Multiparametric wavelet transforms

The main goal of the paper is to show that wavelet transforms and packets have the multiparametric representation in the form of a product of the rotation Jacobi matrices. These representations we call the third and the fourth canonical multiparametric form. Each multiparametric wavelet transform (MPWT) depends on several free Jacobi parameters. When parameters are changed multiparametric transform is changed too taking form of all known and unknown orthogonal wavelet transforms. It gives unified approach to describing a wide set of cyclic orthogonal wavelet transforms and endows with adaptive properties of those transforms.Главная цель статьи — вывод принципиально нового представления быстрых циклических вейвлет-преобразований в виде произведения слабозаполненных матриц вращения, зависящих от конечного числа свободных параметров. При изменении параметров меняется вид преобразования, принимая облик как всех известных, так и неизвестных в настоящее время вейвлет-преобразований и вейвлет-пакетов, что дает практическую основу для их унифицированного описания

High-Dimensional Data Reduction, Image Inpainting and their Astronomical Applications

Technological advances are revolutionizing multispectral astrophysics as well as the detection and study of transient sources. This new era of multitemporal and multispectral data sets demands new ways of data representation, processing and management thus making data dimension reduction instrumental in efficient data organization, retrieval, analysis and information visualization. Other astrophysical applications of data dimension reduction which require new paradigms of data analysis include knowledge discovery, cluster analysis, feature extraction and object classification, de-correlating data elements, discovering meaningful patterns and finding essential representation of correlated variables that form a manifold (e.g. the manifold of galaxies), tagging astronomical images, multiscale analysis synchronized across all available wavelengths, denoising, etc. The second part of this paper is dedicated to a new, active area of image processing: image inpainting that consists of automated methods for filling in missing or damaged regions in images. Inpainting has multiple astronomical applications including restoring images corrupted by instrument artifacts, removing undesirable objects like bright stars and their halos, sky estimating, and pre-processing for the Fourier or wavelet transforms. Applications of high-dimensional data reduction and mitigation of instrument artifacts are demonstrated on images taken by the Spitzer Space Telescope

The Data Big Bang and the Expanding Digital Universe: High-Dimensional, Complex and Massive Data Sets in an Inflationary Epoch

Recent and forthcoming advances in instrumentation, and giant new surveys, are creating astronomical data sets that are not amenable to the methods of analysis familiar to astronomers. Traditional methods are often inadequate not merely because of the size in bytes of the data sets, but also because of the complexity of modern data sets. Mathematical limitations of familiar algorithms and techniques in dealing with such data sets create a critical need for new paradigms for the representation, analysis and scientific visualization (as opposed to illustrative visualization) of heterogeneous, multiresolution data across application domains. Some of the problems presented by the new data sets have been addressed by other disciplines such as applied mathematics, statistics and machine learning and have been utilized by other sciences such as space-based geosciences. Unfortunately, valuable results pertaining to these problems are mostly to be found only in publications outside of astronomy. Here we offer brief overviews of a number of concepts, techniques and developments, some "old" and some new. These are generally unknown to most of the astronomical community, but are vital to the analysis and visualization of complex datasets and images. In order for astronomers to take advantage of the richness and complexity of the new era of data, and to be able to identify, adopt, and apply new solutions, the astronomical community needs a certain degree of awareness and understanding of the new concepts. One of the goals of this paper is to help bridge the gap between applied mathematics, artificial intelligence and computer science on the one side and astronomy on the other.Comment: 24 pages, 8 Figures, 1 Table. Accepted for publication: "Advances in Astronomy, special issue "Robotic Astronomy

Zero-DeepSub: Zero-Shot Deep Subspace Reconstruction for Rapid Multiparametric Quantitative MRI Using 3D-QALAS

Purpose: To develop and evaluate methods for 1) reconstructing 3D-quantification using an interleaved Look-Locker acquisition sequence with T2 preparation pulse (3D-QALAS) time-series images using a low-rank subspace method, which enables accurate and rapid T1 and T2 mapping, and 2) improving the fidelity of subspace QALAS by combining scan-specific deep-learning-based reconstruction and subspace modeling. Methods: A low-rank subspace method for 3D-QALAS (i.e., subspace QALAS) and zero-shot deep-learning subspace method (i.e., Zero-DeepSub) were proposed for rapid and high fidelity T1 and T2 mapping and time-resolved imaging using 3D-QALAS. Using an ISMRM/NIST system phantom, the accuracy of the T1 and T2 maps estimated using the proposed methods was evaluated by comparing them with reference techniques. The reconstruction performance of the proposed subspace QALAS using Zero-DeepSub was evaluated in vivo and compared with conventional QALAS at high reduction factors of up to 9-fold. Results: Phantom experiments showed that subspace QALAS had good linearity with respect to the reference methods while reducing biases compared to conventional QALAS, especially for T2 maps. Moreover, in vivo results demonstrated that subspace QALAS had better g-factor maps and could reduce voxel blurring, noise, and artifacts compared to conventional QALAS and showed robust performance at up to 9-fold acceleration with Zero-DeepSub, which enabled whole-brain T1, T2, and PD mapping at 1 mm isotropic resolution within 2 min of scan time. Conclusion: The proposed subspace QALAS along with Zero-DeepSub enabled high fidelity and rapid whole-brain multiparametric quantification and time-resolved imaging.Comment: 17 figures, 3 table

Generalized classical and quantum signal theories on hypergroups. Part 1. Clasical signal theory

In this paper we develop generalized nonharmonic analysis of signals and images on commu­tative hypergroups, associated with arbitrary unitary (orthogonal) transforms. We introduce general­ized convolutions, correlations, Wigner-Ville distributions, and ambiguity functions. All theorems and properties of ordinary classical Fourier harmonic analysis are transferred on nonharmonic analysis Fourier on arbitrary Abelian hypergroups

Efficient linear local features of digital signals and images: computational and qualitative properties

The paper presents the analysis of efficiency of two original approaches to the construction of the sets of linear local features (LLF), which are used for digital signal and image processing. The first approach is based on generating of LLF set, which consists of separately constructed efficient LLFs, each of which has its own algorithm for feature calculation. The second approach assumes the construction of an efficient LLF set, which has a single algorithm for joint simultaneous computation of all features. The analysis is carried out by several indicators that characterize the computational and qualitative properties of the constructed LLFs

Machine learning classification of OARSI-scored human articular cartilage using magnetic resonance imaging

SummaryObjectiveThe purpose of this study is to evaluate the ability of machine learning to discriminate between magnetic resonance images (MRI) of normal and pathological human articular cartilage obtained under standard clinical conditions.MethodAn approach to MRI classification of cartilage degradation is proposed using pattern recognition and multivariable regression in which image features from MRIs of histologically scored human articular cartilage plugs were computed using weighted neighbor distance using compound hierarchy of algorithms representing morphology (WND-CHRM). The WND-CHRM method was first applied to several clinically available MRI scan types to perform binary classification of normal and osteoarthritic osteochondral plugs based on the Osteoarthritis Research Society International (OARSI) histological system. In addition, the image features computed from WND-CHRM were used to develop a multiple linear least-squares regression model for classification and prediction of an OARSI score for each cartilage plug.ResultsThe binary classification of normal and osteoarthritic plugs yielded results of limited quality with accuracies between 36% and 70%. However, multiple linear least-squares regression successfully predicted OARSI scores and classified plugs with accuracies as high as 86%. The present results improve upon the previously-reported accuracy of classification using average MRI signal intensities and parameter values.ConclusionMRI features detected by WND-CHRM reflect cartilage degradation status as assessed by OARSI histologic grading. WND-CHRM is therefore of potential use in the clinical detection and grading of osteoarthritis

Breast dynamic contrast-enhanced-magnetic resonance imaging and radiomics: State of art

Breast cancer represents the most common malignancy in women, being one of the most frequent cause of cancer-related mortality. Ultrasound, mammography, and magnetic resonance imaging (MRI) play a pivotal role in the diagnosis of breast lesions, with different levels of accuracy. Particularly, dynamic contrast-enhanced MRI has shown high diagnostic value in detecting multifocal, multicentric, or contralateral breast cancers. Radiomics is emerging as a promising tool for quantitative tumor evaluation, allowing the extraction of additional quantitative data from radiological imaging acquired with different modalities. Radiomics analysis may provide novel information through the quantification of lesions heterogeneity, that may be relevant in clinical practice for the characterization of breast lesions, prediction of tumor response to systemic therapies and evaluation of prognosis in patients with breast cancers. Several published studies have explored the value of radiomics with good-to-excellent diagnostic and prognostic performances for the evaluation of breast lesions. Particularly, the integrations of radiomics data with other clinical and histopathological parameters have demonstrated to improve the prediction of tumor aggressiveness with high accuracy and provided precise models that will help to guide clinical decisions and patients management. The purpose of this article in to describe the current application of radiomics in breast dynamic contrast-enhanced MRI