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We propose to learn non-convex regularizers with a prescribed upper bound on their weak-convexity modulus. Such regularizers give rise to variational denoisers that minimize a convex energy. They rely on few parameters (less than 15,000) and offer a signal-processing interpretation as they mimic handcrafted sparsity-promoting regularizers. Through numerical experiments, we show that such denoisers outperform convex-regularization methods as well as the popular BM3D denoiser. Additionally, the learned regularizer can be deployed to solve inverse problems with iterative schemes that provably converge. For both CT and MRI reconstruction, the regularizer generalizes well and offers an excellent tradeoff between performance, number of parameters, guarantees, and interpretability when compared to other data-driven approaches. [ABSTRACT FROM AUTHOR]

The Lizorkin space is well suited to the study of operators like fractional Laplacians and the Radon transform. In this paper, we show that the space is unfortunately not complemented in the Schwartz space. In return, we show that it is dense in C 0 (ℝ d) , a property that is shared by the larger Schwartz space and that turns out to be useful for applications. Based on this result, we investigate subspaces of Lizorkin distributions that are Banach spaces and for which a continuous representation operator exists. Then, we introduce a variational framework that involves these spaces and that makes use of the constructed operator. By investigating two particular cases of this framework, we are able to strengthen existing results for fractional splines and 2-layer ReLU networks. [ABSTRACT FROM AUTHOR]

We formulate as an inverse problem the construction of sparse parametric continuous curve models that fit a sequence of contour points. Our prior is incorporated as a regularization term that encourages rotation invariance and sparsity. We prove that an optimal solution to the inverse problem is a closed curve with spline components. We then show how to efficiently solve the task using B-splines as basis functions. We extend our problem formulation to curves made of two distinct components with complementary smoothness properties and solve it using hybrid splines. We illustrate the performance of our model on contours of different smoothness. Our experimental results show that we can faithfully reconstruct any general contour using few parameters, even in the presence of imprecisions in the measurements. [ABSTRACT FROM AUTHOR]

We propose a novel sparse dictionary learning method for planar shapes in the sense of Kendall, namely configurations of landmarks in the plane considered up to similitudes. Our shape dictionary method provides a good trade-off between algorithmic simplicity and faithfulness with respect to the nonlinear geometric structure of Kendall's shape space. Remarkably, it boils down to a classical dictionary learning formulation modified using complex weights. Existing dictionary learning methods extended to nonlinear spaces map the manifold either to a reproducing kernel Hilbert space or to a tangent space. The first approach is unnecessarily heavy in the case of Kendall's shape space and causes the geometrical understanding of shapes to be lost, while the second one induces distortions and theoretical complexity. Our approach does not suffer from these drawbacks. Instead of embedding the shape space into a linear space, we rely on the hyperplane of centered configurations, including preshapes from which shapes are defined as rotation orbits. In this linear space, the dictionary atoms are scaled and rotated using complex weights before summation. Furthermore, our formulation is more general than Kendall's original one: it applies to discretely defined configurations of landmarks as well as continuously defined interpolating curves. We implemented our algorithm by adapting the method of optimal directions combined to a Cholesky-optimized order recursive matching pursuit. An interesting feature of our shape dictionary is that it produces visually realistic atoms, while guaranteeing reconstruction accuracy. Its efficiency can mostly be attributed to a clear formulation of the framework with complex numbers. We illustrate the strong potential of our approach for the characterization of datasets of shapes up to similitudes and the analysis of patterns in deforming two-dimensional shapes. [ABSTRACT FROM AUTHOR]

We study one-dimensional continuous-domain inverse problems with multiple generalized total-variation regularization, which involves the joint use of several regularization operators. Our starting point is a new representer theorem that states that such inverse problems have hybrid-spline solutions with a total sparsity bounded by the number of measurements. We show that such continuous-domain problems can be discretized in an exact way by using a union of B-spline dictionary bases matched to the regularization operators. We then propose a multiresolution algorithm that selects an appropriate grid size that depends on the problem. Finally, we demonstrate the computational feasibility of our algorithm for multiple-order derivative regularization operators. [ABSTRACT FROM AUTHOR]

This paper deals with the resolution of inverse problems in a periodic setting or, in other terms, the reconstruction of periodic continuous-domain signals from their noisy measurements. We focus on two reconstruction paradigms: variational and statistical. In the variational approach, the reconstructed signal is solution to an optimization problem that establishes a tradeoff between fidelity to the data and smoothness conditions via a quadratic regularization associated with a linear operator. In the statistical approach, the signal is modeled as a stationary random process defined from a Gaussian white noise and a whitening operator; one then looks for the optimal estimator in the mean-square sense. We give a generic form of the reconstructed signals for both approaches, allowing for a rigorous comparison of the two. We fully characterize the conditions under which the two formulations yield the same solution, which is a periodic spline in the case of sampling measurements. We also show that this equivalence between the two approaches remains valid on simulations for a broad class of problems. This extends the practical range of applicability of the variational method. [ABSTRACT FROM AUTHOR]

Splines come in a variety of flavors that can be characterized in terms of some differential operator L. The simplest piecewise-constant model corresponds to the derivative operator. Likewise, one can extend the traditional notion of total variation by considering more general operators than the derivative. This results in the definitions of a generalized total variation seminorm and its corresponding native space, which is further identified as the direct sum of two Banach spaces. We then prove that the minimization of the generalized total variation (gTV), subject to some arbitrary (convex) consistency constraints on the linear measurements of the signal, admits nonuniform L-spline solutions with fewer knots than the number of measurements. This shows that nonuniform splines are universal solutions of continuous-domain linear inverse problems with LASSO, L1, or total-variationlike regularization constraints. Remarkably, the type of spline is fully determined by the choice of L and does not depend on the actual nature of the measurements. [ABSTRACT FROM AUTHOR]

We present a generic method to construct orthogonal projectors for two-dimensional landmark-based parametric spline curves. We construct vector spaces that define a geometric transformation (e.g., affine, similarity, and scaling) that is applied to a reference curve. These vector spaces contain all parametric curves up to the chosen transformation. We define the vector spaces implicitly through an orthogonal projection operator and present a theorem that characterizes the projector for landmark-based spline curves, which are popular for the user-interactive analysis of biomedical images. Finally, we show how shape priors are constructed with the spline projector and provide an example of application for the segmentation of microscopy images in biology. [ABSTRACT FROM AUTHOR]

We present a new trigonometric basis function that is capable of perfectly reproducing circles, spheres and ellipsoids while at the same time being interpolatory. Such basis functions have the advantage that they allow to construct shapes through a sequence of control points that lie on their contour (2-D) or surface (3-D) which facilitates user-interaction, especially in 3-D. Our piecewise exponential basis function has finite support, which enables local control for shape modification. We derive and prove all the necessary properties of the kernel to represent shapes that can be smoothly deformed and show how idealized shapes such as ellipses and spheres can be constructed. [ABSTRACT FROM PUBLISHER]

B-splines are attractive basis functions for the continuous-domain representation of biomedical images and volumes. In this paper, we prove that the extended family of box splines are closed under the Radon transform and derive explicit formulae for their transforms. Our results are general; they cover all known brands of compactly-supported box splines (tensor-product B-splines, separable or not) in any number of dimensions. The proposed box spline approach extends to non-Cartesian lattices used for discretizing the image space. In particular, we prove that the 2-D Radon transform of an N-direction box spline is generally a (nonuniform) polynomial spline of degree N-1. The proposed framework allows for a proper discretization of a variety of tomographic reconstruction problems in a box spline basis. It is of relevance for imaging modalities such as X-ray computed tomography and cryo-electron microscopy. We provide experimental results that demonstrate the practical advantages of the box spline formulation for improving the quality and efficiency of tomographic reconstruction algorithms. [ABSTRACT FROM PUBLISHER]

The fractional Laplacian $(-\triangle)^{\gamma/2}$ commutes with the primary coordination transformations in the Euclidean space ℝ: dilation, translation and rotation, and has tight link to splines, fractals and stable Levy processes. For 0 < γ < d, its inverse is the classical Riesz potential I which is dilation-invariant and translation-invariant. In this work, we investigate the functional properties (continuity, decay and invertibility) of an extended class of differential operators that share those invariance properties. In particular, we extend the definition of the classical Riesz potential I to any non-integer number γ larger than d and show that it is the unique left-inverse of the fractional Laplacian $(-\triangle)^{\gamma/2}$ which is dilation-invariant and translation-invariant. We observe that, for any 1 ≤ p ≤ ∞ and γ ≥ d(1 − 1/ p), there exists a Schwartz function f such that I f is not p-integrable. We then introduce the new unique left-inverse I of the fractional Laplacian $(-\triangle)^{\gamma/2}$ with the property that I is dilation-invariant (but not translation-invariant) and that I f is p-integrable for any Schwartz function f. We finally apply that linear operator I with p = 1 to solve the stochastic partial differential equation $(-\triangle)^{\gamma/2} \Phi=w$ with white Poisson noise as its driving term w. [ABSTRACT FROM AUTHOR]

We introduce an extended family of continuous-domain stochastic models for sparse, piecewise-smooth signals. These are specified as solutions of stochastic differential equations, or, equivalently, in terms of a suitable innovation model; the latter is analogous conceptually to the classical interpretation of a Gaussian stationary process as filtered white noise. The two specific features of our approach are 1) signal generation is driven by a random stream of Dirac impulses (Poisson noise) instead of Gaussian white noise, and 2) the class of admissible whitening operators is considerably larger than what is allowed in the conventional theory of stationary processes. We provide a complete characterization of these finite-rate-of-innovation signals within Gelfand's framework of generalized stochastic processes. We then focus on the class of scale-invariant whitening operators which correspond to unstable systems. We show that these can be solved by introducing proper boundary conditions, which leads to the specification of random, spline-type signals that are piecewise-smooth. These processes are the Poisson counterpart of fractional Brownian motion; they are nonstationary and have the same 1/\omega-type spectral signature. We prove that the generalized Poisson processes have a sparse representation in a wavelet-like basis subject to some mild matching condition. We also present a limit example of sparse process that yields a MAP signal estimator that is equivalent to the popular TV-denoising algorithm. [ABSTRACT FROM AUTHOR]

The efficient realization of linear space-variant (non-convolution) filters is a challenging computational problem in image processing. In this paper, we demonstrate that it is possible to filter an image with a Gaussian-like elliptic window of varying size, elongation and orientation using a fixed number of computations per pixel. The associated algorithm, which is based upon a family of smooth compactly supported piecewise polynomials, the radially-uniform box splines, is realized using preintegration and local finite-differences. The radially-uniform box splines are constructed through the repeated convolution of a fixed number of box distributions, which have been suitably scaled and distributed radially in an uniform fashion. The attractive features of these box splines are their asymptotic behavior, their simple covariance structure, and their quasi-separability. They converge to Gaussians with the increase of their order, and are used to approximate anisotropic Gaussians of varying covariance simply by controlling the scales of the constituent box distributions. Based upon the second feature, we develop a technique for continuously controlling the size, elongation and orientation of these Gaussian-like functions. Finally, the quasi-separable structure, along with a certain scaling property of box distributions, is used to efficiently realize the associated space-variant elliptical filtering, which requires O(1) computations per pixel irrespective of the shape and size of the filter. [ABSTRACT FROM AUTHOR]

Interpolation is the means by which a continuously defined model is fit to discrete data samples. When the data samples are exempt of noise, it seems desirable to build the model by fitting them exactly. In medical imaging, where quality is of paramount importance, this ideal situation unfortunately does not occur. In this paper, we propose a scheme that improves on the quality by specifying a tradeoff between fidelity to the data and robustness to the noise. We resort to variational principles, which allow us to impose smoothness constraints on the model for tackling noisy data. Based on shift-, rotation-, and scale-invariant requirements on the model, we show that the Lp-norm of an appropriate vector derivative is the most suitable choice of regularization for this purpose. In addition to Tikhonov-like quadratic regularization, this includes edge-preserving total-variation-like (TV) regularization. We give algorithms to recover the continuously defined model from noisy samples and also provide a data-driven scheme to determine the optimal amount of regularization. We validate our method with numerical examples where we demonstrate its superiority over an exact fit as well as the benefit of TV-like nonquadratic regularization over Tikhonov-like quadratic regularization. [ABSTRACT FROM AUTHOR]

We consider the problem of optimizing the parameters of a given denoising algorithm for restoration of a signal corrupted by white Gaussian noise. To achieve this, we propose to minimize Stein's unbiased risk estimate (SURE) which provides a means of assessing the true mean-squared error (MSE) purely from the measured data without need for any knowledge about the noise-free signal. Specifically, we present a novel Monte-Carlo technique which enables the user to calculate SURE for an arbitrary denoising algorithm characterized by some specific parameter setting. Our method is a black-box approach which solely uses the response of the denoising operator to additional input noise and does not ask for any information about its functional form. This, therefore, permits the use of SURE for optimization of a wide variety of denoising algorithms. We justify our claims by presenting experimental results for SURE-based optimization of a series of popular image-denoising algorithms such as total-variation denoising, wavelet soft-thresholding, and Wiener filtering/smoothing splines. In the process, we also compare the performance of these methods. We demonstrate numerically that SURE computed using the new approach accurately predicts the true MSE for all the considered algorithms. We also show that SURE uncovers the optimal values of the parameters in all cases. [ABSTRACT FROM AUTHOR]

In a companion paper (see Self-Similarity: Part I—Splines and Operators), we characterized the class of scale-invariant convolution operators: the generalized fractional derivatives of order γ. We used these operators to specify regularization functionals for a series of Tikhonov-like least-squares data fitting problems and proved that the general solution is a fractional spline of twice the order. We investigated the deterministic properties of these smoothing splines and proposed a fast Fourier transform (FFT)-based implementation. Here, we present an alternative stochastic formulation to further justify these fractional spline estimators. As suggested by the title, the relevant processes are those that are statistically self-similar; that is, fractional Brownian motion (Him) and its higher order extensions. To overcome the technical difficulties due to the nonstationary character of Him, we adopt a distributional formulation due to Gel'fand. This allows us to rigorously specify an innovation model for these fractal processes, which rests on the property that they can be whitened by suitable fractional differentiation. Using the characteristic form of the fBm, we then derive the conditional probability density function (PDF) p(BH(t)∣Y), where Y = {BH(k) + n[k]}k∈z are the noisy samples of the Him BH(t) with Hurst exponent H. We find that the conditional mean is a fractional spline of degree 2H, which proves that this class of functions is indeed optimal for the estimation of fractal-like processes. The result also yields the optimal [minimum mean-square error (MMSE)] parameters for the smoothing spline estimator, as well as the connection with kriging and Wiener filtering. [ABSTRACT FROM AUTHOR]

The central theme of this pair of papers (Parts I and II in this issue) is self-similarity, which is used as a bridge for connecting splines and fractals. The first part of the investigation is deterministic, and the context is that of L-splines; these are defined in the following terms: s(t) is a cardinal L-spline if L {s(t)} = Σk∈z a[k]δ(t - k), where L is a suitable pseudodifferential operator. Our starting point for the construction of "self-similar" splines is the identification of the class of differential operators L that are both translation and scale invariant. This results into a two-parameter family of generalized fractional derivatives, ∂τγ, where γ is the order of the derivative and τ is an additional phase factor. We specify the corresponding L-splines, which yield an extended class of fractional splines. The operator ∂τγ is used to define a scale-invariant energy measure—the squared L2 -norm of the γth derivative of the signal—which provides a regularization functional for interpolating or fitting the noisy samples of a signal. We prove that the corresponding variational (or smoothing) spline estimator is a cardinal fractional spline of order 2γ, which admits a stable representation in a B-spline basis. We characterize the equivalent frequency response of the estimator and show that it closely matches that of a classical Butterworth filter of order 2γ. We also establish a formal link between the regularization parameter A and the cutoff frequency of the smoothing spline filter: λ0 ω λ-2γ. Finally, we present an efficient computational solution to the fractional smoothing spline problem: It uses the fast Fourier transform and takes advantage of the multiresolution properties of the underlying basis functions. [ABSTRACT FROM AUTHOR]

In this paper, an approach is introduced based on differential operators to construct wavelet-like basis functions. Given a differential operator L with rational transfer function, elementary building blocks are obtained that are shifted replicates of the Green's function of L. It is shown that these can be used to specify a sequence of embedded spline spaces that admit a hierarchical exponential B-spline representation. The corresponding B-splines are entirely specified by their poles and zeros; they are compactly supported, have an explicit analytical form, and generate multiresolution Riesz bases. Moreover, they satisfy generalized refinement equations with a scale-dependent filter and lead to a representation that is dense in L2. This allows us to specify a corresponding family of semi-orthogonal exponential spline wavelets, which provides a major extension of earlier polynomial spline constructions. These wavelets are completely characterized, and it is proven that they satisfy the following remarkable properties: 1) they are orthogonal across scales and generate Riesz bases at each resolution level; 2) they yield unconditional bases of L2—either compactly supported (B-spline-type) or with exponential decay (orthogonal or dual-type); 3) they have N vanishing exponential moments, where N is the order of the differential operator; 4) they behave like multiresolution versions of the operator L from which they are derived; and 5) their order of approximation is (N - M), where N and M give the number of poles and zeros, respectively. Last but not least, the new wavelet-like decompositions are as computationally efficient as the classical ones. They are computed using an adapted version of Mallat's filter bank algorithm, where the filters depend on the decomposition level. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper deals with fast image and video segmentation using active contours. Region-based active contours using level sets are powerful techniques for video segmentation, but they suffer from large computational cost. A parametric active contour method based on B-Spline interpolation has been proposed in [26] to highly reduce the computational cost, but this method is sensitive to noise. Here, we choose to relax the rigid interpolation constraint in order to robustify our method in the presence of noise: by using smoothing splines, we trade a tunable amount of interpolation error for a smoother spline curve. We show by experiments on natural sequences that this new flexibility yields segmentation results of higher quality at no additional computational cost. Hence, real-time processing for moving objects segmentation is preserved. [ABSTRACT FROM AUTHOR]

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 pixel wise 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. [ABSTRACT FROM AUTHOR]

We introduce local weighted geometric moments that are computed from an image within a sliding window at multiple scales. When the window function satisfies a two-scale relation, we prove that lower order moments can be computed efficiently at dyadic scales by using a multiresolution wavelet-like algorithm. We show that B-splines are well-suited window functions because, in addition to being refinable, they are positive, symmetric, separable, and very nearly isotropic (Gaussian shape). We present three applications of these multiscale local moments. The first is a feature-extraction method for detecting and characterizing elongated structures in images. The second is a noise-reduction method which can be viewed as a multiscale extension of Savitzky-Golay filtering. The third is a multiscale optical-flow algorithm that uses a local affine model for the motion field, extending the Lucas-Kanade optical-flow method. The results obtained in all cases are promising. [ABSTRACT FROM AUTHOR]

We present an algorithm for fast elastic multidimensional intensity-based image registration with a parametric model of the deformation. It is fully automatic in its default mode of operation. In the case of hard real-world problems, it is capable of accepting expert hints in the form of soft landmark constraints. Much fewer landmarks are needed and the results are far superior compared to pure landmark registration. Particular attention has been paid to the factors influencing the speed of this algorithm. The B-spline deformation model is shown to be computationally more efficient than other alternatives. The algorithm has been successfully used for several two-dimensional (2-D) and three-dimensional (3-D) registration tasks in the medical domain, involving MPA, SPECT, CT, and ultrasound image modalities. We also present experiments in a controlled environment, permitting an exact evaluation of the registration accuracy. Test deformations are generated automatically using a random hierarchical fractional wavelet-based generator. [ABSTRACT FROM AUTHOR]

Presents a study that developed an explicit formula for B-spline convolution kernels. Review of the classical B-splines and derive explicit formulas for the multiple B-spline convolution kernels; Introduction of a Radon-based version of a filtered back-projection algorithm that uses spine kernels; Other instances where splines have been used for tomographic reconstruction.

Provides information on a study which expanded the l2 construction of spline pyramids that may be shifted with respect to the finer grid. Theoretical background; Spline approximation; Relation of projection operator and pyramid characterization.

Many feedforward neural networks (NNs) generate continuous and piecewise-linear (CPWL) mappings. Specifically, they partition the input domain into regions on which the mapping is affine. The number of these so-called linear regions offers a natural metric to characterize the expressiveness of CPWL NNs. The precise determination of this quantity is often out of reach in practice, and bounds have been proposed for specific architectures, including for ReLU and Maxout NNs. In this work, we generalize these bounds to NNs with arbitrary and possibly multivariate CPWL activation functions. We first provide upper and lower bounds on the maximal number of linear regions of a CPWL NN given its depth, width, and the number of linear regions of its activation functions. Our results rely on the combinatorial structure of convex partitions and confirm the distinctive role of depth which, on its own, is able to exponentially increase the number of regions. We then introduce a complementary stochastic framework to estimate the average number of linear regions produced by a CPWL NN. Under reasonable assumptions, the expected density of linear regions along any 1D path is bounded by the product of depth, width, and a measure of activation complexity (up to a scaling factor). This yields an identical role to the three sources of expressiveness: no exponential growth with depth is observed anymore. [ABSTRACT FROM AUTHOR]

Rectified-linear-unit (ReLU) neural networks, which play a prominent role in deep learning, generate continuous and piecewise-linear (CPWL) functions. While they provide a powerful parametric representation, the mapping between the parameter and function spaces lacks stability. In this paper, we investigate an alternative representation of CPWL functions that relies on local hat basis functions and that is applicable to low-dimensional regression problems. It is predicated on the fact that any CPWL function can be specified by a triangulation and its values at the grid points. We give the necessary and sufficient condition on the triangulation (in any number of dimensions and with any number of vertices) for the hat functions to form a Riesz basis, which ensures that the link between the parameters and the corresponding CPWL function is stable and unique. In addition, we provide an estimate of the ℓ 2 → L 2 condition number of this local representation. As a special case of our framework, we focus on a systematic parameterization of R d with control points placed on a uniform grid. In particular, we choose hat basis functions that are shifted replicas of a single linear box spline. In this setting, we prove that our general estimate of the condition number is exact. We also relate the local representation to a nonlocal one based on shifts of a causal ReLU-like function. Finally, we indicate how to efficiently estimate the Lipschitz constant of the CPWL mapping. [ABSTRACT FROM AUTHOR]

We study the problem of one-dimensional regression of data points with total-variation (TV) regularization (in the sense of measures) on the second derivative, which is known to promote piecewise-linear solutions with few knots. While there are efficient algorithms for determining such adaptive splines, the difficulty with TV regularization is that the solution is generally non-unique, an aspect that is often ignored in practice. In this paper, we present a systematic analysis that results in a complete description of the solution set with a clear distinction between the cases where the solution is unique and those, much more frequent, where it is not. For the latter scenario, we identify the sparsest solutions, i.e., those with the minimum number of knots, and we derive a formula to compute the minimum number of knots based solely on the data points. To achieve this, we first consider the problem of exact interpolation which leads to an easier theoretical analysis. Next, we relax the exact interpolation requirement to a regression setting, and we consider a penalized optimization problem with a strictly convex data-fidelity cost function. We show that the underlying penalized problem can be reformulated as a constrained problem, and thus that all our previous results still apply. Based on our theoretical analysis, we propose a simple and fast two-step algorithm, agnostic to uniqueness, to reach a sparsest solution of this penalized problem. [ABSTRACT FROM AUTHOR]

In this paper, we formally investigate two mathematical aspects of Hermite splines that are relevant to practical applications. We first demonstrate that Hermite splines are maximally localized, in the sense that the size of their support is minimal among pairs of functions with identical reproduction properties. Then, we precisely quantify the approximation power of Hermite splines for the reconstruction of functions and their derivatives. It is known that the Hermite and B-spline approximation schemes have the same approximation order. More precisely, their approximation error vanishes as O (T 4) when the step size T goes to zero. In this work, we show that they actually have the same asymptotic approximation error constants, too. Therefore, they have identical asymptotic approximation properties. Hermite splines combine optimal localization and excellent approximation power, while retaining interpolation properties and closed-form expression, in contrast to existing similar functions. These findings shed a new light on the convenience of Hermite splines in the context of computer graphics and geometrical design. [ABSTRACT FROM AUTHOR]

Generalized sampling consists in the recovery of a function f , from the samples of the responses of a collection of linear shift-invariant systems to the input f. The reconstructed function is typically a member of a finitely generated integer-shift-invariant space that can reproduce polynomials up to a given degree M. While this property allows for an approximation power of order (M + 1) , it comes with a tradeoff on the length of the support of the basis functions. Specifically, we prove that the sum of the length of the support of the generators is at least (M + 1). Following this result, we introduce the notion of shortest basis of degree M , which is motivated by our desire to minimize computational costs. We then demonstrate that any basis of shortest support generates a Riesz basis. Finally, we introduce a recursive algorithm to construct the shortest-support basis for any multi-spline space. It provides a generalization of both polynomial and Hermite B-splines. This framework paves the way for novel applications such as fast derivative sampling with arbitrarily high approximation power. [ABSTRACT FROM AUTHOR]