Central and Periodic Multi-Scale Discrete Radon Transforms

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×4N, instead of 3N×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× speedup over conventional DRT, probably becoming the faster Radon transform algorithm available, at the cost of ignoring 15% of the summations in the corners.

Support theorems for Funk-type isodistant Radon transforms on constant curvature spaces

A connected maximal submanifold in a constant curvature space is called isodistant if its points are in equal distances from a totally geodesic of codimension 1. The isodistant Radon transform of a suitable real function f on a constant curvature space is the function on the set of the isodistants that gives the integrals of f over the isodistants using the canonical measure. Inverting the isodistant Radon transform is severely overdetermined because the totally geodesic Radon transform, which is a restriction of the isodistant Radon transform, is invertible on some large classes of functions. This raises the admissibility problem that is about finding reasonably small subsets of the set of the isodistants such that the associated restrictions of the isodistant Radon transform are injective on a reasonably large set of functions. One of the main results of this paper is that the Funk-type sets of isodistants are admissible, because the associated restrictions of the isodistant Radon transform, we call them Funk-type isodistant Radon transforms, satisfy appropriate support theorems on a large set of functions. This unifies and sharpens several earlier results for the sphere, and brings to light new results for every constant curvature space.

Riesz potentials and orthogonal radon transforms on affine Grassmannians

We establish intertwining relations between Riesz potentials associated with fractional powers of minus-Laplacian and orthogonal Radon transforms 𝓡 j,k of the Gonzalez-Strichartz type. The latter take functions on the Grassmannian of j-dimensional affine planes in ℝ n to functions on a similar manifold of k-dimensional planes by integration over the set of all j-planes that meet a given k-plane at a right angle. The main results include sharp existence conditions of 𝓡 j,k f on L p -functions, Fuglede type formulas connecting 𝓡 j,k with Radon-John k-plane transforms and Riesz potentials, and explicit inversion formulas for 𝓡 j,k f under the assumption that f belongs to the range of the j-plane transform. The method extends to another class of Radon transforms defined on affine Grassmannians by inclusion.

Automated Detection, Classification, and Tracking of Internal Wave Signatures using X-Band Radar in the Inner Shelf

A method based on machine learning and image processing techniques has been developed to track the surface expression of internal waves in near-real time. X-Band radar scans are first preprocessed and averaged to suppress surface wave clutter and enhance the signal to noise ratio of persistent backscatter features driven by gradients in surface currents. A machine learning algorithm utilizing a support vector machine (SVM) model is then used to classify whether or not the image contains an internal solitary wave (ISW) or internal tide bore (bore). The use of machine learning is found to allow rapid assessment of the large data set, and provides insight on characterizing optimal environmental conditions to allow for radar illumination and detection of ISWs and bores. Radon transforms and local maxima detections are used to locate these features within images that are determined to contain an ISWor bore. The resulting time series of locations is used to create a map of propagation speed and direction that captures the spatiotemporal variability of the ISW or bore in the coastal environment. This technique is applied to 2 months of data collected near Point Sal, California and captures ISW and bore propagation speed and direction information that currently cannot be measured with instruments such as moorings and synthetic aperture radar (SAR).

An lp-space Matching Pursuit algorithm and its application to robust seismic data denoising via time-domain Radon transforms

Sparse solutions of linear systems of equations are important in many applications of seismic data processing. These systems arise in many denoising algorithms, such as those that use Radon transforms. We propose a robust Matching Pursuit algorithm for the retrieval of sparse Radon domain coefficients. The algorithm is robust to outliers and, hence, applicable for seismic data deblending. The classical Matching Pursuit algorithm is often adopted to approximate data by a small number of basis functions. It performs effectively for data contaminated with well-behaved, typically Gaussian, random noise.On the other hand, Matching Pursuit tends to identify the wrong basis functions when erratic noise contaminates our data. Incorporating a lp space inner product into the Matching Pursuit algorithm significantly increases its robustness to erratic signals. Our work describes a Robust Matching Pursuit algorithm that includes lp space inner products. We also provide a detailed description of steps required to implement the proposed lp space Robust Matching Pursuit algorithm when the basis functions are not given in an explicit form, such as is the case with the time-domain Radon transform. Finally, we test the proposed algorithm with deblending problems. Both synthetic and field data examples show a significant denoising improvement compared to deblending via the standard Matching Pursuit algorithm.

Tomography bounds for the Fourier extension operator and applications

We explore the extent to which the Fourier transform of an $$L^p$$ L p density supported on the sphere in $$\mathbb {R}^n$$ R n can have large mass on affine subspaces, placing particular emphasis on lines and hyperplanes. This involves establishing bounds on quantities of the form $$X(|\widehat{gd\sigma }|^2)$$ X ( | g d σ ^ | 2 ) and $$\mathcal {R}(|\widehat{gd\sigma }|^2)$$ R ( | g d σ ^ | 2 ) , where X and $$\mathcal {R}$$ R denote the X-ray and Radon transforms respectively; here $$d\sigma $$ d σ denotes Lebesgue measure on the unit sphere $$\mathbb {S}^{n-1}$$ S n - 1 , and $$g\in L^p(\mathbb {S}^{n-1})$$ g ∈ L p ( S n - 1 ) . We also identify some conjectural bounds of this type that sit between the classical Fourier restriction and Kakeya conjectures. Finally we provide some applications of such tomography bounds to the theory of weighted norm inequalities for $$\widehat{gd\sigma }$$ g d σ ^ , establishing some natural variants of conjectures of Stein and Mizohata–Takeuchi from the 1970s. Our approach, which has its origins in work of Planchon and Vega, exploits cancellation via Plancherel’s theorem on affine subspaces, avoiding the conventional use of wave-packet and stationary-phase methods.