nils-herrmann/clean_scirepeval huggingface dataset:
Theoretical analysis of complex-conjugate-ambiguity suppression in frequency-domain optical-coherence tomography

New phase-shifting techniques have recently been proposed to suppress the complex-conjugate ambiguity in frequency- domain optical-coherence tomography. A phase shift is introduced, in an elegant fashion, by incorporating a small beam offset at the scanning mirror. The tomogram is then computed by using a combination of Hilbert and Fourier transforms. This is a marked deviation from the conventional approaches, wherein each A-scan is reconstructed independently of the others. In this paper, we formulate the problem in a signal processing framework and provide theoretical proofs for maximal and partial suppression of complex-conjugate ambiguity. To supplement the theoretical derivations, we provide experimental results on in vivo measurements of a human finger nail.
Self-Similarity: Part I—Splines and Operators

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 iff L{s(t)}=Sigma kisinZa[k]delta(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, parttau gamma, where gamma is the order of the derivative and tau is an additional phase factor. We specify the corresponding L-splines, which yield an extended class of fractional splines. The operator parttau gamma is used to define a scale-invariant energy measure-the squared L2-norm of the gammath 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 2gamma, 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 2gamma. We also establish a formal link between the regularization parameter lambda and the cutoff frequency of the smoothing spline filter: omega0aplambda-2gamma. 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
Nonlinear operators for improving texture segmentation based on features extracted by spatial filtering

An unsupervised texture segmentation system using texture features obtained from a combination of spatial filters and nonlinear operators is described. Local texture features are evaluated in parallel by a succession of four basic operations: (1) a convolution for local structure detection (local linear transform); (2) a first nonlinearity of the form f(x)= mod x mod /sup alpha /; (3) an iterative smoothing operator; and (4) a second nonlinearity g(x). The Karhunen-Loeve transform is used to reduce the dimensionality of the resulting feature vector, and segmentation is achieved by thresholding or clustering in feature space. The combination of nonlinearities f(x)= mod x mod /sup alpha / (in particular, alpha =2) and g(x)=log x maximizes texture discrimination, and results in a description with variances approximately constant for all feature components and texture regions. This latter property improves the performance of both feature reduction and clustering algorithms significantly.
Nonideal Sampling and Regularization Theory

Shannon's sampling theory and its variants provide effective solutions to the problem of reconstructing a signal from its samples in some "shift-invariant" space, which may or may not be bandlimited. In this paper, we present some further justification for this type of representation, while addressing the issue of the specification of the best reconstruction space. We consider a realistic setting where a multidimensional signal is prefiltered prior to sampling, and the samples are corrupted by additive noise. We adopt a variational approach to the reconstruction problem and minimize a data fidelity term subject to a Tikhonov-like (continuous domain) L2 -regularization to obtain the continuous-space solution. We present theoretical justification for the minimization of this cost functional and show that the globally minimal continuous-space solution belongs to a shift-invariant space generated by a function (generalized B-spline) that is generally not bandlimited. When the sampling is ideal, we recover some of the classical smoothing spline estimators. The optimal reconstruction space is characterized by a condition that links the generating function to the regularization operator and implies the existence of a B-spline-like basis. To make the scheme practical, we specify the generating functions corresponding to the most popular families of regularization operators (derivatives, iterated Laplacian), as well as a new, generalized one that leads to a new brand of Matern splines. We conclude the paper by proposing a stochastic interpretation of the reconstruction algorithm and establishing an equivalence with the minimax and minimum mean square error (MMSE/Wiener) solutions of the generalized sampling problem.
Trends in Bioimaging and Signal Processing [In the Spotlight]

The area of bioimaging and signal processing (BISP) is concerned with the development of dedicated tools for biomedical and bioinformatics applications and with fostering interdisciplinary and integrative approaches of biomedical topics by bridging the biology, medicine, and SP communities . Increasingly sophisticated imaging devices and protocols are now being developed for biological and medical imaging. Understanding functional and pathological mechanisms in patients or model organisms indeed requires being able to visualize and measure in vivo and in situ at scales ranging from subcellular to whole body. A similar trend occurs in the area of genomics and proteomics, where sophisticated bioinformatics methods are required to assemble, decipher, and compare data from large-scale sequencing data . This article focuses on the latest developments in biomedical imaging that has sparked a full array of challenging topics for the signal and image processing community.
Affine transformations of images: a least squares formulation

We present a general framework for the design of discrete geometrical transformation operators, including rotations and scaling. The first step is to fit the discrete input image with a continuous model that provides an exact interpolation at the pixel locations. The corresponding image model is selected within a certain subspace V(/spl phi/)/spl sub/L/sub 2/(R/sup P/) that is generated from the integer translates of a generating function /spl phi/; particular examples of this construction include polynomial spline and bandlimited signal representations. Next, the geometrical transformation is applied to the fitted model, and the result is re-projected onto the representation space. This procedure yields a solution that is optimal in the least squares sense. We show that this method can be implemented exactly using a combination of digital filters and a re-sampling step that uses a modified sampling kernel. We then derive explicit implementation formulas for the piecewise constant and cubic spline image models. Finally, we consider image processing examples and show that the present method compares very favorably with a standard interpolation that uses the same model.<<ETX>>
Automatic tracking of individual fluorescence particles: application to the study of chromosome dynamics

We present a new, robust, computational procedure for tracking fluorescent markers in time-lapse microscopy. The algorithm is optimized for finding the time-trajectory of single particles in very noisy dynamic (two- or three-dimensional) image sequences. It proceeds in three steps. First, the images are aligned to compensate for the movement of the biological structure under investigation. Second, the particle's signature is enhanced by applying a Mexican hat filter, which we show to be the optimal detector of a Gaussian-like spot in 1//spl omega//sup 2/ noise. Finally, the optimal trajectory of the particle is extracted by applying a dynamic programming optimization procedure. We have used this software, which is implemented as a Java plug-in for the public-domain ImageJ software, to track the movement of chromosomal loci within nuclei of budding yeast cells. Besides reducing trajectory analysis time by several 100-fold, we achieve high reproducibility and accuracy of tracking. The application of the method to yeast chromatin dynamics reveals different classes of constraints on mobility of telomeres, reflecting differences in nuclear envelope association. The generic nature of the software allows application to a variety of similar biological imaging tasks that require the extraction and quantitation of a moving particle's trajectory.
A Box Spline Calculus for the Discretization of Computed Tomography Reconstruction Problems

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 -direction box spline is generally a (nonuniform) polynomial spline of degree . 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.
From differential equations to the construction of new wavelet-like bases

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 L/sub 2/. 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 L/sub 2/-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.
Spline kernels for continuous-space image processing

We present an explicit formula for spline kernels; these are defined as the convolution of several B-splines of variable widths h/sub i/ and degrees n/sub i/. The spline kernels are useful for continuous signal processing algorithms that involve B-spline inner-products or the convolution of several spline basis functions. We apply our results to the derivation of spline-based algorithms for two classes of problems. The first is the resizing of images with arbitrary scaling factors. The second is the computation of the Radon transform and of its inverse. In particular, we present a new spline-based version of the filtered back-projection algorithm for tomographic reconstruction. In both cases, our explicit kernel formula allows for the use of high-degree splines; these offer better approximation performance than the conventional lower-order formulations (e.g., piecewise constant or piece-wise linear models).
Variational image reconstruction from arbitrarily spaced samples: a fast multiresolution spline solution

We propose a novel method for image reconstruction from nonuniform samples with no constraints on their locations. We adopt a variational approach where the reconstruction is formulated as the minimizer of a cost that is a weighted sum of two terms: 1) the sum of squared errors at the specified points and 2) a quadratic functional that penalizes the lack of smoothness. We search for a solution that is a uniform spline and show how it can be determined by solving a large, sparse system of linear equations. We interpret the solution of our approach as an approximation of the analytical solution that involves radial basis functions and demonstrate the computational advantages of our approach. Using the two-scale relation for B-splines, we derive an algebraic relation that links together the linear systems of equations specifying reconstructions at different levels of resolution. We use this relation to develop a fast multigrid algorithm. We demonstrate the effectiveness of our approach on some image reconstruction examples.
Analytical Footprints: Compact Representation of Elementary Singularities in Wavelet Bases

We introduce a family of elementary singularities that are point-Hölder α-regular. These singularities are self-similar and are the Green functions of fractional derivative operators; i.e., by suitable fractional differentiation, one retrieves a Dirac δ function at the exact location of the singularity. We propose to use fractional operator-like wavelets that act as a multiscale version of the derivative in order to characterize and localize singularities in the wavelet domain. We show that the characteristic signature when the wavelet interacts with an elementary singularity has an asymptotic closed-form expression, termed the analytical footprint. Practically, this means that the dictionary of wavelet footprints is embodied in a single analytical form. We show that the wavelet coefficients of the (nonredundant) decomposition can be fitted in a multiscale fashion to retrieve the parameters of the underlying singularity. We propose an algorithm based on stepwise parametric fitting and the feasibility of the approach to recover singular signal representations.
Wavelet-regularized reconstruction for rapid MRI

We propose a reconstruction scheme adapted to MRI that takes advantage of a sparsity constraint in the wavelet domain. We show that artifacts are significantly reduced compared to conventional reconstruction methods. Our approach is also competitive with Total Variation regularization both in terms of MSE and computation time. We show that ℓ1 regularization allows partial recovery of the missing k-space regions. We also present a multi-level version that significantly reduces the computational cost.
Minimum support interpolators with optimum approximation properties

We investigate the functions of given approximation order L that have the smallest support. Those are shown to be linear combinations of the B-spline of degree L-1 and its L-1 first derivatives. We then show how to find the functions that minimize the asymptotic approximation constant among this finite dimension space; in particular, a tractable induction relation is worked out. Using these functions instead of splines, we observe that the approximation error is dramatically reduced, not only in the limit when the sampling step tends to zero, but also for higher values up to the Shannon rate. Finally, we show that those optimal functions satisfy a scaling equation, although less simple than the usual two-scale difference equation.
Generalized sampling: a variational approach .I. Theory

We consider the problem of reconstructing a multidimensional vector function f/sub in/: /spl Ropf//sup m//spl rarr//spl Ropf//sup n/ from a finite set of linear measures. These can be irregularly sampled responses of several linear filters. Traditional approaches reconstruct in an a priori given space, e.g., the space of bandlimited functions. Instead, we have chosen to specify a reconstruction that is optimal in the sense of a quadratic plausibility criterion J. First, we present the solution of the generalized interpolation problem. Later, we also consider the approximation problem, and we show that both lead to the same class of solutions. Imposing generally desirable properties on the reconstruction largely limits the choice of the criterion J. Linearity leads to a quadratic criterion based on bilinear forms. Specifically, we show that the requirements of translation, rotation, and scale-invariance restrict the form of the criterion to essentially a one-parameter family. We show that the solution can be obtained as a linear combination of generating functions. We provide analytical techniques to find these functions and the solution itself.
Continuous localization using sparsity constraints for high-density super-resolution microscopy

Super-resolution localization microscopy relies on sparse activation of photo-switchable probes. Such activation, however, introduces limited temporal resolution. High-density imaging overcomes this limitation by allowing several neighboring probes to be activated simultaneously. In this work, we propose an algorithm that incorporates a continuous-domain sparsity prior into the high-density localization problem. We use a Taylor approximation of the PSF, and rely on a fast proximal gradient optimization procedure. Unlike currently available methods that use discrete-domain sparsity priors, our approach does not restrict the estimated locations to a pre-defined sampling grid. Experimental results of simulated and real data demonstrate significant improvement over these methods in terms of accuracy, molecular identification and computational complexity.
Isotropic polyharmonic B-splines: scaling functions and wavelets

In this paper, we use polyharmonic B-splines to build multidimensional wavelet bases. These functions are nonseparable, multidimensional basis functions that are localized versions of radial basis functions. We show that Rabut's elementary polyharmonic B-splines do not converge to a Gaussian as the order parameter increases, as opposed to their separable B-spline counterparts. Therefore, we introduce a more isotropic localization operator that guarantees this convergence, resulting into the isotropic polyharmonic B-splines. Next, we focus on the two-dimensional quincunx subsampling scheme. This configuration is of particular interest for image processing because it yields a finer scale progression than the standard dyadic approach. However, up until now, the design of appropriate filters for the quincunx scheme has mainly been done using the McClellan transform. In our approach, we start from the scaling functions, which are the polyharmonic B-splines and, as such, explicitly known, and we derive a family of polyharmonic spline wavelets corresponding to different flavors of the semi-orthogonal wavelet transform; e.g., orthonormal, B-spline, and dual. The filters are automatically specified by the scaling relations satisfied by these functions. We prove that the isotropic polyharmonic B-spline wavelet converges to a combination of four Gabor atoms, which are well separated in the frequency domain. We also show that these wavelets are nearly isotropic and that they behave as an iterated Laplacian operator at low frequencies. We describe an efficient fast Fourier transform-based implementation of the discrete wavelet transform based on polyharmonic B-splines.
Mate/spl acute/rn B-splines and the optimal reconstruction of signals

Starting from the power spectral density of Mateacutern stochastic processes, we introduce a new family of splines that is defined in terms of the whitening operator of such processes. We show that these Mateacutern splines admit a stable representation in a B-spline-like basis. We specify the Mateacutern B-splines (causal and symmetric) and identify their key properties; in particular, we prove that these generate a Riesz basis and that they can be written as a product of an exponential with a fractional polynomial B-spline. We also indicate how these new functions bridge the gap between the fractional polynomial splines and the cardinal exponential ones. We then show that these splines provide the optimal reconstruction space for the minimum mean-squared error estimation of Mateacutern signals from their noisy samples. We also propose a digital Wiener-filter-like algorithm for the efficient determination of the optimal B-spline coefficients
A Sampling Theory Approach for Continuous ARMA Identification

The problem of estimating continuous-domain autoregressive moving-average processes from sampled data is considered. The proposed approach incorporates the sampling process into the problem formulation while introducing exponential models for both the continuous and the sampled processes. We derive an exact evaluation of the discrete-domain power-spectrum using exponential B-splines and further suggest an estimation approach that is based on digitally filtering the available data. The proposed functional, which is related to Whittle's likelihood function, exhibits several local minima that originate from aliasing. The global minimum, however, corresponds to a maximum-likelihood estimator, regardless of the sampling step. Experimental results indicate that the proposed approach closely follows the Cramér-Rao bound for various aliasing configurations.
Variational Phase Imaging Using the Transport-of-Intensity Equation

We introduce a variational phase retrieval algorithm for the imaging of transparent objects. Our formalism is based on the transport-of-intensity equation (TIE), which relates the phase of an optical field to the variation of its intensity along the direction of propagation. TIE practically requires one to record a set of defocus images to measure the variation of intensity. We first investigate the effect of the defocus distance on the retrieved phase map. Based on our analysis, we propose a weighted phase reconstruction algorithm yielding a phase map that minimizes a convex functional. The method is nonlinear and combines different ranges of spatial frequencies - depending on the defocus value of the measurements - in a regularized fashion. The minimization task is solved iteratively via the alternating-direction method of multipliers. Our simulations outperform commonly used linear and nonlinear TIE solvers. We also illustrate and validate our method on real microscopy data of HeLa cells.