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Computer Tomography in medical imaging provides human internal body pictures in the digital form. The more quality images it provides, the better information we get. Normally, medical imaging can be constructed by projection data from several perspectives. In this paper, our research challenges and describes a numerical method for refining the image of a Region of Interest (ROI) by constructing support within a standard CT image. It is obvious that the quality of tomographic slice is affected by artifacts. CT using filter and
* K*-means clustering provides a way to reconstruct an ROI with minimal artifacts and improve the degree of the spatial resolution. Experimental results are presented for improving the reconstructed images, showing that the approach enhances the overall resolution and contrast of ROI images. Our method provides a number of advantages: robustness with noise in projection data and support construction without the need to acquire any additional setup.

For a long time, research regarding image field has become popular. A digitalized image can be analyzed and manipulated in various meanings. The more image quality is improved, the more image explains and provides details. In order to get a better representation of the taken picture or to improve its quality, it is indisputably that taking more pictures from different views or angles is the simple way. This principle made practical use in medical imaging field also, where an accurate internal image is obtained by combining pictures from different views. This well-known method is used not only in nuclear medicine, but also in many fields such as scientific field, engineering field, military field and so on, for example, taking or scanning shape of extremely tiny object. These pictures can be acquired from data called projections and its procedure to combine the projections together to obtain an image called image reconstruction. In this imaging system, it is required to obtain an image when the number of projection profiles is restricted in angle under some situations. One of the most widely used techniques is Computed Tomography. Computed Tomography (CT) is the technique that can generate the internal structure of a target object by using the projections of various angles. Two examples of the limited angle problem are illustrated in

form technique), which transforms repeatedly between object domain and Fourier domain using a prior data in each iteration. The filtered back projection (FBP) [

The projection data is observed from the limited-view angles in the cases of the metallic implants in patients in medical, ocean acoustic tomography, electron microscopy of macromolecules, and so on. Then the problem is to become a kind of ill-posed type. An iterative Fourier method is to use the relationship between the given sections and prior information in Fourier and object domains. Although many kind developments for the algorithms for solving the limited view CT, the stagnation and fail to converge still are unsolved. Goal of our research is to construct support in order to find black region (pixel intensity is 0) in image reconstructed by projection profiles. Consequently, it is necessary to receive a good image as well as possible.

In this paper, we consider the combination of conventional methods for CT and focus on recent developments of the noise filtering method [

The total variation function is popular in several fields of mathematical image processing. The idea of the total variation has been firstly introduced as a denoising technique by Rudin, Osher and Fatemi [

In this paper, Total Variation Denoising (TVD), also known as total variation regularization was used to be an approach for noise reduction which is defined in terms of an optimisation problem. In order to find the output of the TV denoising, output is obtained by minimizing a particular cost function. Although the algorithm can be solved in several different ways, the derivation is based on the min-max property and the majorization-minimization procedure given in [

Let us consider a discrete real-valued in projection pro-file of N-point signal x ( i ) defined on 1 ≤ i ≤ N . The total variation measures how much the signal changes between signal values. There are many ways to define discrete TV by means of finite different signal value, but we used absolute values (l^{1} norm) because it is one of the simplest ways which is defined as

TV ( x ) = ∑ i = 2 N | x ( i ) − x ( i − 1 ) | (1)

Suppose y is a signal consisted by original signal x and additive white Gaussian noise n as the following.

y = x + n (2)

We want to estimate the target x by using iterative clipping algorithm [

x min = arg min x { y − x 2 2 + λ TV ( x ) } (3)

Smoothness of the signal is controlled by the regularization parameter λ .

To find parameter x , the iterative clipping algorithm for TV denoising was applied and it was clearly explained in [

An iterative FT presented in this paper is used for combining analytical and algebraic reconstruction techniques. An iterative methods such as Algebraic Reconstruction Technique (ART) and analytical Fourier methods such as Filtered Back projection (FBP) have been considered by several applications. There are many approaches in image reconstruction field using projection data, and almost of those methods were developed from ART algorithm because of its simple principle and good effectiveness. The fundamental equation is presented by the following.

∑ j = 1 N A i j f j = p i i = 1 , ⋯ , M (4)

where each p i is a projection along i th ray; M is the total number of rays in all projections; and A i j is a weight of every pixel for all the different rays in the projection; f i represents a column vector which contains the values of pixels and N is the total number of pixels. However, in most cases the ray width is often approximately equal to the cell width of the image and a line integral is called a ray sum. The implementation of ART [

f j k + 1 = f j k + α { p i − ∑ n = 1 N A i n f n k ∑ n = 1 N A i n 2 A i j } (5)

where f j k and f j k + 1 are the current and updated vector respectively; ∑ n = 1 N A i n f n k is the sum of pixels along i th ray in k th iteration; and p i is the sum of projection for the i th ray; and α is a coefficient.

Next, we will explain regarding algorithm of FBP solution. The FBP method uses a relationship between the Fourier transform of the projections of a target image and the correspond sections in the Fourier domain. It is based on the Radon transform. ( r , θ ) is a polar coordinate of the object domain, and F is the Fourier transform of the target image. The relationship is presented as the following.

F ( ρ cos θ , ρ sin θ ) = ∫ − ∞ ∞ g ( r , θ ) exp { − j 2 π r ρ } d r (6)

where g ( r , θ ) is the projection with angle θ and site r , then it is represented by ∫ f ( r cos θ − s sin θ , r sin θ + s cos θ ) exp { − j 2 π r ρ } d s , and non-italic j presents imaginary unit.

We started to construct image using ART as an initial input before iterative FT was applied in next step to enhance the quality of image. The concept of iterative revision model was first provided by Gerchberg and Saxton in order to solve the problem of structure in science imaging. A plausible result is provided by the popular methods, ART and FBP, with enough constraints. However, the problem becomes to be an ill-posed type in the case of the limited view. As a good method using the Fourier domain for such the situation, an iterative Fourier transform method was presented in the GP (Gerchberg-Papoulis) algorithm [

The sections are overwritten by the given projections in 2), and a prior knowledge of the target is embittered to the ρ ′ in 4). The following is well used as one of the updated method in 4),

ρ ( r ) → { ρ ′ ( r ) r ∉ D 0 r ∉ D (7)

where D is the region at which ρ ′ n violates the object-domain constraint such the prior knowledge.

Such the updating procedure has been developed in the challenging the phase retrieval problem [

tions, and falling to the global solution. Therefore, in this presentation, we show a good hybrid of ART, FBP and iterative Fourier transform for the limited view constraint. The algorithm can be explained by following operations:

1) Construct 2D image as initial input which is constructed by traditional method (ART).

2) Construct sinogram from projection project of each angle θ .

3) Take Fourier transform of r to obtain 1D-FT.

4) Revise F by replacing 1D-FT from step 2) by 1DFT of original projection value and ramp filtering are applied.

5) Obtain inverse 1D-FT for the filtered projection for each θ .

6) Take inverse FT and obtain reconstructed image by back projection method.

7) Revise reconstructed image by applying a prior data and support in image space.

8) Go to step 2).

These eight steps are repeatedly calculated until the process is stopped under some suitable condition of convergence. The criterion which is used to terminate process is the ratio of error between distance of original profiles and profiles of reconstructed image to become sufficiently small. Unfortunately, the number of iteration can be easily fixed but ε or value makes loop terminated cannot be fixed or set only one constant. Different data provide different distance. Thus, appropriate way is to select the result that provides the smallest distance of a set of projection in fixed iteration. In our presentation, the iterative method is connectively used with ART and FBP.

The definition of object support (outline of the object) in real space is a region of interest where a target object is located. In order to construct support of reconstructed image, K-means clustering for support construction in diffractive imaging approach [

1) Calculate | ρ ′ | from ρ .

2) Set an initial cluster presented in

3) Calculate center of C1.

a) center _ c 1 = center value of C1.

b) center _ c 1 = 1 N _ c 1 ∑ r ∈ c 1 | ρ ′ ( r ) | .

where N _ c 1 is number of pixel in region C1.

4) Calculate center of C2.

a) center _ c 2 = center value of C2.

b) center _ c 2 = 1 N _ c 2 ∑ r ∈ c 2 | ρ ′ ( r ) | .

where N _ c 2 is number of pixel in region C2.

5) For r < N : N is number of all pixel.

If d ( | ρ ′ ( r ) | , center _ c 1 ) < d ( | ρ ′ ( r ) | , center _ c 2 )

r ∈ C 1 , r ∉ C 2

Else

r ∈ C 1 , r ∉ C 2

6) Consider region of object by comparing value between C1 and C2.

7) Expand 1 pixel region around object.

8) Go to step 3 and repeat until value of C1 and C2 are not changed.

A simple schematic diagram of K-means clustering method is illustrated in

In image segmentation field, information entropy that occasionally called Shanon’s entropy [

For the single threshold, an algorithm can be summarized by the following. Suppose that value of a normalized histogram is shown in term of h ( i ) which i takes integer values from 0 to N and result is converted to 8 bits gray scale image, that is: ∑ i = 0 255 h ( i ) = 1 . By using t as a threshold value, an entropy of black pixels is defined by

H B t = − ∑ i = 0 t h ( i ) ∑ j = 0 t h ( j ) log h ( i ) ∑ j = 0 t h ( j ) (8)

and an entropy of white pixels is given by

H W t = − ∑ i = t + 1 255 h ( i ) ∑ j = t + 1 255 h ( j ) log h ( i ) ∑ j = t + 1 255 h ( j ) (9)

The optimal threshold maximizing the sum of above two entropies is presented as the following. An example of the result of optimal threshold is shown in

t max = arg max t ( H W t + H B t ) (10)

The performance of the proposed algorithm for an ex-ample of a test signal (projection profile at 0˚) with SNR of 30 dB is shown in

Angles | Error (Non-filtering) | Error (Proposed method) |
---|---|---|

32 | 4.34e9 | 8.37e8 |

64 | 8.55e9 | 1.69e9 |

180 | 2.38e10 | 4.79e9 |

360 | 4.72e10 | 9.22e9 |

There are typical methods for the reconstruction of an unknown object using the constraints of the object and Fourier domains. For presenting the effectiveness of a hybrid procedure mixing method referred to previous section, a numerical example is settled in the following. The Shepp-Logan phantom (256 × 256) is used for a target object in

In the next example from Figures 13-16, to test the robustness of the algo-

rithm various noise attack are presented to the reconstructed image. Then, we showed the appearance of support constructed by K-means method compare with maximum entropy thresholding and example of projection profiles.

From experimental results, we can deduce that traditional method does not give a plausible image. However, our proposed method provides the better results than traditional method. When the support results are used to combine with the results from iterative FT, these results are a little refined, and such the hybrid gives a stable process for the reconstruction from the limited angles projections. The proposed method will be successful as compare with other traditional methods with noise reduction. However, the results of iterative methods are different in the situation of the stagnation and fail of convergence under the limited constraint of projections. The more refinement of unifying the reconstruction algorithms and developed computation for the ill-posed imaging problems is our future work.

To summarize, in this paper we tried to show the results after reconstructing an image using projections from different angles under various noise conditions assumed as Gaussian noise. In order to achieve, we divided into 3 steps for our experiment. Firstly, the topical filter method proposed by [

The mathematical analysis and simulation system of our work are supported by MEXT/JSPS Kakenshi 16K00222, JST A-step AS26Z02472H and MP27115663145.

Dhammatorn, W. and Shioya, H. (2017) A Support Construction for CT Image Based on K-Means Clustering. Journal of Computer and Com- munications, 5, 137-151. http://dx.doi.org/10.4236/jcc.2017.51011