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Bioimaging 
Biomedical Image Reconstruction 

M. Unser 


12th European Molecular Imaging Meeting, 57 April 2017, Cologne, Germany. 

A fundamental component of the imaging pipeline is the reconstruction algorithm. In this educational session, we review the physical and mathematical principles that underlie the design of such algorithms. We argue that the concepts are fairly universal and applicable to a majority of (bio)medical imaging modalities, including magnetic resonance imaging and fMRI, xray computer tomography, and positronemission tomography (PET). Interestingly, the paradigm remains valid for modern cellular/molecular imaging with confocal/superresolution fluorescence microscopy, which is highly relevant to molecular imaging as well. In fact, we believe that the huge potential for crossfertilization and mutual reenforcement between imaging modalities has not been fully exploited yet. The prerequisite to image reconstruction is an accurate physical description of the imageformation process: the socalled forward model, which is assumed to be linear. Numerically, this translates into the specification of a system matrix, while the reconstruction of images conceptually boils down to a stable inversion of this matrix. The difficulty is essentially twofold: (i) the system matrix is usually much too large to be stored/inverted directly, and (ii) the problem is inherently illposed due to the presence of noise and/or bad conditioning of the system. Our starting point is an overview of the modalities in relation to their forward model. We then discuss the classical linear reconstruction methods that typically involve some form of backpropagation (CT or PET) and/or the fast Fourier transform (in the case of MRI). We present stabilized variants of these methods that rely on (Tikhonov) regularization or the injection of prior statistical knowledge under the Gaussian hypothesis. Next, we review modern iterative schemes that can handle challenging acquisition setups such as parallel MRI, nonCartesian sampling grids, and/or missing views. In particular, we discuss sparsitypromoting methods that are supported by the theory of compressed sensing. We show how to implement such schemes efficiently using simple combinations of linear solvers and thresholding operations. The main advantage of these recent algorithms is that they improve the quality of the image reconstruction. Alternatively, they allow a substantial reduction of the radiation dose and/or acquisition time without noticeable degradation in quality. This behavior is illustrated practically. In the final part of the tutorial, we discuss the current challenges and directions of research in the field; in particular, the necessity of dealing with large data sets in multiple dimensions: 2D or 3D space combined with time (in the case of dynamic imaging) and/or multispectral/multimodal information.




Challenges and Opportunities in Biological Imaging 

M. Unser, Professor, Ecole Polytechnique Fédérale de Lausanne, Biomedical Imaging Group 


Plenary. IEEE International Conference on Image Processing (ICIP), 2730 September, 2015, Québec City, Canada. 

While the major achievements in medical imaging can be traced back to the end the 20th century, there are strong indicators that we have recently entered the golden age of cellular/biological imaging. The enabling modality is fluorescence microscopy which results from the combination of highly specific fluorescent probes (Nobel Prize 2008) and sophisticated optical instrumentation (Nobel Prize 2014). This has led to the emergence of modern microscopy centers that are providing biologists with unprecedented amounts of data in 3D + time. To address the computational aspects, two nascent fields have emerged in which image processing is expected to play a significant role. The first is "digital optics" where the idea is to combine optics with advanced signal processing in order to increase spatial resolution while reducing acquisition time. The second area is "bioimage informatics" which is concerned with the development of image analysis software to make microscopy more quantitative. The key issue here is reliable image segmentation as well as the ability to track structures of interest over time. We shall discuss specific examples and describe stateoftheart solutions for bioimage reconstruction and analysis. This will help us build a list of challenges and opportunities to guide further research in bioimaging. 



Sparse stochastic processes: A statistical framework for compressed sensing and biomedical image reconstruction 

M. Unser 


4 hours tutorial, Inverse Problems and Imaging Conference, Institut Henri Poincaré, Paris, April 711, 2014. 

We introduce an extended family of continuousdomain sparse processes that are specified by a generic (nonGaussian) innovation model or, equivalently, as solutions of linear stochastic differential equations driven by white Lévy noise. We present the functional tools for their characterization. We show that their transformdomain probability distributions are infinitely divisible, which induces two distinct types of behavior‐Gaussian vs. sparse‐at the exclusion of any other. This is the key to proving that the nonGaussian members of the family admit a sparse representation in a matched wavelet basis. Next, we apply our continuousdomain characterization of the signal to the discretization of illconditioned linear inverse problems where both the statistical and physical measurement models are projected onto a linear reconstruction space. This leads the derivation of a general class of maximum a posteriori (MAP) signal estimators. While the formulation is compatible with the standard methods of Tikhonov and l1type regularizations, which both appear as particular cases, it open the door to a much broader class of sparsitypromoting regularization schemes that are typically nonconvex. We illustrate the concept with the derivation of algorithms for the reconstruction of biomedical images (deconvolution microscopy, MRI, Xray tomography) from noisy and/or incomplete data. The proposed framework also suggests alternative Bayesian recovery procedures that minimize t he estimation error. Reference




The Colored Revolution of Bioimaging 

C. Vonesch, F. Aguet, J.L. Vonesch, M. Unser 

IEEE Signal Processing Magazine, vol. 23, no. 3, pp. 2031, May 2006. 

With the recent development of fluorescent probes and new highresolution microscopes, biological imaging has entered a new era and is presently having a profound impact on the way research is being conducted in the life sciences. Biologists have come to depend more and more on imaging. They can now visualize subcellular components and processes in vivo, both structurally and functionally. Observations can be made in two or three dimensions, at different wavelengths (spectroscopy), possibly with timelapse imaging to investigate cellular dynamics. The observation of many biological processes relies on the ability to identify and locate specific proteins within their cellular environment. Cells are mostly transparent in their natural state and the immense number of molecules that constitute them are optically indistinguishable from one another. This makes the identification of a particular protein a very complex task—akin to finding a needle in a haystack. 

Wavelets, sparsity and biomedical image reconstruction 

M. Unser 

Imaging Seminar, University of Bern, Inselspital November 13, 2012. 

Our purpose in this talk is to advocate the use of wavelets for advanced biomedical imaging. We start with a short tutorial on wavelet bases, emphasizing the fact that they provide a sparse representation of images. We then discuss a simple, but remarkably effective, imagedenoising procedure that essentially amounts to discarding small wavelet coefficients (softthresholding). The crucial observation is that this type of “sparsitypromoting” algorithm is the solution of a l1norm minimization problem. The underlying principle of wavelet regularization is a powerful concept that has been used advantageously for compressed sensing and for reconstructing images from limited and/or noisy measurements. We illustrate the point by presenting waveletbased algorithms for 3D deconvolution microscopy, and MRI reconstruction (with multiple coils and/or nonCartesian kspace sampling). These methods were developed at the EPFL in collaboration with imaging scientists and are, for the most part, providing stateoftheart performance. 



Recent Advances in Biomedical Imaging and Signal Analysis 

M. Unser 

Proceedings of the Eighteenth European Signal Processing Conference (EUSIPCO'10), Ålborg, Denmark, August 2327, 2010, EURASIP Fellow inaugural lecture. 

Wavelets have the remarkable property of providing sparse representations of a wide variety of "natural" images. They have been applied successfully to biomedical image analysis and processing since the early 1990s. In the first part of this talk, we explain how one can exploit the sparsifying property of wavelets to design more effective algorithms for image denoising and reconstruction, both in terms of quality and computational performance. This is achieved within a variational framework by imposing some ℓ_{1}type regularization in the wavelet domain, which favors sparse solutions. We discuss some corresponding iterative skrinkagethresholding algorithms (ISTA) for sparse signal recovery and introduce a multilevel variant for greater computational efficiency. We illustrate the method with two concrete imaging examples: the deconvolution of 3D fluorescence micrographs, and the reconstruction of magnetic resonance images from arbitrary (nonuniform) kspace trajectories. In the second part, we show how to design new wavelet bases that are better matched to the directional characteristics of images. We introduce a general operatorbased framework for the construction of steerable wavelets in any number of dimensions. This approach gives access to a broad class of steerable wavelets that are selfreversible and linearly parameterized by a matrix of shaping coefficients; it extends upon Simoncelli's steerable pyramid by providing much greater wavelet diversity. The basic version of the transform (higherorder Riesz wavelets) extracts the partial derivatives of order N of the signal (e.g., gradient or Hessian). We also introduce a signaladapted design, which yields a PCAlike tight wavelet frame. We illustrate the capabilities of these new steerable wavelets for image analysis and processing (denoising). 



Sampling and Interpolation for Biomedical Imaging 

M. Unser 
Part I Part II 

2006 IEEE International Symposium on Biomedical Imaging, April 69, 2006, Arlington, Virginia, USA. 

This tutorial will explain the modern, Hilbertspace approach for the discretization (sampling) and reconstruction (interpolation) of images
(in two or higher dimensions). The emphasis will be on quality and optimality, which are important considerations for biomedical applications.




Image Processing with ImageJ 

M. Abramoff, P. Magalhães, S. Ram 

Biophotonics International, vol. 11, no. 7, pp. 3642, July 2004. 

As the popularity of the ImageJ opensource, Javabased imaging program grows, its capabilities increase, too. It is now being used for imaging applications ranging from skin analysis to neuroscience. 

A Review of Wavelets in Biomedical Applications 

M. Unser, A. Aldroubi 

Proceedings of the IEEE, vol. 84, no. 4, pp. 626638, April 1996. 

In this paper, we present an overview of the various uses of the wavelet transform (WT) in medicine and biology. We start by describing the wavelet properties that are the most important for biomedical applications. In particular, we provide an interpretation of the continuous WT as a prewhitening multiscale matched filter. We also briefly indicate the analogy between the WT and some of the biological processing that occurs in the early components of the auditory and visual system. We then review the uses of the WT for the analysis of onedimensional physiological signals obtained by phonocardiography, electrocardiography (ECG), and electroencephalography (EEG), including evoked response potentials. Next, we provide a survey of recent wavelet developments in medical imaging. These include biomedical image processing algorithms (e.g., noise reduction, image enhancement, and detection of microcalcifications in mammograms); image reconstruction and acquisition schemes (tomography, and magnetic resonance imaging (MRI)); and multiresolution methods for the registration and statistical analysis of functional images of the brain (positron emission tomography (PET), and functional MRI). In each case, we provide the reader with some general background information and a brief explanation of how the methods work. The paper also includes an extensive bibliography. 

Wavelets in Medicine and Biology 

A. Aldroubi, M.A. Unser, Eds. 

ISBN 084939483X, CRC Press, Boca Raton FL, USA, 1996, 616 p. 

For the first time, the field's leading international experts have come together to produce a complete guide to wavelet transform applications in medicine and biology. This book provides guidelines for all those interested in learning about waveletes and their applications to biomedical problems. The introductory material is written for nonexperts and includes basic discussions of the theoretical and practical foundations of wavelet methods. This is followed by contributions from the most prominent researchers in the field, giving the reader a complete survey of the use of wavelets in biomedical engineering. The book consists of four main sections:



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