Search Results (11)

We investigate classical continuous systems characterized by singular velocity distributions, where the corresponding Radon measures are defined over the entire space with infinite mass. These singular distributions are used to model particle velocities in systems where traditional velocity distributions do not apply. As a result, the particle positions in such systems no longer conform to conventional configurations in physical space. This necessitates the development of novel analytical tools to understand the underlying models. To address this, we introduce a new conceptual framework that redefines particle configurations in phase space, where each particle is represented by its spatial position and a velocity vector. The key idea is the construction of the Plato space, which is designed to represent idealized particle configurations where the total velocity remains bounded within any compact subset of phase space. This space serves as a crucial bridge to the space of vector-valued discrete Radon measures, where each measure captures the velocity distribution over the entire system. Given the inherent complexity of analyzing infinite-dimensional spaces, we tackle the problem by reformulating it onto a finite-dimensional configuration space. This is achieved by decomposing the infinite space into smaller, more manageable components. A central tool in this reformulation is the K-transform, which is pivotal in enabling harmonic analysis of the space. The K-transform allows us to represent the system in terms of components that are more amenable to analysis, thus simplifying the study of the system’s dynamics. Furthermore, we extend previous results in the study of correlation functions by developing correlation measures tailored for these vector-valued Radon measures. These generalized functions provide deeper insights into the correlations between particle positions and velocities, expanding the range of analysis to systems with singular velocity distributions. Through this approach, we develop a robust mathematical framework that sheds light on the structure and dynamics of complex particle systems, especially those characterized by singular velocity distributions. Our results offer a new perspective on systems with non-traditional velocity distributions, advancing the theory and methodology of particle systems in both classical and modern contexts. Full article

Addressing contemporary challenges in computed tomography (CT) demands precise and efficient reconstruction. This necessitates the optimization of CT methods, particularly by improving the algorithmic efficiency of the most computationally demanding operators—forward projection and backprojection. Every measurement setup requires a unique pair of these operators. While fast algorithms for calculating forward projection operators are adaptable across various setups, they fall short in three-dimensional scanning scenarios. Hence, fast algorithms are imperative for backprojection, an integral aspect of all established reconstruction methods. This paper introduces a general method for the calculation of backprojection operators in any measurement setup. It introduces a versatile method for transposing summation-based algorithms, which rely exclusively on addition operations. The proposed approach allows for the transformation of algorithms designed for forward projection calculation into those suitable for backprojection, with the latter maintaining asymptotic algorithmic complexity. Employing this method, fast algorithms for both forward projection and backprojection have been developed for the 2D few-view parallel-beam CT as well as for the 3D cone-beam CT. The theoretically substantiated complexity values for the proposed algorithms align with their experimentally derived estimates. Full article

The Hough transform, interpreted as the discretization of the Radon transform, is a widely used tool in image processing and machine vision. The primary way to speed it up is to employ the Brady–Yong algorithm. However, the accuracy of the straight line discretization utilized in this algorithm is limited. In this study, we propose a novel algorithm called $ASD2$ that offers fast computation of the Hough transform for images of arbitrary sizes. Our approach adopts a computation scheme similar to the Brady–Yong algorithm but incorporates the best possible line discretization for improved accuracy. By employing the Method of Four Russians, we demonstrate that for an image of size $n\times n$ where $n=8^q$ and $q\in \mathbb{N}$, the computational complexity of the $ASD2$ algorithm is $O\left(n^{8/3}\right)$ when summing over $O\left(n^2\right)$ digital straight line segments. Full article

In this work, two methods are proposed for solving the problem of one-dimensional barcode segmentation in images, with an emphasis on augmented reality (AR) applications. These methods take the partial discrete Radon transform as a building block. The first proposed method uses overlapping tiles for obtaining good angle precision while maintaining good spatial precision. The second one uses an encoder–decoder structure inspired by state-of-the-art convolutional neural networks for segmentation while maintaining a classical processing framework, thus not requiring training. It is shown that the second method’s processing time is lower than the video acquisition time with a 1024 × 1024 input on a CPU, which had not been previously achieved. The accuracy it obtained on datasets widely used by the scientific community was almost on par with that obtained using the most-recent state-of-the-art methods using deep learning. Beyond the challenges of those datasets, the method proposed is particularly well suited to image sequences taken with short exposure and exhibiting motion blur and lens blur, which are expected in a real-world AR scenario. Two implementations of the proposed methods are made available to the scientific community: one for easy prototyping and one optimised for parallel implementation, which can be run on desktop and mobile phone CPUs. Full article

Many previous research studies have shown how local and even regional earthquakes can significantly affect the release of radon in the soil. The aim of this work is to investigate the relationship between radon measurements and the daily seismic activity rate and develop a methodology that allows estimating the seismic activity rate using only radon measurements. To carry out this study, the earthquake catalogue of the Vrancea region (Romania) has been used to estimate the daily seismic activity rate during a given time period, in which radon measurements were also recorded, from January 2016 to September 2020. The Vrancea zone represents the most active seismic zone in Europe and is located on the eastern edge of the strongly bent Carpathian arc. In the case of the radon measurements, seasonal behaviours and linear trends due to non-seismic factors have been identified and subsequently removed. The discrete wavelet transform has been used to analyse the radon signal at two different scales: long and short periods. From the analysis carried out on a long-period scale, an approximate linear relationship has been obtained between the radon series and the daily seismic activity rate, which provides insights into the behaviour of the seismic activity in the study region with only the radon information. In addition, the study reveals certain characteristics that could be used as precursors of earthquakes at different scales: weeks in the case of the estimated daily seismic activity rate, and days in the case of the short-period signal obtained by the wavelet analysis. The results obtained for this region allow us to hope that the analysis of the radon time series can become an effective complement to the conventional seismic analysis used in operational earthquake forecasting. Full article

We present an overview of a beam-based approach to ultra-wide band (UWB) tomographic inverse scattering, where beam-waves are used for local data-processing and local imaging, as an alternative to the conventional plane-wave and Green’s function approaches. Specifically, the method utilizes a phase–space set of iso-diffracting beam-waves that emerge from a discrete set of points and directions in the source domain. It is shown that with a proper choice of parameters, this set constitutes a frame (an overcomplete generalization of a basis), termed “beam frame”, over the entire propagation domain. An important feature of these beam frames is that they need to be calculated once and then used for all frequencies, hence the method can be implemented either in the multi-frequency domain (FD), or directly in the time domain (TD). The algorithm consists of two phases: in the processing phase, the scattering data is transformed to the beam domain using windowed phase–space transformations, while in the imaging phase, the beams are backpropagated to the target domain to form the image. The beam-domain data is not only localized and compressed, but it is also physically related to the local Radon transform (RT) of the scatterer via a local Snell’s reflection of the beam-waves. This expresses the imaging as an inverse local RT that can be applied to any local domain of interest (DoI). In previous publications, the emphasis has been set on TD data processing using a special class of localized space–time beam-waves (wave-packets). The goal of the present paper is to present the imaging scheme in the UWB FD, utilizing simpler Fourier-based data-processing tools in the space and time domains. Full article

The multi-scale discrete Radon transform (DRT) calculates, with linearithmic complexity, the summation of pixels, through a set of discrete lines, covering all possible slopes and intercepts in an image, exclusively with integer arithmetic operations. An inversion algorithm exists and is exact and fast, in spite of being iterative. In this work, the DRT forward and backward pair is evolved to propose two faster algorithms: central DRT, which computes only the central portion of intercepts; and periodic DRT, which computes the line integrals on the periodic extension of the input. Both have an output of size $N\times 4N$, instead of $3N\times 4N$, as in the original algorithm. Periodic DRT is proven to have a fast inversion, whereas central DRT does not. An interesting application of periodic DRT is its use as building a block of discrete curvelet transform. Central DRT can provide almost a $2\times$ speedup over conventional DRT, probably becoming the faster Radon transform algorithm available, at the cost of ignoring 15% of the summations in the corners. Full article

Cork stoppers were shown to have unique characteristics that allow their use for authentication purposes in an anti-counterfeiting effort. This authentication process relies on the comparison between a user’s cork image and all registered cork images in the database of genuine items. With the growth of the database, this one-to-many comparison method becomes lengthier and therefore usefulness decreases. To tackle this problem, the present work designs and compares hashing-assisted image matching methods that can be used in cork stopper authentication. The analyzed approaches are the discrete cosine transform, wavelet transform, Radon transform, and other methods such as difference hash and average hash. The most successful approach uses a 1024-bit hash length and difference hash method providing a 98% accuracy rate. By transforming the image matching into a hash matching problem, the approach presented becomes almost 40 times faster when compared to the literature. Full article

The Radon transform is a valuable tool in inverse problems such as the ones present in electromagnetic imaging. Up to now the inversion of the multiscale discrete Radon transform has been only possible by iterative numerical methods while the continuous Radon transform is usually tackled with the filtered backprojection approach. In this study, we will show, for the first time, that the multiscale discrete version of Radon transform can as well be inverted with filtered backprojection, and by doing so, we will achieve the fastest implementation until now of bidimensional discrete Radon inversion. Moreover, the proposed method allows the sacrifice of accuracy for further acceleration. It is a well-conditioned inversion that exhibits a resistance against noise similar to that of iterative methods. Full article

The contribution of this paper is to show the opportunities for using the compressive sensing (CS) technique for detecting harmonics in a frequency sparse signal. The signal in a ship’s electrical network, polluted by harmonic distortions, can be modeled as a superposition of a small number of sinusoids and the discrete Fourier transform (DFT) basis forms its sparse domain. According to the theory of CS, a signal may be reconstructed from under-sampled incoherent linear measurements. This paper highlights the use of the discrete Radon transform (DRT) techniques in the CS scheme. In the reconstruction algorithm section, a fast algorithm based on the inverse DRT is presented, in which a few randomly sampled projections of the input signal are used to correctly reconstruct the original signal. However, DRT requires a very large set of measurements that can defeat the purpose of compressive data acquisition. To acquire the wideband data below the Nyquist frequency, the K-rank-order filter is applied in the sparse transform domain to extract the most significant components and accelerate the convergence of the solution. While most CS research efforts focus on random Gaussian measurements, the Bernoulli matrix with different values of the probability of ones is applied in the presented algorithm. Preliminary results of numerical simulation confirm the effectiveness of the algorithm used, but also indicate its limitations. A significant advantage of the proposed approach is the speed of analysis, which uses fast Fourier transform (FFT) and inverse FFT (IFFT) algorithms widely available in programming environments. Moreover, the data processing algorithm is quite simple, and therefore memory usage and burden of the data processing load are relatively low. Full article

Climatological changes occur globally but have local impacts. Increased storminess, sea level rise and more powerful waves are expected to batter the coastal zone more often and more intense. To understand climate change impacts, regional bathymetry information is paramount. A major issue is that the bathymetries are often non-existent or if they do exist, outdated. This sparsity can be overcome by space-borne satellite techniques to derive bathymetry. Sentinel-2 optical imagery is collected continuously and has a revisit-time around a few days depending on the orbital-position around the world. In this work, Sentinel-2 imagery derived wave patterns are extracted using a localized radon transform. A discrete fast-Fourier (DFT) procedure per direction in Radon space (sinogram) is then applied to derive wave spectra. Sentinel-2 time-lag between detector bands is employed to compute the spectral wave-phase shift and depth using the gravity wave linear dispersion. With this novel technique, regional bathymetries are derived at the test-site of Capbreton, France with an root mean squared (RMS)-error of 2.58 m and a correlation coefficient of 0.82 when compared to the survey for depths until 30 m. With the proposed method, the 10 m Sentinel-2 resolution is sufficient to adequately estimate bathymetries for a wave period of 6.5 s or greater. For shorter periods, the pixel resolution does not allow to detect a stable celerity. In addition to the wave-signature enhancement, the capability of the Radon Transform to augment Sentinel-2 20 m resolution imagery to 10 m is demonstrated, increasing the number of suitable bands for the depth inversion. Full article