From the Radon Transform to Deep Learning in Medical Image Reconstruction

Abstract

Medical image reconstruction began with an exact mathematical result. In 1917, Radon showed how to recover a function from its line integrals, providing the mathematical basis for computed tomography, positron emission tomography, and single-photon emission computed tomography. This review follows the development of reconstruction from analytical inversion to modern deep learning. We first present the Radon transform, its attenuated and three-dimensional generalizations, and the corresponding inversion formulae, including filtered backprojection. We then review iterative and optimization-based methods, including maximum-likelihood expectation-maximization, variational regularization, and compressed sensing. The discussion proceeds to deep learning, covering convolutional and transformer architectures, algorithm unrolling, physics-informed networks, neural operators, and generative diffusion priors. Our central argument is that analytical, iterative, and learned reconstruction are different formulations of the same inverse problem rather than competing paradigms. Many learned components derive from classical formulations, and many classical operators admit learnable counterparts. We conclude by discussing open challenges related to theoretical guarantees, trustworthiness, data efficiency, federated learning, and regulatory validation. The most reliable learned systems are likely to retain the mathematical and physical structure of established reconstruction methods.

1. Introduction

Medical imaging is among the most significant achievements of applied mathematics, physics, and engineering in the twentieth century. The non-invasive visualization of the interior of the human body is made possible through the solution of complex inverse problems. This enables physicians to locate tumors and internal organs, monitor the response of tumors to treatment, map blood flow, quantify metabolic activity, and guide surgery. Whether involving X-rays, gamma rays, or radiofrequency pulses interacting with tissue, each imaging modality generates projection or frequency-domain measurements from which an image must be reconstructed. The accuracy, stability, and computational cost of this reconstruction determine the diagnostic utility of the resulting image.

At the heart of computerized tomography (CT), positron emission tomography (PET), and single-photon emission computed tomography (SPECT) lies the celebrated Radon transform, which, in the two-dimensional case, maps a so-called “point function” (Punktfunktion), $f\left(P\right)$, where $P=(x_1,x_2)$ denotes an arbitrary point in the plane, to a “line function” (Geradenfunktion), following Radon’s original terminology [1,2]. A line function is simply a function defined on the set of straight lines. In modern notation, this line function is written as

$$F\left(G\right)={\int }_{G}f\left(P\right)\mathrm{d}s,$$

(1)

where G is an arbitrary straight line in the plane. The mathematical challenge of inverting the integral transform defined in Equation (1) involves recovering, or reconstructing, f from its projections. It was solved analytically by Johann Radon in 1917 [1]. Radon followed unpublished work by Hendrik Antoon Lorentz from 1905, in which the original problem was formulated, and a three-dimensional function was recovered from its integrals over corresponding planes [3]. The inversion of the three-dimensional transform was later presented by Hermann Bockwinkel in his study of light propagation in biaxial crystals, where the inversion formula was explicitly attributed to Lorentz [4,5].

Little could Radon have anticipated that this abstract mathematical construction would eventually become one of the foundations of modern medical imaging. He died in 1956, before the invention of X-ray CT transformed his theory into clinical reality. Decades later, Cormack established the mathematical foundations of modern X-ray tomography [6,7], while Hounsfield developed the first practical X-ray CT scanner [8], initially unaware that both Radon and Cormack had already demonstrated, at least theoretically, that images could be reconstructed from projection data [9]. Their contributions were recognized with the 1979 Nobel Prize in Physiology or Medicine. These foundational developments paved the way for practical analytical reconstruction algorithms. Derived from the Fourier slice theorem [10], filtered backprojection (FBP) became the classical reconstruction algorithm in CT and emission tomography [11,12,13].

Over the following decades, a series of important advances emerged in image reconstruction, including the maximum-likelihood expectation-maximization (MLEM) algorithm [14] and its ordered-subset expectation-maximization acceleration (OSEM) [15], which provided a principled statistical framework for modeling of photon-counting noise. Furthermore, the theory of compressed sensing [16,17] demonstrated that sparse images can be reconstructed accurately from highly undersampled measurements, beyond what would be predicted by the Shannon–Nyquist sampling theorem [18,19]. In parallel, iterative regularization methods, particularly total variation (TV) minimization [20], emerged as powerful tools for suppressing reconstruction artifacts while preserving important image features such as edges. Analytical approaches to image reconstruction also continued to evolve, including the Tretiak–Metz reconstruction approach for emission tomography [21], advances in the inversion of the attenuated Radon transform [22,23], and reconstruction methods based on piecewise-continuous polynomial approximations for PET and SPECT [24,25,26,27]. Such algorithms have also been implemented within STIR (Software for Tomographic Image Reconstruction) [28].

More recently, deep learning has become an important paradigm in medical imaging and image analysis [29,30]. The U-Net architecture [31] demonstrated the effectiveness of convolutional encoder–decoder networks with skip connections for biomedical image segmentation. Since then, deep learning methods have been applied to a wide range of tasks in medical image analysis [32], namely: image reconstruction [33,34], segmentation [35], registration [36], super-resolution [37], and multi-modal fusion [38]. The introduction of transformer architectures [39] and vision transformers [40] has further expanded available methods, thus enabling the modeling of long-range spatial dependencies that complement the local feature extraction of convolutional networks. Hybrid approaches have sought to connect analytical reconstruction theory with deep learning [41,42,43]; however, such efforts remain relatively limited.

Despite this progress, a significant gap persists in the literature. Analytical and deep learning approaches have largely evolved independently, with limited integration of ideas and methodologies. As a result, the relationship between analytical and modern learning-based methodologies remains insufficiently explored. In this review, we argue that these approaches are more closely related than is often acknowledged. Both emerge from the same underlying mathematical foundations, including inverse problems, functional and complex analysis, approximation theory, and optimization. Figure 1 summarizes the principal milestones discussed in this review along a single timeline spanning the analytical, iterative, and deep-learning eras.

Figure 1. A century of medical imaging mathematics. Selected milestones from the Radon transform (1917) to contemporary foundation models, partitioned into an analytical era, an iterative/sparse era, and a deep-learning era.

Several comprehensive surveys cover parts of this landscape. Deep learning for tomographic reconstruction is reviewed by Wang et al. [44], deep learning for inverse problems in imaging by Ongie et al. [45], and artificial intelligence for PET reconstruction by Reader and Pan [46]. These surveys are organized by application domain or by architecture, and none develop the operator-level correspondence between classical and learned reconstruction pursued here. Classical analytical and iterative methods are treated thoroughly in standard texts [10,47]. The present review differs in three respects. First, it treats analytical, iterative, and learned reconstruction within a single mathematical framework. Learned methods often derive from classical formulations, and the correspondences between classical and learned operators are synthesized explicitly in tabular form later in the manuscript. Second, it covers the emission-tomography thread, including the attenuated Radon transform and its analytic inversion, which falls outside the scope of the CT- and magnetic-resonance-imaging (MRI) focused surveys above. Third, it closes with a concrete research direction arising from this viewpoint: the embedding of analytical inversion operators beyond FBP into learned architectures.

This is a narrative review, organized by mathematical structure rather than by a systematic search protocol. The literature was selected to trace the development of reconstruction methodology across the analytical, iterative, and learned paradigms, from Radon’s inversion formula of 1917 to work published in 2026. Foundational works were included for the analytical and iterative eras, while deep-learning methods were included if they introduced a major class of approaches or made an explicit connection to classical reconstruction. Emphasis is placed on tomographic modalities, namely CT, PET, and SPECT, with methods from other modalities included where they introduced techniques subsequently adopted in tomography. Within the learned paradigm, post-processing, unrolled, physics-informed, operator-learning, and generative methods are each represented for CT, and the emission-tomography counterparts are included where such methods have been reported. Where a class of methods has been developed only for CT or MRI, this is stated. The literature is synthesized comparatively rather than as a sequence of individual studies. The comparison of paradigms and the correspondence between classical and learned methods are summarized in tabular form in Section 6, and Table 1 places the principal milestones on a common timeline.

Table 1. Chronological milestones in mathematical and computational methods for medical image reconstruction.

Year	Contribution	Paradigm	Refs.
1917	Radon transform and analytical inversion	Analytical	[1,2]
1963	Mathematical foundations of X-ray CT	Analytical	[6,7]
1970	Algebraic reconstruction technique (ART)	Iterative	[48]
1973	First practical X-ray CT scanner	Analytical	[8]
1974	FBP	Analytical	[13]
1980	Tretiak–Metz reconstruction method	Analytical	[21]
1982	MLEM	Iterative	[14]
1984	Cone-beam FBP (Feldkamp algorithm)	Analytical	[49]
1990	Bayesian image reconstruction in PET	Iterative	[50]
1992	Total variation regularization	Iterative	[20]
1994	OSEM	Iterative	[15]
1995	Wavelet shrinkage and sparse denoising	Iterative	[51]
1998	Separable paraboloidal surrogates algorithm	Iterative	[52]
2002	Analytic inversion of the attenuated Radon transform	Analytical	[22]
2006	Compressed sensing	Iterative	[16,17]
2015	U-Net for biomedical image segmentation	Deep learning	[31]
2018	Learned Primal–Dual reconstruction	Deep learning	[41]
2018	Model-based deep learning (MoDL)	Deep learning	[42]
2018	Deep image prior (untrained networks)	Deep learning	[53]
2019	Physics-informed neural networks	Deep learning	[54]
2021	Fourier neural operator	Deep learning	[55]
2022	Diffusion priors for reconstruction	Deep learning	[56]

2. Mathematical Foundations: The Radon Transform and Its Generalizations

This section introduces the mathematical foundations underlying tomographic imaging. We begin with the classical two-dimensional Radon transform, before discussing important generalizations, including the attenuated and three-dimensional Radon transforms.

2.1. The Classical Radon Transform

Let $f:{\mathbb{R}}^2\to \mathbb{R}$ be a compactly supported function. The two-dimensional Radon transform maps f to its line integrals over sets of parallel lines in the plane [57]. More generally, the Radon transform in ${\mathbb{R}}^n$ associates a function with its integrals over hyperplanes [10].

The line integral of a continuous function $f:D\subset {\mathbb{R}}^2\to \mathbb{R}$ along a differentiable curve $C:[a,b]\to D\subset {\mathbb{R}}^2$ is defined by:

$${\int }_{C}f\mathrm{d}s={\int }_a^{b}f\left(\mathbf{r}\left(\tau \right)\right){∥{\mathbf{r}}^{\prime }\left(\tau \right)∥}_2\mathrm{d}\tau ,$$

(2)

where $\mathbf{r}:[a,b]\to C$ is a bijective map, namely the parameterization of the curve C, and $\left|\right|·{\left|\right|}_2$ denotes the Euclidean norm in ${\mathbb{R}}^2$. In tomographic imaging, the curves of interest are straight lines. The Radon transform is obtained by restricting the above construction to straight lines. To parameterize families of parallel lines, we adopt the following coordinate system.

A line L on the $x_1{x}_2$-plane can be uniquely specified by the signed distance from the origin, $\rho$ with $-\infty <\rho <\infty$, and the angle with the $x_1$-axis, $\theta$ with $0⩽\theta <2\pi$, see Figure 2. If we denote the corresponding unit vectors parallel and perpendicular to L by ${\mathbf{e}}_1$ and ${\mathbf{e}}_2$, respectively:

$${\mathbf{e}}_1:={\mathbf{e}}_1\left(\theta \right)=\left(\cos \theta ,\sin \theta \right)\mathrm{and}{\mathbf{e}}_2:={\mathbf{e}}_2\left(\theta \right)=\left(-\sin \theta ,\cos \theta \right),$$

(3)

then every point $\mathbf{x}=(x_1,x_2)$ on L in Cartesian coordinates can be expressed in terms of the line coordinates ($\rho ,\tau$) via

$$\mathbf{x}=\tau {\mathbf{e}}_1+\rho {\mathbf{e}}_2,$$

where $\tau$ denotes the arc length, namely,

$$x_1=\tau \cos \theta -\rho \sin \theta ,\mathrm{and}{x}_2=\tau \sin \theta +\rho \cos \theta .$$

(4)

Equations (4) yield the parameterization

$$\mathbf{r}\left(\tau \right)=\left[\begin{array}{c}\tau \cos \theta -\rho \sin \theta \\ \tau \sin \theta +\rho \cos \theta \end{array}\right],$$

(5)

where $\tau$ represents the arc length along L. It is straightforward to deduce that ${∥{\mathbf{r}}^{\prime }\left(\tau \right)∥}_2=1$, thus Equation (5) is a natural parameterization of the line L. For a straight line L parameterized by its arc length $\tau$, Equation (2) reduces to

$${\int }_{C}f\mathrm{d}s={\int }_{-\infty }^{\infty }f\left(\tau \cos \theta -\rho \sin \theta ,\tau \sin \theta +\rho \cos \theta \right)\mathrm{d}\tau .$$

(6)

Solving Equations (4) for $(\rho ,\tau )$ yields:

$$\rho =x_2\cos \theta -x_1\sin \theta \mathrm{and}\tau =x_2\sin \theta +x_1\cos \theta .$$

(7)

Figure 2. Radon transform: A two-dimensional function, $f(x_1,x_2)$, and its corresponding projections, $g(\rho ,\theta )$, in Cartesian $(x_1,x_2)$ and local $(\rho ,\tau )$ coordinates.

We denote the space of rapidly decreasing (or Schwartz) functions on ${\mathbb{R}}^n$ by $\mathcal{S}\left({\mathbb{R}}^n\right)$, namely:

$$\mathcal{S}\left({\mathbb{R}}^n\right)=\left\{f\in C^{\infty }\left({\mathbb{R}}^n\right){:\left|\right|f\left|\right|}_{\alpha ,\beta }<\infty \right\}\subset C^{\infty }\left({\mathbb{R}}^n\right),$$

(8)

where

$${\left|\right|f\left|\right|}_{\alpha ,\beta }=\underset{x\in {\mathbb{R}}^n}{\sup }\left|x^{\alpha }{D}^{\beta }f\left(x\right)\right|,\forall \mathrm{multi}\text{-}\mathrm{index}\alpha ,\beta ,\left|x^{\alpha }{D}^{\beta }f\left(x\right)\right|\to 0,\mathrm{as}\left|x\right|\to \infty .$$

(9)

Then, the two-dimensional Radon transform ${\mathcal{R}}_2$ of f, $f\in \mathcal{S}\left({\mathbb{R}}^2\right)$, denoted by g, is defined as

$$\begin{array}{c}g(\rho ,\theta )=\left({\mathcal{R}}_{2}f\right)(\rho ,\theta )={\int }_{-\infty }^{\infty }f(\tau \cos \theta -\rho \sin \theta ,\tau \sin \theta +\rho \cos \theta )\mathrm{d}\tau ,\\ 0⩽\theta <2\pi ,-\infty <\rho <\infty .\hspace{1em} \end{array}$$

(10)

Here, f is introduced as compactly supported, which suffices for the line integral in Equation (2) to be well defined. From Equation (10) onward, we work in the Schwartz class $\mathcal{S}\left({\mathbb{R}}^2\right)$, which guarantees the rapid decay required by the Fourier slice theorem, the inversion formula, and the range conditions. The resulting function $g={\mathcal{R}}_{2}f$ is referred to as the sinogram, the measured data in CT and emission tomography. Using Dirac’s delta function $\delta (·)$ (line impulse), the Radon transform may be written as [25]:

$$g(\rho ,\theta )={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }f(x_1,x_2)\delta (\rho +x_1\sin \theta -x_2\cos \theta )\mathrm{d}{x}_1\mathrm{d}{x}_2,0⩽\theta <2\pi ,-\infty <\rho <\infty .$$

(11)

The Fourier slice theorem provides the mathematical foundation of analytical tomographic reconstruction [13]. It establishes a direct relationship between the Radon transform and the Fourier transform [58] and may be expressed as follows:

$${\mathcal{F}}_2\left\{f\right\}={\mathcal{F}}_1\left\{{\mathcal{R}}_2\left\{f\right\}\right\},$$

(12)

where ${\mathcal{F}}_2$ is defined by

$$\left({\mathcal{F}}_2\left\{f\right\}\right)({\xi }_1,{\xi }_2)={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }f(x_1,x_2)e^{-2\pi \mathrm{i}({\xi }_1{x}_1+{\xi }_2{x}_2)}\mathrm{d}{x}_1\mathrm{d}{x}_2,$$

(13)

${\mathcal{F}}_1$ is defined by

$$\left({\mathcal{F}}_1\left\{g\right\}\right)\left(r\right)={\int }_{-\infty }^{\infty }g\left(\rho \right)e^{-2\pi \mathrm{i}r\rho }\mathrm{d}\rho ,$$

(14)

${\mathcal{R}}_2$ is defined in Equation (10), where the variables are related through

$${\xi }_1=-r\sin \theta \mathrm{and}{\xi }_2=r\cos \theta .$$

(15)

Thus, the one-dimensional Fourier transform of a projection coincides with the restriction of the two-dimensional Fourier transform of the image to the corresponding radial line in frequency space [10]. This relationship forms the basis of FBP, enabling the reconstruction of an image from its projections (sinogram).

The mathematical analysis of tomographic reconstruction is closely linked to the properties of the Radon transform. Questions of uniqueness, stability, and invertibility are central to understanding the effects of noise, incomplete data, and reconstruction artifacts [59]. Explicit singular value decompositions of the Radon transform provide valuable insight into the structure of the inverse problem and the role of filtering in analytical reconstruction [60]. In many imaging modalities, however, the measured data are affected by additional physical phenomena, requiring extensions of the classical Radon transform.

2.2. The Attenuated Radon Transform

In SPECT, gamma rays emitted by a radiolabeled tracer are attenuated as they propagate through tissue before reaching the detector. Consequently, the measured data are described not by the classical Radon transform but by the attenuated Radon transform. Unlike SPECT, in PET, two annihilation photons are emitted in approximately opposite directions and detected in coincidence, defining a “line of response.” Since the photons traverse complementary segments of this line, their combined attenuation is independent of the annihilation position and appears as a multiplicative factor of the Radon transform. If $\mu (x_1,x_2)$ denotes the spatially varying attenuation map, the attenuated Radon transform ${\mathcal{R}}_{\mu }$ is defined by

$$\begin{array}{c}{g}_{\mu }(\rho ,\theta )=\left({\mathcal{R}}_{\mu }f\right)(\rho ,\theta )={\int }_{-\infty }^{\infty }\exp \left(-{\int }_{\tau }^{\infty }\mu (s\cos \theta -\rho \sin \theta ,s\sin \theta +\rho \cos \theta )\mathrm{d}s\right)\hfill \\ \hfill \times f(\tau \cos \theta -\rho \sin \theta ,\tau \sin \theta +\rho \cos \theta )\mathrm{d}\tau ,0⩽\theta <2\pi ,-\infty <\rho <\infty .\end{array}$$

(16)

The exponential factor in Equation (16) accounts for the attenuation accumulated along the photon path between the emission point and the detector. Unlike the classical Radon transform, the attenuation map $\mu$ enters explicitly into the forward model, making the corresponding inverse problem substantially more challenging. The development of explicit inversion formulae for the attenuated Radon transform [22,23,61,62] established the mathematical foundation for quantitative SPECT reconstruction.

2.3. The Three-Dimensional Radon Transform and PET

In three space dimensions, the Radon transform integrates over planes rather than lines. The three-dimensional Radon transform ${\mathcal{R}}_3$ of f, $f\in \mathcal{S}\left({\mathbb{R}}^3\right)$, denoted by g, is defined as

$$g(\rho ,\theta ,\varphi )=\left({\mathcal{R}}_{3}f\right)(\rho ,\theta ,\varphi )={\int \int }_{\Pi }f\mathrm{d}s,$$

(17)

where $\Pi$ represents parallel planes, $\mathrm{d}s$ denotes an area differential, and $S\left({\mathbb{R}}^3\right)$ is defined in Equation (8) for $n=3$. Each plane $\Pi \subset {\mathbb{R}}^3$ is uniquely characterized by a vector and a scalar, namely, its unit normal vector $\mathbf{n}$,

$$\mathbf{n}=\left[\begin{array}{c}{n}_1\\ n_2\\ n_3\end{array}\right],\mathrm{with}\sqrt{n_1^2+n_2^2+n_3^2}=1,$$

(18)

and its signed distance from the origin $\rho$. The equation of the plane $\Pi$ is

$$\rho -\mathbf{n}·\mathbf{x}=0,\forall \mathbf{x}\in \Pi .$$

(19)

Furthermore, we suppose that $\Pi$ intersects the $x_1{x}_2$-plane in an angle $\theta$, and the $x_2{x}_3$-plane in an angle $\varphi$ (spherical angles). This implies that we can rewrite the normal vector in spherical angles as follows:

$$\mathbf{n}(\theta ,\varphi )=\left[\begin{array}{c}\sin \theta \cos \varphi \\ \sin \theta \sin \varphi \\ \cos \theta \end{array}\right],0⩽\theta ⩽\pi ,0\le \varphi \le 2\pi .$$

(20)

Inserting Equation (20) in Equation (19) yields

$$\rho -x_1\sin \theta \cos \varphi -x_2\sin \theta \sin \varphi -x_3\cos \theta =0,\forall \mathbf{x}={(x_1,x_2,x_3)}^T\in \Pi .$$

(21)

According to Equation (21), an alternative way to express the three-dimensional Radon transform of a function $f:{\mathbb{R}}^3\to \mathbb{R}$ involves a Dirac delta, or a “plane impulse”, namely

$$g(\rho ,\theta ,\varphi )={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }f(x_1,x_2,x_3)\delta (\rho -x_1\sin \theta \cos \varphi -x_2\sin \theta \sin \varphi -x_3\cos \theta )\mathrm{d}{x}_1\mathrm{d}{x}_2\mathrm{d}{x}_3.$$

(22)

PET scanners operating in three-dimensional mode measure coincidence events along lines of response connecting pairs of detectors, so the measured data are samples of the three-dimensional X-ray transform, not of the plane-integral Radon transform. The three-dimensional Radon transform enters as an intermediate representation. Plane integrals can be formed from line-integral data by integrating the X-ray transform over a family of parallel lines that foliate each plane. This operation requires adequate sampling of those lines and is therefore restricted by the finite axial acceptance of the scanner, an assumption underlying rebinning schemes that pass through plane-integral data [63]. The spline-based inversion of the three-dimensional Radon transform [64] applies to plane-integral data obtained in this way and thus provides an analytically explicit route from three-dimensional PET data to the image through the intermediate Radon representation, rather than a direct inversion of the measured line integrals.

3. Analytical Reconstruction

The starting point of analytical reconstruction is the inversion formula derived by Radon in his seminal 1917 paper [1]. Radon parameterized straight lines by their normal angle $\varphi$, rather than the angle $\theta$ adopted throughout this review. He considered the mean value of the line-function data over all tangent lines to the circle centered at the reconstruction point P with radius q, denoted by $\overline{F_P}\left(q\right)$, namely,

$$\overline{F_P}\left(q\right)={\displaystyle \frac{1}{2\pi }}{\int }_0^{2\pi }g\left(x_1\cos \varphi +x_2\sin \varphi +q,\varphi \right)\mathrm{d}\varphi ,$$

(23)

where g denotes the Radon transform defined in Equation (10). The parameterizations are related by

$$\varphi =\theta +{\displaystyle \frac{\pi }{2}},$$

(24)

since Radon measured the angle of the line normal, whereas throughout this review, $\theta$ is defined according to the convention introduced in Equation (7) and Figure 2. Using the parameterization introduced in Section 2, Equation (23) becomes

$$\overline{F_P}\left(q\right)={\displaystyle \frac{1}{2\pi }}{\int }_0^{2\pi }g\left(\rho +q,\theta \right)\mathrm{d}\theta$$

(25)

where $\rho$ is defined in the first of Equations (7). Radon then obtained an explicit inversion formula for the transform defined in Equation (1), which would later become known as the Radon transform:

$$f\left(P\right)=-{\displaystyle \frac{1}{\pi }}{\int }_0^{\infty }{\displaystyle \frac{\mathrm{d}\overline{F_P}\left(q\right)}{q}},$$

(26)

where the integral is understood in the Stieltjes sense. Equivalently,

$$f\left(P\right)={\displaystyle \frac{1}{\pi }}\underset{ϵ\to 0}{\lim }\left[{\displaystyle \frac{\overline{F_P}\left(ϵ\right)}{ϵ}}-{\int }_{ϵ}^{\infty }{\displaystyle \frac{\overline{F_P}\left(q\right)}{q^2}}\mathrm{d}q\right].$$

(27)

The key idea of Radon’s derivation was to relate the averaged line-function data to the circular means of the unknown function, thereby reducing the reconstruction problem to an Abel integral equation. To simplify the analysis, Radon first considered the special case in which the reconstruction point coincides with the origin. In this case, the averaged projection data satisfy

$$\overline{F_0}\left(q\right)=2{\int }_q^{\infty }{\displaystyle \frac{\overline{f_0}\left(r\right)r}{\sqrt{r^2-q^2}}}\mathrm{d}r,$$

(28)

where $\overline{f_0}\left(r\right)$ denotes the circular mean of f over the circle of radius r centered at the origin. Introducing the change of variables

$$u=q^2,v=r^2,$$

reduces this relation to the classical Abel integral equation [65,66], whose explicit inversion recovers the circular means of the unknown function. Substituting these into the averaged line-function formula yields Radon’s inversion Formula (26).

3.1. Filtered Backprojection

Although Radon’s original derivation proceeded through Abel inversion, later analyses established an equivalent formulation of the inverse Radon transform in terms of the Hilbert transform ($\mathcal{H}$). The Hilbert transform of a function $g\in L^2\left(\mathbb{R}\right)$ is defined by the Cauchy principal value integral

$$\left(\mathcal{H}g\right)\left(\rho \right)={\displaystyle \frac{1}{\pi }}\mathrm{P}.\mathrm{V}.{\int }_{-\infty }^{\infty }{\displaystyle \frac{g\left(t\right)}{\rho -t}}\mathrm{d}t.$$

(29)

In this formulation, the reconstruction is expressed in terms of the derivative of the Hilbert transform of the sinogram. The Fourier transform of the Hilbert transform is given by [58]

$$\mathcal{F}\left\{Hg\right\}\left(\omega \right)=-\mathrm{i}\mathrm{sgn}\left(\omega \right)\mathcal{F}\left\{g\right\}\left(\omega \right),$$

(30)

where H denotes the Hilbert transform. Since

$$\mathcal{F}\left\{{\partial }_{\rho }g\right\}\left(\omega \right)=2\pi \mathrm{i}\omega \mathcal{F}\left\{g\right\}\left(\omega \right),$$

(31)

applying the Fourier differentiation property then yields

$$\mathcal{F}\left\{{\partial }_{\rho }Hg\right\}\left(\omega \right)=2\pi \left|\omega \right|\mathcal{F}\left\{g\right\}\left(\omega \right).$$

(32)

Thus, the operator ${\partial }_{\rho }H$ corresponds to multiplication by the so-called ramp filter, $\left|\omega \right|$, in the Fourier domain. The Fourier Slice Theorem identifies the one-dimensional Fourier transform of each projection with the corresponding radial slice of the two-dimensional Fourier transform of the image. Combined with the ramp-filter representation above, the reconstruction problem reduces to filtering each projection and subsequently backprojecting the filtered projections over all projection angles. This procedure constitutes the classical FBP method, which has remained the predominant analytical reconstruction algorithm in clinical CT for several decades and is also widely used in PET and SPECT because of its computational efficiency [10,47,58].

The FBP reconstruction of a two-dimensional function f from its projections g is given by

$$f(x_1,x_2)={\int }_0^{\pi }(Q\ast g)(\rho ,\theta )\mathrm{d}\theta ,$$

(33)

where $\rho$ is defined in the first of Equations (7), ∗ denotes convolution with respect to the detector variable $\rho$, and Q denotes the reconstruction kernel, whose one-dimensional Fourier transform is given by

$$\mathcal{F}\left\{Q\right\}\left(\omega \right)=\left|\omega \right|,$$

(34)

where Q is understood as a tempered distribution, and the convolution in Equation (33) is interpreted accordingly. In practice, the ideal ramp filter is commonly multiplied by a low-pass apodization window, such as the Hann, Hamming, Butterworth, or Shepp-Logan window, thereby providing a trade-off between noise suppression and spatial resolution [10,47]. Among these, the Shepp-Logan filter is one of the most widely used choices in clinical practice.

The integration over $\theta$ in Equation (33) is the backprojection operator, namely the adjoint ${\mathcal{R}}_2^{*}$ of the Radon transform. It is defined by the identity $⟨{\mathcal{R}}_{2}f,u⟩=⟨f,{\mathcal{R}}_2^{*}u⟩$ with respect to the $L^2$ inner products on image and sinogram space. Backprojection assigns each projection value to all points on the corresponding line and integrates over all angles. The adjoint alone does not invert the transform, since ${\mathcal{R}}_2^{*}{\mathcal{R}}_2$ is a smoothing operator rather than the identity. The ramp filter provides the correction required for inversion.

Despite its computational efficiency and widespread clinical adoption, FBP assumes an ideal line-integral acquisition model and therefore cannot explicitly account for physical effects such as detector response, photon attenuation, scatter, or Poisson counting statistics. Moreover, its sensitivity to measurement noise and angular undersampling often leads to high-frequency noise amplification and streak artifacts. These limitations have motivated the development of the iterative reconstruction methods discussed in Section 4, and more recently, deep learning-based approaches presented in Section 5.

3.2. Spline-Based Reconstruction

Spline-based methods represent the image or projection data by smooth piecewise-polynomial functions rather than piecewise-constant pixels. They extend earlier work on reconstruction from projections [67,68] and on spline discretizations of the Radon transform [69,70]. Further developments include box-spline formulations of Radon projection [71,72] and optimized spline and Kaiser-Bessel projectors for few-view CT [73,74].

The spline reconstruction technique (SRT) employs splines not to discretize the forward projector, but to evaluate the inverse Radon transform. The inverse can be written in terms of the Hilbert transform as

$$f(x_1,x_2)={\displaystyle \frac{1}{4\pi }}{\int }_0^{2\pi }{\left.{\displaystyle \frac{\partial }{\partial \rho }}\mathcal{H}\left\{g(\rho ,\theta )\right\}\right|}_{\rho =x_2\cos \theta -x_1\sin \theta }\mathrm{d}\theta .$$

(35)

The essential operation is the derivative with respect to $\rho$ of the Hilbert transform of the sinogram. In SRT, the sinogram is approximated at each angle by cubic splines in the detector variable $\rho$, and this operation is then carried out analytically on the spline representation rather than by numerical differentiation of noisy data [25]. Spline approximation thus enters the inversion formula itself, rather than serving only as an interpolation step. It should be stressed that this is not a different inversion formula. By Equation (32), the operator ${\partial }_{\rho }H$ corresponds, up to a constant factor, to multiplication by $\left|\omega \right|$ in the Fourier domain, so Equation (35) and the FBP Formula (33) are equivalent representations of the same inversion identity. The differences between SRT and FBP arise entirely from where and how the continuous formula is approximated. FBP discretizes the ramp filter, applies it to the sampled sinogram, typically in the frequency domain, and controls the resulting amplification of high-frequency noise using an apodization window chosen by the user. SRT instead replaces the sampled sinogram by a cubic spline in $\rho$ and evaluates ${\partial }_{\rho }H$ on the spline in closed form, so that the derivative acts on a smooth interpolant rather than directly on the noisy samples, and no explicit apodization window is required. The two methods therefore differ in their noise behavior, treatment of detector sampling, and computational structure, not in the underlying inversion identity. This distinguishes SRT from FBP, where reconstruction is performed by ramp filtering followed by backprojection.

The attenuated spline reconstruction technique (aSRT) [27] applies the same idea to the inverse attenuated Radon transform of Novikov [22] and of Fokas et al. [62]. In this case, the relevant one-dimensional functions in the inversion formula are represented by splines. These include the Hilbert transforms of the attenuation map and of two sinusoidal functions of the attenuated sinogram. As a result, attenuation correction enters the formula directly, rather than as a post-processing step. The same piecewise-polynomial strategy extends to three dimensions, where the Radon transform integrates over planes [64].

3.3. Orthogonal Polynomial Reconstruction

Orthogonal polynomial methods provide another analytical route to Radon inversion. The object, or sometimes the projection data, is expanded in an orthogonal polynomial basis adapted to the geometry of the problem. Reconstruction, then, is reduced to recovering the expansion coefficients, rather than directly reconstructing pointwise image values. Writing $f\simeq {\sum }_k{a}_k{\psi }_k$ gives, by linearity, $\mathcal{R}f\simeq {\sum }_k{a}_k\mathcal{R}{\psi }_k$, hence the unknowns are the coefficients $a_k$, obtained by matching the measured sinogram against the known projected basis functions $\mathcal{R}{\psi }_k$. This idea goes back to Cormack, who used circular harmonics and Zernike-type expansions to reconstruct a function from its line integrals [6,7]. The singular value decomposition of the Radon transform can be written in terms of orthogonal polynomial bases [60,75]. Louis later clarified how orthogonal expansions describe the null space of the transform [76]. Related reconstruction formulae based on multivariate orthogonal polynomials were subsequently developed as part of approximation theory [77,78]. Chebyshev decomposition combined with backprojection has also been used for fast reconstruction directly from sinogram data [79].

For Chebyshev polynomials of the first kind, the sinogram is expanded, for each angle $\theta$, as

$$g(\rho ,\theta )\simeq {\displaystyle \frac{1}{2}}{c}_0\left(\theta \right)+{\displaystyle \sum _{n=1}^N}{c}_n\left(\theta \right)T_n\left(\rho \right),T_n\left(\rho \right)=\cos \left(n\arccos \rho \right),\rho \in [-1,1].$$

(36)

Substituting the Chebyshev expansion into the Hilbert transform term of Equation (35) allows that term to be evaluated analytically. In the weighted finite Hilbert-transform setting, each $T_n$ maps, up to normalization, to a Chebyshev polynomial of the second kind, so that the derivative required in the inversion formula is computed term by term. As in SRT, the approximation enters the inversion formula itself. The difference is that splines give a local representation, whereas Chebyshev polynomials give a global one. A Chebyshev-based scheme of this type was introduced for analytical PET reconstruction [80]. A practical issue is that standard Chebyshev interpolation uses non-uniform nodes, whereas tomographic detectors are almost always uniformly spaced. Interpolation on almost equally spaced points addresses this mismatch and reduces the Runge phenomenon [81].

3.4. Further Analytical Reconstruction Methods

The method of FBP, along with the spline and orthogonal polynomial methods discussed above, are not the only analytical inversion techniques for the Radon transform. The most direct alternative is Fourier inversion, which follows from the Fourier slice theorem, see Section 2. In this approach, the image is reconstructed by applying the inverse Fourier transform to the one-dimensional Fourier transforms of its projections. The FBP algorithm can be viewed as a particular implementation of this inversion, obtained by expressing the inverse Fourier transform as a filtering (convolution) step followed by backprojection [10,47].

A different approach is provided by moment methods. The projection moments

$$m_k\left(\theta \right)={\int }_{-\infty }^{\infty }{\rho }^{k}g(\rho ,\theta )\mathrm{d}\rho ={\int }_{{\mathbb{R}}^2}{\left(\mathbf{x}·{\mathbf{e}}_2\left(\theta \right)\right)}^{k}f\left(\mathbf{x}\right)\mathrm{d}\mathbf{x},$$

(37)

where ${\mathbf{e}}_2\left(\theta \right)$ is as defined in Equation (3), relate explicitly to the image moments, by the definition of the Radon transform. This yields a means of recovering the low-order structure of the object, or of building finite-dimensional approximations from moment data [82].

Furthermore, analytical inversion extends beyond the classical Radon transform. In SPECT, attenuation is part of the forward model, so the corresponding operator is the attenuated Radon transform. For this specific operator, explicit analytical reconstruction is provided both by the inversion formulae of Novikov [22] and Kunyansky [23], and by the unified-transform approach of Fokas [83]. Their advantage is that attenuation correction enters the inverse directly, rather than as a separate correction after reconstruction. Exact FBP-type formulae have also been derived for fan-beam [47], cone-beam [84,85], and helical [86] geometries, while the approximate but widely used Feldkamp algorithm handles circular cone-beam scans [49]. Together, these methods show that analytical reconstruction is not limited to standard parallel-beam FBP. It can also use the specific structure of the Radon transform, or of a related transform, instead of solving the inverse problem by general numerical optimization.

4. Iterative and Optimization-Based Reconstruction

As noted at the end of Section 3, FBP assumes an idealized acquisition model based on line-integrals, thus it cannot explicitly account for detector response, photon attenuation, scatter, or Poisson counting statistics. Iterative reconstruction methods address these limitations by explicitly modeling the physics and statistics of the acquisition process within an optimization process. However, this comes at the expense of substantially greater computational cost than FBP.

4.1. Algebraic and Statistical Iterative Methods

The earliest iterative approaches to tomographic reconstruction were purely algebraic. In this direction, the algebraic reconstruction technique (ART) [48] applies Kaczmarz’s classical row-action method for linear systems [87] to the discretized imaging equations, updating the current image estimate by sequentially projecting onto the hyperplane defined by each measured ray-sum. While historically important, ART does not account for the Poisson-distributed nature of photon-counting measurements in emission tomography, motivating a statistical reformulation of the reconstruction problem [88].

The transition to the discrete setting is obtained by expanding the image in a pixel or voxel basis, so that $f_j$ denotes the value associated with the j-th pixel or voxel. Sampling the sinogram along m rays and collecting the n image values into a vector turns the continuous model $g={\mathcal{R}}_{2}f$ into the linear system $y=Af$. The matrix $A\in {\mathbb{R}}^{m\times n}$ is the discrete counterpart of ${\mathcal{R}}_2$, and its entry $a_{ij}$ represents the contribution of pixel j to measurement i. The transpose, $A^{\top }$, is the discrete counterpart of the adjoint ${\mathcal{R}}_2^{*}$. Thus, backprojection in the continuous setting corresponds to multiplication by $A^{\top }$ in the discrete setting.

In PET and SPECT, the number of photons detected along each line of response is well modeled as a Poisson random variable whose mean is given by a linear forward model $Af$, where $A=\left(a_{ij}\right)$ is the system matrix and $a_{ij}$ denotes the probability that an emission from voxel j is detected in bin i. Maximizing the corresponding Poisson log-likelihood leads to the MLEM algorithm [14], with multiplicative update

$$f_j^{(n+1)}={\displaystyle \frac{f_j^{\left(n\right)}}{{\displaystyle \sum _i}{a}_{ij}}}{\displaystyle \sum _i}{a}_{ij}{\displaystyle \frac{y_i}{{\displaystyle \sum _k}{a}_{ik}{f}_k^{\left(n\right)}}},$$

(38)

where $y_i$ denotes the measured counts in bin i (the sampled entries of the sinogram g of Section 2) and $f_j^{\left(n\right)}$ the current estimate of voxel j at iteration n. In contrast to FBP, each MLEM iteration is guaranteed to increase the log-likelihood monotonically, and the algorithm preserves non-negativity of the image estimate automatically. This monotonicity is not specific to tomography. It follows from the structure of MLEM as an instance of the general expectation-maximization (EM) framework of Dempster, Laird, and Rubin [89], in which each iteration maximizes a lower bound on the observed-data log-likelihood that is tight at the current iterate. The statistical properties in the case of emission tomography, including the behavior of the maximum-likelihood estimator itself, were analyzed by Vardi, Shepp, and Kaufman [90], following the closely related EM formulation of Lange and Carson [91]. A well-known drawback is that MLEM converges relatively slowly and, without early stopping or regularization, the image estimate becomes increasingly noisy as iterations progress. This behavior motivated maximum a posteriori (MAP) formulations incorporating smoothness priors employed on the image, such as the Bayesian reconstruction [50] and the one-step-late algorithm [92], which modifies the EM update to accommodate a Gibbs prior.

The OSEM algorithm [15] accelerates MLEM. In particular, it partitions the projection data into disjoint subsets and accordingly updates the image estimate using one subset per sub-iteration rather than the full data set. This yields an order-of-magnitude reduction in the number of full iterations required for a given image quality. As a consequence, OSEM has become the iterative reconstruction method of choice in emission tomography, implemented on essentially all commercial PET and SPECT systems [93,94]. It is worth noting that the acceleration comes at a cost, since each sub-iteration uses only part of the data, and OSEM therefore does not converge to a fixed point in general, but cycles among several limit points, one per subset. In practice, it is terminated after a fixed number of iterations rather than run to convergence, which turns the iteration count itself into an implicit form of regularization. Convergent alternatives exist, namely the separable paraboloidal surrogates (SPS) family, which replaces the EM surrogate with a tighter quadratic surrogate admitting explicit penalized-likelihood maximization [52], and its relaxed ordered-subsets variant, which recovers subset-based acceleration without sacrificing that guarantee by introducing a diminishing relaxation parameter across iterations [95].

4.2. Variational and Regularization-Based Methods

The discrete forward operator A inherits the ill-posedness of the continuous transform it discretizes, whether the classical Radon transform, its attenuated counterpart, or the associated X-ray transform of Section 2. Its singular values decay rapidly toward zero, and A is consequently severely ill-conditioned. The least-squares solution, $f_{\mathrm{LS}}=\arg \underset{f}{\min }{\parallel Af-y\parallel }_2^2$, is therefore unstable, since the noise components associated with the smallest singular values enter the solution amplified by the reciprocals of those values. Regularization restores stability by augmenting the data-fidelity term with a certain penalty, leading to the variational problem

$$f^{\star }=\arg \underset{f}{\min }{\displaystyle \frac{1}{2}}{\parallel Af-y\parallel }_2^2+\lambda R\left(f\right),$$

(39)

where y denotes the discretized measurement data, that is, the sinogram g sampled on a finite set of rays, $R\left(f\right)$ is a regularization functional incorporating prior assumptions about the image, and $\lambda >0$ balances the two terms. Equation (39) is the generic form of the variational problem, with a quadratic data-fidelity term appropriate to additive Gaussian noise. For emission tomography, where the data follow the Poisson model of Section 4.1, the quadratic term is replaced by the negative Poisson log-likelihood, yielding the penalized-likelihood objective addressed by the surrogate algorithms discussed there [52]. The quadratic form remains relevant to emission tomography in the high-count limit, where the Poisson likelihood is well approximated by a weighted least-squares term, and as the form in which most of the algorithms in this section are stated. The classical choice is the quadratic Tikhonov functional $R\left(f\right)=\frac{1}{2}{\parallel Lf\parallel }_2^2$ [96,97], with L denoting a discrete gradient operator, for which (39) admits the closed-form normal equations

$$(A^{\top }A+\lambda L^{\top }L)f^{\star }=A^{\top }y.$$

(40)

The penalty term limits the influence of the small singular values of A on the reconstruction, stabilizing it but, by penalizing gradient energy, oversmoothing edges [98].

Total variation (TV) regularization [20] instead sets $R\left(f\right)={\parallel \nabla f\parallel }_1$, penalizing the ${\mathsf{\ell }}_1$-norm of the image gradient. Since TV penalizes gradient magnitude rather than gradient energy, it suppresses noise while preserving sharp edges [99]. The functional is convex, however non-smooth, so its minimizer is characterized not by a vanishing gradient but by the subdifferential optimality condition, namely

$$0\in A^{\top }(A{f}^{\star }-y)+\lambda \partial {\parallel \nabla f^{\star }\parallel }_1,$$

(41)

where ∂ denotes the subdifferential. This non-smoothness rules out direct gradient descent, and specialized first-order algorithms are therefore required. The iterative shrinkage-thresholding algorithm addresses the composite objective $F\left(f\right)+G\left(f\right)$, with $F\left(f\right)=\frac{1}{2}{\parallel Af-y\parallel }_2^2$ smooth and $G\left(f\right)={\lambda \parallel \nabla f\parallel }_1$ convex but non-smooth, by alternating a gradient step on F with a proximal step on G, namely:

$$f^{(n+1)}={\mathrm{prox}}_{tG}\left(f^{\left(n\right)}-t\nabla F\left(f^{\left(n\right)}\right)\right),$$

(42)

where

$${\mathrm{prox}}_{\gamma G}\left(v\right)=\arg \underset{u}{\min }{\displaystyle \frac{1}{2}}{\parallel u-v\parallel }_2^2+\gamma G\left(u\right),$$

(43)

where ${\mathrm{prox}}_{\gamma G}$ denotes the proximal mapping of Moreau [100], and the step size satisfies $0<t⩽1/{\parallel A\parallel }_2^2$, since ${\parallel A\parallel }_2^2$ is a Lipschitz constant for $\nabla F$. Its accelerated variant, the fast iterative shrinkage-thresholding algorithm (FISTA) [101], applies the same proximal step at an extrapolated point built from the two previous iterates, improving the rate of convergence of the objective from $O(1/n)$ to $O(1/n^2)$. The alternating direction method of multipliers (ADMM) [102,103,104,105], the closely related split Bregman method [106,107], and first-order primal–dual methods such as the Chambolle–Pock algorithm [108] solve the associated saddle-point problem without requiring differentiability of the objective. In tomographic imaging specifically, TV-regularized reconstruction has been demonstrated for constrained cone-beam CT [109] and, combined directly with the expectation-maximization update of Section 4, for penalized reconstruction under Poisson statistics [110,111,112]. A further refinement penalizes the total variation of the sinogram in addition to that of the image, for which existence, uniqueness, and stability of the resulting variational problem have been established [113], illustrating how the variational and statistical iterative viewpoints combine in practice.

A well-known artifact of standard TV regularization is that, because the penalty favors piecewise-constant images by design, it can introduce a staircasing effect in regions that are genuinely smoothly varying [114]. Higher-order functionals address this by incorporating derivatives beyond the first into the penalty. Total generalized variation (TGV) [115] regularizes selectively across regularity levels, thereby avoiding staircasing while retaining edge preservation, and related models combine the total variation with the total variation of the first derivatives [116] or with infimal-convolution constructions [117,118]. In the same spirit, an augmented TV functional has recently been introduced for Radon-type inverse problems [119], where analysis of the resulting functional’s subdifferential yields reconstructions that suppress staircasing while preserving edges. These are members of a large family of TV-type functionals, surveyed by Burger and Osher [120].

4.3. Compressed Sensing and Sparse Recovery

Compressed sensing [16,17] established that signals admitting a sparse representation in a suitable basis can be recovered with high probability from far fewer linear measurements than predicted by the celebrated Shannon-Nyquist sampling principle [18,19]. Under appropriate conditions on the measurement operator, most commonly, the restricted isometry property (RIP) [121], this recovery is guaranteed. A matrix A satisfies the RIP of order s with constant ${\delta }_s\in (0,1)$ if

$$(1-{\delta }_s){\parallel f\parallel }_2^2⩽{\parallel Af\parallel }_2^2⩽(1+{\delta }_s){\parallel f\parallel }_2^2$$

(44)

for every s-sparse vector f. Thus, A approximately preserves the Euclidean norm of all sparse vectors. If ${\delta }_{2s}$ is sufficiently small, then the ${\mathsf{\ell }}_1$ relaxation recovers the same solution as the combinatorial sparsest-solution problem. In tomographic reconstruction, this theory motivates image recovery from angularly undersampled, or otherwise incomplete, projection data. A standard constrained formulation is the following:

$$f^{\star }=\arg \underset{f}{\min }{\parallel \mathrm{\Psi }f\parallel }_1,\mathrm{subject}\mathrm{to}{\parallel Af-y\parallel }_2⩽ϵ,$$

(45)

where $\mathrm{\Psi }$ is a sparsifying transform [51] and $ϵ$ is an upper bound on the expected measurement noise. By Lagrangian duality, Equation (45) is equivalent, for a suitable multiplier $\lambda \left(ϵ\right)$, to the unconstrained problem

$$f^{\star }=\arg \underset{f}{\min }\frac{1}{2}{\parallel Af-y\parallel }_2^2+\lambda {\parallel \mathrm{\Psi }f\parallel }_1,$$

(46)

which has the same composite form as the variational problem (39), and is solved by the same first-order algorithms. Setting $\mathrm{\Psi }=\nabla$ recovers TV regularization. In this case, sparsity is imposed on the image gradient rather than on the image itself. Recovery guarantees analogous to those of standard compressed sensing have been established for this setting [122,123]. In CT, TV-based compressed-sensing reconstructions have proved particularly effective for sparse-view and low-dose imaging. Diagnostic-quality images can often be reconstructed from fewer projection angles than would be required by classical sampling arguments [109,124]. Related sparsity priors extend the same principle to emission tomography, where reduced scan time or injected activity leads to incomplete or noisy projection data [125,126].

4.4. Regularization Parameter Selection

Both formulations (39) and (46) depend on a regularization parameter $\lambda$. This parameter is rarely known a priori and must instead be selected from the data (or calibrated empirically). Since $\lambda$ controls the trade-off between data fidelity and prior regularity, its choice strongly affects image quality, and is, in practice, as important as the choice of the regularizer R itself.

When an estimate $\delta$ of the noise level in y is available, Morozov’s discrepancy principle [127] selects the largest $\lambda$ whose residual matches the noise magnitude, namely,

$$\parallel A{f}_{\lambda }^{\star }{-y\parallel }_2=\tau \delta ,\tau ⩾1,$$

(47)

where $\tau$ is a fixed constant close to unity. When no noise estimate is available, data-driven criteria are employed. The L-curve method [128,129] uses a log-log plot of the solution norm against the residual norm. For ill-posed problems, this curve often has an L-shape. The selected $\lambda$ lies at the corner of this curve, where a branch dominated by noise meets a branch dominated by regularization bias. Under generalized cross-validation [130], $\lambda$ is chosen to minimize a rotation-invariant prediction error, which requires no knowledge of the noise level. Under Stein’s unbiased risk estimate [131], $\lambda$ minimizes an unbiased estimate of the mean-squared error, applicable when the noise variance is known. Adaptive schemes go further by updating $\lambda$ during the iterative process [132] or allowing it to vary spatially across the image, with stronger regularization in smooth regions and weaker regularization near edges. In practice, no single criterion is uniformly preferred across imaging modalities. The choice often depends on whether an accurate noise estimate is available.

Despite their improved noise robustness and explicit physical modeling, iterative and optimization-based methods share a common limitation. The regularizer $R\left(f\right)$, or the sparsifying transform $\mathrm{\Psi }$, must be chosen by hand, and any single fixed choice imposes a generic notion of image regularity that may not match the structure of a specific anatomical target. This observation motivates a data-driven alternative, in which the regularizer itself, or the entire reconstruction operator, is learned directly from example data, as discussed in Section 5.

5. Deep Learning for Medical Image Reconstruction

The iterative methods of Section 4 improve analytical reconstruction by modeling the physics and statistics of acquisition. However, they retain one structural limitation, namely that the regularizer is chosen by hand. Deep learning removes this constraint. It replaces the hand-designed prior with one learned from data. Rather than assuming sparsity in a fixed transform domain, a network learns the regular structure of images from representative examples. The learned prior can post-process an analytical reconstruction, act as a regularizer inside an iterative scheme, or approximate the inverse map directly. This section reviews the architectures behind this transition. It also explains how each architecture connects to the mathematical structure of the reconstruction problem.

The material in this section falls into three groups. Section 5.1 and Section 5.2 cover general-purpose architectures, including convolutional encoder–decoders, transformers, and state-space models, which serve as building blocks for reconstruction. Section 5.3 and Section 5.4 cover methods developed specifically for reconstruction, in which the forward operator appears explicitly. Section 5.5 covers learned priors, namely generative models trained independently of the measurement process and coupled to it at inference. Throughout, architectures developed for other imaging tasks are discussed only insofar as they contribute to reconstruction, for example, as encoder–decoder backbones in unrolled schemes, as the network used in the deep image prior, or as sources of learned priors.

5.1. Convolutional Architectures and the U-Net

The first group consists of general-purpose architectures. Convolutional neural networks (CNNs) remain the natural architecture for image data [30,133]. They apply the same learnable filters at every spatial location. This gives CNNs translation equivariance while using far fewer parameters than fully connected networks [134]. The most influential convolutional architecture in medical imaging is the U-Net, introduced by Ronneberger, Fischer, and Brox [31]. It consists of an encoder and a decoder linked by symmetric skip connections. The encoder progressively reduces the spatial resolution through convolution and pooling, while extracting increasingly abstract features. The decoder then restores the original resolution via upsampling. At each scale, skip connections pass high-resolution encoder features directly to the decoder. This helps preserve fine spatial detail that would otherwise be lost during downsampling. These same encoder–decoder structures form the backbone of the learned reconstruction methods discussed in Section 6, where the U-Net serves as the denoiser in unrolled schemes and as the network in the deep image prior [42,53].

The U-Net was initially introduced for cell segmentation but has since been applied much more broadly [135]. The nnU-Net framework [136] automatically configures the architecture, preprocessing, and training procedure for a given dataset. However, a persistent difficulty is the limited availability of labeled medical data. Expert annotation is certainly costly, and fully labeled datasets are often small. Few-shot meta-learning addresses this problem by training models to generalize from only a few examples. A prototypical-network strategy combined with a U-Net achieved competitive lung-lesion segmentation in PET/CT using only a few labeled cases per class [137], a setting in which conventional fully supervised training may fail to generalize. The same scarcity of labeled data motivates the training-data-free reconstruction methods discussed later in this review.

5.2. Residual Learning, Attention, and Transformer Models

Adding more layers to a plain deep network eventually increases its training error rather than reducing it. This effect, known as the degradation problem, is not caused by overfitting, since it raises the training error and not only the test error. Residual learning [138] addresses this issue by reformulating each block to compute $\mathcal{F}\left(x\right)+x$, where $\mathcal{F}\left(x\right)$ is a residual relative to the block input x, so that near-identity mappings are approached by driving $\mathcal{F}\left(x\right)$ toward zero rather than by fitting an identity through nonlinear layers. Residual connections are now near-universal in deep image models, including U-Net variants [35]. Attention mechanisms [139] let a network reweight feature responses adaptively and emphasize informative channels and regions.

Attention becomes the central primitive in the transformer [39]. The transformer replaces convolution with self-attention. Each element of the input is updated by a weighted average over all elements. The weights come from similarities between learned query and key projections. Unlike a convolution of fixed kernel size, which captures only local interactions, this mechanism captures long-range dependencies directly. The vision transformer [40] applies the idea to images by treating patches as tokens. It matches, or even exceeds, convolutional performance given enough data. Because that data requirement is rarely met in the medical setting, architectures that combine convolutional locality with global attention are often preferred. TransUNet [140] places a transformer at the U-Net bottleneck and combines convolutional locality with global attention. Swin UNETR [141] extends this to three dimensions using the shifted-window transformer [142]. State-space models such as Mamba offer a linear-time alternative for long-range modeling [143,144]. Self-attention admits a precise mathematical interpretation. It can be read as a discretized integral operator whose kernel is computed from the input itself [145]. Each output is a weighted average of value projections, with weights given by the softmax of the query-key similarities. This kernel is therefore data-dependent, unlike the fixed kernel of a convolution. This relates transformers to the integral transforms of Section 2. Furthermore, it connects them to the theory of reproducing kernel Hilbert spaces [146].

5.3. Learned Image Reconstruction

The second group consists of methods developed specifically for reconstruction. Deep learning enters the image reconstruction problem in three main ways, with increasing use of the forward model [44,45,46].

The first approach treats reconstruction as post-processing. An image is first reconstructed by FBP, and then a CNN is trained to remove noise and streak artifacts. This is effective for low-dose and sparse-view CT, where learned denoising may outperform hand-crafted filters [147]. For this purpose, FBPConvNet [148] introduced a general U-Net framework. Since the network does not use the raw projection data, its output is not guaranteed to be consistent with the measurements.

The second approach incorporates the forward operator through algorithm unrolling. An iterative scheme is truncated to a fixed number of steps, and its hand-designed proximal or regularization operator is replaced by a learned network. The entire cascade is then trained end-to-end. The learned primal-dual method [41] unrolls the Chambolle–Pock iteration of Section 4. It alternates learned updates in image and measurement space with exact applications of the Radon operator and its adjoint, outperforming both FBP and TV-based reconstruction for CT. Similarly, model-based deep learning [42] combines learned denoising with explicit data-consistency steps. Unrolling, therefore, provides a direct synthesis of analytical and learned reconstruction. The network preserves the forward model and the structure of the underlying algorithm, while learning the regularizer from data. This idea traces back to the learned iterative shrinkage-thresholding algorithm (ISTA) [149], which showed that a truncated learned iteration may adequately approximate sparse codes much faster than its hand-tuned counterpart. Examples extend across imaging modalities. For example, in MRI, the variational network [150], the deep cascade of Schlemper et al. [151], and the deep ADMM network [152] unroll iterative reconstruction algorithms, while AUTOMAP [153] learns the full mapping from k-space to the image. In PET, DeepPET [154] learns the analogous sinogram-to-image mapping, whereas FBSEM-Net [155] unrolls the expectation-maximization update of Section 4. Direct mappings of this kind have two structural costs. The fully connected layers of AUTOMAP scale with the product of the measurement and image dimensions, so memory grows rapidly with matrix size and restricts the method to modest image sizes. In addition, such networks are tied to the acquisition geometry seen in training, so a change of scanner or protocol requires retraining. The unrolled schemes above, in turn, depend on paired training data whose reference images are themselves reconstructions, so the learned mapping inherits the characteristics, and potentially the biases, of the reference method.

The third approach incorporates the analytical inverse directly into the network architecture. The iRadonMAP framework [43] consists of a learnable sinogram-domain filter, a learnable backprojection layer, and a convolutional refinement module. Together, these components mirror the FBP formula in Equation (33). Each has a clear analytical interpretation, so the network retains the inductive bias of FBP while gaining the flexibility of a learned model. It improves low-dose CT reconstruction with fewer parameters than a generic CNN. Encoding the known inverse operator into the architecture is therefore one of the most direct links between classical and learned reconstruction [156]. We return to this connection in Section 6.

Learned reconstruction in emission tomography follows several distinct strategies that reflect its statistical and physical structure. An early SPECT study used a direct projection-to-image network [157], later extended with deep convolutional models [158]. In PET, the corresponding direct approach is DeepPET, while FastPET maps a time-of-flight histo-image directly to the reconstructed image [159]. Other methods retain the Poisson likelihood and modify the image model instead. Kim et al. [160] use a trained denoiser within penalized-likelihood reconstruction, whereas Gong et al. [161] parameterize the image by a convolutional network and alternate likelihood updates with network fitting.

A further class unrolls iterative reconstruction algorithms. Examples in PET include FBSEM-Net, the block-coordinate network of Lim et al. [162], and learned primal-dual reconstruction [163]. In SPECT, Li et al. [164] embed a network-regularized expectation-maximization scheme in a fixed number of layers and train it end-to-end through an exact adjoint projector. Training-free per-image reconstruction has also been extended to list-mode PET [165]. Direct mappings are fast at inference but depend strongly on the training geometry. Unrolled methods retain the forward model and, in EM-based schemes, the Poisson statistics, at the cost of greater computation. Compared with CT, emission-tomography studies more often prioritize quantitative accuracy and count statistics, and substantially fewer learned-reconstruction studies have been reported for SPECT than for PET.

Across these studies, the substantive differences lie in three design choices rather than in the network architecture. The first is the domain in which learning acts, which may be the image domain after an analytical reconstruction, the measurement domain before it, or both alternately, as in unrolled schemes. The second is how the forward operator enters, whether not at all, as a fixed layer applied exactly, or as a learnable component. The third is the form of supervision, which ranges from paired reference reconstructions, through measurement-only self-supervision, to no training data at all. These choices, rather than depth or parameter count, largely determine the data requirements, the consistency guarantees, and the transferability of each method, and they are the axes along which the representative methods are compared below.

5.4. Physics-Informed Networks and Neural Operators

A complementary line of work enforces the physical model through the training objective rather than the architecture. Physics-informed neural networks (PINNs) [54] add a term to the data-fidelity loss that penalizes violation of the governing equations, an approach now surveyed across medical imaging modalities [166]. A Kolmogorov–Arnold formulation of this framework has recently been developed for forward and inverse problems [167,168,169]. In tomography, the forward model supplies exactly such a constraint. Reconstruction networks of the PINN type are trained to minimize a composite loss of the form

$$\mathcal{L}={\mathcal{L}}_{\mathrm{image}}+\lambda {\parallel \mathcal{R}{f}_{\theta }-g\parallel }_2^2,$$

(48)

where $f_{\theta }$ denotes the network output, $\mathcal{R}$ the Radon operator of Section 2, g the measured sinogram, $\lambda >0$ a weight balancing image fidelity against data consistency, and ${\mathcal{L}}_{\mathrm{image}}$ an image-domain loss term. Data consistency is imposed as a soft constraint during training.

Neural operators provide a related approach. They learn mappings between function spaces rather than fixed grids [170]. The Fourier neural operator [55] represents such mappings through learned multiplication in the frequency domain, similar to the filtering step of FBP. Since neural operators learn mappings between function spaces rather than grid-specific image representations, they can, in principle, generalize across spatial resolutions.

5.5. Generative Priors: Diffusion and Foundation Models

The third group consists of learned priors. The newest class of methods replaces the explicit regularizer $R\left(f\right)$ of Section 4 with a learned generative model of the image distribution. Generative adversarial networks (GANs) [171] provided an earlier approach to such a learned prior. Denoising diffusion models learn to reverse a gradual noising process and thereby model the prior distribution of images [172,173], whereas latent diffusion models made high-resolution generation practical [174]. Once trained as an unconditional prior, such a model can be combined with the measurement model at inference time. The sampling then draws from the posterior $p(f\mid g)$ without paired training data. Each step alternates a learned denoising update with a data-consistency update that enforces agreement with g. This construction generalizes the plug-and-play priors paradigm from classical inverse-problem theory. Song et al. [56] applied it to sparse-view CT with a fully unsupervised score-based sampler, and Chung et al. [175] extended diffusion priors to fully three-dimensional CT by combining a pretrained two-dimensional diffusion model with a model-based prior along the third axis, reconstructing accurately even in the two-view regime. These methods decouple the learned prior from the measurement process, so a single trained model adapts to different acquisition geometries at test time. That is a marked advantage over end-to-end supervised networks. Both studies, however, were evaluated on simulated projection data derived from public CT datasets, and their samplers require many network evaluations per volume, so their accuracy and runtime on clinical raw data remain to be demonstrated. The stochasticity of sampling raises questions of reconstruction variability and probable hallucination that remain active concerns for extensive clinical use. Diffusion-based reconstruction has been surveyed across CT, PET, and MRI by Webber and Reader [176]. Across these modalities, the recurring limitations are the computational cost of iterative sampling and the risk of hallucinated detail, examined further in Section 7.

Foundation models extend the same idea to its largest scale. Rather than training a prior for one task or modality, a single model is pretrained on a very large corpus and then adapted downstream. For reconstruction, the most direct instance is the use of pretrained generative models as priors. A diffusion model trained once on images can be combined with any measurement model at inference time, as in the samplers described above, and this decoupling is what gives such methods their flexibility across acquisition geometries [56,175,176]. Self-supervised pretraining through masked autoencoders similarly yields transferable initializations [177]. Segmentation-oriented foundation models such as the Segment Anything Model illustrate both the promise of natural-image pretraining and its limits in the medical domain [178,179]. For reconstruction, the open question is whether priors learned at scale from natural images are correct priors for medical data, a question that turns on the inductive-bias considerations examined in Section 6.

6. A Mathematical Perspective: Bridges Between Classical and Learned Methods

The previous sections examined two approaches to image reconstruction. Analytical and iterative reconstruction use explicit operators and regularizers specified in advance, whereas deep learning uses parameterized mappings learned from data. These approaches are often presented as competing alternatives, but they are closely connected mathematically. Several learned components have classical counterparts, and, at the same time, certain classical operators can be made learnable. Table 2 compares the three paradigms, while Table 3 summarizes the main correspondences between classical and learned methods.

Table 2. Representative methods from the three reconstruction paradigms, organized according to their functional role.

Role	Analytical	Iterative/Variational	Deep Learning
Direct reconstruction	[2,10,11,47,49]	—	[153,154]
Post-processing	—	—	[147,148]
Statistical iterative reconstruction	—	[14,15,89]	—
Algorithm unrolling	—	—	[41,42,149,150,151,152,155]
Regularization	—	[16,17,20,112,121]	[180,181,182,183,184,185]
Physical or inverse operator encoded explicitly	[21,22,84]	—	[43,54,156]
Fourier and operator representations	[11]	—	[55,170]
Learned image prior	—	—	[53,56,172,174,175,186]

Table 3. Classical reconstruction constructs and their learned counterparts, grouped by their shared mathematical structure. The guarantees of the classical constructs carry over to their learned counterparts only under additional conditions.

Classical Construct	Learned Counterpart	Shared Structure	Refs.
Ramp filter (fixed Fourier multiplier)	Learned sinogram filter	A frequency-domain multiplier on each projection, fixed at $\left|\omega \right|$ classically and learned from data	[43,156]
Backprojection operator ${\mathcal{R}}^{*}$	Learnable backprojection layer	Mapping from projection to image space fixed as ${\mathcal{R}}^{*}$ classically and parameterized by trainable weights when learned	[43,156]
Data-fidelity term or forward-model constraint	Data-consistency or physics-informed loss	The agreement between the reconstructed image and the measured data, imposed explicitly in the objective classically and included in the training loss when learned	[54,156]
Prescribed regularizer $R\left(f\right)$	Learned regularizer	The penalty term of the variational problem, written by hand classically and fitted to data when learned	[180,185]
Proximal operator	Learned proximal denoiser	The proximal step of a splitting scheme, in closed form classically and a trained denoiser when learned	[42,182]
Iterative optimization scheme	Unrolled network	One first-order iteration, run to convergence classically and truncated to a trainable network when learned	[41,149]
Fixed image prior	Generative or diffusion prior	A prior over images, modeled explicitly classically and drawn from a generative model when learned	[56,172,174]
Operator between function spaces	Neural operator	A mapping between functions, specified analytically in the classical setting and parameterized from data in the learned setting	[55,170]
Attenuation map (attenuated Radon)	Learned attenuation map	The attenuation term of the emission model, measured from CT classically and predicted by a network when learned	[187]

The three paradigms differ in practical terms, not only mathematical ones. The FBP algorithm reconstructs in a single pass and remains the default when speed is critical. Iterative methods require one application of A and $A^{\top }$ per iteration, so reconstruction time increases with the number of iterations. However, they require no training, and their parameters can transfer across scanners. Learned methods shift much of the computational cost to training. Direct networks reconstruct quickly at inference but may require retraining when the acquisition protocol changes. Unrolled networks inherit the per-iteration cost of the operators they embed, while diffusion samplers require multiple network evaluations per image. Memory requirements, latency, and retraining costs therefore differ substantially across the three paradigms, and the appropriate choice depends on the constraints of the clinical application.

Quantitative comparison across the three paradigms requires caution. Reported values of the peak signal-to-noise ratio (PSNR) and the structural similarity index (SSIM) depend on the dose level, the data split, the training configuration, and the implementation, so values taken from different publications are not directly comparable [188]. Pixel-level metrics also measure agreement with a reference image rather than diagnostic reliability. A learned method can attain high PSNR while hallucinating or suppressing clinically relevant structure [189,190], whereas iterative methods may achieve lower scores yet fail in more visible and predictable ways, such as noise or blur. Reader studies and task-based measures, therefore, remain important for clinical assessment, while implementation-controlled benchmarks on experimental data provide a more appropriate basis for quantitative comparison. Table 4 compares representative methods along the axes that can be compared across studies, namely modality, training-data requirement, measurement consistency, and the type of validation reported. The comparison shows that these properties do not vary together in a simple hierarchy. Classical methods require no training data and have the longest record of clinical use, whereas learned methods differ substantially in both their data requirements and the way measurement consistency is imposed. Post-processing networks operate only in the image domain, unrolled methods retain the forward model explicitly, and diffusion posterior-sampling methods impose consistency during inference. At the same time, much of the reported validation of learned reconstruction still relies on simulated measurements derived from clinical images rather than on measured clinical raw data.

Table 4. Representative reconstruction methods compared by modality, training-data requirement, measurement-consistency mechanism, and reported validation.

Method	Modality	Training Data	Measurement-Consistency Mechanism	Reported Validation
FBP [13]	CT, PET, SPECT	None	Analytical inversion of the ideal forward model; no separate data-consistency step	Decades of clinical use
SRT and aSRT [25,27]	PET (SRT), SPECT (aSRT)	None	Analytical inversion of the ideal forward model; no separate data-consistency step	Measured preclinical and clinical data
MLEM and OSEM [14,15]	PET, SPECT	None	Poisson-likelihood data fidelity evaluated through the forward model at every iteration	Decades of clinical use
TV and compressed sensing [16,17,20]	CT, PET	None	Explicit data-fidelity term or constraint at every iteration	Simulated and measured data
Post-processing networks [147,148]	CT	Paired low-dose and normal-dose images	Not enforced at inference; projection data are not used by the image-domain network	Simulated low-dose data from clinical CT images
Learned primal–dual [41]	CT	Paired sinogram and image data	Explicit measurement-domain updates using the forward and adjoint operators in every layer	Simulated projections from clinical CT images
FBSEM-Net [155]	PET	Paired low-count and high-count PET	Poisson-likelihood EM update embedded in every layer	Simulated PET brain phantoms
iRadonMAP [43]	CT	Paired, with ImageNet pretraining and clinical fine-tuning	No explicit consistency constraint, FBP-like layers encode the acquisition geometry	Clinical low-dose CT
Deep image prior [53,186]	PET	None, per-image optimization	Explicit data-fidelity term during per-image optimization	Simulated and measured PET
Diffusion posterior sampling [56,175]	CT, MRI	Unpaired clinical images	Data-consistency updates during sampling	Simulated measurements from clinical images
Neural-operator reconstruction [191]	CT	Paired sparse-view and full-view data	No explicit inference-time consistency step; the reconstruction mapping is learned from paired training data	Simulated sparse-view CT

6.1. Regularization as Inductive Bias

The variational problem of Equation (39) shows how the regularizer influences reconstruction. The term $R\left(f\right)$ encodes prior assumptions about admissible images. Tikhonov regularization promotes smoothness, whereas TV regularization favors piecewise-constant structure [192]. The deep image prior illustrates this effect clearly. An untrained convolutional network fitted to a single noisy image recovers image structure before fitting the noise [53]. The same idea has been applied to PET, where an untrained network fitted to a single scan acts as a regularizer [186]. In both cases, the regularizing effect comes from the network architecture rather than from a training dataset. Avoiding training data shifts the computational cost to per-image optimization at reconstruction time. The result also depends on when the iteration is stopped, since the network eventually fits the noise, and no principled stopping criterion is currently available.

Variational and neural regularization, therefore, constrain the reconstruction in different ways. A variational prior is expressed through a functional, whereas a neural prior is induced by the network parameterization. Mathematical theory also informs the architecture at a finer level. Kolmogorov–Arnold networks replace fixed activation functions with learnable univariate functions on their edges [167], drawing on the Kolmogorov–Arnold representation theorem [193,194]. Similarly, the expressive power of multilayer perceptrons is connected to the universal approximation theorem [195]. In both cases, an existence theorem provides a mathematical basis for a modern architecture.

6.2. Unrolled Algorithms and the Mathematics of Deep Networks

Algorithm unrolling gives the tightest formal link between classical and learned reconstruction [196]. Consider the proximal-gradient iteration of Equation (42) [101]. Truncating it to K steps and replacing each proximal operator by a learned one yields a feedforward network of depth K. The correspondence is structural rather than exact. Network depth corresponds to iteration count and each layer to one first-order step, while the forward operator A and its adjoint appear explicitly within every layer. What is not preserved, in general, is the variational interpretation of the learned component. A trained denoiser need not be the proximal operator of any regularizer unless additional structural conditions are imposed [197]. Conditions such as nonexpansiveness or averagedness can instead provide convergence guarantees for the resulting plug-and-play iteration [198]. Without such conditions, the unrolled network is a truncated iteration in form, but not in its theoretical guarantees.

This viewpoint explains several empirical observations. Unrolled networks need far less training data than generic post-processing networks, because the physics is built in rather than inferred [46,196]. They generalize better across acquisition settings for the same reason. The learned primal-dual scheme [41] and model-based deep learning [42] realize this construction for the Radon and Fourier forward models, respectively. The cost is computational. Training backpropagates through every layer, and each layer applies both A and its adjoint, so memory grows linearly with K. This constraint limits the depth that can be trained in practice. At inference, each layer again applies A and its adjoint. Unrolled networks are therefore slower than direct networks, with the computational cost increasing with the number of iterations embedded in the architecture. Letting K grow without bound leads instead to a fixed-point formulation, discussed in Section 6.5 below. In this sense, deep networks and iterative solvers are the same object read at different levels of abstraction [199].

6.3. Spectral Methods and Neural Operators

Classical analytical reconstruction has a strong spectral basis. The ramp filter of Equation (34) acts as multiplication in the Fourier domain, while the Fourier slice theorem relates each projection to a radial slice of the image spectrum [10,11,47]. Neural operators extend this operator-based viewpoint to data-driven models. The Fourier neural operator [55] alternates learned Fourier multipliers with pointwise nonlinearities, whereas DeepONet [200] represents an operator through separate branch and trunk networks.

The connection between FBP and the Fourier neural operator is structural. The FBP algorithm applies the fixed multiplier $\left|\omega \right|$, whereas the Fourier neural operator learns frequency-dependent multipliers from data. This principle has been applied to tomography through learnable multiscale Fourier interpolation for sparse-view CT [201], and through operator-learning reconstruction in function space [191]. Reported results for operator-learning reconstruction remain limited to simulated sparse-view studies, and clinical validation has not yet been undertaken. Since neural operators learn mappings between function spaces rather than fixed-dimensional arrays [170], their parameters can, at least in principle, be used across different discretizations. Grid-based convolutional networks, by contrast, are usually tied more closely to the resolution used during training [202].

6.4. The Radon Transform as a Learnable Layer

The most direct connection between classical and learned reconstruction places the Radon transform inside the network [156]. The iRadonMAP architecture [43] implements filtering, backprojection, and image refinement as differentiable layers that follow the structure of Equation (33). The backprojection layer represents the discrete adjoint ${\mathcal{R}}^{*}$, whereas the filtering layer replaces the fixed ramp filter with a learnable one [203]. Training adjusts the filter and refinement module, while the acquisition geometry remains fixed. The network therefore retains the structure of analytical inversion and learns its adaptable components. The fixed geometry is also the framework’s main restriction, since a network trained for one scanner configuration does not transfer to another without retraining the embedded operators.

Reconstruction-informed learning applies a related principle in emission tomography. A deep network was used to generate attenuation maps for SPECT myocardial perfusion imaging, which were then used to reconstruct attenuation-corrected images within the training loop [187]. Supervision was applied in both the domains of the attenuation map and of the reconstructed image. In external validation, this indirect approach outperformed direct image-to-image mappings. Incorporating the physics of the attenuated Radon transform in Equation (16) improved both accuracy and generalization. Both approaches incorporate known imaging operators into the network instead of treating them as separate preprocessing or post-processing steps [156,204].

6.5. Theoretical Guarantees for Learned Reconstruction

Classical regularization theory provides guarantees of convergence and stability. Comparable results are now emerging for learned reconstruction [205]. Adversarial regularizers learn functionals that distinguish clean images from reconstructions corrupted by artifacts, and can be analyzed within a variational framework [180,206]. Deep equilibrium models define the reconstruction as the fixed point of a learned operator, allowing convergence to be established under suitable fixed-point assumptions [184]. Regularization by denoising and plug-and-play methods insert a learned denoiser into an iterative scheme, with convergence guarantees available under appropriate conditions on the denoiser and the algorithm [181,182,198]. The network Tikhonov (NETT) framework provides convergence results under source conditions analogous to those of classical regularization theory [183]. These developments and the conditions required for stability are reviewed by Mukherjee et al. [185]. The strongest guarantees currently apply to convex models or methods based on a single and well-controlled operator. They are more limited for highly nonconvex learned regularizers, despite their strong empirical performance, though recent work extends convergence guarantees into certain nonconvex settings [197]. This gap is discussed further in Section 7.

7. Open Challenges and Future Directions

Several challenges must be addressed before learned reconstruction can be used routinely in clinical practice. We group most of these into theoretical, practical, and regulatory issues, and conclude with several emerging directions.

7.1. Theoretical Foundations of Learned Reconstruction

The guarantees surveyed in Section 6 apply only to a certain class of methods, while many of the networks that perform best in practice lie outside this rather narrow class [207]. Three questions stand out. First, what stability guarantees hold for a deep unrolled network viewed as a mapping from sinogram to image [190]? Second, how well does the network generalize to anatomies outside the training distribution [208]? Third, how does the loss function determine the implicit prior learned by the network [209]? Answering these questions will require tools from approximation theory, optimization, and statistical learning. The aim is a theory that predicts when a learned reconstruction can be trusted.

7.2. Analytical Inversion Operators in Learned Reconstruction

An immediate direction is to replace the FBP layers commonly used in learned reconstruction with certain other analytical inversion operators. In PET, SRT offers a natural alternative, since the inversion is evaluated analytically on a spline representation rather than by numerical differentiation of noisy data [25]. Since ${\partial }_{\rho }H$ and the ramp filter are equivalent representations of the same operator, replacing an FBP layer with an SRT layer changes only the implementation. Differentiation of a spline interpolant replaces frequency-domain filtering of sampled data, while the underlying inversion formula remains unchanged. In SPECT, the inversion formula for the attenuated Radon transform, implemented through aSRT, incorporates attenuation correction directly into the reconstruction [27,62]. Embedding these operators into learnable architectures, as in Section 6, would retain the structure of analytical inversion while allowing selected components, such as the spline interpolation, to be learned from data.

7.3. Interpretability and Trustworthiness

Learned reconstruction introduces failure modes that may be difficult to detect. Small structured perturbations can produce severe reconstruction artifacts [190]. Networks may also fabricate features unsupported by the measurements or suppress structures that are present (hallucinations) [189]. Such errors may remain concealed in images that appear visually plausible. The problem is therefore one of clinical safety, not only reconstruction accuracy. Uncertainty quantification can identify regions in which reconstructions are unreliable. Bayesian and ensemble methods estimate the variability of the reconstructed image [210,211]. Uncertainty estimates are useful only if they are calibrated, and their calibration must itself be validated. On simulated data, this can be assessed directly by checking whether reported credible intervals contain the known truth at the nominal coverage rate. On measured data, where the truth is unavailable, calibration can be probed only indirectly, for example, by comparing the predicted uncertainty with the empirical variability across repeated acquisitions or ensembles of reconstructions [210,212]. Data consistency constraints provide a complementary mechanism by restricting the output to images compatible with the measurements. This mechanism has a limitation of its own. It constrains only the component of the image that is determined by the measurements. When the data are incomplete, as in sparse-view or limited-angle acquisition, the forward operator leaves substantial image components unobserved or only weakly constrained, so many reconstructions may remain consistent with the measurements. Data consistency cannot distinguish among them, and the choice is then determined by the prior, learned or otherwise. Hallucinated structures can arise precisely in these unconstrained or weakly constrained components. Measurement consistency is necessary for trustworthy reconstruction, but it is not sufficient. Reliable clinical use, therefore, requires uncertainty estimates, measurement consistency, and interpretable failure mechanisms.

7.4. Multi-Modal Data Fusion

Several clinical imaging systems combine modalities, including PET/CT, SPECT/CT, and PET/MRI. In classical reconstruction, anatomical information can guide emission reconstruction through Bayesian priors that align structural boundaries [213], or through joint anato-functional penalties [214]. Learned methods extend this idea to the joint reconstruction of multiple modalities [215]. The attenuated Radon transform in Equation (16) provides another form of coupling, since the attenuation map enters the emission forward model directly. A learned attenuation map can therefore replace the co-registered CT image. Hajianfar et al. [187] estimated the map from emission data alone, and the resulting attenuation-corrected images were not distinguished from the CT-based reference by nuclear medicine specialists. Such methods may be proven useful for reducing radiation dose and further simplifying acquisition protocols. However, errors in the estimated attenuation map propagate directly into the reconstructed image. Reliable multi-modal fusion, therefore, requires uncertainty quantification and data-consistency constraints in addition to reconstruction accuracy [212].

7.5. Data Efficiency, Generalization, and Foundation Models

Supervised reconstruction networks require large paired datasets, which are often unavailable in medical imaging. This has motivated three directions: (i) self-supervised methods that reduce the need for labeled examples [216,217], (ii) per-instance methods, such as the deep image prior, that avoid a separate training dataset, while diffusion samplers reuse pretrained generative priors [53,56], and (iii) foundation models pretrained on large datasets provide representations that can be adapted to new tasks [178,179]. Each direction replaces one requirement with another. Self-supervision replaces labels with unlabeled data and a suitable measurement structure [218]. Per-instance and generative methods increase the computational cost at inference. Foundation models replace task-specific data with large-scale pretraining. Test-time training can close the natural distribution shift performance gap in deep learning-based compressed sensing [219].

7.6. Federated Learning at Scale

Learned methods require large and diverse training datasets. However, medical data are distributed among institutions and restricted by privacy requirements. Federated learning addresses this problem by training a shared model across multiple sites without transferring local data [220,221]. Each site computes model updates locally, and a central server aggregates them. The aggregation is most commonly a weighted average of the local updates, as in federated averaging [220]. Under heterogeneous data distributions, proximal approaches such as FedProx modify the local optimization problem to reduce the effects of client heterogeneity [222]. Differential privacy can provide an additional safeguard by adding calibrated noise to the shared updates and limiting the information revealed about individual participants [223]. Combining federated learning with privacy mechanisms has already been proposed for large-scale medical-imaging studies [224,225,226]. For image reconstruction, a learned regularizer or unrolled network can be trained at several institutions while each dataset remains local [227,228,229]. This increases the diversity of the training data along patients, scanners, and acquisition protocols. The trade-off is that stronger privacy guarantees require more noise, which may reduce reconstruction accuracy [230], and the relation between privacy budget and image quality is not yet well understood for reconstruction. Scaling across several institutions raises further difficulties. Data distributions vary substantially because acquisition settings and patient cohorts are inherently nonuniform [222,231]. Such differences can slow convergence and lead to uneven performance between sites. Communication is another limitation, particularly for large unrolled networks. Future work must improve robustness across sites, develop privacy mechanisms suited to imaging applications, and personalize to local data distributions [232]. The aim is a shared reconstruction model that adapts to each site, while ultimately protecting patient privacy. Regulatory validation of such a model raises an additional question regarding which institutions and data distributions the validation set represents, since a model that performs well in aggregate may still fail at a site whose population or scanner is under-represented in training.

7.7. Regulatory Validation

A learned reconstruction method used clinically must be validated as a medical device. This differs from the benchmark evaluation common in research [233]. Validation requires external and prospective testing, with particular attention to performance under distribution shift [234]. Reconstruction poses an additional difficulty because its output is interpreted by a clinician. Failure must therefore be assessed in terms of its diagnostic consequences, not only through global image-quality metrics. Standardized benchmarks and clinically meaningful failure criteria remain relatively underdeveloped, although open evaluations using real experimental CT data have begun to address this [188]. They are essential for regulatory approval and clinical adoption.

7.8. Brain-Inspired and Neuromorphic Approaches

Deep reconstruction is computationally demanding, particularly for full 3D volumes. Neuromorphic computing offers a possible approach to more energy-efficient inference. Spiking neural networks represent information through discrete spikes and their timing rather than continuous activations [235]. When implemented on event-driven hardware, they may substantially reduce energy consumption [236,237]. Reconstruction in emission tomography fits this paradigm naturally, since measurements in list-mode PET and SPECT are recorded as individual detection events [238]. However, the mathematical foundations remain largely undeveloped. An important direction is whether the inversion and backprojection structure of the Radon transform can be implemented efficiently in a spiking architecture.

8. Conclusions

This review followed a single mathematical thread from the Radon transform to deep learning. Analytical reconstruction provided explicit inversion formulae and a spectral understanding of the problem. Iterative reconstruction introduced statistical noise models and explicit regularization. Deep learning introduced learned priors and parameterized reconstruction components. We argued that these paradigms are not rivals but different formulations of the same inverse problem. Unrolled networks are truncated optimization schemes with learnable components. Fourier neural operators use learned spectral multipliers. Physics-embedded architectures place the Radon transform inside the network. In each case, classical mathematical structure is retained, and what is learned is the part that classical theory could not specify in advance.

The synthesis is promising but incomplete. Learned reconstruction lacks the stability and convergence theory available for classical methods, and its guarantees hold only under conditions that the best-performing networks do not always satisfy. Its failures can be difficult to detect, so accuracy alone does not establish clinical trust. Access to training data remains constrained by privacy and institutional separation, and clinical adoption requires validation as a medical device rather than benchmarking as an algorithm. These are the questions that will decide how widely learned reconstruction enters routine practice.

One direction follows directly from the viewpoint of this review. If learned systems benefit from preserving known reconstruction structure, then the analytical inversion operators embedded in them need not be limited to filtered backprojection. Analytical methods that evaluate the same inversion formula through different implementations, including those that incorporate attenuation directly into the inversion, are natural candidates for the fixed layers of learned architectures in emission tomography. The development from Radon to deep learning, therefore, points not toward the replacement of classical methods but toward their closer integration with data-driven models.