 150th volume is an event to be celebrated and, to mark the occasion, the editor has brought together leaders of some of the main themes of past and hopefully of future volumes: electron microscopy, since Ladislaus Marton was one of the pioneers, mathematical morphology, which has often appeared in this series and also fills a supplement, so often cited that it usually appears just as Academic Press, 1994 (H.J.A.M. Heijmans, Morphological Image Operators, Supplement 25, 1994) with no mention of the Advances, ptychography, a highly original approach to the phase problem, the latter also the subject of a much cited Supplement (W.O. Saxton, 'Computer Techniques for Image Processing in Electron Microscopy', Supplement 10, 1978), and wavelets, which have become a subject in their own right, not just a tool in image processing.

* Updated with contributions from leading international scholars and industry experts
* Discusses hot topic areas and presents current and future research trends
* Invaluable reference and guide for physicists, engineers and mathematicians

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Advances in Imaging and Electron Physics merges two long-running serials-Advances in Electronics and Electron Physics and Advances in Optical and Electron Microscopy. This series features extended articles on the physics of electron devices (especially semiconductor devices), particle optics at high and low energies, microlithography, image science and digital image processing, electromagnetic wave propagation, electron microscopy, and the computing methods used in all these domains. Publication of this 150th volume is an event to be celebrated and, to mark the occasion, the editor has brought together leaders of some of the main themes of past and hopefully of future volumes: electron microscopy, since Ladislaus Marton was one of the pioneers; mathematical morphology, which has often appeared in this series and also fills a supplement, so often cited that it usually appears just as "e;Academic Press, 1994 (H.J.A.M. Heijmans, Morphological Image Operators, Supplement 25, 1994) with no mention of the Advances; ptychography, a highly original approach to the phase problem, the latter also the subject of a much cited Supplement (W.O. Saxton, 'Computer Techniques for Image Processing in Electron Microscopy', Supplement 10, 1978); and wavelets, which have become a subject in their own right, not just a tool in image processing. - Updated with contributions from leading international scholars and industry experts- Discusses hot topic areas and presents current and future research trends- Invaluable reference and guide for physicists, engineers and mathematicians

Inhalt

Front cover 1
Advances in Imaging and Electron Physics 4
Copyright page 5
Contents 6
Contributors 8
Preface 10
Future Contributions 14
Chapter 1. On Some Iterative Concepts for Image Restoration 20
I. Introduction 21
II. Simultaneous Decomposition, Deblurring, and Denoising of Images by Means of Wavelets 22
III. Vector-Valued Regimes and Mixed Constraints 40
IV. Image Restoration with General Convex Constraints 49
V. Hybrid Wavelet-PDE Image Restoration Schemes 61
References 67
Chapter 2. Significant Advances in Scanning Electron Microscopes (1965-2007) 72
I. Introduction 72
II. Brief Review of SEM Operation 74
III. Contrast 77
IV. Improvements since 1965 80
V. Tabletop SEM 94
VI. Aberration Correction 94
VII. Digital Image Processing 98
VIII. Ultrahigh Vacuum 100
IX. Future Developments 100
References 102
Chapter 3. Ptychography and Related Diffractive Imaging Methods 106
I. Introduction 106
II. Ptychography in Context 107
III. Coordinates, Nomenclature, and Scattering Approximations 138
IV. The Variants: Data, Data Processing, and Experimental Results 157
V. Conclusions 197
References 199
Chapter 4. Advances in Mathematical Morphology: Segmentation 204
I. Introduction 204
II. Criteria, Partitions, and Segmentation 207
III. Connective Segmentation 210
IV. Examples of Connective Segmentations 215
V. Partial Connections and Mixed Segmentations 223
VI. Iterated Jumps and Color Image Segmentation 225
VII. Connected Operators 228
VIII. Hierarchies and Connected Operators 231
IX. Conclusion 234
References 234
Cumulative Author Index 240
Contents of Volumes 100–149 260
Subject Index 276

Textauszug

On Some Iterative Concepts for Image Restoration

Ingrid Daubechies1; Gerd Teschke2; Luminita Vese3    1 Princeton University, PACM, Washington Road, Princeton, NJ 08544-1000, USA
2 Konrad-Zuse Institute Berlin, Takustr. 7, D-14195 Berlin-Dahlem, Germany
3 Department of Mathematics, UCLA, 405 Hilgard Avenue, Los Angeles, CA 90095-1555, USA

Publisher Summary

This chapter discusses several iterative strategies for solving inverse problems in the context of signal and image processing. The chapter focuses on problems for which it is reasonable to assume that the solution has a sparse expansion with respect to a wavelet basis or frame. In each case, a variational formulation of the problem is presented, and an iteration scheme for which iterates approximate the solution is constructed. Surrogate functionals are applied; the corresponding strategy is shown to converge in norm and to regularize the problem. The chapter begins with the concrete problem of simultaneously denoising, decomposing, and deblurring a given image. The associated variational formulation of the problem contains terms that promote sparsity and smoothness. The process to transform problem is presented in the chapter, such as the basic method of Daubechies and all. In a second example, a natural extension to vector-valued inverse problems is discussed. Potential applications include seismic or astrophysical data decomposition/reconstruction and color image reconstruction. The illustration presented contains audio data coding. In the linear case, and under fairly general assumptions on the constraint, it is proved that weak convergence of the iterative scheme always holds. In certain cases (i.e, for special families of convex constraints), this weak convergence implies norm convergence. The presented technique covers a wide range of problems. The chapter also discusses image restoration problems in which Besov- or bounded variation (BV) constraints are involved. The chapter concludes with a sketch design of hybrid wavelet-partial differential equation (PDE) image restoration schemes (i.e, with variational problems that contain wavelet and BV constraints).

I INTRODUCTION

This chapter discusses several iterative strategies for solving inverse problems in the context of signal and image processing. We essentially focus on problems for which it is reasonable to assume that the solution has a sparse expansion with respect to a wavelet basis or frame. In each case, we consider a variational formulation of the problem and construct an iteration scheme for which the iterates approximate the solution. To this end, we apply surrogate functionals; the corresponding strategy was shown to converge in norm and to regularize the problem (see Daubechies et al., 2004). We discuss special cases and generalizations.

The surrogate functional method in its initial setup as described in Daubechies et al. (2004) amounts to a combination of Landweber's method and a shrinkage operation, applied in each iteration step. The shrinkage is due to the presence of the 1-penalization term in the functional. Recent developments in the field of signal and image processing have shown the importance of sparse representations for various tasks in inverse problems (such as compression, denoising, deblurring, decomposition, texture analysis); 1-constraints select for such sparsity. Here we limit ourselves to illustrating a small number of concrete inverse problems for which we show in detail the variational formulations and the resulting expressions for the iteration. For all cases we discuss convergence and give detailed numerical illustrations. In addition to these case studies, we also present strategies for more general constraints.

We begin with the concrete problem of simultaneously denoising, decomposing, and deblurring a given image. The associated variational formulation of the problem contains terms that promote sparsity and smoothness. We show how to transform the problem such that the basic method of Daubechies et al. (2004) applies. In a second example, we discuss a natural extension to vector-valued inverse problems. Potential applications include seismic or astrophysical data decomposition/reconstruction and color image reconstruction. The illustration presented here contains audio data coding. After these two case studies, we turn to more general formulations. We allow the constraint to be some other positive, homogeneous, and convex functional than the 1-norm. In the linear case, and under fairly general assumptions on the constraint, we prove that weak convergence of the iterative scheme always holds. In certain cases (i.e., for special families of convex constraints) this weak convergence implies norm convergence. The presented technique covers a wide range of problems. Here we discuss in greater detail image restoration problems in which Besov- or bounded variation (BV) constraints are involved. We close with sketching the design of hybrid wavelet-partial differential equation (PDE) image restoration schemes (i.e., with variational problems that contain wavelet and BV constraints).

II SIMULTANEOUS DECOMPOSITION, DEBLURRING, AND DENOISING OF IMAGES BY MEANS OF WAVELETS

This section is devoted to wavelet-based treatments of variational problems in the field of image processing. In particular, we follow approaches presented in Meyer (2002), Vese and Osher (2003, 2004), and Osher et al. (2003) and discuss a special class of variational functionals that induce a decomposition of images into oscillating and cartoon components and possibly an appropriate “noise” component. In the setting of Vese and Osher (2003) and Osher et al. (2003), the cartoon component of an image is modeled by a BV function; the corresponding incorporation of BV penalty terms in the variational functional leads to PDE schemes that are numerically intensive. By replacing the BV penalty term by a 11(L1) term (which amounts to a slightly stronger constraint on the minimizer) and writing the problem in a wavelet framework, elegant and numerically efficient schemes are obtained with results very similar to those obtained in Osher et al. (2003) and superior to those from Rudin et al. (1992). This approach allows incorporating bounded linear blur operators into the problem so that the minimization leads to a simultaneous decomposition, deblurring, and denoising.

A Wavelet-Based Reformulation of the Variational Problem

As mentioned previously, we focus on a special class of variational problems that induce a decomposition of images into “texture” and “cartoon” components. Ideally, the cartoon part is piecewise smooth with possibly abrupt edges and contours; the texture part fills in the smooth regions in the cartoon with typically oscillating features. Inspired by Meyer (2002), Vese and Osher (2003) and Osher et al. (2003) propose to model the cartoon component by the space BV; this induces a penalty term that allows edges and contours in the reconstructed cartoon images, leading to a numerically intensive PDE-based scheme.

Our main goal is to provide a computationally thriftier algorithm by using a wavelet-based scheme that solves not the same but a very similar variational problem, in which the BV constraint, which cannot easily be expressed in the wavelet domain, is replaced by a 11(L1)-term (i.e., a slightly stricter constraint, since 11(L1)⊂BV in two dimensions). Moreover, we can easily incorporate the action of linear bounded blur operators; we also show convergence of the proposed scheme.

To provide a brief description of the underlying variational problems, recall the methods proposed in Vese and Osher (2003) and Osher et al. (2003). They follow the idea of Meyer (2002), proposed as an improvement on the total variation framework of Rudin et al. (1992). In principle, the models can be understood as a decomposition of an image f into =u+v, where u represents the cartoon part and v the texture part. In the model by Vese and Osher (2003, 2004), the decomposition is induced by solving

u,g1,g2Gp(u,g1,g2),whereGp(u,g1,g2)=∫Ω|∇u|+λ‖f−(u+divg)‖L2(Ω)2+μ‖|g|‖Lp(Ω),

  (1)

with ∈L2(Ω), ⊂R2, and =divg=div(g1,g2). The first term is the total variation of u. If ∈L1 and ∇u| is a finite measure on Ω, then ∈BV(Ω). This space allows discontinuities; therefore edges and contours generally appear in u. The second term represents the restoration discrepancy; to penalize v, the third term approximates (by taking p finite) the norm of the space G of oscillating functions introduced by Meyer (with =∞), which is in some sense dual to (Ω). (For details, refer to Meyer, 2002.) Setting...

Produkt-Details

• Erscheint lt. Verlag: 6.9.2011
• Mitarbeit: Herausgeber (Serie): Peter W. Hawkes
• Sprache: englisch
• Themenwelt: Sachbuch/Ratgeber
• Themenwelt: Informatik ► Grafik / Design ► Digitale Bildverarbeitung
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Elektrodynamik
• Themenwelt: Technik ► Bauwesen
• Themenwelt: Technik ► Elektrotechnik / Energietechnik
• Themenwelt: Technik ► Maschinenbau
• ISBN-10: 0-08-056912-9 / 0080569129
• ISBN-13: 978-0-08-056912-3 / 9780080569123