Archive

Posts Tagged ‘denoising’

Buy my book!

July 20, 2012 Leave a comment

Well, ok, it is not my book technically, but I am one of the authors of one of the chapters. And no, as far as I know, I don’t get a dime of the sales in concept of copyright or anything else.

As the title suggests (Modeling Nanoscale Imaging in Electron Microscopy), this book presents some recent advances that have been made using mathematical methods to resolve problems in electron microscopy. With improvements in hardware-based aberration software significantly expanding the nanoscale imaging capabilities of scanning transmission electron microscopes (STEM), these mathematical models can replace some labor intensive procedures used to operate and maintain STEMs. This book, the first in its field since 1998, covers relevant concepts such as super-resolution techniques (that’s my contribution!), special de-noising methods, application of mathematical/statistical learning theory, and compressed sensing.

We even got a nice review in Physics Today by Les Allen, no less!

Imaging with electrons, in particular scanning transmission electron microscopy (STEM), is now in widespread use in the physical and biological sciences. And its importance will only grow as nanotechnology and nano-Biology continue to flourish. Many applications of electron microscopy are testing the limits of current imaging capabilities and highlight the need for further technological improvements. For example, high throughput in the combinatorial chemical synthesis of catalysts demands automated imaging. The handling of noisy data also calls for new approaches, particularly because low electron doses are used for sensitive samples such as biological and organic specimens.

Modeling Nanoscale Imaging in Electron Microscopy addresses all those issues and more. Edited by Thomas Vogt and Peter Binev at the University of South Carolina (USC) and Wolfgang Dahmen at RWTH Aachen University in Germany, the book came out of a series of workshops organized by the Interdisciplinary Mathematics Institute and the NanoCenter at USC. Those sessions took the unusual but innovative approach of bringing together electron microscopists, engineers, physicists, mathematicians, and even a philosopher to discuss new strategies for image analysis in electron microscopy.

In six chapters, the editors tackle the ambitious challenge of bridging the gap between high-level applied mathematics and experimental electron microscopy. They have met the challenge admirably. I believe that high-resolution electron microscopy is at a point where it will benefit considerably from an influx of new mathematical approaches, daunting as they may seem; in that regard Modeling Nanoscale Imaging in Electron Microscopy is a major step forward. Some sections present a level of mathematical sophistication seldom encountered in the experimentally focused electron-microscopy literature.
The first chapter, by philosopher of science Michael Dickson, looks at the big picture by raising the question of how we perceive nano-structures and suggesting that a Kantian approach would be fruitful. The book then moves into a review of the application of STEM to nanoscale systems, by Nigel Browning, a leading experimentalist in the field, and other well-known experts. Using case studies, the authors show how beam-sensitive samples can be studied with high spatial resolution, provided one controls the beam dose and establishes the experimental parameters that allow for the optimum dose.

The third chapter, written by image-processing experts Sarah Haigh and Angus Kirkland, addresses the reconstruction, from atomic-resolution images, of the wave at the exit surface of a specimen. The exit surface wave is a fundamental quantity containing not only amplitude (image) information but also phase information that is often intimately related to the atomic-level structure of the specimen. The next two chapters, by Binev and other experts, are based on work carried out using the experimental and computational resources available at USC. Examples in chapter four address the mathematical foundations of compressed sensing as applied to electron microscopy, and in particular high-angle annular dark-field STEM. That emerging approach uses randomness to extract the essential content from low-information signals. Chapter five eloquently discusses the efficacy of analyzing several low-dose images with specially adapted digital-image-processing techniques that allow one to keep the cumulative electron dose low and still achieve acceptable resolution.

The book concludes with a wide-ranging discussion by mathematicians Amit Singer and Yoel Shkolnisky on the reconstruction of a three-dimensional object via projected data taken at random and initially unknown object orientations. The discussion is an extension of the authors’ globally consistent angular reconstitution approach for recovering the structure of a macromolecule using cryo-electron microscopy. That work is also applicable to the new generation of x-ray free-electron lasers, which have similar prospective applications, and illustrates nicely the importance of applied mathematics in the physical sciences.

Modeling Nanoscale Imaging in Electron Microscopy will be an important resource for graduate students and researchers in the area of high-resolution electron microscopy.

(Les J. Allen, Physics Today, Vol. 65 (5), May, 2012)

Table of contents Preface Sample chapter
Categories: Approximation Theory, Books, Geometry, Imaging, Linear Algebra, Scientific Computing, Statistics Tags: , , , , , , , , , , , , , , , , , , , , , , , ,

The Nonlocal-means Algorithm

April 28, 2010 Leave a comment

The nonlocal-means algorithm [Buades, Coll, Morel] was designed to perform noise reduction on digital images, while preserving the main geometrical configurations, as well as finer structures, details and texture. The algorithm is consistent under the condition that one can find many samples of every image detail within the same image.

Barbara Noise added, std=30 Denoised image, h=93

The algorithm has the following closed form: Given a finite grid \Lambda \subset \mathbb{Z}^2 of the form \Lambda = \Omega \cap \mathbb{Z}^2 for some compact set \Omega \subset \mathbb{R}^2, a signal f \in L_2(\Lambda,\mathbb{R}^+), and a family of windows \{ \mathcal{R}_k \}_{k \in \Lambda} satisfying the conditions

  1. k \in \mathcal{R}_k for all k \in \Lambda.
  2. If j \in \mathcal{R}_k, then k \in \mathcal{R}_j,

the nonlocal-means operator \text{NL}_h\colon \ell_2(\Lambda,\mathbb{R}) \to \ell_2(\Lambda,\mathbb{R}) with filtering parameter h>0, is defined by

\displaystyle{\text{NL}_h f(k) = \sum_{j \in \Lambda} \omega_h(j,k) f(j)},

where the weights \{ \omega_h(j,k) \}_{j,k \in \Lambda} are defined by

\displaystyle{ \omega_h(j,k) = \frac{ \exp \bigg( -\frac{\left\lVert f(\mathcal{R}_j) - f(\mathcal{R}_k) \right\rVert_{2,a}^2}{h^2} \bigg) }{ \sum_{j \in \Lambda} \exp \bigg( - \frac{\left\lVert f(\mathcal{R}_j) - f(\mathcal{R}_k) \right\rVert_{2,a}^2}{h^2} \bigg)}. }

Here, f(\mathcal{R}) denotes a patch of the image f supported on the window \mathcal{R}.

Notice that the similarity check between patches is nothing but a simple Gaussian weighted Euclidean distance, which accounts for difference of grayscales alone. Efros and Leung prove that this distance is a reliable measure for the comparison of texture patches, and at the same time copes very well with additive white noise; in particular, if f and g are respectively the noisy and original images, and \sigma^2 is the noise variance, then the most similar patches in the noisy image are also expected to be the most similar in the original:

\mathbb{E} \left\lVert f(\mathcal{R}_j) - f(\mathcal{R}_k) \right\rVert_{2,a}^2 = \left\lVert g(\mathcal{R}_j) - g(\mathcal{R}_k) \right\rVert_{2,a}^2 + 2\sigma^2 .

Categories: Imaging, Scientific Computing Tags: , ,