nils-herrmann/clean_scirepeval huggingface dataset:
Polyharmonic smoothing splines and the multidimensional Wiener filtering of fractal-like signals

Motivated by the fractal-like behavior of natural images, we develop a smoothing technique that uses a regularization functional which is a fractional iterate of the Laplacian. This type of functional was initially introduced by Duchon for the approximation of nonuniformily sampled, multidimensional data. He proved that the general solution is a smoothing spline that is represented by a linear combination of radial basis functions (RBFs). Unfortunately, this is tedious to implement for images because of the poor conditioning of RBFs and their lack of decay. Here, we present a much more efficient method for the special case of a uniform grid. The key idea is to express Duchon's solution in a fractional polyharmonic B-spline basis that spans the same space as the RBFs. This allows us to derive an algorithm where the smoothing is performed by filtering in the Fourier domain. Next, we prove that the above smoothing spline can be optimally tuned to provide the MMSE estimation of a fractional Brownian field corrupted by white noise. This is a strong result that not only yields the best linear filter (Wiener solution), but also the optimal interpolation space, which is not bandlimited. It also suggests a way of using the noisy data to identify the optimal parameters (order of the spline and smoothing strength), which yields a fully automatic smoothing procedure. We evaluate the performance of our algorithm by comparing it against an oracle Wiener filter, which requires the knowledge of the true noiseless power spectrum of the signal. We find that our approach performs almost as well as the oracle solution over a wide range of conditions
Multigrid image reconstruction from arbitrarily spaced samples

We propose a novel multiresolution-multigrid based signal reconstruction method from arbitrarily spaced samples. The signal is reconstructed on a uniform grid using B-splines basis functions. The computation of spline weights is formulated as a variational problem. Specifically, we minimize a cost that is a weighted sum of two terms: (i) the sum of squared errors at the specified points; (ii) a quadratic functional that penalizes the lack of smoothness. The problem is equivalent to solving a very large system of linear equations, with the dimension equal to the number of grid points. We develop a computationally efficient multiresolution-multigrid scheme for solving the system. We demonstrate the method with image reconstruction from contour points.
The Pairing of a Wavelet Basis With a Mildly Redundant Analysis via Subband Regression

A distinction is usually made between wavelet bases and wavelet frames. The former are associated with a one-to-one representation of signals, which is somewhat constrained but most efficient computationally. The latter are over-complete, but they offer advantages in terms of flexibility (shape of the basis functions) and shift-invariance. In this paper, we propose a framework for improved wavelet analysis based on an appropriate pairing of a wavelet basis with a mildly redundant version of itself (frame). The processing is accomplished in four steps: 1) redundant wavelet analysis, 2) wavelet-domain processing, 3) projection of the results onto the wavelet basis, and 4) reconstruction of the signal from its nonredundant wavelet expansion. The wavelet analysis is pyramid-like and is obtained by simple modification of Mallat's filterbank algorithm (e.g., suppression of the down-sampling in the wavelet channels only). The key component of the method is the subband regression filter (Step 3) which computes a wavelet expansion that is maximally consistent in the least squares sense with the redundant wavelet analysis. We demonstrate that this approach significantly improves the performance of soft-threshold wavelet denoising with a moderate increase in computational cost. We also show that the analysis filters in the proposed framework can be adjusted for improved feature detection; in particular, a new quincunx Mexican-hat-like wavelet transform that is fully reversible and essentially behaves the (gamma/2)th Laplacian of a Gaussian.
One-Bit Measurements With Adaptive Thresholds

We introduce a new method for adaptive one-bit quantization of linear measurements and propose an algorithm for the recovery of signals based on generalized approximate message passing (GAMP). Our method exploits the prior statistical information on the signal for estimating the minimum-mean-squared error solution from one-bit measurements. Our approach allows the one-bit quantizer to use thresholds on the real line. Given the previous measurements, each new threshold is selected so as to partition the consistent region along its centroid computed by GAMP. We demonstrate that the proposed adaptive-quantization scheme with GAMP reconstruction greatly improves the performance of signal and image recovery from one-bit measurements.
On the approximation power of convolution-based least squares versus interpolation

There are many signal processing tasks for which convolution-based continuous signal representations such as splines and wavelets provide an interesting and practical alternative to the more traditional sine-based methods. The coefficients of the corresponding signal approximations are typically obtained by direct sampling (interpolation or quasi-interpolation) or by using least squares techniques that apply a prefilter prior to sampling. We compare the performance of these approaches and provide quantitative error estimates that can be used for the appropriate selection of the sampling step h. Specifically, we review several results in approximation theory with a special emphasis on the Strang-Fix (1971) conditions, which relate the general O(h/sup L/) behavior of the error to the ability of the representation to reproduce polynomials of degree n=L-1. We use this theory to derive pointwise error estimates for the various algorithms and to obtain the asymptotic limit of the L/sub 2/-error as h tends to zero. We also propose a new improved L/sub 2/-error bound for the least squares case. In the process, we provide all the relevant bound constants for polynomial splines. Some of our results suggest the existence of an intermediate range of sampling steps where the least squares method is roughly equivalent to an interpolator with twice the order. We present experimental examples that illustrate the theory and confirm the adequacy of our various bound and limit determinations.
Mathematical properties of the JPEG2000 wavelet filters

The LeGall 5/3 and Daubechies 9/7 filters have risen to special prominence because they were selected for inclusion in the JPEG2000 standard. We determine their key mathematical features: Riesz bounds, order of approximation, and regularity (Hölder and Sobolev). We give approximation theoretic quantities such as the asymptotic constant for the L2 error and the angle between the analysis and synthesis spaces which characterizes the loss of performance with respect to an orthogonal projection. We also derive new asymptotic error formulae that exhibit bound constants that are proportional to the magnitude of the first nonvanishing moment of the wavelet. The Daubechies 9/7 stands out because it is very close to orthonormal, but this turns out to be slightly detrimental to its asymptotic performance when compared to other wavelets with four vanishing moments.
Trigonometric Interpolation Kernel to Construct Deformable Shapes for User-Interactive Applications

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.
Generalized total variation denoising via augmented Lagrangian cycle spinning with Haar wavelets

We consider the denoising of signals and images using regularized least-squares method. In particular, we propose a simple minimization algorithm for regularizers that are functions of the discrete gradient. By exploiting the connection of the discrete gradient with the Haar-wavelet transform, the n-dimensional vector minimization can be decoupled into n scalar minimizations. The proposed method can efficiently solve total-variation (TV) denoising by iteratively shrinking shifted Haar-wavelet transforms. Furthermore, the decoupling naturally lends itself to extensions beyond ℓ1 regularizers.
The analog formulation of sparsity implies infinite divisibility and rules out Bernoulli-Gaussian priors

Motivated by the analog nature of real-world signals, we investigate continuous-time random processes. For this purpose, we consider the stochastic processes that can be whitened by linear transformations and we show that the distribution of their samples is necessarily infinitely divisible. As a consequence, such a modeling rules out the Bernoulli-Gaussian distribution since we are able to show in this paper that it is not infinitely divisible. In other words, while the Bernoulli-Gaussian distribution is among the most studied priors for modeling sparse signals, it cannot be associated with any continuous-time stochastic process. Instead, we propose to adapt the priors that correspond to the increments of compound Poisson processes, which are both sparse and infinitely divisible.
Hessian Schatten-Norm Regularization for Linear Inverse Problems

We introduce a novel family of invariant, convex, and non-quadratic functionals that we employ to derive regularized solutions of ill-posed linear inverse imaging problems. The proposed regularizers involve the Schatten norms of the Hessian matrix, which are computed at every pixel of the image. They can be viewed as second-order extensions of the popular total-variation (TV) semi-norm since they satisfy the same invariance properties. Meanwhile, by taking advantage of second-order derivatives, they avoid the staircase effect, a common artifact of TV-based reconstructions, and perform well for a wide range of applications. To solve the corresponding optimization problems, we propose an algorithm that is based on a primal-dual formulation. A fundamental ingredient of this algorithm is the projection of matrices onto Schatten norm balls of arbitrary radius. This operation is performed efficiently based on a direct link we provide between vector projections onto norm balls and matrix projections onto Schatten norm balls. Finally, we demonstrate the effectiveness of the proposed methods through experimental results on several inverse imaging problems with real and simulated data.
Steerable Wavelet Machines (SWM): Learning Moving Frames for Texture Classification

We present texture operators encoding class-specific local organizations of image directions (LOIDs) in a rotation-invariant fashion. The LOIDs are key for visual understanding, and are at the origin of the success of the popular approaches, such as local binary patterns (LBPs) and the scale-invariant feature transform (SIFT). Whereas, LBPs and SIFT yield hand-crafted image representations, we propose to learn data-specific representations of the LOIDs in a rotation-invariant fashion. The image operators are based on steerable circular harmonic wavelets (CHWs), offering a rich and yet compact initial representation for characterizing natural textures. The joint location and orientation required to encode the LOIDs is preserved by using moving frames (MFs) texture representations built from locally-steered image gradients that are invariant to rigid motions. In a second step, we use support vector machines to learn a multi-class shaping matrix for the initial CHW representation, yielding data-driven MFs called steerable wavelet machines (SWMs). The SWM forward function is composed of linear operations (i.e., convolution and weighted combinations) interleaved with non-linear steermax operations. We experimentally demonstrate the effectiveness of the proposed operators for classifying natural textures. Our scheme outperforms recent approaches on several test suites of the Outex and the CUReT databases.
Higher-order riesz transforms and steerablewavelet frames

We introduce an Nth-order extension of the Riesz transform in d dimensions. We prove that this generalized transform has the following remarkable properties: shift-invariance, scale-invariance, inner-product preservation, and steerability. The pleasing consequence is that the transform maps any primary wavelet frame (or basis) of L2(ℝd) into another “steerable” wavelet frame, while preserving the frame bounds. The concept provides a rigorous functional counterpart to Simoncelli's steerable pyramid whose construction was entirely based on digital filter design. The proposed mechanism allows for the specification of wavelets with any order of steerability in any number of dimensions; it also yields a perfect reconstruction filterbank algorithm. We illustrate the method using a Mexican-hat-like polyharmonic spline wavelet transform as our primary frame.
Fractional Brownian models for vector field data

In this note we introduce a vector generalization of fractional Brownian motion. Our definition takes into account directional properties of vector fields—such as divergence, rotational behaviour, and interactions with coordinate transformations—that have no counterpart in the scalar setting. Apart from the Hurst exponent which dictates the scale-dependent structure of the field, additional parameters of the new model control the balance between solenoidal and irrotational behaviour. This level of versatility makes these random fields potentially interesting candidates for the stochastic modelling of physical phenomena in various fields of application such as fluid dynamics, field theory, and medical image processing.
SUB-RESOLUTION MAXIMUM-LIKELIHOOD BASED LOCALIZATION OF FLUORESCENT NANOPARTICLES IN THREE DIMENSIONS

Several recent studies have shown that fluorescent particles can be localized with an accuracy that is well beyond traditional resolution limits. Using a theoretical model of the image formation process that accounts for possible sources of noise, Cramer-Rao bounds have been used to define the theoretical limits. A crucial influence on these bounds is the mismatch of refractive indices that is usually present between immersion medium and specimen. This results in an axially shift-variant point spread function, meaning that the bounds change as a function of the particle's position in the z-direction. We investigate the theoretical bounds for this shift-variant model, and propose a maximum-likelihood estimator for the particle position in 3D (XYZ position). Using this estimator, sub-resolution localization at the nanometer scale is demonstrated on experimental data. The results provide optimal conditions for particle tracking and localization experiments.
Shape Projectors for Landmark-Based Spline Curves

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.
Recursive filtering for splines on hexagonal lattices

Hex-splines are a novel family of bivariate splines which are well suited to handle hexagonally sampled data. Similar to classical 1D B-splines, the spline coefficients need to be computed by a prefilter. Unfortunately, the elegant implementation of this prefilter by causal and anti-causal recursive filtering is not applicable for the (non-separable) hex-splines. Therefore, in this paper we introduce a novel approach from the viewpoint of approximation theory. We propose three different recursive filters and optimize their parameters such that a desired order of approximation is obtained. The results for third and fourth order hex-splines are discussed. Although the proposed solutions provide only quasi-interpolation, they tend to be very close to the interpolation prefilter.
WSPM or How to Obtain Statistical Parametric Maps using Shift-Invariant Wavelet Processing

Recently, we have proposed a new framework for detecting brain activity from fMRI data, which is based on the spatial discrete wavelet transform. The standard wavelet-based approach performs a statistical test in the wavelet domain, and therefore fails to provide a rigorous statistical interpretation in the spatial domain. The new framework provides an "integrated" approach: the data is processed in the wavelet domain (by thresholding wavelet coefficients), and a suitable statistical testing procedure is applied afterwards in the spatial domain. This method is based on conservative assumptions only and has a strong type-I error control by construction. At the same time, it has a sensitivity comparable to that of SPM. Here, we discuss the extension of our algorithm to the redundant discrete wavelet transform, which provides a shift-invariant detection scheme. The key features of our technique are illustrated with experimental results. An implementation of our framework is available as a toolbox (WSPM) for the SPM2 software
Deep Convolutional Neural Network for Inverse Problems in Imaging

In this paper, we propose a novel deep convolutional neural network (CNN)-based algorithm for solving ill-posed inverse problems. Regularized iterative algorithms have emerged as the standard approach to ill-posed inverse problems in the past few decades. These methods produce excellent results, but can be challenging to deploy in practice due to factors including the high computational cost of the forward and adjoint operators and the difficulty of hyperparameter selection. The starting point of this paper is the observation that unrolled iterative methods have the form of a CNN (filtering followed by pointwise non-linearity) when the normal operator (<inline-formula> <tex-math notation="LaTeX">$H^{*}H$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$H^{*}$ </tex-math></inline-formula> is the adjoint of the forward imaging operator, <inline-formula> <tex-math notation="LaTeX">$H$ </tex-math></inline-formula>) of the forward model is a convolution. Based on this observation, we propose using direct inversion followed by a CNN to solve normal-convolutional inverse problems. The direct inversion encapsulates the physical model of the system, but leads to artifacts when the problem is ill posed; the CNN combines multiresolution decomposition and residual learning in order to learn to remove these artifacts while preserving image structure. We demonstrate the performance of the proposed network in sparse-view reconstruction (down to 50 views) on parallel beam X-ray computed tomography in synthetic phantoms as well as in real experimental sinograms. The proposed network outperforms total variation-regularized iterative reconstruction for the more realistic phantoms and requires less than a second to reconstruct a <inline-formula> <tex-math notation="LaTeX">$512\times 512$ </tex-math></inline-formula> image on the GPU.
Sparsity and Infinite Divisibility

We adopt an innovation-driven framework and investigate the sparse/compressible distributions obtained by linearly measuring or expanding continuous-domain stochastic models. Starting from the first principles, we show that all such distributions are necessarily infinitely divisible. This property is satisfied by many distributions used in statistical learning, such as Gaussian, Laplace, and a wide range of fat-tailed distributions, such as student's-t and α-stable laws. However, it excludes some popular distributions used in compressed sensing, such as the Bernoulli-Gaussian distribution and distributions, that decay like exp (- O(|x|p)) for 1 <; p <; 2. We further explore the implications of infinite divisibility on distributions and conclude that tail decay and unimodality are preserved by all linear functionals of the same continuous-domain process. We explain how these results help in distinguishing suitable variational techniques for statistically solving inverse problems like denoising.
A second-order extension of TV regularization for image deblurring

In this paper, we propose a novel second-order regularizer based on the maximum response of the second-order directional derivative, assuming that the image under consideration belongs to the class of piecewise-linear signals. Compared to total-variation regularization that preserves edges but transforms piecewise-smooth regions into piecewise-constant regions, the proposed model is able to restore piecewise-linear regions and finer details. Deconvolution experiments demonstrate the performance of our approach in terms of the quality of reconstruction.
Bayesian denoising of generalized poisson processes with finite rate of innovation

We investigate the problem of the optimal reconstruction of a generalized Poisson process from its noisy samples. The process is known to have a finite rate of innovation since it is generated by a random stream of Diracs with a finite average number of impulses per unit interval. We formulate the recovery problem in a Bayesian framework and explicitly derive the joint probability density function (pdf) of the sampled signal. We compare the performance of the optimal Minimum Mean Square Error (MMSE) estimator with common regularization techniques such as ℓ1 and Log penalty functions. The simulation results indicate that, under certain conditions, the regularization techniques can achieve a performance close to the MMSE method.
Sparse Stochastic Processes and Discretization of Linear Inverse Problems

We present a novel statistically-based discretization paradigm and derive a class of maximum a posteriori (MAP) estimators for solving ill-conditioned linear inverse problems. We are guided by the theory of sparse stochastic processes, which specifies continuous-domain signals as solutions of linear stochastic differential equations. Accordingly, we show that the class of admissible priors for the discretized version of the signal is confined to the family of infinitely divisible distributions. Our estimators not only cover the well-studied methods of Tikhonov and l1-type regularizations as particular cases, but also open the door to a broader class of sparsity-promoting regularization schemes that are typically nonconvex. We provide an algorithm that handles the corresponding nonconvex problems and illustrate the use of our formalism by applying it to deconvolution, magnetic resonance imaging, and X-ray tomographic reconstruction problems. Finally, we compare the performance of estimators associated with models of increasing sparsity.