Computed Tomography Reconstruction Using Only One Projection Angle

Let <inline-formula> <tex-math notation="LaTeX">$F$ </tex-math></inline-formula> represent a digitized version of an image <inline-formula> <tex-math notation="LaTeX">$f\left ({x,y }\right)$ </tex-math></inline-formula>. Assume that the image fits inside a rectangular region and this region is subdivided into <inline-formula> <tex-math notation="LaTeX">$M\,\,\times \,\,N$ </tex-math></inline-formula> squares. We call these squares the shifted box functions. Thus <inline-formula> <tex-math notation="LaTeX">$f\left ({x,y }\right)$ </tex-math></inline-formula> is approximated by <inline-formula> <tex-math notation="LaTeX">$M\,\,\times \,\,N$ </tex-math></inline-formula> matrix <inline-formula> <tex-math notation="LaTeX">$F$ </tex-math></inline-formula>. This paper proofs that <inline-formula> <tex-math notation="LaTeX">$F$ </tex-math></inline-formula> can be recovered exactly and uniquely from the Radon transform of <inline-formula> <tex-math notation="LaTeX">$f$ </tex-math></inline-formula> using only one selected view angle with a well selected family of <inline-formula> <tex-math notation="LaTeX">$MN$ </tex-math></inline-formula> lines. The paper also proposes a precise method for computing the Radon transform of an image. The approach can be categorized as an algebraic reconstruction, but it is merely a theoretical contribution for the field of limited data tomography.


I. INTRODUCTION
In this paper, we propose a new algebraic reconstruction algorithm for tomography using only one selected view angle of projection with a selected family of projection lines. Our proposed approach is a nontraditional algebraic approach, recursive, but a theoretical solution.
Tomography is defined as the process of recovering an object from measurements that are line integrals of that object at a known complete set of view angles. In 1917 Radon, [1], showed that it is possible to recover a suitably regular real valued function f from the set of all projections f ∨ (p, ϕ) for angles of projections ϕ, and all lines p = x cos ϕ + y sinϕ. Radon's work found applications in the field of computed tomography [2], [3], [4], [5]. In this context, f (x, y) corresponds to the density of tissue at a point (x, y) in some plane slice through a human body and the Radon transform is a measure of the logarithm of the absorption of an x-ray beam that passes through the body along the line L. In addition to its use in the medical imaging; tomography has been successfully applied in various applications such as Digital Rock Physics, for instance [6], [7]. Additionally, image registration techniques utilize the integral transforms approach including the Radon transform, and others. Different approaches are proposed to recover spatial transformations relating two The associate editor coordinating the review of this manuscript and approving it for publication was Yiming Tang . images using their Radon transforms [8], [9], [10]. Another major approach related to Radon transform uses invariant pattern recognition [11], [12], [13].
In the field of tomography, there are several categories of reconstruction algorithms for approximating the inversion of Radon transform [2]; this includes: signal space convolution and frequency space filtering in which the filtered back projection is one possible approach; series methods and orthogonal functions; Fourier methods, that comes from the Fourier slice theorem; algebraiciterative methods and others. The Algebraic Reconstruction Technique (ART), Simultaneous Iterative Reconstruction Technique (SIRT), and, the Simultaneous Algebraic Reconstruction Technique (SART); all were developed for tomography applications [14], [15], [16], [17], [18]. In fact, the ART methods are based on Kaczmarz' method [19], [20]. Modern approaches for the low-dose computed tomography problem and denoising includes what is known today as ''the machine learning for image reconstruction'' [21], [22], [23], [24], such as batch gradient descent, stochastic gradient descent, and many others.
The original inversion formula given by Radon [1], is valid when f is continuous with compact support and that the projections f ∨ (p, ϕ) are given for all lines-infinite set of projections, not a discrete one. In the traditional tomography, a full range of projections are available. These projections are given over 0 to π with a fine uniformly spaced radial values defining the lines of integration . The term limited  set of Radon projections takes different forms due to different applications or mathematical assumptions associated  with it. Examples of variations of mathematical assumptions  can include, only few projections are available, or projections are available on a limited range of angles, or other conditions. The problem of recovering a function f (x, y) from a limited set of Radon projections has been dealt with in the mathematical and engineering literatures, for instance, [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41]. Examples of different approaches include interpolating the missing views such as [25], spectrum analysis [24], statistical iterative methods [27]. A common approach is the theory of using different moments to recover the function, [28], [29], [30].
In this paper, we present a new theoretical algorithm for tomography using only one selected view angle of projection with a well selected family of MN lines. In fact, we propose a direct algebraic reconstruction solution by recovering the image piece by piece, for example column by column. In doing so, we solve small linear systems recursively. We use the unit box function to represent each pixel of the image which allows us to use a closed form formula for the Radon transform of the box; so that the system matrix is exact. The result of this work is a pure computation and may not be applied for a real tomography problem. The work is organized as follows: In section II, we give background material regarding the Radon transform, image representation, and a review of the algebraic reconstruction method. In section III, we present our theorem for algebraic reconstruction from one view angle of projection as well as a demonstration. A numerical evaluation is presented in section IV. Finally, the conclusion is written in section V.

II. BACKGROUND MATERIAL
This section presents a review of three mathematical tools and notation that will be used in this work. These tools are: a review of the Radon transform, image representation using the box function, and a review of the algebraic reconstruction method.
Let (x, y) designate the coordinates of points in the plane and consider a function f defined on some domain D of R 2 . Consider the line L in the plane whose equation in the normal form is given by One way to write the Radon transform of f along the line L is: We refer to f ∨ (p, ϕ) or f ∨ ϕ (p) as the Radon projection of f . More precisely, the term Radon projection, in this work, means a single non-negative number equals to the line integral of f along one line specified by the pair (p, ϕ).
Among the basic properties of Radon transform is the shifting property that is relevant to our discussion: If g is obtained from f by the real translation parameters x 0 , y 0 so that Another common tool that we need is the ''Box Function'', as shown in Fig 1a. By direct calculations [2], we obtain the analytical form of the Radon transform of this function In this paper, we will be working with a function that is made of shifted box functions. Let M , N be two positive integers and let represents a shifted box function, as shown in Fig. 1b. VOLUME 11, 2023 Define the function, where F ij are real positive values. In this way, the function f is a non-negative real valued function defined on the compact We work with both spatial coordinate system (x, y), as well as the pixel indexing system (i, j), as suggested in Fig 1c. Let f ∨ (p, ϕ) be the Radon transform of f in (7), using (3) and (5) to obtain: One can always map a given M × N image to f with F ij being the grey levels of the image. Exact calculations can then be performed. Finally, we review the basics of the algebraic reconstruction approach. Algebraic methods basically work with a linear system Ax = b that is generated from the raw projections. Given k views of an image at the angles ϕ 1, ϕ 2 , . . . , ϕ k . The system is built so that the entries of matrix A are the Radon transforms of boxes B ∨ ij (p, ϕ). These entries are nonzero only if the line of projection passed through the pixel (i, j), as in Fig. 2. Thus, matrix A is a sparse matrix. The column vector x is an MN × 1 column of the MN unknows {F ij }, written as: where T is for the transpose. Assuming m projections (lines) for each angle, then we have mk equations. In this way, the size of matrix A is (mk × MN ). The vector b is an (mk × 1) column vector of the measurement f ∨ (p, ϕ). Every single choice of a fixed angle ϕ at a fixed radial value p generates one equation for the above system. A typical equation of this system would be of the form: In matrix from, we have, Each row of the matrix A corresponds to a new pair p, ϕ, and each entry of b is the Radon transform at that new pair p, ϕ.
The following constructive proof demonstrates an algorithm to determine f . Proof: The approach here is an algebraic reconstruction. Consider the required family of lines (12). From (1), the line p i,j passes through the point (j − 1, M − i + 1). Let us recover the first column of f by backward recursive calculations using the data f ∨ p i,1 , ϕ 0 , i = 1, . . . , M .
Observe that: Notice that 1 sinα is the length of the segment of the line p i,j that lies within the box.
In general, we solve for F M −I ,1 , using the equation So far, we have recovered the first column of f . To recover the next column, we update the data. Let Accordingly, The first column of g 2 is now zeros. To recover the second column of g 2 , which is the second column of f , we use the lines with This completes the proof.

B. DEMONSTRATION
To illustrate the idea, as shown in Fig 3; assume that M = N = 4, and Assuming one angle of projection, and the family of 16 lines: The first four lines, p 1,1 , p 2,1, p 3,1 , and p 4,1, are shown in Fig 3a. These four lines pass through the first column of f . To recover the first column of f , apply (13) to build the following system: In this way, coulmn one is now determined . Fig 3b illustrates g 2 (x, y) from (14) and the four lines p 1,2 , p 2,2, p 3,2 , and p 4,2 that are involved in (16).
A summary of the above arguments is as follows: 1. The image in question is assumed to be of the form (7) For instance, Fig 4 shows a 4 × 4 image of this class. Consider g ∨ J from (19). For (24), we can write,  On the other hand, consider (19) and ( For I = 1, . . . , M − 1; we use (19) to write Simplify the left side of (27), and apply (25) on the right side, we obtain, From (26) and (28)  : : In this way, the J th column of the desired image, (flipped upside down), is recovered by solving (29).
To verify the overall effectiveness of the proposed method, experiments were performed on different images. For the first example, consider the 128 ×128 brain image in Fig 5a. the required view angle of projection ϕ 0 as well as the required lines of projections p i,j are defined by (12). Specifically, ϕ 0 = π 2 − α; with α = tan −1 (128), and the family of 128 ×128 = 16384 lines: . . , 128, j = 1, 2, . . . , 128. Figure 5b shows the profile f ∨ p i,j , ϕ 0 , which is the raw data that assumed to be available. This graph consists of 16384 points. Now, observe that g ∨ J p i,J , ϕ 0 in (29) is extracted from f ∨ p i,j , ϕ 0 as given in (18). Observe also that g ∨ J is calculated only for the lines p i,J , i = 1, 2 . . . , M . For instance, Fig. 5(c-d) show the projections g ∨ 2 p i,2 , ϕ 0 and g ∨ 20 p i,20 , ϕ 0 , i. = 1, 2 . . . , M . Figure 5e shows g ∨ J p i,J , ϕ 0 for all J and Fig. 5f displays the reconstructed image that was recovered column by column from left to right by solving (29), M times.
In addition to the visual comparisons between the original and reconstructed images; we use two quantitative measures: The normalized average absolute distance, And the root-mean-square error, where F ij and F * ij represent the pixel values corresponding to the original image and the reconstructed image, respectively.
The values (30)(31) for Fig. 5f are: D 1 = 3.8 × 0 −14 and D 2 = 9.5 × 10 −13 . As observed, the reconstruction in Fig. 5f is exact. This is because of two factors: equation (8) that calculates the transform exactly, and equation (19) implemented in (29) that returns the image column by column precisely.  For further testing, let us use 128 × 128 Shepp-Logan image with values of E range from 0.1% to 0.5%. Fig. 7 shows the visual results and table 1 displays the measures D 1 and D 2 . In fact, Fig. 7b-6f are the reconstructed images after the perturbation of the Radon projections f ∨ p i,j , ϕ 0 with a noise level 0.1%, 0.2%, 0.3%, 0.4%, and 0.5%, respectively.
From these experiments we observe the following: First, with the absence of noise in f ∨ p i,j , ϕ 0 the reconstruction is exact as in Fig 5f. As said, this accuracy of reconstruction is due to equation (8) that calculates the transform exactly, and equation (19) implemented in (29) that returns the image column by column precisely.
Second, from Fig. 6b and Fig. 7b-f, the noise on any of these reconstructed images increases horizontally from left to right. That is due to the recursive nature of this method. We recover the image column by column, from left to right, and the error caused by the added noise accumulated. This observation can also be seen in the plots of Fig 6 c-j where a larger J means a larger deviation of the distorted projections h ∨ p i,J , ϕ 0 from the exact projections g ∨ p i,J , ϕ 0 .
Third, from table 1, we see that the errors D 1 and D 2 increase linearly against the level of the added noise, this is suggested by Fig 8.

V. CONCLUSION
In this paper, we obtained two results; one is practical and the other is theoretical. Our practical result is described in equations (7) and (8) where it possible to compute the Radon transform of a given image precisely. For the second result, we have seen that a function of the form (7) can be reconstructed from its Radon transform using only one view angle with lines of projection described in equation (12). The approach can be categorized as an algebraic reconstruction, but it is merely a theoretical contribution for the field of limited data tomography.