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Biomedical Image Reconstruction

M. Unser

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12th European Molecular Imaging Meeting, 5-7 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, x-ray computer tomography, and positron-emission tomography (PET). Interestingly, the paradigm remains valid for modern cellular/molecular imaging with confocal/super-resolution fluorescence microscopy, which is highly relevant to molecular imaging as well. In fact, we believe that the huge potential for cross-fertilization and mutual re-enforcement between imaging modalities has not been fully exploited yet.

The prerequisite to image reconstruction is an accurate physical description of the image-formation process: the so-called 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 ill-posed 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, non-Cartesian sampling grids, and/or missing views. In particular, we discuss sparsity-promoting 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.

Splines, Noise, Fractals and Optimal Signal Reconstruction

M. Unser

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Plenary talk, Seventh International Workshop on Sampling Theory and Applications (SampTA'07), Thessaloniki, Greece, June 1-5, 2007.

We consider the generalized sampling problem with non-ideal acquisition device. The task is to “optimally” reconstruct a continuously-varying input signal from its discrete, noisy measurements in some integer-shift-invariant space. We propose three formulations of the problem—variational/Tikhonov, minimax, and minimum mean square error estimation—and derive the corresponding solutions for a given reconstruction space. We prove that these solutions are also globally-optimal, provided that the reconstruction space is matched to the regularization operator (deterministic signal) or, alternatively, to the whitening operator of the process (stochastic modeling). Moreover, the three formulations lead to the same generalized smoothing spline reconstruction algorithm, but only if the reconstruction space is chosen optimally. We then show that fractional splines and fractal processes (fBm) are solutions of the same type of differential equations, except that the context is different: deterministic versus stochastic. We use this link to provide a solid stochastic justification of spline-based reconstruction algorithms. Finally, we propose a novel formulation of vector-splines based on similar principles, and demonstrate their application to flow field reconstruction from non-uniform, incomplete ultrasound Doppler data.

This is joint work with Yonina Eldar, Thierry Blu, and Muthuvel Arigovindan.

The Colored Revolution of Bioimaging

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

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IEEE Signal Processing Magazine, vol. 23, no. 3, pp. 20-31, May 2006.

With the recent development of fluorescent probes and new high-resolution 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 time-lapse 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.

Full story ...

Wavelet Demystified

M. Unser

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Invited presentation, Technical University of Eindhoven, The Netherlands, May 31, 2006.

This 2-hour tutorial focuses on wavelet bases : it covers the concept of multi-resolution analysis, the construction of wavelets, filterbank algorithms, as well as an in-depth discussion of fundamental wavelet properties. The presentation is progressive starting with the example of the Haar transform and essentially self contained.
We emphasize the crucial role of splines in wavelet theory, presenting a non-standard point of view that simplifies the mathematical formulation. The key point is that any wavelet (or scaling function) can be expressed as the convolution of a (fractional) B-spline and a singular distribution, and that all fundamental spline properties (reproduction of polynomials, regularity, order of approximation, etc.) are preserved through the convolution operation. A direct implication is that the wavelets have vanishing moments and that they behave like multiscale differentiators. These latter two properties are the key for understanding why wavelets yield sparse representations of piecewise-smooth signals.

Sampling—50 Years After Shannon

M. Unser

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Proceedings of the IEEE, vol. 88, no. 4, pp. 569-587, April 2000.

This paper presents an account of the current state of sampling, 50 years after Shannon's formulation of the sampling theorem. The emphasis is on regular sampling where the grid is uniform. This topic has benefited from a strong research revival during the past few years, thanks in part to the mathematical connections that were made with wavelet theory. To introduce the reader to the modern, Hilbert-space formulation, we re-interpret Shannon's sampling procedure as an orthogonal projection onto the subspace of bandlimited functions. We then extend the standard sampling paradigm for a representation of functions in the more general class of "shift-invariant" functions spaces, including splines and wavelets. Practically, this allows for simpler—and possibly more realistic—interpolation models, which can be used in conjunction with a much wider class of (anti-aliasing) pre-filters that are not necessarily ideal lowpass. We summarize and discuss the results available for the determination of the approximation error and of the sampling rate when the input of the system is essentially arbitrary; e.g., non-bandlimited. We also review variations of sampling that can be understood from the same unifying perspective. These include wavelets, multi-wavelets, Papoulis generalized sampling, finite elements, and frames. Irregular sampling and radial basis functions are briefly mentioned.

Wavelet Games

M. Unser, M. Unser

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The Wavelet Digest, vol. 11, Issue 4, April 1, 2003.

Summary: A dialogue between father and son about wavelets, Legos, and other existential questions.

Fractional Splines and Wavelets

M. Unser, T. Blu

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SIAM Review, vol. 42, no. 1, pp. 43-67, March 2000.

We extend Schoenberg's family of polynomial splines with uniform knots to all fractional degrees α > -1. These splines, which involve linear combinations of the one-sided power functions x+α = max(0, x)α, belong to L1 and are α-Hölder continuous for α > 0. We construct the corresponding B-splines by taking fractional finite differences and provide an explicit characterization in both time and frequency domains. We show that these functions satisfy most of the properties of the traditional B-splines, including the convolution property, and a generalized fractional differentiation rule that involves finite differences only. We characterize the decay of the B-splines which are not compactly supported for non-integral α's. Their most astonishing feature (in reference to the Strang-Fix theory) is that they have a fractional order of approximation α + 1 while they reproduce the polynomials of degree [α]. For α > 1/2, they satisfy all the requirements for a multiresolution analysis of L2 (Riesz bounds, two scale relation) and may therefore be used to build new families of wavelet bases with a continuously-varying order parameter. Our construction also yields symmetrized fractional B-splines which provide the connection with Duchon's general theory of radial (m,s)-splines (including thin-plate splines). In particular, we show that the symmetric version of our splines can be obtained as solution of a variational problem involving the norm of a fractional derivative.

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