Non-integral geometry: additional term $f_A$ as a regularizing term

Abstract: In the present paper, we first describe the principal basis of non-integral geometry. Non-integral geometry is a new field of generalized function (distribution) theory where the effects breaking the symmetry of integration measure have been investigated. In turn, the non-symmetric integration measure (the non-invariant measure) leads to the complex form of the universal, dimension-independent inverse operator with the additional contributions compared to the methods of integral geometry. The additional term with the complex integration measure serves to the extension that improves the image reconstruction procedure. Then, we proof that this additional term $f_A$ in the universal inverse Radon transforms plays a role of the regularizing contribution. In particular, we show that owing to the presence of $f_A$ the corresponding complex singularities can be eliminated in the image reconstruction process.

• The paper introduces a universal inverse Radon transform that incorporates an additional term f_A to cancel complex singularities.
• The methodology shows that symmetry-breaking and non-invariant support require modified angular measures for accurate image reconstruction.
• The results enable robust reconstructions in tomography, seismology, and related fields by generalizing classical integral geometry.

Non-integral Geometry and the Role of the Additional Term $f_A$ as a Regularizer

Introduction and Motivation

This paper develops the theoretical foundations of non-integral geometry within the context of generalized function (distribution) theory, with particular focus on the regularization of inverse Radon transforms. The author explores the impact of symmetry-breaking in integration measures, departing from classical integral geometry where the invariance of the measure is central. The primary contribution is the introduction and analysis of an additional term $f_A$ in the universal inverse Radon transform, which arises naturally under asymmetric or inhomogeneous function supports and acts as a regularizer to eliminate complex singularities in image reconstruction.

Universal Inversion and Symmetry Breaking

Traditional approaches to Radon transform inversion depend crucially on the parity of space dimensions, leading to separate procedures for even and odd dimensional spaces. This dependency traces back to Courant-Hilbert identities, which enforce integration over the full angular domain $(0, 2\pi)$. The paper showcases how symmetry-breaking—either via angular restrictions or inhomogeneous function supports—necessitates a non-invariant measure, thereby violating the standard principles of integral geometry and prompting the emergence of non-integral geometry.

The author derives a universal inverse Radon transform (IRT) representation applicable to arbitrary dimensions, circumventing dimensional parity issues. The regularized inversion yields:

$$f_\epsilon(\vec{\bf x}) = f_S(\vec{\bf x}) + f_A(\vec{\bf x})$$

where $f_S$ is the standard term (real measure) and $f_A$ is the additional term (complex measure), both defined through explicit non-invariant angular domains and reflecting the asymmetries in the outset function.

Direct Radon Transform with Non-Symmetric Support

A detailed exposition is provided for $\mathbb{R}^2$, using an outset function with asymmetric support (restricted to specific quadrants). The calculation of the direct Radon transform (DRT) of such functions demonstrates a strong, nontrivial dependence on the angular variable $\varphi$. Notably, this dependence persists unless the support is extended to the full disk with symmetric density. This illustrates the inherent necessity of adapting the integration measure when dealing with real-world applications, where symmetry is not guaranteed.

Universal Inverse Transform: Structure and Complexity

The paper presents the universal IRT in $\mathbb{R}^2$, detailing the forms for both $f_S$ and $f_A$0. The latter is associated with an imaginary prefactor $f_A$1, signifying its origin from the complex measure introduced by regularization procedures. Crucially, as shown through direct integration and analysis, the computation of the standard term $f_A$2 can yield complex singularities due to cuts in logarithmic and polylogarithmic functions. These singularities are both $f_A$3-dependent and independent, and pose significant challenges for robust image reconstruction.

Regularizing Role of the Additional Term $f_A$4

The central result is a rigorous demonstration that the inclusion of $f_A$5 cancels these complex singularities arising from $f_A$6, achieving regularization. The cancellation occurs for a wide class of singularities, including both location-dependent and location-independent types, and is backed by explicit algebraic analysis and parameter set construction. The paper provides several alternative parameterizations that accomplish this cancellation, underscoring the generality of the approach.

Importantly, the paper asserts that this regularizing property of $f_A$7 is not limited to the simple exemplary outset function used in the analysis but extends to arbitrary, practical functions with non-symmetric or inhomogeneous supports. Thus, $f_A$8 serves as a universal regularizer in the context of image reconstruction using Radon inversion.

Implications and Prospective Developments

Practically, the findings enable improved image reconstruction algorithms, particularly in fields such as computerized tomography, astrophysics, and seismology, where the assumption of symmetric object supports rarely holds. Theoretically, the work opens new directions in the analysis of generalized functions and inverse problems, emphasizing the necessity of considering non-invariant measures and the utility of complex-valued regularizing terms.

The universality of the inversion method advocated here, which is dimension-independent and not reliant on the full angular integration, points towards future developments in optimal regularization schemes for ill-posed inverse problems. Further investigation into non-integral geometry could yield deeper understanding of how asymmetries in physical systems influence reconstruction, as well as broader mathematical applications in distribution theory and functional analysis.

Conclusion

The paper establishes non-integral geometry as a formal extension of classical integral geometry, characterized by a non-invariant measure and a universal, dimension-independent approach to Radon inversion. The additional term $f_A$9 is rigorously shown to act as a regularizer, cancelling complex singularities and enabling robust image reconstruction from asymmetric supports. This framework is general and applicable to a wide range of practical problems, suggesting fertile ground for both theoretical advancement and algorithmic innovation in inverse problem solving.

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For full technical details and analytic expressions, see "Non-integral geometry: additional term $(0, 2\pi)$0 as a regularizing term" (2605.28670).