Inversions of the windowed ray transform

Abstract: The windowed ray transform is a natural generalization of the "Analytic-Signal Transform" which is developed to extend arbitrary functions from $\RR^n$ to $\CC^n$. We present several inversion formulas here.

• The paper presents multiple explicit inversion formulas that relax admissibility constraints using Fourier, Mellin, and harmonic analysis.
• It employs shift-invariant and axial techniques to reconstruct functions from windowed ray data, broadening the class of admissible window functions.
• The results extend traditional Radon transform methods, offering promising applications in computed tomography and signal processing.

Inversion Formulas for the Windowed Ray Transform

Introduction

The paper "Inversions of the windowed ray transform" (1309.3343) addresses the inversion problem for the windowed ray transform, a functional integral that generalizes both the Analytic-Signal Transform and the classical X-ray (Radon) transform. The windowed ray transform $Phf(u, v)$ introduces a smooth and rapidly decaying window function $h$, which brings additional flexibility compared to the Radon transform and is particularly relevant for theoretical extensions in harmonic analysis, signal processing, and mathematical tomography.

The core contribution comprises several new explicit inversion formulas for the windowed ray transform, some of which relax previously held admissibility conditions and others that exploit alternative harmonic analysis tools (Fourier, Mellin, circular harmonics). The theoretical advancements have ramifications for inverse problems in analysis and potentially for advanced modalities in image reconstruction.

Definitions and Analytical Framework

The windowed ray transform of a Schwartz-class function $f$ is given by

$$Phf(u, v) = \int_\mathbb{R} f(u + tv) h(t) dt, \quad (u, v) \in \mathbb{R}^n \times (\mathbb{R}^n \setminus \{0\}),$$

where $h \in S(\mathbb{R})$ is the window. For $h=1$ and $|v|=1$, $Phf$ recovers the classic X-ray transform. The analyticity assumptions and rapid decay of $h$ enable rigorous Fourier and Mellin analysis throughout the procedures.

The inversion problem is overdetermined, as $Phf$ depends on $2n$ variables, exceeding the $n$ variables of $f$. Consequently, the inversion is not unique, and multiple reconstructive procedures may coexist.

Main Inversion Theorems

Fourier-Based Inversion (General Dimension)

A primary inversion formula (Theorem 1) leverages the Fourier transform with respect to the positional variable $u$. Notably, the derived formula does not require the so-called admissibility condition on $h$ (e.g., $h(0) = 0$), which was a constraint in previous work:

$$f(x) = \mathbf{C}_h \iint_{\mathbb{R}^n \times \mathbb{R}^n} (\widehat{Phf}(x-vt, v))\, \mathcal{K}(t, v)\, dt dv, \quad \mathcal{K}(t, v) = I^{-1}h(-t) |v|^{-n}$$

Here, $\mathbf{C}_h$ is an explicit normalization constant depending on powers of $\pi$ and integrals over $|h|^2$.

The proof demonstrates that Fourier analysis, combined with polar co-ordinates over directional parameters and careful use of Plancherel, leads to the inversion. The approach outperforms earlier methods by relaxing regularity constraints on $h$, broadening the class of admissible window functions.

Shift-Invariant and Axial Inversion

Theorem 3 and Theorem 4 introduce inversion techniques based on invariance under translations along the integration direction. In particular, Theorem 4 provides a formula requiring only that $h$ be non-vanishing at a single point $a$, drastically weakening regularity requirements on the window:

$$|\omega|\, \widehat{Phf}(\omega, a\omega', v') = 2\pi\, \widehat{f}(\omega, a v' + u') h(a)$$

This explicit connection allows, for certain coordinate axes and directions, the reconstruction of $f$ from line-restricted or marginal data, further emphasizing the redundancy and flexibility of the windowed ray data.

The accompanying remarks clarify that this procedure requires only local information about $h$ and works even if data is limited to lines or certain projections, highlighting its applicability in limited-angle inversion.

Harmonic and Mellin Techniques in Dimension Two

For functions on $\mathbb{R}^2$, the paper derives a series inversion formula using polar coordinates and expansions in circular harmonics. The Mellin transform is applied to the angular Fourier coefficients, yielding:

$$\mathcal{M}g_l(s) = \mathcal{M}f_l(s+1)\, \mathcal{M}H(s)$$

Here, the Mellin transforms establish an explicit algebraic relationship between the radial harmonic coefficients of the data and the original function. This approach is particularly elegant in the context of compactly supported functions and provides a rigorous analytic framework for recovery from windowed projections.

Theoretical Significance and Numerical Implications

The set of inversion formulas broadens the theoretical understanding of generalized Radon-type transforms and their invertibility in various analytic settings. The relaxation of the admissibility condition, the use of directional and marginal restrictions, and the harmonic/Mellin decomposition equip analysts and practitioners with new tools for tackling inverse problems with incomplete or noisy data and for designing tailored windows in signal recovery and tomography.

From a numerical and practical viewpoint, these formulas imply that inversion schemes can be formed for broader classes of windows, potentially improving stability or robustness, and that partial or undersampled data settings may be addressed by exploiting the redundancy and dimensional freedom in the transform.

Prospects for Future Work

Future directions include analyzing stability and error bounds for the presented inversion formulas, particularly when $h$ is not highly regular or the measurements are corrupted by noise. Extension to distributions or more general function spaces and connections to microlocal analysis and partial differential operators pose natural problems. Application-driven research in computed tomography, non-destructive evaluation, and signal denoising where windows are used to enforce physical or measurement constraints would benefit from these analytic advancements.

Conclusion

This work rigorously addresses the inversion of the windowed ray transform by deriving several explicit inversion formulas under varying regularity assumptions on the window function and the data. These results consolidate and extend the analytic foundation for generalized Radon transforms and pave the way for both theoretical and applied developments in inverse problems, harmonic analysis, and imaging sciences.