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Splines 
Beyond the digital divide: Ten good reasons for using splines 

M. Unser 


Seminars of Numerical Analysis, EPFL, May 9, 2010. 

"Think analog, act digital" is a motto that is relevant to scientific computing and algorithm design in a variety of disciplines,
including numerical analysis, image/signal processing, and computer graphics. 



Splines: A Unifying Framework for Image Processing 

M. Unser 

Plenary talk, 2005 IEEE International Conference on Image Processing (ICIP'05), Genova, Italy, September 1114, 2005. 

Our purpose is to justify the use splines in imaging applications, emphasizing their ease of use, as well as their fundamental properties. Modeling images with splines is painless: it essentially amounts to replacing the pixels by Bspline basis functions, which are piecewise polynomials with a maximum order of differentiability. The spline representation is flexible and provides the best cost/quality tradeoff among all interpolation methods: by increasing the degree, one shifts from a simple piecewise linear representation to a higher order one that gets closer and closer to being bandlimited. We will describe efficient digital filterbased algorithms for interpolating and processing images within this framework. We will also discuss the multiresolution properties of splines that make them especially attractive for multiscale processing.




Calculer le charme discret de la continuité 

P. Roth 

Horizon, le magazine suisse de la recherche scientifique, no. 79, Décembre 2008. 

Le traitement des images médicales a pu être fortement amélioré et accéléré grâce à des fonctions mathématiques appelées splines. Michael Unser a apporté une contribution essentielle dans ce secteur, tant sur le plan de la théorie que des applications. 



Sparsity and the optimality of splines for inverse problems: Deterministic vs. statistical justifications 

M. Unser 


Invited talk: Mathematics and Image Analysis (MIA'16), 1820 January, 2016, Institut Henri Poincaré, Paris, France. 

In recent years, significant progress has been achieved in the resolution of illposed linear inverse problems by imposing l1/TV regularization constraints on the solution. Such sparsitypromoting schemes are supported by the theory of compressed sensing, which is finite dimensional for the most part. In this talk, we take an infinitedimensional point of view by considering signals that are defined in the continuous domain. We claim that nonuniform splines whose type is matched to the regularization operator are optimal candidate solutions. We show that such functions are global minimizers of a broad family of convex variational problems where the measurements are linear and the regularization is a generalized form of total variation associated with some operator L. We then discuss the link with sparse stochastic processes that are solutions of the same type of differential equations.The pleasing outcome is that the statistical formulation yields maximum a posteriori (MAP) signal estimators that involve the same type of sparsitypromoting regularization, albeit in a discretized form. The latter corresponds to the loglikelihood of the projection of the stochastic model onto a finitedimensional reconstruction space.




Splines, Noise, Fractals and Optimal Signal Reconstruction 

M. Unser 

Plenary talk, Seventh International Workshop on Sampling Theory and Applications (SampTA'07), Thessaloniki, Greece, June 15, 2007. 

We consider the generalized sampling problem with nonideal acquisition device. The task is to “optimally” reconstruct a continuouslyvarying input signal from its discrete, noisy measurements in some integershiftinvariant 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 globallyoptimal, 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 splinebased reconstruction algorithms. Finally, we propose a novel formulation of vectorsplines based on similar principles, and demonstrate their application to flow field reconstruction from nonuniform, incomplete ultrasound Doppler data. This is joint work with Yonina Eldar, Thierry Blu, and Muthuvel Arigovindan. 



Vers une théorie unificatrice pour le traitement numérique/analogique des signaux 

M. Unser 

Twentieth GRETSI Symposium on Signal and Image Processing (GRETSI'05), LouvainlaNeuve, Belgium, September 69, 2005. 

We introduce a Hilbertspace framework, inspired by wavelet theory, that provides an exact link between the traditional—discrete and analog—formulations of signal processing. In contrast to Shannon's sampling theory, our approach uses basis functions that are compactly supported and therefore better suited for numerical computations. The underlying continuoustime signal model is of exponential spline type (with rational transfer function); this family of functions has the advantage of being closed under the basic signalprocessing operations: differentiation, continuoustime convolution, and modulation. A key point of the method is that it allows an exact implementation of continuoustime operators by simple processing in the discrete domain, provided that one updates the basis functions appropriately. The framework is ideally suited for hybrid signal processing because it can jointly represent the effect of the various (analog or digital) components of the system. This point will be illustrated with the design of hybrid systems for improved AtoD and DtoA conversion. On the more fundamental front, the proposed formulation sheds new light on the striking parallel that exists between the basic analog and discrete operators in the classical theory of linear systems. 



Splines: on Scale, Differential Operators and Fast Algorithms 

M. Unser 

5th International Conference on Scale Space and PDE Methods in Computer Vision, Hofgeismar, Germany, April 610, 2005. 



Think Analog, Act Digital 

M. Unser 


Plenary talk, Seventh Biennial Conference, 2004 International Conference on Signal Processing and Communications (SPCOM'04), Bangalore, India, December 1114, 2004. 

By interpreting the Greenfunction reproduction property of exponential splines in signalprocessing terms, we uncover a fundamental relation that connects the impulse responses of allpole analog filters to their discrete counterparts. The link is that the latter are the Bspline coefficients of the former (which happen to be exponential splines). Motivated by this observation, we introduce an extended family of cardinal splines—the generalized Esplines—to generalize the concept for all convolution operators with rational transfer functions. We construct the corresponding compactly supported Bspline basis functions which are characterized by their poles and zeros, thereby establishing an interesting connection with analogfilter design techniques. We investigate the properties of these new Bsplines and present the corresponding signalprocessing calculus, which allows us to perform continuoustime operations such as convolution, differential operators, and modulation, by simple application of the discrete version of these operators in the Bspline domain. In particular, we show how the formalism can be used to obtain exact, discrete implementations of analog filters. We also apply our results to the design of hybrid signalprocessing systems that rely on digital filtering to compensate for the nonideal characteristics of realworld AtoD and DtoA conversion systems. 



Splines: A Perfect Fit for Signal and Image Processing 

M. Unser 

IEEE Signal Processing Magazine, vol. 16, no. 6, pp. 2238, November 1999. IEEE Signal Processing Society's 2000 Magazine Award 

The goals of this article are threefold:


Image Interpolation and Resampling 

P. Thévenaz, T. Blu, M. Unser 

Handbook of Medical Imaging, Processing and Analysis, I.N. Bankman, Ed., Academic Press, San Diego CA, USA, pp. 393420, 2000. 

This chapter presents a survey of interpolation and resampling techniques in the context of exact, separable interpolation of regularly sampled data. In this context, the traditional view of interpolation is to represent an arbitrary continuous function as a discrete sum of weighted and shifted synthesis functions—in other words, a mixed convolution equation. An important issue is the choice of adequate synthesis functions that satisfy interpolation properties. Examples of finitesupport ones are the square pulse (nearestneighbor interpolation), the hat function (linear interpolation), the cubic Keys' function, and various truncated or windowed versions of the sinc function. On the other hand, splines provide examples of infinitesupport interpolation functions that can be realized exactly at a finite, surprisingly small computational cost. We discuss implementation issues and illustrate the performance of each synthesis function. We also highlight several artifacts that may arise when performing interpolation, such as ringing, aliasing, blocking and blurring. We explain why the approximation order inherent in the synthesis function is important to limit these interpolation artifacts, which motivates the use of splines as a tunable way to keep them in check without any significant cost penalty. 


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