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15 pages, 2449 KiB

Open AccessArticle

Sinogram Upsampling via Sub-Riemannian Diffusion with Adaptive Weighting

by JaKeoung Koo

Electronics 2023, 12(21), 4503; https://doi.org/10.3390/electronics12214503 - 1 Nov 2023

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Computed tomography is a versatile imaging technique used to enable seeing internal structures of objects without opening or destroying them. This is possible through a process called tomographic reconstruction, which reconstructs images from projections of the object that are obtained by penetrating the object with beams of radiation, such as X-rays, from different angles. These projection data are often limited to low-resolution data in terms of projection angles. These limited or subsampled data make it difficult to obtain high-quality reconstruction results. Hence, upsampling projection data is necessary. In this paper, we propose a sinogram upsampling method via the sub-Riemannian diffusion process. We first lift the data into a feature space, and we fill in the missing angle parts by propagating information from the observed data to the missing parts. We observe that the sinogram with limited angle data has high directional dependency, and based on this observation, we suggest an adaptive weighting scheme to keep information propagating toward the missing regions. This adaptive weighting allows for diffusing toward the desired directions. The experimental results show the effectiveness of the proposed method in some scenarios regarding inpainting fine details, when compared to the existing model-based methods, such as Plug-and-Play and total generalized variation. Full article

(This article belongs to the Special Issue Advances in Biomedical Imaging and Processing)

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76 pages, 654 KiB

Open AccessArticle

Entropy Dissipation for Degenerate Stochastic Differential Equations via Sub-Riemannian Density Manifold

by Qi Feng and Wuchen Li

Entropy 2023, 25(5), 786; https://doi.org/10.3390/e25050786 - 11 May 2023

Cited by 3 | Viewed by 2124

Abstract

We studied the dynamical behaviors of degenerate stochastic differential equations (SDEs). We selected an auxiliary Fisher information functional as the Lyapunov functional. Using generalized Fisher information, we conducted the Lyapunov exponential convergence analysis of degenerate SDEs. We derived the convergence rate condition by generalized Gamma calculus. Examples of the generalized Bochner’s formula are provided in the Heisenberg group, displacement group, and Martinet sub-Riemannian structure. We show that the generalized Bochner’s formula follows a generalized second-order calculus of Kullback–Leibler divergence in density space embedded with a sub-Riemannian-type optimal transport metric. Full article

(This article belongs to the Special Issue Information Geometry and Its Applications)

21 pages, 2435 KiB

Open AccessArticle

Multi-Frequency Image Completion via a Biologically-Inspired Sub-Riemannian Model with Frequency and Phase

by Emre Baspinar

J. Imaging 2021, 7(12), 271; https://doi.org/10.3390/jimaging7120271 - 9 Dec 2021

Cited by 2 | Viewed by 2589

Abstract

We present a novel cortically-inspired image completion algorithm. It uses five-dimensional sub-Riemannian cortical geometry, modeling the orientation, spatial frequency and phase-selective behavior of the cells in the visual cortex. The algorithm extracts the orientation, frequency and phase information existing in a given two-dimensional corrupted input image via a Gabor transform and represents those values in terms of cortical cell output responses in the model geometry. Then, it performs completion via a diffusion concentrated in a neighborhood along the neural connections within the model geometry. The diffusion models the activity propagation integrating orientation, frequency and phase features along the neural connections. Finally, the algorithm transforms the diffused and completed output responses back to the two-dimensional image plane. Full article

(This article belongs to the Special Issue Mathematical Modeling of Human Vision and Its Application to Image Processing)

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6 pages, 533 KiB

Open AccessEditorial

Joseph Fourier 250th Birthday: Modern Fourier Analysis and Fourier Heat Equation in Information Sciences for the XXIst Century

by Frédéric Barbaresco and Jean-Pierre Gazeau

Entropy 2019, 21(3), 250; https://doi.org/10.3390/e21030250 - 6 Mar 2019

Cited by 2 | Viewed by 4344

Abstract

For the 250th birthday of Joseph Fourier, born in 1768 at Auxerre in France, this MDPI special issue will explore modern topics related to Fourier analysis and Fourier Heat Equation. Fourier analysis, named after Joseph Fourier, addresses classically commutative harmonic analysis. The modern development of Fourier analysis during XXth century has explored the generalization of Fourier and Fourier-Plancherel formula for non-commutative harmonic analysis, applied to locally compact non-Abelian groups. In parallel, the theory of coherent states and wavelets has been generalized over Lie groups (by associating coherent states to group representations that are square integrable over a homogeneous space). The name of Joseph Fourier is also inseparable from the study of mathematics of heat. Modern research on Heat equation explores geometric extension of classical diffusion equation on Riemannian, sub-Riemannian manifolds, and Lie groups. The heat equation for a general volume form that not necessarily coincides with the Riemannian one is useful in sub-Riemannian geometry, where a canonical volume only exists in certain cases. A new geometric theory of heat is emerging by applying geometric mechanics tools extended for statistical mechanics, for example, the Lie groups thermodynamics. Full article

(This article belongs to the Special Issue Joseph Fourier 250th Birthday: Modern Fourier Analysis and Fourier Heat Equation in Information Sciences for the XXIst century)

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