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Image Reconstruction Algorithms For Computer Tomography

On Different Projections

B.S.Sathishkumar

1

, Dr.G.Nagarajan

2

and Dr.P.Elavarasan

3

1

Research Scholar, 2Professor and 3Associate Professor ,

1,2Department of ECE, Pondicherry Engineering College, Puducherry - 605 014, India

3Department of ECE, Rajiv Gandhi College of Engineering and Technology, Kirumampakkam, Puducherry - 605 014, India 1bssathishkumar.79@gmail.com

2

nagarajanpec@pec.edu

3elavarasan_2000@yahoo.co.in

Abstract—The Rapid system and hardware development of X-ray Tomography techniques have been escorted by equal scope in image reconstruction algorithms also. Image reconstruction is a mathematical process that converts the projection data into image space. The adequate result, amount of input and convergence time of reconstruction algorithm are major factors in developing algorithm. In this paper, landweber iteration algorithm is simulated and its performance is compared with other iterative algorithms. The quality of image is verified with the help of correlation coefficient and Mean squared error.

Keywords— Analytical and Iterative Reconstruction Algorithm Computer Tomography, Projections, Radiation Dose.

I. INTRODUCTION

X-ray radiography has offered a valuable non-invasive means of diagnosis [1]. There are many situations of real world, when it is desired to determines some structural (internal) properties of an object or substance, using the measurements, called data, obtained by methods that do not damage or disturb the conditions of the object or substance under investigation. X-ray is used as tool for investigation of internal information of the object in medical imaging [2].

X-ray Tomography is a method in medical imaging technique that enables the reconstruction of cross-sections of an object, using a series of X-ray measurements taken from many angles around the object in noninvasive order [3,4]. To accomplish this, the narrow X-ray beam sent through the object from X-ray source and other side the detectors are placed which is used to detect the intensity [5]. Then, change the position of the source and detector around the object and the measurement process is repeated. The system requirement for the sensor arrangements and imaging modality are different in industrial tomography than medical tomography [6]. The intensity of X-ray beam is detected in one direction called projection.

The procedure to create the cross-sectional information of object is called image reconstruction from projections [7]. In 1963, image reconstruction was introduced and its benefits lead this technique widely used in diagnostic medical field. i.e., provide the inner information of object without harmful.

Simultaneously acquired the sonogram data in different directions and each have unique X-ray energy distribution in CT [8]. Fig: 1 shows the basic CT image.

Fig: 1. Simplest CT scan involves measuring the intensity of a pencil beam of X-rays from

many different angular positions

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namely; Parallel beam, Fan beam and Cone beam. Reconstruction in Tomography is a process to convert the projection space into image space. Image reconstruction from projections is the field that lays the foundations for computed tomography [9].

Attenuation distribution from the objection in all the directions of projections are defined into image space is called reconstruction. In MR images, it is required to combine the field map calculation and image reconstruction procedures [10].

There are two methods of inversion or recoveries from the samples are set of line-integral projections (Transform method) or set of integrals on planes (Radon Transform) of an object density distribution. Both will be same in 2D case. The algorithm which reconstruct image from the density distribution in the measured data through line integral [11].

The procedure to reconstruct the image, based on the many projections at different angles is made with a reconstruction algorithm. The main usage of reconstruction is pointed out the variation of attenuation coefficients. Thousands of equations must be solved to determine the linear attenuation coefficient of all pixels in the image domain. Several algorithms are developed to form an image from a set of projections. Two major classification of reconstruction algorithms are Analytical and Iterative methods [12] as shown in Fig: 2.

Fig: 2. Classification of Reconstruction Algorithms

II. ANALYTICAL RECONSTRUCTION ALGORITHM

Analytical reconstruction algorithm is a single step algorithm. It is noise free in reconstruction process. It attempts to find a direct mathematical solution for image from unknown projections [13]. It has computationally more efficient. It consists of two types, Filtered back projection and Back projection filtered methods. In an image, there are many frequency components.

Spatial domain does not reflect the characteristics of image. So it is transformed into frequency domain through Fourier transform [14]. An iterative reconstruction algorithm uses a multiple repetition in which current solution convergences towards a better solution [15]. As consequences, the computational demands are much higher.

III. ITERATIVERECONSTRUCNTIONALGORITHM

Due to exponential growth of computer and computational capacities available in processor, the usages of IR methods have become a reliable option in reconstruction time acceptable for clinical applications [16]. IR algorithms are proved to be stable in removing the problems in the analytical method [17]. IR tries to formulate the final result as a solution either set of equation or solution of an optimized problem which is solved in iterative fashion. It can improve image quality and reduce the metal artifact and noise during image reconstruction process.

The basic implementation of iterative algorithm begins with initial guess may be empty space or image acquired through FBP method and iteratively apply corrections to current estimation of image. The correction is a difference between the known projection data and the projection calculated from the current estimate. This process will be repeated until this comparison is zero or within the limits. Several IR algorithms developed for clinical applications and some of the techniques in IR are Simultaneous Iterative Reconstruction Algorithm (SIRT), Iterative Least Square Technique (ILST), etc., The difference among these techniques are way in which corrections are carried in every iteration.

Fig: 3. Schematic Representations of the Steps of Iterative Reconstruction.

As shown in Fig: 3, the basic operation of iterative reconstruction consists of the following three steps repeated:

 Forward projection of the examined object with creation of artificial raw data.

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 Back projection of the correction term onto the volumetric object estimate.

IV. SIMULATIONRESULTS

There are variety of image reconstruction algorithms existing in the medical tomography, it is very hard to select an optimal method for a specific application. Also reconstruction algorithms are depending on number of projections, sensor geometry, angle projection, radiation dose etc., The issue here with an example that shows the differences in images of different iteration algorithm like SART, CAV, Cimmino, DROP and landweber iteration algorithm. A Matlab generated test phantom image is taken as input and reconstructed image of different algorithm for fixed number of projection and iterations shown. The image obtained from different algorithms are appear blurred except landweber iteration. The size of the input image of all algorithms is taken as 256 X 256.

The simulated projections are 2D Shepp–Logan head phantom with well-known properties which are widely used in order to validate reconstruction algorithms. The Shepp–Logan phantom contain’s ellipses with different absorption properties that resemble the outline of a head.

The number of projections and iterations are fixed for above algorithm, the resultant reconstructed image has a good quality obtained from the landweber compared to other algorithms.

Fig: 4. Test and Reconstructed Image (Parallel Beam)

Fig: 5. Test and Reconstructed Image (Fan Beam)

Fig: 4. and Fig: 5 show that the effect of image quality in terms of mean absolute error Vs Number of iteration and Correlation Coefficient Vs Number of iterations between original and various iteration algorithms. The number of projections increases proportionally to difference between the original and reconstructed image which is minimized. However, the correlation coefficients are much deviated from original and reconstructed image. To evaluate the effect of thersholding on image quality, they are calculated for each type of coefficient selection, the average image error per pixel:

,

f(x,y) h(x,y)

(1) MN

x y

Mean Squared Error

  

where, f(x,y) and h(x,y) are the reference and reconstructed images respectively with dimensions M X N.

0 10 20 30 40 50 60

0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16

Number of iteration

M

ea

n

ab

so

lu

te

e

rr

or

Number of iteration Vs.MAE

land sart cav cimmino drop

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0 10 20 30 40 50 60

Fig: 7. Correlation Coefficient between Original and Reconstructed

Figs: 6 to 7 show the performance comparison between the proposed and other iterative algorithms. The mean square error and correlation coefficient are measured in original and reconstructed image for different iteration in Test phantom simulated with different iterative algorithms. The proposed landweber iteration algorithm produces much better results in lower iteration itself with quality of image.

V. CONCLUSION

The growth of technology is made in the development of image reconstruction algorithm will accelerate the accuracy and also exhibits other benefits in tomography. From the advantages of iterative reconstruction algorithms over analytical, the simulation carried out on landweber, CIMMNO, DROP, CAV and SART algorithms. The performances of above algorithms are noted in terms of correlation coefficient and Mean squared error. Among this, the visual appearance of reconstructed image is higher in landweber algorithm than others in case of limited and projection and iterations. The level of error is also less.

REFERENCES

[1] P. Phillippe and Bruyant, "Analytical and Iterative Reconstruction Algorithms in SPECT", Medicine Journal of Nuclear, Volume. 43, No. 10, 2002, pp. 1343-1358.

[2] Mohammed A.Al-Masni, et al., ―A rapid algebraic 3D volume image reconstruction technique for cone beam computed tomography‖, Biocybernetics and Biomedical Engineering, Volume. 37, Issue .4, 2017, pp. 619–629. [3] M. Bertero and P. Boccacci, Inverse Problems in

Imaging. Philadelphia, PA: Inst. Physical Publishing, 1998.

[4] P. Alan et al, ―Analytic and Iterative Reconstruction Algorithms in SPECT‖, Journal of Nuclear Medicine, Volume. 43, 2002, pp. 1343-1358.

[5] D. F. Jackson and D. J. Hawkes, ―X-ray attenuation coefficients of elements and mixtures‖, Physics Reports, Volume. 70(3), pp. 169-233.

[6] E. P. A. Constantino and K. B. Ozanyan, ―Tomographic Imaging of Surface Deformation From Scarce Measurements via Sinogram Recovery‖, IEEE Sensors Journal, Volume. 9, Issue. 4, 2009, pp. 399 – 410. [7] A. C. Kak and M. Slaney, ―Principles of Computerized

Tomographic Imaging‖, IEEE Press, 1988, Chapter 2. [8] S. S. Gleason, H. Sari-Sarraf, et al, "Reconstruction of

multienergy X-ray computer tomography images of laboratory mice", IEEE Transactions on Nuclear Science, Volume. 46, Aug. 1999, pp. 1081-1086. [9] Rolf Clackdoyle and Michel Defrise, ―Tomographic

Reconstruction in the 21st Century‖, IEEE Signal Processing Magazine, Volume. 27, Issue.4, 2010, pp. 60-80.

[10]B. P. Sutton, D. C. Noll, et al, ―Fast, iterative image reconstruction for MRI in the presence of field in homogeneities‖, IEEE Transactions on Medical Imaging, Volume. 22, Issue. 2, 2003, pp. 178 – 188.

[11] L. A. Shepp and B. F. Logan, ―The Fourier reconstruction of a head section‖, IEEE Transactions on Nuclear Science, Volume. NS21(3): 1974, pp. 21-34. [12] J. Ni, X. Li, et al, ―Review of Parallel Computing

Techniques for Computed Tomography Image Reconstruction‖, Current Medical Imaging Reviews, 2006, Volume. 2, Issue 4, pp. 405-414.

[13] R. Gordon, R. Bender, et al, ―Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and X-ray Photography‖, Journal of Theoretical Biomedical, Volume. 29, pp. 471–482, 1970.

[14] Wilkinson, Allen, ―Parallel programming: techniques and applications using networked workstations and parallel computers‖, Prentice Hall Press, 2005.

[15] C. Rafael González, Richard Eugene Woods (2008). Digital image processing, Third Edition. Chapter 5. [16] Y. Saad and H. A. van der Vorst, ―Iterative solution of

linear systems in the 20th century‖, J. Computer Journal of computational and Applied Mathematics, Volume. 123, 2000, pp. 1–33.

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References

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