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- X-Ray Computed Tomography in Biomedical Engineering - Robert Cierniak - Google книги
- X-Ray Computed Tomography in Biomedical Engineering
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Computed Tomography will appeal to a wide audience: students and researchers alike will appreciate its relevance to such disciplines as biomedical engineering, medical electronics, computer science and medicine. Robert Cierniak. From to he conducted research at the Czestochowa University of Technology, Poland.
This was based in part on micro-CT reconstruction. His team enhanced interior tomography via compressed sensing in a highly cited paper, challenging the conventional wisdom that there is no unique solution to the interior problem.
X-Ray Computed Tomography in Biomedical Engineering - Robert Cierniak - Google книги
Also, Ge Wang performed micro-tomography research to enable cone-beam image reconstruction of either elongated or planar samples. In service to his former institution, Virginia Tech, Ge Wang led the multi-scale CT facility openly available to colleagues and industry. BE, , Dept. China the best radar signal processing specialty in China. MS, , Dept. PhD, , Dept.
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X-Ray Computed Tomography in Biomedical Engineering
We can then formulate the following model:. In the classical least squares LS approach [ 15 , 16 ], the estimator is chosen to minimize noise:. However, in this work the matrix H is ill-conditioned, thus the LS solution usually has a huge norm and is meaningless. We then follow the Bayesian approach [ 17 ] whose optimization target in this work is to maximize the posterior probability of x given y , which is defined as:.
In this framework, the target x is regarded as a stochastic quantity with a prior density, and:. Actually, this imposes the prior information onto the estimation. Now, Eq. In this formulation 7 , the first part measures the difference between x and y , and the second part restricts x to the prior information.
In this work, with the decrease of noise, y is approximately equal to Hx , so the first term in 7 is close to zero. Next, the problem defined in 7 is redefined as a constrained optimization problem:. To solve the problem of 8 , we bring in the diffusion equation: [ 16 ]. I represents the diffusion target, and I s is the result after diffusion. According to [ 18 ], the diffusion process can be observed as a gradient descent of the prior energy, and the gradient descent algorithm is:.
In Eq. After the step of gradient descent, the algebraic reconstruction technique ART is applied to reduce the projection onto the convex set POCS : [ 21 ]. The matrix H is exploited to remove the metal trace, and it is constructed according to the position of the missing projections: the value in the position of the metal in the projection domain is set to zero, and the value in other areas is set to one. What calls for special attention is that the operator H multiplies others is multiplying the corresponding elements between two matrices. Step 1 and step 2 are quoted from the literature [ 16 ].
In this diagram, the corresponding sinograms are produced by forward projection. Diagram of the proposed Gaussian diffusion sinogram inpainting algorithm. From the uncorrected image, the prior image and metal-only image are obtained. The corresponding sinograms are yielded via forward projection. The gradient descent algorithm and POCS process are applied to restore the sinogram. The final corrected image is reconstructed via the FBP method from the corrected sinogram. Concerning the prior image, Li et al. First, they segmented the metal, and then a prior image was produced via an edge-preserving filter and a recovery procedure of the adjacent anatomical structures [ 22 ].
Meyer et al. Bone pixels kept their values because of the inherent variability of the bone densities. Different from Bal and Spies [ 13 ] who set the values of air and soft tissue with their own average CT numbers, in this work we obtained the prior image by the method of [ 14 ]. The performance of the proposed algorithm is tested on both the simulated and clinical datasets, and the results are compared with two conventional algorithms LI and NMAR.
In the simulation, we suppose that the process of the detector receiving photons can be modeled by the Poisson distribution [ 23 ]:. After logarithmic processing for the result of 13 , the projection that contains noise and beam hardening is obtained. The clinical dataset includes four patients: a patient with a single metal tooth, a patient implanted with two metal teeth, a patient with a single hip prosthesis and a patient with multiple metals implanted in the vertebrae.
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- Gaussian diffusion sinogram inpainting for X-ray CT metal artifact reduction.
The performance of the Gaussian diffusion sinogram inpainting algorithm was compared with the LI and NMAR algorithms subjectively and objectively. We subjectively evaluated the image quality for the proposed algorithm and the two other compared algorithms. We objectively evaluated the performance using the signal-to-noise ratio SNR and the normalized mean absolute deviation NMAD for the corrected images:. Generally, SNR measures the anti-noise performance of the algorithm, and its value is greater, the algorithm produces a better quality image; NMAD measures the difference between result image and ideal image, and its value is smaller, the algorithm produces a result that is much closer to the ideal image.
Besides, the mean pixel value and the standard deviation of different region of interests ROIs in each resulted image are analyzed. For the clinic datasets, the ROI is defined on uncorrected images containing the metallic implants, and it is magnified for observing clearly. Besides, the reference images of the clinic datasets is hard to define, thus, we evaluated the performance only by the image quality. In this section, the results corrected by the LI, NMAR and proposed algorithms for the simulation and clinical datasets are presented.
We first analyzed the simulated results, and the clinical part was analyzed subsequently. However, Fig.
Clearly, the original metal artifacts are successfully reduced by each method. Unfortunately, the LI MAR algorithm causes severe secondary artifacts, which blur the tissue around the metal. NMAR and the proposed algorithms reduce the artifacts with fewer secondary artifacts than LI, and with regard to the entire picture, the proposed algorithm has a better result than NMAR, especially in the area between the teeth as indicated by the arrows in Fig. However, in the proposed algorithm result, some slight artifacts are introduced around the metals, indicated as arrows in Fig. Images of the jaw phantom.
The display window is [0 0. From the table, the NMAD of the proposed algorithm is smaller than those of the LI and NMAR algorithms, indicating that the image corrected by the proposed algorithm is closer to that of the true image. While the largest SNR indicates that the proposed algorithm has a better effect on suppressing the metal artifact and noise, thus producing a higher quality image.
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All of the MAR algorithms have a good effect on reducing the metal artifacts, while the LI method introduces many residual artifacts and blurs the bone structures. Regarding the proposed algorithm, the image only introduces slight secondary artifacts, which are indicated by the arrows in Fig.
Thus, from subjective judgment, the proposed algorithm performs the best. Images of the hip phantom. The ROIs of different phantoms are defined in Figs. Generally, the proposed method has the mean pixel that is closest to the ideal, and it also has the smallest standard deviation in most ROIs. The results imply that the proposed algorithm has the best performance in suppressing the artifacts.
The mean pixel value and the standard deviation in different ROIs for the two phantoms. The top one are the results of the jaw phantom and the bottom one belong to the hip phantom. Eight RIOs are defined in Figs. These two groups of phantom experiments show that whether the bone structure in the image is simple or complex, the proposed algorithm has a good effect on suppressing metal artifacts and reducing the noise level. In the LI MAR result, the image shows severe secondary artifacts, and some structures are even more blurred.
Both NMAR and the proposed algorithm successfully reduce the artifacts and introduce hardly any secondary artifacts, but the proposed algorithm performs better. As indicated by the arrows in Fig. Thus, in this dataset, the proposed method performs the best, and the performance has proven that the algorithm can be used in complicated bone structure images containing a single implanted metal. Uncorrected image and results corrected via the different algorithms for the patient with a single metal dental filling. The display window width and window center are and HU, respectively.
In this dataset, the LI reduces the original metal artifacts existing in the uncorrected image but introduces severe secondary artifacts, making the image quality even lower than that of the uncorrected image. Compared with LI, NMAR produces a better result and does not cause artifacts between teeth; however, secondary artifacts still blur the structures between the two metals.