## 1 Introduction
### Projection-Like Geometry of Radon  
Tomography is based on the principle that an object is determined by the set of its plane projections taken at different angles [31; 32; 33]. The definition of tomography is based on the Radon transform inversion theorem. There is a bijection between the determination of an object in a Euclidean space and its determination in the associated Radon space \(S\): \(\mathbb{R}^{+}\) x \(S\) for Euclidean space \(\mathbb{R}^{2}\). A function defined in Radon space is called a sinogram. A function defined in the Euclidean space is called the image, the elements of which are pixels (2D). The reconstruction of an image is the synthesis operation of the projections as a grayscale or color map. The well-known definition of the Radon transform of the function \(f\), in the spatial domain, is as follows:  
\[\mathcal{R}:\ L^{1}\Big{(}\mathbb{R}^{2}\Big{)}\rightarrow\ L^{1}\big{(} \mathbb{R}^{+}\,\text{x}\,S\big{)} \tag{1}\]  
\[\mathcal{R}f=g(\mathbf{\beta},\theta)=\int_{\mathbb{R}^{2}}f(\mathbf{\pi})\delta(\mathbf{ \beta}-\mathbf{\pi}.\theta)\ d\mathbf{\pi} \tag{2}\]where \(\overline{\alpha}\in\mathbb{R}^{2}\), \(\beta\in\mathbb{R}^{+}\), and \(\theta\in S\) are the angle of diffraction or observation. Its inverse is obtained by the operation (inversion theorem):  
\[f(\overline{\alpha})=\frac{1}{2\pi}\int_{S}\int_{\mathbb{R}^{+}}g(\beta,\theta) \otimes\dot{\delta}(\beta-\overline{\alpha}.\theta)d\beta d\theta \tag{3}\]  
\[\dot{\delta}(\beta)=\frac{1}{2\pi}\int_{\mathbb{R}}|k|e^{ik\beta}dk \tag{4}\]  
where \(k\) is the wave number, and \(\otimes\) denotes the convolution operation. Usually, the operator between the two function spaces considered is described by a kernel denoted \(K\):  
\[K(\overline{\alpha},\beta,\theta)=\delta(\beta-\overline{\alpha}.\theta) \tag{5}\]  
\[K^{-1}(\overline{\alpha},\beta,\theta)=\delta(\beta-\overline{\alpha}.\theta) \tag{6}\]  
The Radon transform describes the ideal case in which the information contained in the sinogram is taken independently of any frequency bias. In practice, the information (i.e., signals) is available in a limited frequency spectrum. By introducing a frequency bias \(S(k)\) (\(|S|<\infty\)), independent of the direction of observation \(\theta\), and generally characterized by the width of the impulse response \(s(\beta)\), the finite-frequency Radon transform is defined as follows:  
\[\tilde{K}(\overline{\alpha},\beta,\theta)=s(\beta)\otimes\delta(\beta- \overline{\alpha}.\theta) \tag{7}\]  
Its inversion involves a deconvolution operation. \[\tilde{K}^{-1}(\overline{\alpha},\beta,\theta)=s^{-1}(\beta)\otimes\dot{ \delta}(\beta-\overline{\alpha}.\theta) \tag{8}\]  
is the inverse Radon kernel, and \(s^{-1}(\beta)\) is the spatial domain deconvolution filter. Diffraction tomography is based on the principle that an object is determined by the information given by the diffracted field under the illumination of an incident wave field. Thus, an object to be imaged is a perturbation of a reference medium in which waves propagate. Let us first present the ideal linear scalar case in an infinite homogeneous medium. In the Euclidean space \(\mathbb{R}^{2}\), the incident and object-diffracted fields are decomposed into plane waves, and identified respectively by the normal vector \(\mathbf{n}_{0}\in S\), associated with the angle of incidence \(\phi\), and the normal vector \(\mathbf{n}\in S\), associated with the angle of observation \(\theta\). The diffraction transfer function of the object \(\dot{\beta}\) relates the diffraction field to the incident field.