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Regularization addresses the ill-posedness of the training problem in machine learning or the reconstruction of a signal from a limited number of measurements. The method is applicable whenever the problem is formulated as an optimization task. The standard strategy consists in augmenting the original cost functional by an energy that penalizes solutions with undesirable behavior. The effect of regularization is very well understood when the penalty involves a Hilbertian norm. Another popular configuration is the use of an [Formula omitted]-norm (or some variant thereof) that favors sparse solutions. In this paper, we propose a higher-level formulation of regularization within the context of Banach spaces. We present a general representer theorem that characterizes the solutions of a remarkably broad class of optimization problems. We then use our theorem to retrieve a number of known results in the literature such as the celebrated representer theorem of machine leaning for RKHS, Tikhonov regularization, representer theorems for sparsity promoting functionals, the recovery of spikes, as well as a few new ones., Author(s): Michael Unser [sup.1] Author Affiliations: (1) grid.5333.6, 0000000121839049, Biomedical Imaging Group, Ecole polytechnique fédérale de Lausanne (EPFL), , Station 17, CH-1015, Lausanne, Switzerland Introduction A recurrent problem in science [...]

We present experimental validation of a new reconstruction method for off-axis digital holographic microscopy (DHM). This method effectively suppresses the object autocorrelation, namely, the zero-order term, from holographic data, thereby improving the reconstruction bandwidth of complex wavefronts. The algorithm is based on nonlinear filtering and can be applied to standard DHM setups with realistic recording conditions. We study the robustness of the technique under different experimental configurations, and quantitatively demonstrate its enhancement capabilities on phase signals. [c] 2009 Optical Society of America OCIS codes: 090.1995, 100.2000, 100.3010, 100.3175, 120.5050, 180.3170.

A novel Monte-Carlo technique is presented for the calculation of Stein's unbiased risk estimate (SURE) for an arbitrary denoising algorithm characterized by some specific parameter setting. Results suggest the accurate prediction of the true mean-squared error (MSE) for all the considered algorithms.

Tomographic reconstruction from positron emission tomography (PET) data is an ill-posed problem that requires regularization. An attractive approach is to impose an l1 -regularization constraint, which favors sparse solutions in the wavelet domain. This can be achieved quite efficiently thanks to the iterative algorithm developed by Daubechies et al., 2004. In this paper, we apply this technique and extend it for the reconstruction of dynamic (spatio-temporal) PET data. Moreover, instead of using classical wavelets in the temporal dimension, we introduce exponential-spline wavelets (E-spline wavelets) that are specially tailored to model time activity curves (TACs) in PET. We show that the exponential-spline wavelets naturally arise from the compartmental description of the dynamics of the tracer distribution. We address the issue of the selection of the 'optimal' E-spline parameters (poles and zeros) and we investigate their effect on reconstruction quality. We demonstrate the usefulness of spatio-temporal regularization and the superior performance of E-spline wavelets over conventional Battle-Lemarie wavelets in a series of experiments: the 1-D fitting of TACs, and the tomographic reconstruction of both simulated and clinical data. We find that the E-spline wavelets outperform the conventional wavelets in terms of the reconstructed signal-to-noise ratio (SNR) and the sparsity of the wavelet coefficients. Based on our simulations, we conclude that replacing the conventional wavelets with E-spline wavelets leads to equal reconstruction quality for a 40% reduction of detected coincidences, meaning an improved image quality for the same number of counts or equivalently a reduced exposure to the patient for the same image quality. Index Terms--Differential system, E-spline wavelets, [l.sub.1]-regularization, spatio-temporal positron emission tomography (PET) reconstruction, time-activity-curves.

A novel method is proposed for the joint estimation of the texture and topography of a specimen from a series of brightfield optical sections. Findings reveal the potential and efficiency of the proposed method.

The details of the design process of a snakuscule, the simplest active contour designed with the quintessence of traditional snake, energy terms governed by the data, and a regularization term, are discussed. The effectiveness and application of the snakuscule for segmenting the aorta and for cell counting is described.

A fast variational thresholded Landweber (TL) method for wavelet-regularized deconvolution, based on the application of a Shannon wavelet basis, which permits for a subband-adapted minimization strategy, is presented. The extraordinary empirical results in 3-D deconvolution microscopy are presented, where the size of typical data sets does not allow more than a few tens of iterations.

The generalized sampling problem is considered in multiple dimensions and the issue of the best reconstruction space in the shift-invariant framework is addressed. A stochastic interpretation is provided for the reconstruction algorithm and equivalence is established with the minimax and minimum mean square error solutions of the generalized sampling problem.

The article characterizes the generalized orthonormal Daubechies wavelets by scaling functions reproducing exponential polynomials. The signal types ensure approximation and sparsity properties.

A consistent sampling theorem for preserving certain preferential signals is discussed. Findings reveal that the theorem offers a unique and well-defined solution in the absence of one-to-one mapping between the measurements and the reconstructed signal.

Magnetic resonance spectroscopy imaging (MRSI) is an attractive tool for medical imaging. However, its practical use is often limited by the intrinsic low spatial resolution and long acquisition time. Spectral localization by imaging (SLIM) has been proposed as a non-Fourier reconstruction algorithm that incorporates spatial a priori information about spectroscopically uniform compartments. Unfortunately, the influence of the magnetic field inhomogeneity--in particular, the susceptibility effects at tissues' boundaries--undermines the validity of the compartmental model. Therefore, we propose BSLIM as an extension of SLIM with field inhomogeneity compensation. A [B.sub.0]-field inhomogeneity map, which can be acquired rapidly and at high resolution, is used by the new algorithm as additional a priori information. We show that the proposed method is distinct from the generalized SLIM (GSLIM) framework. Experimental results of a two-compartment phantom demonstrate the feasibility of the method and the importance of inhomogeneity compensation. Index Terms--Chemical shift imaging, constrained reconstruction, magnetic field inhomogeneity, magnetic resonance spectroscopy imaging (MRSI).

The identification of the class of differential operators L that are both translation and scale invariant is the starting point for the construction of self-similar splines. The fractional smoothing spline estimators are characterized, which are optimal in a deterministic and variational sense.

An alternative stochastic formulation is presented to justify the fractional spline estimators and a formal connection is established between splines and stochastic fractal processes).The results have yielded the optimal minimum mean-square error (MMSE) parameters for the smoothing spline estimator, as well as the connection with kriging and Wiener filtering.

A new approach is described for orthonormal wavelet image denoising that does not require any prior statistical modelization of the wavelet coefficients. An interscale orthonormal wavelet thresholding algorithm is described based on Stein's unbiased risk estimator (SURE) approach.

We present a new computational method for reconstructing a vector velocity field from scattered, pulsed-wave ultrasound Doppler data. The main difficulty is that the Doppler measurements are incomplete, for they do only capture the velocity component along the beam direction. We thus propose to combine measurements from different beam directions. However, this is not yet sufficient to make the problem well posed because 1) the angle between the directions is typically small and 2) the data is noisy and nonuniformly sampled. We propose to solve this reconstruction problem in the continuous domain using regularization. The reconstruction is formulated as the minimizer of a cost that is a weighted sum of two terms: 1) the sum of squared difference between the Doppler data and the projected velocities 2) a quadratic regularization functional that imposes some smoothness on the velocity field. We express our solution for this minimization problem in a B-spline basis, obtaining a sparse system of equations that can be solved efficiently. Using synthetic phantom data, we demonstrate the significance of tuning the regularization according to the a priori knowledge about the physical property of the motion. Next, we validate our method using real phantom data for which the ground truth is known. We then present reconstruction results obtained from clinical data that originate from 1) blood flow in carotid bifurcation and 2) cardiac wall motion. Index Terms--Color Doppler imaging, color flow imaging, echocardiography, nonuniform sampling, projected sampling, pulsed wave Doppler, regularized reconstruction, shift-invariant spaces, tissue Doppler imaging, ultrasound Doppler, variational reconstruction, vector field reconstruction, velocity field reconstruction.

A smoothing technique is developed motivated by the fractal-like behavior of natural images, enabling the derivation of an algorithm to express Duchon's solution in a fractional polyharmonic B-spline basis that spans the same space as radial basis functions (RBFs). The smoothing is performed by filtering in the Fourier domain, and the algorithm is compared with a multidimensional oracle Wiener filter, requiring the knowledge of the true noiseless power spectrum of the signal, and is found to perform as well as the oracle solution in a wide range of conditions.

The recovery of a continuous-time signal given samples of its filtered version is addressed, where the signal reconstruction is performed in a shift-invariant space, addressing the noisy data interpolation and the deconvolution problems simultaneously. The classical deconvolution solutions are adapted to the situation and some concrete examples of reconstruction filters are presented, as also simple guidelines for selecting the free parameters, that is, regularization of the various algorithms.

An approach based on differential operators is introduced to construct wavelet-like basis functions. The wavelets are characterized and it is found that the proposed wavelet-like decompositions are as computationally efficient as the classical ones and are computed using an adapted version of Mallat's filter bank algorithm, where the filters depend on the decomposition level.

A three-dimensional (3-D) parametric active contour algorithm for the shape estimation of DNA molecules from stereo cryo-electron micrographs is introduced. The shape is estimated by matching the projection of a 3-D global shape model with the micrographs.

Polyharmonic B-splines are used for building multidimensional wavelet bases. The B-splines-type wavelets converge to a sum of four Gabor atoms with the increase in order.

A new, computational procedure for tracking fluorescent markers in time-lapse microscopy is presented. The algorithm is optimized for finding the time-trajectory of single particles in very noisy dynamic image sequences.

We propose a new spatio-temporal elastic registration algorithm for motion reconstruction from a series of images. The specific application is to estimate displacement fields from two-dimensional ultrasound sequences of the heart. The basic idea is to find a spatio-temporal deformation field that effectively compensates for the motion by minimizing a difference with respect to a reference frame. The key feature of our method is the use of a semi-local spatio-temporal parametric model for the deformation using splines, and the reformulation of the registration task as a global optimization problem. The scale of the spline model controls the smoothness of the displacement field. Our algorithm uses a multiresolution optimization strategy to obtain a higher speed and robustness. We evaluated the accuracy of our algorithm using a synthetic sequence generated with an ultrasound simulation package, together with a realistic cardiac motion model. We compared our new global multiframe approach with a previous method based on pairwise registration of consecutive frames to demonstrate the benefits of introducing temporal consistency. Finally, we applied the algorithm to the regional analysis of the left ventricle. Displacement and strain parameters were evaluated showing significant differences between the normal and pathological segments, thereby illustrating the clinical applicability of our method. Index Terms--Cardiac motion, elastic registration, parametric models, splines, temporal models.

A parametric active contour method based on B-Spline interpolation is proposed, to highly reduce the computational cost, but this method is sensitive to noise. The rigid interpolation constraint in order to robustify the method in the presence of noise is relaxed, by using smoothing splines and a tunable amount of interpolation error for a smoother spline curve is proposed.

A series of mathematical arguments is presented to justify the use of splines in signal processing applications. The proposed formalism yields the optimal discretization of the classical Wiener filter, together with a fast recursive algorithm.

A new optical flow-based method for estimating heart motion from two-dimensional echocardiographic sequences is presented. The radial component of the velocity with respect to a reference point is color coded and visualized inside a time-varying region of interest, in order to characterize myocardial contractility and to simplify the detection of myocardial dysfunction.

A new family of two-dimensional and three-dimensional orthogonal wavelets, which uses quincunx sampling, is presented. A fast Fourier transform-based implementation that turns out to be surprisingly fast is proposed.

A novel method image reconstruction from nonuniform samples with no constraints on their locations is proposed. A variational approach where the reconstruction is formulated is adopted.

An extended family of cardinal splines is introduced for generalization of the concept for all convolution operators with rational transfer functions. B-spline representation, continuous-time signal processing, ease of manipulation and generality are the advantages of using cardinal E-splines for signal processing.

A complete and self-contained signal processing formulation of exponential splines defined on a uniform grid is proposed. An underlying multiresolution spline structure, which does not involve simple dilations of the B-spline basis function, is observed.

We present an elastic registration algorithm for the alignment of biological images. Our method combines and extends some of the best techniques available in the context of medical imaging. We express the deformation field as a B-spline model, which allows us to deal with a rich variety of deformations. We solve the registration problem by minimizing a pixelwise mean-square distance measure between the target image and the warped source. The problem is further constrained by way of a vector-spline regularization which provides some control over two independent quantities that are intrinsic to the deformation: its divergence, and its curl. Our algorithm is also able to handle soft landmark constraints, which is particularly useful when parts of the images contain very little information or when its repartition is uneven. We provide an optimal analytical solution in the case when only landmarks and smoothness considerations are taken into account. We have applied our approach to perform the elastic registration of images such as electrophoretic gels and fly embryos. The validation of the results by experts has been favorable in all cases.

General parametric snakes for image segmentation are focused, their limitations are illustrated and a new image energy term is proposed to improve performance. This general energy term is arrived at by unifying different energy approaches to obtain a general algorithm that easily tunes in to the problems.

We propose a general approach for the design of 2D feature detectors from a class of steerable functions based on the optimization of a Canny-like criterion. In contrast with previous computational designs, our approach is truly 2D and provides filters that have closed-form expressions. It also yields operators that have a better orientation selectivity than the classical gradient or Hessian-based detectors. We illustrate the method with the design of operators for edge and ridge detection. We present some experimental results that demonstrate the performance improvement of these new feature detectors. We propose computationally efficient local optimization algorithms for the estimation of feature orientation. We also introduce the notion of shape-adaptable feature detection and use it for the detection of image corners. Index Terms--Steerable, feature, edge, detection, ridge, contours, boundary, lines.

A mathematical analysis of the new bivariate spline family is presented and a closed form for a hex-spline of arbitrary order is defined. Some important properties such as their Fourier transform and the fact they form a Riesz basis is discussed.

A method to improve the standard piecewise-linear interpolation with uniform knots by shifting the sampling knots by an optimal nontrivial shift value, close to 1/5 for linear interpolation is presented. Results of the cumulative rotation and zoom experiments show that the shifted linear method is perceptually sharper than the result obtained using Keys' cubic kernel.

The proposal of a fast multiresolution wavelet-like algorithm to calculate multiscale local geometric moments of different orders with a dyadic scale progression and B-spline-weighted is presented. Three applications of these multiscale local moments namely the feature-extraction method, noise-reduction method and multiscale optical-flow algorithm are studied.

An algorithm is presented for fast elastic multidimensional intensity-based image registration with a parametric model of the deformation.

A procedure is proposed to help designers evaluate the complete space of piecewise-polynomial interpolation kernels.

Mathematical features of LeGall 5/3 and Daubechies 9/7 filters are determined and reported.

The isosurface of a continuous model of volumetric discrete and regular data is rendered by means of ray tracing.

A spline-based interpretation and derivation of wavelet properties is presented.

A wavelet basis is proposed for processing and reconstruction of digital holograms. It exploits the mathematical properties of the Fresnel transform.

Wavelet-based methods have become most popular for the compression of two-dimensional medical images and sequences. The standard implementations consider data sizes that are powers of two. There is also a large body of literature treating issues such as the choice of the 'optimal' wavelets and the performance comparison of competing algorithms. With the advent of telemedicine, there is a strong incentive to extend these techniques to higher dimensional data such as dynamic three-dimensional (3-D) echocardiography [four-dimensional (4-D) datasets]. One of the practical difficulties is that the size of this data is often not a multiple of a power of two, which can lead to increased computational complexity and impaired compression power. Our contribution in this paper is to present a genuine 4-D extension of the well-known zerotree algorithm for arbitrarily sized data. The key component of our method is a one-dimensional wavelet algorithm that can handle arbitrarily sized input signals. The method uses a pair of symmetric/antisymmetric wavelets (10/6) together with some appropriate midpoint symmetry boundary conditions that reduce border artifacts. The zerotree structure is also adapted so that it can accommodate noneven data splitting. We have applied our method to the compression of real 3-D dynamic sequences from clinical cardiac ultrasound examinations. Our new algorithm compares very favorably with other more ad hoc adaptations (image extension and tiling) of the standard powers-of-two methods, in terms of both compression performance and computational cost. It is vastly superior to slice-by-slice wavelet encoding. This was seen not only in numerical image quality parameters but also in expert ratings, where significant improvement using the new approach could be documented. Our validation experiments show that one can safely compress 4-D data sets at ratios of 128:1 without compromising the diagnostic value of the images. We also display some more extreme compression results at ratios of 2000:1 where some key diagnostically relevant key features are preserved. Index Terms--Echocardiography, image coding, multidimensional images, zerotree algorithm.

A general theory for reconstructing a signal, given a finite set of linear measurements, was developed in a previous paper. Here, those results are used to compute the solution of a specific sampling problem.

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A theoretical framework for obtaining multiresolution image approximations with non-Euclidian norms is described. An efficient iterative algorithm for calculating these approximations is proposed.