Selected Mathematical Tools for Modeling in Tomography and Image Processing with Some Applications

In this theoretical paper, we propose several useful mathematical tools for conducting research in the fields of tomography and image processing. Several mathematical models are presented including analytical formulation of the Radon transform for three elementary functions. A family of block binary images for testing ideas is explained. Representation of an image using the box function is suggested. This representation allows exact computations for the Radon transform, Fourier transform, moments, and many others. This representation can also be involved in the mathematical arrangements of the iterative reconstruction algorithms of tomography. Overall, we believe that these tools are helpful for researchers in these fields. We show accurate results in several calculations on synthetic images as well the algebraic reconstruction.


Introduction
In this theoretical paper, we propose several useful mathematical tools that are related to Radon transform, tomography and image processing fields.For these fields, it is useful to work with some elementary known objects (Functions and Images) to evaluate, analyze, and validate the performance of various imaging approaches.We review the Box, the Disk, and the Gaussian functions with analytical formulation of their Radon transform.It is also convenient to represent a digital image in terms of the unit box function.We gain several benefits from this representation including several computations on the given image.This representation can also be involved in the mathematical arrangements of the iterative reconstruction algorithms of tomography so that the coefficient matrix of the linear system is exact.
We start with a brief revision on Radon transform and its inverse problem.Radon transform found its application in the field of computed tomography [1][2][3][4][5][6] that motivated researchers for different reconstruction methods to serve medical modalities such as Emission CT, Ultrasound CT, Magnetic Resonance, and others.There are many other applications involved with tomography such as synthetic aperture radar, Cryo-electron microscopy for structuring viruses, Digital Rock Physics, for example [5][6][7].Image registration techniques also utilize integral transforms approach which include the Radon, Fourier transform, Hough transform, Trace transform, and many others [8][9][10][11][12][13][14].
This work is presented in the following order: in section 2, background material regarding Radon transform is reviewed.Three elementary functions and phantoms are presented.In section 3, a representation of an image via the Box function with several computations are performed.In section 4, a brief review of the reconstruction problem is given with a focus on iterative methods using the Gradient descent approach.Experiments and demonstrations are presented in section 5, conclusion is written in section 6.

Radon Transform
Let (, ) be a function on  2 , and consider the line  in the plane: The Radon Transform of  along the line  is: shown in figure 1.
One can write a code to provide the suitable translation parameters for each of the letters A…Z.Once we have the basic tool of displaying these letters, we then can involve any of rules (3-9) to display different combinations of block images at different scales and orientations.Moreover, the Radon transform of these images is calculated precisely from (11) and the shifting rule.

Representation of an Image Via the Box Function
The following image representation can be useful in two major computations: the first is the use of the analytical tools for direct manipulation of certain aspects of an image such as the Radon transform, that can detect any linear transformation applied on the image, the computation of image moments, projection moments, and many other important measures and features of a given image.The second, is the use of this representation in some tomography problems such as algebraic and iterative reconstruction algorithms.We now describe this method.
Then, using (25) to find   () at the unknown angles .The first two moments and the centroid of an image are frequently used in many applications such as linear transformation of images, segmentation, texture analysis, and many others.In particular, the image centroid can be calculated by:

The Gradient Descent
Using the Gradient Descent method [18][19][20], let  be a function of the vector .The gradient of  at   is written as (  ).One can show that  =   − (  ), where  > 0 is the learning rate.We minimize () by updating  in the equation: .The gradient descent algorithm will simultaneously update   , The vectorized version is efficient for large scale calculations:

Demonstration
The following examples highlight certain computations in our presentation.

Example 2: Image Formation and Transformation
Consider the block image (16).It is a straightforward calculation to form the image, test transformations, and compute features as needed.Figure 6 shows the block with the sequence of transformation of reflection, scaling, translation, and rotation.Interested readers can build software for this class of images.This may include the disapply of images, implementing the transformation rules (3-9), computations of transforms, moments, and others; and trying tomography applications.18) with the projection  ∨ (, 0) from (19).

Example 4: Limited Data Tomography Using Algebraic Method
Consider figure 8: a 32 × 32 image where the sum of its gray values is 1.Using 90 angles of projections to form the system (29) with a total of 3717 equations for 1024 unknowns.Simulation of the Gradient Descent above, with α=.5 and 2000 iterations.The result is presented in figure 8b.A plot of the cost function is shown in figure 8c.Notice that this is just an example to demonstrate the application of (33) in solving the system (29) and is not necessarily the optimal reconstruction for this image.

Conclusion
In this review, we presented three elementary functions that are useful examples in trying image processing calculations.The family of block binary images is a good class of testing images that can be used for image registration which is an important area of image processing field.We have seen a way of image representation using the box function so that the image is more amenable to further manipulation.In particular, the computations regarding the Radon transform, Fourier transform, and moments.We have seen that it is possible for interested reader to develop a software based on this class of images that would perform several types of calculations such as displaying images, studying transformation rules (3)(4)(5)(6)(7)(8)(9), computing transforms, moments, centroids, and many.Moreover, reconstruction algorithms for tomography can be implemented and tested.We have used the (images representation using boxes) in the mathematical arrangements of the algebraic reconstruction algorithms of tomography.We have seen that the linear system (29) has an exact coefficient matrix due to the use of the box function for each pixel.The Gradient descent approach is very powerful tool to handle this system, this allows an accurate image reconstruction with limited data, which is a critical problem in tomography applications.

Figure 3 .
Figure 3. (a-b) Box and translated Box, (c) spatial coordinates and the pixel indexing.

Example 1 :Figure 5 .
Image Representation Consider the symmetric unit Gaussian (14-15) on the support[−2,2] × [−2,2] with  = 200 , shown in figure 5a.Applying (18) we display the same image in the first quadrant as 200 × 200 image, as in figure 5b.In figure 5c, we see the Radon transform of image (a) with  = 0 using (15), and in figure 5d, we see the Transform of image (b) with  = 0 using (19).(a) The symmetric unit Gaussian on the support, (b) 200 × 200 image seated in the first quadrant as a representation of image a, (c) Radon Transform of image a with ,  = 0, (d) Radon Transform of image b with ,  = 0.