nils-herrmann/clean_scirepeval huggingface dataset:
Weighted averaging of a set of noisy images for maximum signal-to-noise ratio

The problem of estimating a signal from a weighted average of N registered noisy observations is considered. A set of optimal weighting coefficients is determined by maximizing a signal-to-noise ratio criterion. This solution can be computed by first standardizing each observation with respect to its first and second moments and then evaluating the first eigenvector of the corresponding N*N inner-product matrix. The resulting average is shown to be proportional to the first basis vector of the Karhunen-Loeve transform provided that the data has been standardized in an appropriate fashion. The low sensitivity of this approach to the presence of outliers is illustrated by using real electron micrographs of ostensibly identical virus particles.
A multi-resolution feature reduction technique for image segmentation with multiple components

The authors present a linear feature-reduction technique for multicomponent or textured image segmentation. The transformation matrix is computed by simultaneously diagonalizing scatter matrices evaluated at two different spatial resolutions. Under reasonable conditions, this transform closely approximates the generalized Fisher linear discriminants which are optimal for region separability. Experimental examples suggest that this technique is superior to the Karhunen-Loeve transform for texture segmentation.<<ETX>>
Sampling-50 years after Shannon

This paper presents an account of the current state of sampling, 50 years after Shannon's formulation of the sampling theorem. The emphasis is on regular sampling, where the grid is uniform. This topic has benefitted from a strong research revival during the past few years, thanks in part to the mathematical connections that were made with wavelet theory. To introduce the reader to the modern, Hilbert-space formulation, we reinterpret Shannon's sampling procedure as an orthogonal projection onto the subspace of band-limited functions. We then extend the standard sampling paradigm for a presentation of functions in the more general class of "shift-in-variant" function spaces, including splines and wavelets. Practically, this allows for simpler-and possibly more realistic-interpolation models, which can be used in conjunction with a much wider class of (anti-aliasing) prefilters that are not necessarily ideal low-pass. We summarize and discuss the results available for the determination of the approximation error and of the sampling rate when the input of the system is essentially arbitrary; e.g., nonbandlimited. We also review variations of sampling that can be understood from the same unifying perspective. These include wavelets, multiwavelets, Papoulis generalized sampling, finite elements, and frames. Irregular sampling and radial basis functions are briefly mentioned.
Rapid computation of the continuous wavelet transform by oblique projections

We introduce a fast simple method for computing the real continuous wavelet transform (CWT). The approach has the following attractive features: it achieves O(N) complexity per scale, the filter coefficients can be analytically obtained by a simple integration, and the algorithm is faster than a least squares approach with negligible loss in accuracy. Our method is to use P wavelets per octave and to approximate them with their oblique projection onto a space defined by a compactly supported scaling function. The wavelet templates are expanded to larger sizes (octaves) using the two-scale relation and zero-padded filtering. Error bounds are presented to justify the use of an oblique projection over an orthogonal one. All the filters are FIR with the exception of one filter, which is implemented using a fast recursive algorithm.
Four-dimensional wavelet compression of arbitrarily sized echocardiographic data

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

We investigate a non quadratic regularizer that is based on the Hessian operator for dealing with the restoration of 3-D images in a variational framework. We show that the regularizer under study is a valid extension of the total-variation (TV) functional, in the sense that it retains its favorable properties while following a similar underlying principle. We argue that the new functional is well suited for the restoration of 3-D biological images since it does not suffer from the well-known staircase effect of TV. Furthermore, we present an efficient 3-D algorithm for the minimization of the corresponding objective function. Finally, we validate the overall proposed regularization framework through image deblurring experiments on simulated and real biological data.
Fast continuous wavelet transform

We introduce a general framework for the efficient computation of the continuous wavelet transform (CWT). The method allows arbitrary sampling along the scale axis, and achieves O(N) complexity per scale where N is the length of the signal. Our approach makes use of a compactly supported scaling function to approximate the analyzing wavelet. We derive error bounds on the wavelet approximation and show how to obtain any desired level of accuracy through the use of higher order representations. Finally, we present examples of implementation for different wavelets using polynomial spline approximations.
A Generalized Sampling Theory without bandlimiting constraints

We consider the problem of the reconstruction of a continuous-time function f(x)/spl isin/H from the samples of the responses of m linear shift-invariant systems sampled at 1/m the reconstruction rate. We extend Papoulis' generalized sampling theory in two important respects. First, our class of admissible input signals (typ. H=L/sub 2/) is considerably larger than the subspace of band-limited functions. Second, we use a more general specification of the reconstruction subspace V(/spl psi/), so that the output of the system can take the form of a band-limited function, a spline, or a wavelet expansion. Since we have enlarged the class of admissible input functions, we have to give up Shannon and Papoulis' principle of an exact reconstruction. Instead, we seek an approximation f/spl isin/V(/spl psi/) that is consistent in the sense that it produces exactly the same measurements as the input of the system. This leads to a generalization of Papoulis' sampling theorem and a practical reconstruction algorithm that takes the form of a multivariate filter. In particular, we show that the corresponding system acts as a projector from H onto V(/spl psi/). We then propose two complementary polyphase and modulation domain interpretations of our solution. The polyphase representation leads to a simple understanding of our reconstruction algorithm in terms of a perfect reconstruction filter bank. The modulation analysis, on the other hand, is useful in providing the connection with Papoulis' earlier results for the band-limited case. Finally, we illustrate the general applicability of our theory by presenting new examples of interlaced and derivative sampling using splines.
Wavelets, fractals, and radial basis functions

Wavelets and radial basis functions (RBFs) lead to two distinct ways of representing signals in terms of shifted basis functions. RBFs, unlike wavelets, are nonlocal and do not involve any scaling, which makes them applicable to nonuniform grids. Despite these fundamental differences, we show that the two types of representation are closely linked together ...through fractals. First, we identify and characterize the whole class of self-similar radial basis functions that can be localized to yield conventional multiresolution wavelet bases. Conversely, we prove that for any compactly supported scaling function /spl phi/(x), there exists a one-sided central basis function /spl rho//sub +/(x) that spans the same multiresolution subspaces. The central property is that the multiresolution bases are generated by simple translation of /spl rho//sub +/ without any dilation. We also present an explicit time-domain representation of a scaling function as a sum of harmonic splines. The leading term in the decomposition corresponds to the fractional splines: a recent, continuous-order generalization of the polynomial splines.
B-spline signal processing. II. Efficiency design and applications

For pt.I see ibid., vol.41, no.2, p.821-33 (1993). A class of recursive filtering algorithms for the efficient implementation of B-spline interpolation and approximation techniques is described. In terms of simplicity of realization and reduction of computational complexity, these algorithms compare favorably with conventional matrix approaches. A filtering interpretation (low-pass filter followed by an exact polynomial spline interpolator) of smoothing spline and least-squares approximation methods is proposed. These techniques are applied to the design of digital filters for cubic spline signal processing. An efficient implementation of a smoothing spline edge detector is proposed. It is also shown how to construct a cubic spline image pyramid that minimizes the loss of information in passage from one resolution level to the next. In terms of common measures of fidelity, this data structure appears to be superior to the Gaussian/Laplacian pyramid.
Spline multiresolutions and wavelet transforms

An extension of the family of Battle/Lemarie spline wavelet transforms is presented. By relaxing the intralevel orthogonality constraint, the authors show how to construct generalized polynomial spline scaling functions and wavelets that span the same multiresolution spaces, but can also exhibit very distinct properties. Particular examples in this family include the B-spline wavelets of compact support that are optimally localized in time-frequency; the cardinal spline wavelets that have the fundamental interpolation property; and the dual spline wavelets (the biorthogonal complement of the B-spline wavelets). A full characterization of the digital filters for the corresponding fast wavelet transform algorithms is provided. The asymptotic properties of these representations are discussed, and the link with Shannon's sampling theory and the Gabor transform is indicated.<<ETX>>
B-spline snakes: a flexible tool for parametric contour detection

We present a novel formulation for B-spline snakes that can be used as a tool for fast and intuitive contour outlining. We start with a theoretical argument in favor of splines in the traditional formulation by showing that the optimal, curvature-constrained snake is a cubic spline, irrespective of the form of the external energy field. Unfortunately, such regularized snakes suffer from slow convergence speed because of a large number of control points, as well as from difficulties in determining the weight factors associated to the internal energies of the curve. We therefore propose an alternative formulation in which the intrinsic scale of the spline model is adjusted a priori; this leads to a reduction of the number of parameters to be optimized and eliminates the need for internal energies (i.e., the regularization term). In other words, we are now controlling the elasticity of the spline implicitly and rather intuitively by varying the spacing between the spline knots. The theory is embedded into a multiresolution formulation demonstrating improved stability in noisy image environments. Validation results are presented, comparing the traditional snake using internal energies and the proposed approach without internal energies, showing the similar performance of the latter. Several biomedical examples of applications are included to illustrate the versatility of the method.
Cardinal exponential splines: part II - think analog, act digital

By interpreting the Green-function reproduction property of exponential splines in signal processing terms, we uncover a fundamental relation that connects the impulse responses of allpole analog filters to their discrete counterparts. The link is that the latter are the B-spline coefficients of the former (which happen to be exponential splines). Motivated by this observation, we introduce an extended family of cardinal splines-the generalized E-splines-to generalize the concept for all convolution operators with rational transfer functions. We construct the corresponding compactly supported B-spline basis functions, which are characterized by their poles and zeros, thereby establishing an interesting connection with analog filter design techniques. We investigate the properties of these new B-splines and present the corresponding signal processing calculus, which allows us to perform continuous-time operations, such as convolution, differential operators, and modulation, by simple application of the discrete version of these operators in the B-spline domain. In particular, we show how the formalism can be used to obtain exact, discrete implementations of analog filters. Finally, we apply our results to the design of hybrid signal processing systems that rely on digital filtering to compensate for the nonideal characteristics of real-world analog-to-digital (A-to-D) and D-to-A conversion systems.
Sparsity-Driven Reconstruction for FDOT With Anatomical Priors

In this paper we propose a method based on (2, 1)-mixed-norm penalization for incorporating a structural prior in FDOT image reconstruction. The effect of (2, 1)-mixed-norm penalization is twofold: first, a sparsifying effect which isolates few anatomical regions where the fluorescent probe has accumulated, and second, a regularization effect inside the selected anatomical regions. After formulating the reconstruction in a variational framework, we analyze the resulting optimization problem and derive a practical numerical method tailored to (2, 1)-mixed-norm regularization. The proposed method includes as particular cases other sparsity promoting regularization methods such as ℓ1-norm penalization and total variation penalization. Results on synthetic and experimental data are presented.
The L2-Polynomial Spline Pyramid

The authors are concerned with the derivation of general methods for the L/sub 2/ approximation of signals by polynomial splines. The main result is that the expansion coefficients of the approximation are obtained by linear filtering and sampling. The authors apply those results to construct a L/sub 2/ polynomial spline pyramid that is a parametric multiresolution representation of a signal. This hierarchical data structure is generated by repeated application of a REDUCE function (prefilter and down-sampler). A complementary EXPAND function (up-sampler and post-filter) allows a finer resolution mapping of any coarser level of the pyramid. Four equivalent representations of this pyramid are considered, and the corresponding REDUCE and EXPAND filters are determined explicitly for polynomial splines of any order n (odd). Some image processing examples are presented. It is demonstrated that the performance of the Laplacian pyramid can be improved significantly by using a modified EXPAND function associated with the dual representation of a cubic spline pyramid.
Interpretation of continuous-time autoregressive processes as random exponential splines

We consider the class of continuous-time autoregressive (CAR) processes driven by (possibly non-Gaussian) Lévy white noises. When the excitation is an impulsive noise, also known as compound Poisson noise, the associated CAR process is a random non-uniform exponential spline. Therefore, Poisson-type processes are relatively easy to understand in the sense that they have a finite rate of innovation. We show in this paper that any CAR process is the limit in distribution of a sequence of CAR processes driven by impulsive noises. Hence, we provide a new interpretation of general CAR processes as limits of random exponential splines. We illustrate our result with simulations.
A PSF-based approach to Biplane calibration in 3D super-resolution microscopy

Super-resolution localization microscopy methods such as PALM and STORM have been shown to provide imaging with resolutions up to a few tens of nanometers while using relatively simple setups. Biplane PALM has extended the PALM technique to three-dimensions, by simultaneously using two imaging planes, with different focal depths. A key aspect in achieving good axial localization results is the alignment of the two planes. Currently available approaches assume that misaligned planes only result in scaling and rotation of the PSF pattern. We show in this work that this does not necessarily hold true, especially in the presence of refractive index mismatch between the different optical layers. Instead, we suggest a calibration algorithm that relies on a realistic PSF model and finds the affine transform that relates the two planes with respect to a point source in the object domain. Our calibration algorithm also determines the defocus distance between the planes.
On the multidimensional extension of the quincunx subsampling matrix

The dilation matrix associated with the three-dimensional (3-D) face-centered cubic (FCC) sublattice is often considered to be the natural 3-D extension of the two-dimensional (2-D) quincunx dilation matrix. However, we demonstrate that both dilation matrices are of different nature: while the 2-D quincunx matrix is a similarity transform, the 3-D FCC matrix is not. More generally, we show that is impossible to obtain a dilation matrix that is a similarity transform and performs downsampling of the Cartesian lattice by a factor of two in more than two dimensions. Furthermore, we observe that the popular 3-D FCC subsampling scheme alternates between three different lattices: Cartesian, FCC, and quincunx. The latter one provides a less isotropic sampling density, a property that should be taken into account to properly orient 3-D data before processing using such a subsampling matrix.
Complex Wavelet Bases, Steerability, and the Marr-Like Pyramid

Our aim in this paper is to tighten the link between wavelets, some classical image-processing operators, and David Marr's theory of early vision. The cornerstone of our approach is a new complex wavelet basis that behaves like a smoothed version of the Gradient-Laplace operator. Starting from first principles, we show that a single-generator wavelet can be defined analytically and that it yields a semi-orthogonal complex basis of L 2(R2), irrespective of the dilation matrix used. We also provide an efficient FFT-based filterbank implementation. We then propose a slightly redundant version of the transform that is nearly translation-invariant and that is optimized for better steerability (Gaussian-like smoothing kernel). We call it the Marr-like wavelet pyramid because it essentially replicates the processing steps in Marr's theory of early vision. We use it to derive a primal wavelet sketch which is a compact description of the image by a multiscale, subsampled edge map. Finally, we provide an efficient iterative algorithm for the reconstruction of an image from its primal wavelet sketch.
Polynomial spline signal processing algorithms

The authors describe a novel digital filtering algorithms for the processing and representation of signals using polynomial splines. The classical polynomial spline interpolation problem is considered. It is found that it can be solved efficiently by recursive digital filtering. This result also yields a simple procedure for signal differentiation. Filters that efficiently solve the problem of smoothing spline approximations are derived. This technique is a regularized version of spline interpolation and is therefore less sensitive to noise. It is applied to the design of a robust edge detection algorithm with an adjustable scale parameter. A filtering/sampling algorithm for least squares spline approximation is described. This data reduction technique is applied to the generation of a cubic spline image pyramid that is found to compare favorably with the Gauss/Laplace pyramid.<<ETX>>
Fast $O(1)$  Bilateral Filtering Using Trigonometric Range Kernels

It is well known that spatial averaging can be realized (in space or frequency domain) using algorithms whose complexity does not scale with the size or shape of the filter. These fast algorithms are generally referred to as constant-time or O(1) algorithms in the image-processing literature. Along with the spatial filter, the edge-preserving bilateral filter involves an additional range kernel. This is used to restrict the averaging to those neighborhood pixels whose intensity are similar or close to that of the pixel of interest. The range kernel operates by acting on the pixel intensities. This makes the averaging process nonlinear and computationally intensive, particularly when the spatial filter is large. In this paper, we show how the O(1) averaging algorithms can be leveraged for realizing the bilateral filter in constant time, by using trigonometric range kernels. This is done by generalizing the idea presented by Porikli, i.e., using polynomial kernels. The class of trigonometric kernels turns out to be sufficiently rich, allowing for the approximation of the standard Gaussian bilateral filter. The attractive feature of our approach is that, for a fixed number of terms, the quality of approximation achieved using trigonometric kernels is much superior to that obtained by Porikli using polynomials.
Snakes with tangent-based control and energies for bioimage analysis

We propose a novel active contour for the analysis of filament-like structures and boundaries - features that traditional snakes based on closed curves have difficulties to delineate. Our method relies on a parametric representation of an open curve involving Hermite-spline basis functions. This allows us to impose constraints both on the contour and on its derivatives. The proposed parameterization enables tangential controls and facilitates the design of an energy term that considers oriented features. In this way, our technique can be used to detect edges as well as ridges. The use of the Hermite-spline basis is well suited to a semi-interactive implementation. We developed an ImageJ plugin, and present experimental results on real biological data.