Tomographic Image Reconstruction Using Filtered Back Projection (FBP) And Algebraic Reconstruction Technique (ART) Nabil Chetih*, Zoubeida Messali** *
Welding and NDT Research Center (CSC), BP64, Cheraga, Algiers, Algeria. E-mail :
[email protected] ** Université El Bachir El Ibrahimi de Bordj Bou-Arréridj, Faculté des Sciences et de la Technologie, Département d’Electronique, 34265 El Anassers, Bordj Bou-Arréridj, Algérie. E-mail :
[email protected]
Abstract—This paper presents comparative study and experimentation of Algebraic Reconstruction Technique (ART) and Filter Back Projection (FBP). The ART and FBP methods are used to reconstruct the object from the X-ray projection. The process of creating back the object image from the Radon Transform of the object is known as Image Reconstruction. Image reconstruction is a famous and interesting field which comes under computed tomography. Computed Tomography is used for identifying the hidden or inner defects of objects. In this paper Algebraic Reconstruction technique and Filter Back Projection methods are implemented and the experimented results are compared using performance parameters for various test cases. Projections for the image reconstruction are calculated analytically by defining two phantoms: Shepp-Logan phantom head model and the standard medical image of abdomen. The original images are grayscale images of size 128 × 128, 256 × 256, respectively. Keywords- Computed Tomography; X-ray projection; Filter Back Projection; Algebraic Reconstruction Technique.
I.
INTRODUCTION
Computed tomography (CT) has been extensively studied for years and widely used. Although the filtered backprojection method has been the method of choice by CT manufacturers, efforts are being made to revisit iterative methods [1]–[7]. The word tomography means reconstruction from slices. It is an imaging technique which uses the absorption of X-rays by a number of organs in the body. This creates a shadow picture (projection). The problem of tomographic reconstruction is an inverse problem i.e. to estimate the object starting from its projections. The most effective type of reconstruction technique is able to reconstruct good quality CT images even when the projected data are noisy. The conventional algorithms of image reconstruction for CT are Back Projection BP and Filtered Back Projection FBP reconstruction techniques which are analytical reconstruction methods. In Filtered Back Projection methodology, Fourier Slice Theorem is made into use for Image reconstruction [8].
Although for now the Filtered Back Projection algorithm is most widely used by manufacturers. On the other hand, the algebraic reconstruction methods such as the algebraic reconstruction technique (ART) [9,10,11], the simultaneous algebraic reconstruction technique (SART) [12] and the simultaneous iterative reconstruction technique (SIRT) [13] are a few of those iterative methods. Historically, the algebraic reconstruction technique (ART) was the first algorithm applied in CT [14]. ART can be seen as a linear algebraic problem. It is based on the simple assumption that calculating the crosssection can be done by making algebraic equations for the array of unknowns referring to the measured projection data. This paper is aimed to establish a comparative study of and experimentation of Filter Back Projection (FBP) and Algebraic Reconstruction Technique (ART) in enhance the image quality. Two Phantoms are used to test the results of the research: the Shepp-Logan Phantom and the standard medical image of abdomen. So, physical phenomena such as attenuation, scattering, and so forth are not under consideration here. The original images are grayscale images of size 128x128, 256x256 respectively (see Figure 2 and 3). After introducing the basic concepts and the supporting theory in section 2, we present the reconstruction techniques in sections 3 and 4. Following this, in section 5, we explain how to evaluate the quality of the reconstruction. Section 6 provides some test results and comparisons. Finally, in the last section our conclusions can be read. II. BASIC FORMULATION Fundamentally, tomographic imaging deals with reconstructing an image from its projections. In the strict sense of the word, projections are a set of measurements of the integrated values of some parameter of the object-integrations being along straight lines through the object and being referred to as line integrals. A line integral, as the name implies, represents the integral of some parameter of the image along a line as illustrated in Figure 1. The projections (or line integral), as shown in Figure 1, are defined as:
ሺሻ ൌ ሺǡ୲ሻ ሺǡ ሻ ሺͳሻ
Figure 1.
(1)
An object, ሺǡ ሻ, and its projection, ሺሺሻ, are shown for an angle of Ʌ
The function ሺሻ is known as the Raddon transform of the function ሺǡ ሻ, where ሺǡ ሻ represents thhe 2D image to be reconstructed, and ሺɅǡ ሻ the parameters of eaach line integral. ሺሻ is also known as a sinogram of the imagge. A sinogram as shown in Figure 2 and Figure 3 , is a 2D imaage, in which the horizontal axis represents the count location on the detector, and the vertical axis corresponds to the angulaar position of the detector. Using a delta function, ሺሻ can be rrewritten as: ାஶ
ሺሻ ൌ ିஶ ሺǡ ሻɁሺ
Ʌ Ʌ െ ሻ
(2)
III.
FILTERED BACK-P PROJECTION (FBP) ALGORITH HM Analytical reconstruction metho ods consider continuous tomography. So the problem is: giv ven the sinogram ሺሻ we want to recover the object described d in ሺǡ ሻ coordinates. The most common type of image an nalytical reconstruction is Filtered Back Projection (FBP). It is possible to derive he so-called Fourier slice reconstruction algorithms from th theorem [8]. This theorem relates the Fourier transform of a projection to the two-dimensionall Fourier transform of the object which is to be reconstructeed. Thus given the Fourier transform of a projection at enou ugh angles the projections could be assembled into a comp plete estimate of the two dimensional transform and then sim mply inverted to arrive at an estimate of the object. The algorith hm that is derived by using the Fourier Slice Theorem is thee Filtered Back Projection algorithm. It has been shown to be b extremely accurate and amenable to fast implementation. The FBP mathematically expressed as:
ሺǡ ሻ ൌ ො ሺሻɅ
(3)
Where ො ሺሻ is the filtered version of o ሺሻ with the ramp filter which gives a weight proportional to its frequency to each of the components. So, the relationship between ො ሺሻ and ሺሻ is expressed as ஶ
ො ሺሻ ൌ ିஶ ሺሻ ȉ ȁȁ ȉ ୨ଶ୴୲
(4)
Where ሺሻ is 1D Fourier Transfform of ሺሻ, multiplying with ȁȁ gives ramp filtering, integ grating the expression from െλ to λ gives inverse 1D Fourier Transform. Substituting (4) in (3), yields:
Figure 2. (Left) Shepp-Logan phantom head model ((128 x 128 pixels). (Right) Corresponding sinogram, with coverage angle rannging from 0° to 180
ஶ
ሺǡ ሻ ൌ ൫ିஶ ሺሻ ȉ ȁȁ ȉ ୨ଶ୴୲୲ ൯Ʌ (5) Therefore, the complete Filtered Back Projection algorithm can be viewed as follows: ࣂ ሺ࢚ሻ
FT ऐ
Filter ȁ࢜ȁ
Figure 4. Figure 3. (Left) Standard Medical Image of Abdomen.. (256 x 256 pixels). (Right) Corresponding sinogram, with coverage angle rannging from 0° to 180°
It is well known that, from knowledge of the sinogram ఏ ሺݐሻ, one can readily reconstruct the image ݂ ݂ሺݔǡ ݕሻ by use of computationally efficient and numerically sttable algorithms. The algorithm that is currently being useed in almost all applications of straight ray tomography iss Filtered BackProjection (FBP) algorithm. This algorithm will be described in the following.
IF FT ି ऐି
ෝ ࣂ ሺ࢚ሻ
ࢌሺ࢞ǡ ࢟ሻ Backprojection
Filtered Bacck Projection Algorithm
Notice that analytical reconstruction methods work ber of projections uniformly well when we have a great numb distributed around the object an nd cannot give satisfactory results when the number of projeections is less than four as it will be shown in our simulation results. IV.
ALGEBRAIC RECONSTR RUCTION TECHNIQUE (ART) Mathematically, the reconsttruction problem is posed as a system of linear equations [8 8], which should be solved when reconstructing an image: ଵଵ ଵ ଶଵ ଵ ଷଵ ଵ
ଵଶ ଶ ଶଶ ଶ ଷଶ ଶ
ଵଷ ଷ ڮ ଶଷ ଷ ڮ ଷଷ ଷ ڮ
ଵ ൌ ଶ ൌ ଷସ ൌ
ଵ ଶ ଷ
(6)
ڮ
ଵ ଵ
ଶ ଶ ଷ ଷ ڮ
V. PERFORMANCE EVALUATION ൌ
So we write the set of linear equations for each ray as: σ ୨ୀଵ ୧୨ ୨ ൌ ୧ ǡ ൌ ͳǡʹǡ ڮǡ ሺሻ Where ୧ are projection data along the ୲୦ ray. In other words, the projection at a given angle is the sum of nonoverlapping, equally wide rays covering the figure, is the total number of rays (in all the projection). ୧୨ are the weights that represent the contribution of every pixel for all the different rays in the projection. ୨ are the values of all the pixels in the image and is the total number of the pixels in the image. The ART is sequential method, i.e., each equation is treated at a time, since each equation is dependent on the previous. The equation of ART is given by ሺ୩ାଵሻ
୨
ሺ୩ሻ
ൌ ୨
ሺ୩ሻ
Ƚ
ሺౡሻ ୮ ିσొ సభ మ ొ σసభ
ሺ୩ାଵሻ
Where ୨ ୨
୧୨
(8)
The FBP, ART algorithms are implemented and tested for two test images. These algorithms have been implemented on a PC using Matlab programming language. In order to objectively evaluate the reconstructed results, we have computed four image quality measurement parameters proposed in [11] in addition to two relative norm errors and the visual quality of the reconstructed image. The quality measurements are listed below: 1 The relative norm error of the resulting images [8] is used and defined as: ൌ
2
Measured Projection
Estimated Image ݂ ሺሻ
Forward Projection ܣ
Projection of estimated image
˰˰
ሺሻ
Ƹ ൌ ܣ ݂
update
Correction term in imagespace ܣ ߜ ߜ݂ ൌ σ ܣ
The relative norm error of the simulated projections and defined as
are the current and the new
is the sum weighted pixels estimates, respectively; along ray ; for the ୲୦ iteration; ୧ is the measured projection for the ୲୦ ray , and Ƚ is the relaxation parameter. The second term on the left in equation (8) is the term of correction. The process starts by making an intial guess. Observing equation (8), we see that (a) this correction term is added to the current estimate in order to found the new estimate and (b) the comparison consists in the substraction of the estimated projections from the measured projections. Also, we can easily see, that equation (8) is used to update the value of the ୲୦ pixel on every ray equation. This process can be summarized as follows:
compare
Backprojection ܣ௧
Correction term in projection space ߜ ൌ ˰˰Ƹ
Figure 5. Algebraic Reconstruction Technique
Notice that the role of the relaxation paramemeter Ƚ is to reduce the effects of the noise which is caused by poor approximations of the ray-sums.
(9)
ԡԡమ
Where is the gray level value of the test image and መ is the gray level value of the reconstructed image.
ሺ୩ሻ σ ୬ୀଵ ୧୬ ୬
Initial guess ݂ ሺሻ
మ ฮିመฮ
ൌ
ԡ୮ି୮ ෝ ԡమ
(10)
ԡ୮ԡమ
Where is the measured projection and ො is the simulated projection. 3
Normalized cross-correlation (ܰ)ܥܥ: is one of the
methods used for template matching, a process used for finding incidences of a pattern or object within an image: ଶ
ൌ σ ୨ୀଵൣ୨ െ ൧ ോ σ୨ୀଵൣ୨ ൧
ଶ
(12)
4 Structural content () : it’s the measure of image similarity based on small regions of the images containing significant low level structural information and is defined as: ଶ መଶ ൌ σ (22) ୨ୀଵ ୨ ോ σ୨ୀଵ ୨ Smaller errors ǡ , and , means that the resulting reconstructed image is closer to the test image. Another criterion for the ART is the number of iterations and the number of projections for FBP. VI. RESULTS AND DISCUSSION Comparison of reconstruction techniques such as FBP (analytical), & ART (iterative) with respect to quality of reconstructed images is presented in this section. The reconstruction performances are calculated for 16, 32, 64 and 180 projections. The Figure 6 and 7 shows reconstruction of phantom head model and the standard medical of abdomen by FBP with coverage angle ranging from 0 to 180º with an incremental value of 10º to 2º.
TABLE 1.
QUALITY MEASUREMENTS BY VARYING THE FILTER KEEPING FIXED NUMBER OF PROJECTIONS (32) FBP Test Image: Sheep Logan
Filter Rampe Hanning Hamming Shepp Logan
(a)
(b)
(c )
(d)
Figure 6. Reconstructed image of Sheep logan phantom by FBP using (a)16, (b)32, (c) 64 and (d)180 number of projections.
(a)
(b)
ࢊࢌ
ࢊ
ࡹࡿࡱ
0.4168 0.3584 0.3616 0.4008
0.0880 0.0789 0.0795 0.0858
0.0359 0.0266 0.0270 0.0332
ࡺ
0.1324 0.0979 0.0997 0.1225
From Figure 8, it can be clearly seen that visual quality of the reconstructed images for various filters is almost similar.
(a)
(b)
(c )
(d)
Figure 8. Reconstructed images by FBP using (a) Rampe, (b) Sheep logan, (d) Hamming and (e) Hanning filters, number of projections 64.
The resultant reconstructed images obtained from ART algorithm by varying number of projections for 100 iterations, are shown in Figure 9 and 10.
(c )
(d)
Figure 7. Reconstructed image of standard medical image by FBP using (a)16, (b)32, (c) 64 and (d)180 number of projections.
Figure 6 and 7 clearly reveals that the quality of reconstructed images of both Sheep logan phantom and the standard medical image of abdomen increases as number of projection increases. Table 1 shows the quality measurements by varying the filter used in FBP and keeping common number of projections. Table 1 demonstrates that the quality measurements, from FBP using Hanning filter are slightly better than those of other filters. This means that the choice of the filter has not a significant effect to enhance the image quality.
(a)
(b)
(c )
(d)
Figure 9. Reconstructed image of Sheep logan phantom by ART using (a)16, (b)32, (c) 64 and (d)180 number of projections for 100 iterations
Normalized cross correlation NCC
(a)
(b)
0
10
-1
10
FBP ART -2
10
10
20
30
40
50
60
70
80
90 100 110 120
Number of projections N (c)
(c )
(d)
Figure 10. Reconstructed image of standard medical image by ART using (a)16, (b)32, (c) 64 and (d)180 number of projections for 100 iterations
Relative Norm Error df
Figures 9 and 10 clearly reveal that the quality of reconstructed image slightly increases as number of projections increases. The reconstructed images appear to be very blurry. The graphical representation of quality measurements versus projections of Shepp-Logan image using FBP and ART algorithm for 200 iterations are shown in Figure 11.
FBP ART
0
10
-0.2
10
-0.4
10
10
20
30
40
50
60
70
80
90
100 110 120
Number of projections N (a) Relative Norm Error dP
FBP
0.3
ART
0.25 0.2 0.15 0.1 0.05 0
20
40
60
80
Number of projections N (d)
100
120
Figure 11. Quality measurements v/s number of projections for FBP and ART algorithms. Using 200 iterations (a) , (b), (c) , (d) Test image: Shepp-Logan
We can easily observe that the FBP algorithm is fast and efficient with large number of projections. Also after about 32 projections, trend of errors appear to decrease consistently, unlike which significantly increases after 32 projections. The errors ,ǡ and obtained from the FBP algorithm are less than those of ART. Also, the resultant reconstructed images obtained from FBP are providing better reconstruction than that of ART by. VII. CONCLUSION
0
10
-1
10
FBP ART -2
10
Mean Square Error MSE
0.35
10
20
30
40
50
60
70
80
90 100 110 120
Number of projections N (b)
This paper presents the comparisons of the image reconstruction algorithms using Filtered Back Projection (FBP) analytical method and Algebraic Reconstruction Technique (ART) iterative method. In this work, objective measurement by four performance parameters, led to an ability to subjectively judge the reconstructed image quality. The results show that Filtered Back Projection (FBP) method provides the best image quality, small values of the errors ,, and than those of Algebraic Reconstruction Technique (ART). From the simulated results, we shall conclude that the Filtered Back Projection algorithm is reliable and practical to enhance the quality of reconstructed images.
[1]
Y. Censor, “Finite series-expansion reconstruction methods,” Proc. IEEE, vol. 71, pp. 409–419, 1983.
[10] L. Bao-dong, L. Zeng, and J. Dong-jiang, “Algebraic Reconstruction Technique Class for Linear Scan CT of Long Object, ” In. Proc. the 17th World Conference on Nondestructive Testing, Shanghai, China, October 2008.
[2]
Y. Censor and G. T. Herman, “On some optimization techniques in image reconstruction from projections,” Appl. Numer. Math., vol. 3, pp. 365–391, 1987.
[11] S. D. Desai, and L. Kulkami, “A Quantitative Comparative Study of Analytical and Iterative Reconstruction Techniques”, International Journal of Processing (IJIP), 4(4), pp. 307-319, Oct 2010.
[3]
M. Hanke and P. C. Hansen, “Regularization methods for large scale problems,” Surv. Math. Indust., vol. 3, pp. 235–315, 1993.
[4]
Y. Censor and S. A. Zenios, “Parallel Optimization: Theory, Algorithms, and Applications,” New York: Oxford Univ. Press, 1997.
[5]
M. Bertero and P. Boccacci, “Inverse Problems in Imaging,” Philadelphia, PA: Inst. Physical Publishing, 1998.
[12] M. M. Anishinraj, B. Venkataraman, and V. Vaithiyanathan, “ A Comparative Study and Experimentation of Simultaneous Algebraic Reconstruction Technique (SART) and Ordered Subsets Expectation Maximization (OSEM) ,” Eur. J. Sci. Res. 4(68), 584-590, 2012.
[6]
G. Wang, M. W. Vannier, and P. C. Cheng, “Iterative X-ray cone-beam tomography for metal artifact reduction and local region reconstruction,” Microsc. Microanal., vol. 5, pp. 58–65, 1999.
[7]
Y. Saad and H. A. van der Vorst, “Iterative solution of linear systems in the 20th century,” J. Comput. Appl. Math., vol. 123, pp. 1–33, 2000.
[8]
C. Kak and M Slaney ,“ Principles of computerized Tomographic Imaging,” New york: The Institute of Electrical and electronics Engineers, Inc., IEEE Press, 1988.
[9]
Q. Zhong, W. Junhao , and Y. Pan, “Algebraic Reconstruction Technique in Image Reconstruction with Narrow Fan Beam”. IEEE Trans. Medical Imaging. 52(5), 1227-1235, 2005.
REFERENCES
[13] V. H. Tessa, W. Sarah, G. Maggie, B. K. Joost, and S. Jan, “The implementation of Iterative Reconstruction Algorithms in MATLAB,” Master Thesis, Department of Industrial Sciences and Technology, University College of Antwerp, Belgium ,July 2007. [14] R. Gordon, R. Bender, and G. T. Herman, “Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and X-ray photography,” J. Theor. Biol., vol. 29, pp. 471–482, 1970.
