The atomic Hardy space for a general Bessel operator

We study Hardy spaces associated with a general multidimensional Bessel operator $$\mathbb {B}_\nu $$ B ν . This operator depends on a multiparameter of type $$\nu $$ ν that is usually restricted to a product of half-lines. Here we deal with the Bessel operator in the general context, with no restrictions on the type parameter. We define the Hardy space $$H^1$$ H 1 for $$\mathbb {B}_\nu $$ B ν in terms of the maximal operator of the semigroup of operators $$\exp (-t\mathbb {B}_\nu )$$ exp ( - t B ν ) . Then we prove that, in general, $$H^1$$ H 1 admits an atomic decomposition of local type.

Anisotropic Cosmological Model in a Modified Theory of Gravitation

Current observations indicate that, on a large enough scale, the universe is homogeneous and isotropic. However, this does not preclude the possibility of some anisotropy having occurred during the early stages of the evolution of the universe, which could then have been damped out later. This idea has aroused interest in the Bianchi models, which are homogeneous but anisotropic. Secondly, there is much interest in modified gravity these days due to the problems that the usual ΛCDM model faces in general relativity. Hence, in this paper, a study was conducted on the Bianchi type-I cosmological model in f(R,T)-modified gravity. Following some ideas from cosmography, a specific form of the deceleration parameter was assumed, leading to a model that exhibited a transition from early deceleration to late-time acceleration. The derived model approached isotropy at late times. The physical properties of the model were discussed, and expressions for the various parameters of the model were derived. It is also possible to make progress towards solving the cosmological constant problem, since in this model in f(R,T) gravity, a variable cosmological-type parameter arose, which was large early on but decreased to a constant value in later times.

Dual-polarimetric descriptors from Sentinel-1 GRD SAR data for crop growth assessment

Accurate and high-resolution spatio-temporal information about crop phenology obtained from Synthetic Aperture Radar (SAR) data is an essential component for crop management and yield estimation at a local scale. Crop growth monitoring studies seldom exploit complete polarimetric information contained in dual-pol GRD SAR data. In this study, we propose three polarimetric descriptors: the pseudo scattering-type parameter (θc), the pseudo scattering entropy parameter (Hc), and the co-pol purity parameter (mc) from dual-pol S1 GRD SAR data. We also introduce a novel unsupervised clustering framework using Hc and θc with six clustering zones to represent various scattering mechanisms. We implemented the proposed algorithm on the cloud-based Google Earth Engine (GEE) platform for Sentinel-1 SAR data. We have shown the sensitivity of these descriptors over a time series of data for wheat and canola crops at a test site in Canada. From the leaf development stage to the flowering stage for both crops, the pseudo scattering-type parameter θc changes by approximately 17°. Moreover, within the entire phenology window, both mc and Hc varies by about 0.6. The effectiveness of θc and Hc to cluster the phenological stages for the two crops is also evident from the clustering plot. During the leaf development stage, about 90 % of the sampling points were clustered into the low to medium entropy scattering zone for both the crops. Throughout the flowering stage, the entire cluster shifted into the high entropy vegetation scattering zone. Finally, during the ripening stage, the clusters of sample points were split between the high entropy vegetation scattering zone and the high entropy distributed scattering zone, with > 55 % of the sampling points in the high entropy distributed scattering zone. This innovative clustering framework will facilitate<br>the operational use of S1 GRD SAR data for agricultural applications.<div><b><br></b></div><div>This article is submitted to ISPRS Journal of Photogrammetry and Remote Sensing<br><br></div>

Dual-polarimetric descriptors from Sentinel-1 GRD SAR data for crop growth assessment

Accurate and high-resolution spatio-temporal information about crop phenology obtained from Synthetic Aperture Radar (SAR) data is an essential component for crop management and yield estimation at a local scale. Crop growth monitoring studies seldom exploit complete polarimetric information contained in dual-pol GRD SAR data. In this study, we propose three polarimetric descriptors: the pseudo scattering-type parameter (θc), the pseudo scattering entropy parameter (Hc), and the co-pol purity parameter (mc) from dual-pol S1 GRD SAR data. We also introduce a novel unsupervised clustering framework using Hc and θc with six clustering zones to represent various scattering mechanisms. We implemented the proposed algorithm on the cloud-based Google Earth Engine (GEE) platform for Sentinel-1 SAR data. We have shown the sensitivity of these descriptors over a time series of data for wheat and canola crops at a test site in Canada. From the leaf development stage to the flowering stage for both crops, the pseudo scattering-type parameter θc changes by approximately 17°. Moreover, within the entire phenology window, both mc and Hc varies by about 0.6. The effectiveness of θc and Hc to cluster the phenological stages for the two crops is also evident from the clustering plot. During the leaf development stage, about 90 % of the sampling points were clustered into the low to medium entropy scattering zone for both the crops. Throughout the flowering stage, the entire cluster shifted into the high entropy vegetation scattering zone. Finally, during the ripening stage, the clusters of sample points were split between the high entropy vegetation scattering zone and the high entropy distributed scattering zone, with > 55 % of the sampling points in the high entropy distributed scattering zone. This innovative clustering framework will facilitate<br>the operational use of S1 GRD SAR data for agricultural applications.<div><b><br></b></div><div>This article is submitted to ISPRS Journal of Photogrammetry and Remote Sensing<br><br></div>

Novel Clustering Schemes for Full and Compact Polarimetric SAR Data: An Application for Rice Phenology Characterization

<div>Information on rice phenological stages from Synthetic Aperture Radar (SAR) images is of prime interest for in-season monitoring. Often, prior in-situ measurements of phenology are not available. In such situations, unsupervised clustering of SAR images might help in discriminating phenological stages of a crop throughout its growing period. Among the existing unsupervised clustering techniques using full-polarimetric (FP) SAR images, the eigenvalue-eigenvector based roll-invariant scattering-type parameter, and the scattering entropy parameter are widely used in the literature. In this study, we utilize a unique target scattering-type parameter, which jointly uses the Barakat degree of polarization and the elements of the polarimetric coherency matrix. Likewise, we also utilize an equivalent parameter proposed for compact-polarimetric (CP) SAR data. These scattering-type parameters are analogous to the Cloude-Pottier's parameter for FP SAR data and the ellipticity parameter for CP SAR data. Besides this, we also introduce new clustering schemes for both FP and CP SAR data for segmenting diverse scattering mechanisms across the phenological stages of rice. In this study, we use the RADARSAT-2 FP and simulated CP SAR data acquired over the Indian test site of Vijayawada under the Joint Experiment for Crop Assessment and Monitoring (JECAM) initiative. The temporal analysis of the scattering-type parameters and the new clustering schemes help us to investigate detailed scattering characteristics from rice across its phenological</div><div>stages.</div>

Novel Clustering Schemes for Full and Compact Polarimetric SAR Data: An Application for Rice Phenology Characterization

<div>Information on rice phenological stages from Synthetic Aperture Radar (SAR) images is of prime interest for in-season monitoring. Often, prior in-situ measurements of phenology are not available. In such situations, unsupervised clustering of SAR images might help in discriminating phenological stages of a crop throughout its growing period. Among the existing unsupervised clustering techniques using full-polarimetric (FP) SAR images, the eigenvalue-eigenvector based roll-invariant scattering-type parameter, and the scattering entropy parameter are widely used in the literature. In this study, we utilize a unique target scattering-type parameter, which jointly uses the Barakat degree of polarization and the elements of the polarimetric coherency matrix. Likewise, we also utilize an equivalent parameter proposed for compact-polarimetric (CP) SAR data. These scattering-type parameters are analogous to the Cloude-Pottier's parameter for FP SAR data and the ellipticity parameter for CP SAR data. Besides this, we also introduce new clustering schemes for both FP and CP SAR data for segmenting diverse scattering mechanisms across the phenological stages of rice. In this study, we use the RADARSAT-2 FP and simulated CP SAR data acquired over the Indian test site of Vijayawada under the Joint Experiment for Crop Assessment and Monitoring (JECAM) initiative. The temporal analysis of the scattering-type parameters and the new clustering schemes help us to investigate detailed scattering characteristics from rice across its phenological</div><div>stages.</div>

Novel Clustering Schemes for Full and Compact Polarimetric SAR Data: A Case Study for Rice Phenology Characterization

Information on rice phenological stages from Synthetic Aperture Radar (SAR)images is of prime interest for in-season monitoring. Often, prior in-situ measurements of phenology are not available. In such situations, unsupervised clustering of SAR images might help in discriminating phenological stages of a crop throughout its growing period. Among the existing unsupervised clustering techniques using full-polarimetric (FP) SAR images, the eigenvalue-eigenvector based roll-invariant scattering-type parameter, and the scattering entropy parameter are widely used in the literature. In this study, we utilize a unique target scattering-type parameter, which jointly uses the Barakat degree of polarization and the elements of the polarimetric coherency matrix. In particular, the degree of polarization attributes to scattering randomness from a target. The scattering randomness in crops increases with advancements in its growth stages due to the development of branches and foliage. Hence, the degree of polarization varies with changes in the crop growth stages. Besides, the elements of the coherency matrices are directly related to the crop geometry as well as soil and crop water content. There-fore, this complementarity information captures the scattering randomness at each crop growth stage while taking into account diverse crop morphological characteristics. Likewise, we also utilize an equivalent parameter proposed for compact-polarimetric (CP) SAR data. These scattering-type parameters are analogous to the Cloude-Pottier’s parameter for FP SAR data and the ellipticity parameter for CP SAR data. Besides this, we also introduce new clustering schemes for both FP and CP SAR data for segmenting diverse scattering mechanisms across the phenological stages of rice. In this study, we use the RADARSAT-2 FP and simulated CP SAR data acquired over the Indian test site of Vijayawada under the Joint Experiment for Crop Assessment and Monitoring (JECAM) initiative. The temporal analysis of the scattering-type parameters and the new clustering schemes help us to investigate detailed scattering characteristics from rice across its phenological stages.<div>(Submitted to ISPRS journal)</div>

A PolSAR Scattering Power Factorization Framework and Novel Roll-Invariant Parameter-Based Unsupervised Classification Scheme Using a Geodesic Distance

© 1980-2012 IEEE. We propose a generic scattering power factorization framework (SPFF) for polarimetric synthetic aperture radar (PolSAR) data to directly obtain N scattering power components along with a residue power component for each pixel. Each scattering power component is factorized into similarity (or dissimilarity) using elementary targets and a generalized volume model. The similarity measure is derived using a geodesic distance between pairs of 4× 4 real Kennaugh matrices. In standard model-based decomposition schemes, the 3× 3 Hermitian-positive semi-definite covariance (or coherency) matrix is expressed as a weighted linear combination of scattering targets following a fixed hierarchical process. In contrast, under the proposed framework, a convex splitting of unity is performed to obtain the weights while preserving the dominance of the scattering components. The product of the total power (Span) with these weights provides the nonnegative scattering power components. Furthermore, the framework, along with the geodesic distance (GD) is effectively used to obtain specific roll-invariant parameters such as scattering-type parameter (αGD), helicity parameter (τ GD), and purity parameter (PGD). A PGD/αGD unsupervised classification scheme is also proposed for PolSAR images. The SPFF, the roll invariant parameters, and the classification results are assessed using C-band RADARSAT-2 and L-band ALOS-2 images of San Francisco.