nils-herrmann/clean_scirepeval huggingface dataset:
Representer Theorems for Sparsity-Promoting  $\ell _{1}$  Regularization

We present a theoretical analysis and comparison of the effect of l1 versus l2 regularization for the resolution of ill-posed linear inverse and/or compressed sensing problems. Our formulation covers the most general setting where the solution is specified as the minimizer of a convex cost functional. We derive a series of representer theorems that give the generic form of the solution depending on the type of regularization. We start with the analysis of the problem in finite dimensions and then extend our results to the infinite-dimensional spaces l2(Z) and l1(Z). We also consider the use of linear transformations in the form of dictionaries or regularization operators. In particular, we show that the l2 solution is forced to live in a predefined subspace that is intrinsically smooth and tied to the measurement operator. The l1 solution, on the other hand, is formed by adaptively selecting a subset of atoms in a dictionary that is specified by the regularization operator. Beside the proof that l1 solutions are intrinsically sparse, the main outcome of our investigation is that the use of l1 regularization is much more favorable for injecting prior knowledge: it results in a functional form that is independent of the system matrix, while this is not so in the l2 scenario.
Optimized Wavelet Denoising for Self-Similar $\alpha $ -Stable Processes

We investigate the performance of wavelet shrinkage methods for the denoising of symmetric-<inline-formula> <tex-math notation="LaTeX">${\alpha }$ </tex-math></inline-formula>-stable (<inline-formula> <tex-math notation="LaTeX">$\text{S} {\alpha }\text{S}$ </tex-math></inline-formula>) self-similar stochastic processes corrupted by additive white Gaussian noise (AWGN), where <inline-formula> <tex-math notation="LaTeX">$ {\alpha }$ </tex-math></inline-formula> is tied to the sparsity of the process. The wavelet transform is assumed to be orthonormal and the shrinkage function minimizes the mean-square approximation error (MMSE estimator). We derive the corresponding formula for the expected value of the averaged estimation error. We show that the predicted MMSE is a monotone function of a simple criterion that depends on the wavelet and the statistical parameters of the process. Using the calculus of variations, we then optimize this criterion to find the best performing wavelet within the extended family of Meyer wavelets, which are bandlimited. These are compared with the Daubechies wavelets, which are compactly supported in time. We find that the wavelets that are shorter in time (in particular, the Haar basis) are better suited to denoise the sparser processes (say, <inline-formula> <tex-math notation="LaTeX">$ {\alpha <1.2}$ </tex-math></inline-formula>), while the bandlimited ones (including the Held and Shannon wavelets) offer the best performance for <inline-formula> <tex-math notation="LaTeX">$ {\alpha >1.6}$ </tex-math></inline-formula>, the limit corresponding to the Gaussian case (fBm) with <inline-formula> <tex-math notation="LaTeX">$ {\alpha =2}$ </tex-math></inline-formula>.
Multiresolution moment filters: theory and applications

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.
On the Unique Identification of Continuous-Time Autoregressive Models From Sampled Data

In this work, we investigate the relationship between continuous-time autoregressive (AR) models and their sampled version. We consider uniform sampling and derive criteria for uniquely determining the continuous-time parameters from sampled data; the model order is assumed to be known. We achieve this by removing a set of measure zero from the collection of all AR models and by investigating the asymptotic behavior of the remaining set of autocorrelation functions. We provide necessary and sufficient conditions for uniqueness of general AR models, and we demonstrate the usefulness of this result by considering particular examples. We further exploit our theory and introduce an estimation algorithm that recovers continuous-time AR parameters from sampled data, regardless of the sampling interval. We demonstrate the usefulness of our algorithm for various Gaussian and non-Gaussian AR processes.
Maximum likelihood estimation of liner signal parameters for Poisson processes

The estimation of linear signal parameters is studied under the hypothesis of independently Poisson distributed measurements. A simple interactive maximum-likelihood estimator (MLE) is derived and is optimized for rapid convergence. It is shown to be statistically optimal in the sense of providing unbiased and minimum variance estimates. Experimental conditions are identified where MLE results in significant performance improvement when compared to conventional linear least-squares (LLS).
Poisson Image Reconstruction With Hessian Schatten-Norm Regularization

Poisson inverse problems arise in many modern imaging applications, including biomedical and astronomical ones. The main challenge is to obtain an estimate of the underlying image from a set of measurements degraded by a linear operator and further corrupted by Poisson noise. In this paper, we propose an efficient framework for Poisson image reconstruction, under a regularization approach, which depends on matrix-valued regularization operators. In particular, the employed regularizers involve the Hessian as the regularization operator and Schatten matrix norms as the potential functions. For the solution of the problem, we propose two optimization algorithms that are specifically tailored to the Poisson nature of the noise. These algorithms are based on an augmented-Lagrangian formulation of the problem and correspond to two variants of the alternating direction method of multipliers. Further, we derive a link that relates the proximal map of an lp norm with the proximal map of a Schatten matrix norm of order p. This link plays a key role in the development of one of the proposed algorithms. Finally, we provide experimental results on natural and biological images for the task of Poisson image deblurring and demonstrate the practical relevance and effectiveness of the proposed framework.
Generalized sampling: stability and performance analysis

Generalized sampling provides a general mechanism for recovering an unknown input function f(x)/spl epsiv//spl Hscr/ from the samples of the responses of m linear shift-invariant systems sampled at 1/mth the reconstruction rate. The system can be designed to perform a projection of f(x) onto the reconstruction subspace V(/spl phi/)=span  for example, the family of bandlimited signals with /spl phi/(x)=sinc(x). This implies that the reconstruction will be perfect when the input signal is included in V(/spl phi/): the traditional framework of Papoulis' (1977) generalized sampling theory. Otherwise, one recovers a signal approximation f(x)/spl epsiv/V(/spl phi/) that is consistent with f(x) in the sense that it produces the same measurements. To characterize the stability of the algorithm, we prove that the dual synthesis functions that appear in the generalized sampling reconstruction formula constitute a Riesz basis of V(/spl phi/), and we use the corresponding Riesz bounds to define the condition number of the system. We then use these results to analyze the stability of various instances of interlaced and derivative sampling. Next, we consider the issue of performance, which becomes pertinent once we have extended the applicability of the method to arbitrary input functions, that is, when /spl Hscr/ is considerably larger than V(/spl phi/), and the reconstruction is no longer exact. We show that the generalized sampling solution is essentially equivalent to the optimal minimum error approximation. We then perform a detailed analysis for the case in which the analysis filters are in L/sub 2/ and determine all relevant bound constants explicitly. Finally, we use an interlaced sampling example to illustrate these various calculations.
High-quality image resizing using oblique projection operators

The standard interpolation approach to image resizing is to fit the original picture with a continuous model and resample the function at the desired rate. However, one can obtain more accurate results if one applies a filter prior to sampling, a fact well known from sampling theory. The optimal solution corresponds to an orthogonal projection onto the underlying continuous signal space. Unfortunately, the optimal projection prefilter is difficult to implement when sine or high order spline functions are used. We propose to resize the image using an oblique rather than an orthogonal projection operator in order to make use of faster, simpler, and more general algorithms. We show that we can achieve almost the same result as with the orthogonal projection provided that we use the same approximation space. The main advantage is that it becomes perfectly feasible to use higher order models (e.g. splines of degree n=or>3). We develop the theoretical background and present a simple and practical implementation procedure using B-splines. Our experiments show that the proposed algorithm consistently outperforms the standard interpolation methods and that it provides essentially the same performance as the optimal procedure (least squares solution) with considerably fewer computations. The method works for arbitrary scaling factors and is applicable to both image enlargement and reduction.
Fast parametric snakes for 3D microscopy

We present a new fast active contour for images in 3D microscopy. We introduce a fully parametric design that relies on exponential B-spline bases and allows us to impose a sphere-like topology. The proposed 3D snake can approximate blob-like objects with good accuracy. The optimization process is remarkably fast. Our technique yields successful segmentation results even for a challenging data set where object contours are not well defined. This happens because our parametric approach allows us to favor prior shapes. This work comes with a companion software that allows extensive interactions between the end-user and our snakes through the intuitive manipulation of the few control points that fully characterize them.
Wavelets, statistics, and biomedical applications

This paper emphasizes the statistical properties of the wavelet transform (WT) and discusses some examples of applications in medicine and biology. The redundant forms of the transform (continuous wavelet transform (CWT) and wavelet frames) are well suited for detection tasks (e.g., spikes in EEG, or microcalcifications in mammograms). The CWT, in particular, can be interpreted as a prewhitening multi-scale matched filter. Redundant wavelet decompositions are also very useful for the characterization of singularities, as well as for the time-frequency analysis of non-stationary signals. We discuss some examples of applications in phonocardiography, electrocardiography (EGG), and electroencephalography (EEG). Wavelet bases (WB) provide a similar, non-redundant decomposition of a signal in terms of the shifts and dilations of a wavelet. This makes WB well suited for any of the tasks for which block transforms have been used traditionally. Wavelets, however, may present certain advantages because they can improve the signal-to-noise ratio, while retaining a certain degree of localization in the time (or space) domain. We present three illustrative examples. The first is a denoising technique that applies a soft threshold in the wavelet domain. The second is a more refined version that uses generalized Wiener filtering. The third is a statistical method for detecting and locating patterns of brain activity in functional images acquired using magnetic resonance imaging (MRI). We conclude by describing a wavelet generalization of the classical Karhunen-Loeve transform.
Exponential Hermite splines for the analysis of biomedical images

We present a new exponential B-spline basis that enables the construction of active contours for the analysis of biomedical images. Our functions generalize the well-known polynomial Hermite B-splines and provide us with a direct control over the tangents of the parameterized contour, which is absent in traditional spline-based active contours. Our basis functions have been designed to perfectly reproduce elliptical and circular shapes. Moreover, they can approximate any closed curve up to arbitrary precision by increasing the number of anchor points. They are therefore well-suited to the segmentation of the roundish objects that are commonly encountered in the analysis of bioimages. We illustrate the performance of an active contour built using our functions on some examples of real biological data.
Nonideal sampling and interpolation from noisy observations in shift-invariant spaces

Digital analysis and processing of signals inherently relies on the existence of methods for reconstructing a continuous-time signal from a sequence of corrupted discrete-time samples. In this paper, a general formulation of this problem is developed that treats the interpolation problem from ideal, noisy samples, and the deconvolution problem in which the signal is filtered prior to sampling, in a unified way. The signal reconstruction is performed in a shift-invariant subspace spanned by the integer shifts of a generating function, where the expansion coefficients are obtained by processing the noisy samples with a digital correction filter. Several alternative approaches to designing the correction filter are suggested, which differ in their assumptions on the signal and noise. The classical deconvolution solutions (least-squares, Tikhonov, and Wiener) are adapted to our particular situation, and new methods that are optimal in a minimax sense are also proposed. The solutions often have a similar structure and can be computed simply and efficiently by digital filtering. Some concrete examples of reconstruction filters are presented, as well as simple guidelines for selecting the free parameters (e.g., regularization) of the various algorithms
Polyharmonic smoothing splines for multi-dimensional signals with 1/ /spl par//spl omega//spl par//sup /spl tau//-like spectra [image denoising applications]

Motivated by the fractal-like behavior of natural images, we propose a new smoothing technique that uses a regularization functional which is a fractional iterate of the Laplacian. This type of functional has previously been introduced by Duchon in the context of radial basis functions (RBFs) for the approximation of non-uniform data. Here, we introduce a new solution to Duchon's smoothing problem in multiple dimensions using non-separable fractional polyharmonic B-splines. The smoothing is performed in the Fourier domain by filtering, thereby making the algorithm fast enough for most multi-dimensional real-time applications.
Monte-Carlo Sure: A Black-Box Optimization of Regularization Parameters for General Denoising Algorithms

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.
Pixel intensity distribution models for filtered back-projection

The authors define a pixel intensity distribution model (PIDM) to study the discretization of direct reconstruction schemes in a proper way. For the filtered backprojection (FBP) algorithm, this leads to a derivation of the filter for a B-spline PIDM and its discretization, a simple but exact implementation for this class of functions, and a rule for the sampling ratio relating pixel size to projection cell size according to the device characteristics. Actual phantom reconstructions are presented both for standard and spline FBP schemes. The degree of the B-spline reconstruction is discussed in connection with the angular sampling and the kind of detection task to be applied on the image.<<ETX>>
Fast Space-Variant Elliptical Filtering Using Box Splines

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.
Generalized Daubechies Wavelet Families

We present a generalization of the orthonormal Daubechies wavelets and of their related biorthogonal flavors (Cohen-Daubechies-Feauveau, 9/7). Our fundamental constraint is that the scaling functions should reproduce a predefined set of exponential polynomials. This allows one to tune the corresponding wavelet transform to a specific class of signals, thereby ensuring good approximation and sparsity properties. The main difference with the classical construction of Daubechies is that the multiresolution spaces are derived from scale-dependent generating functions. However, from an algorithmic standpoint, Mallat's fast wavelet transform algorithm can still be applied; the only adaptation consists in using scale-dependent filter banks. Finite support ensures the same computational efficiency as in the classical case. We characterize the scaling and wavelet filters, construct them and show several examples of the associated functions. We prove that these functions are square-integrable and that they converge to their classical counterparts of the corresponding order.
USER-friendly image-based segmentation and analysis of chromosomes

We designed two efficient and user-friendly tools for the segmentation and analysis of images containing chromosomes or, more generally, rod-shaped elements that are spread on microscopic slides. The segmentation tool allows to automatically extract the profile of each chromosome and to sort the collection of profiles in a karyotype image. The analysis tool is interactive and allows to extract quantitative measurements and annotate the relative position of the centromere to the chromosome extremities in a fast and reproducible way. The two methods rely on custom variants of parametric active contours. Both have been designed as user-friendly plug-ins for the open-source software ImageJ.
A spline-based forward model for Optical Diffuse Tomography

Reconstruction algorithms for Optical Diffuse Tomography (ODT) rely heavily on fast and accurate forward models. Arbitrary geometries and boundary conditions need to be handled rigorously since they are the only input to the inverse problem. From this perspective, Finite Element Methods (FEM) are good candidates to implement a forward model. However, these methods require to mesh the domain of interest, which is impractical on a routine basis. The other downside of the FEM is that the basis functions are often not compatible with the ones used for solving the inverse problem, which typically have less degrees of freedom. In this work, we tackle the 2D problem, and propose a forward model that uses a mesh-free discretization based on linear B-Splines. It combines the advantages of the FEM, while offering a fast and much simpler way of handling complex geometries. Another motivation for this work is that the underlying B-spline model is equally suitable for the subsequent reconstruction part of the process (solving the inverse problem). In particular, it is compatible with wavelets and multiresolution-type signal representations.
Region of interest X-ray computed tomography via corrected back projection

We present a new method for region of interest (ROI) reconstruction from high-resolution, nontruncated data in parallel-ray X-ray computed tomography (CT). Many of the approaches to ROI CT reconstruction in the literature rely on a costly forward projection step to form an ROI-only sinogram. Our approach instead relies on a digital filtering implementation of the normal operator (HT H) to compute a back projected version of the ROI-only sinogram that can be used directly for reconstruction, thus eliminating the forward projection step altogether. Results on three synthetic datasets with a variety of experimental conditions show that the method provides accuracy on par with a full reconstruction at a fraction of the time and memory cost.