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We develop a hyperparameter inference method for image reconstruction from Radon transform which often appears in the computed tomography, in the manner of Bayesian inference. Hyperparameters are often introduced in Bayesian inference to control the strength ratio between prior information and the fidelity to the observation. Since the quality of the reconstructed image is controlled by the estimation accuracy of these hyperparameters, we apply Bayesian inference into the filtered back-projection (FBP) reconstruction method with hyperparameters inference and demonstrate that the estimated hyperparameters can adapt to the noise level in the observation automatically. In the computer simulation, at first, we show that our algorithm works well in the model framework environment, that is, observation noise is an additive white Gaussian noise case. Then, we also show that our algorithm works well in the more realistic environment, that is, observation noise is Poissonian noise case. After that, we demonstrate an application for the real chest CT image reconstruction under the Gaussian and Poissonian observation noises.

In the field of medical imaging and noninvasive measurement, computed tomography (CT) plays an important role in diagnosis. The tomography image is reconstructed from a series of projection data, which are transmitted signals throughout an object, such as X-rays, in multiple directions. A lot of algorithms have been proposed to reconstruct tomography images [

In order to improve image quality occurred by noisy observation, several image restoration methods based on the Bayesian inference are discussed in the field of image processing [

In contrast, from the viewpoint of the Bayesian inference, the hyperparameter inference problem can be expressed naturally. For example, in the field of the image restoration, Molina et al. demonstrated several hyperparameter inference methods in the Bayesian manner in the manner of a hierarchical Bayes inference [

In typical conventional methods, which use MAP inference for the computed tomography, a cost function that consists of data-fitting terms and several smoothness constraints has been introduced, and a minimization of the cost function is carried out in order to obtain the reconstructed image from the noisy observation data. Unfortunately, there have been few discussions related to the inference of a proper ratio between the data fitting and the constraints within the MAP framework. On the contrary, from the Bayesian inference point of view, it is natural to discuss the hyperparameter inference for image restoration using an evidence framework [

In our previous work, we proposed a CT image reconstruction in the manner of Bayes inference with a hyperparameter inference method from the noisy Radon-transformed observation by the evidence framework [

Moreover, we apply our reconstruction model into the real CT image data. Shepp and Logan phantom, which is usually used for evaluation of CT/PET image reconstruction, is a simple model of the axial cross-section human body. The internal organ of human body is not so much simple, so we use a real CT image data for reconstruction.

In order to explain our Bayesian inference method, we show the conventional CT reconstruction method using filtered backprojection (FBP) under the formulation of the Radon transform. After that, we introduce Bayesian inference into the reconstruction process.

Briefly, the Radon transform assumes that the observed signals are transmitted through the target object. Figure

Schematic diagram of the Radon transform. Detectors are aligned on the

We describe the density of the target as

Before introducing the Bayes inference, we formulate the conventional filtered backprojection (FBP) method. This reconstruction method is mainly formulated on the frequency domain, so we introduce the 2-dimensional Fourier transform of the reconstruction image

Meanwhile, we can apply a 1-dimensional Fourier transform for the

The FBP method is derived as a coordinate transformation from Cartesian coordinate

Thus, the reconstructed image

In this section, we introduce a stochastic observation process into the FBP method. Of course, it is natural to consider Poissonian noise for observation in a realistic model; however, introducing Poissonian process makes it hard to solve the reconstruction in analytic form. We consider that a solvable model is important for understanding the reconstruction process. So in our theoretical framework, we introduced additive white Gaussian noise for observation on the signal

To reconstruct an image from noisy data, using Bayes inference, we also denote the prior distribution. At first, we introduce the following energy function

From (

In order to calculate the denominator value called partition function, we discretize the integral description in the partition function over polar coordinate in frequency domain. When we denote the sampling width for radial direction and polar angle as

When we discretize the integral

We adopt the marginalized posterior mean

To reconstruct an appropriate tomography image with our Bayesian inference, we need to assign proper values to the hyperparameters

To maximize the marginal log likelihood (

In the computer simulation, we created the Shepp and Logan phantom image in

For hyperparameter inference, we adopt a gradient method that requires initial state of these parameters. In the following simulations, the initial state of

In order to evaluate the performance of the hyperparameter inference, we carry out the simulation in the additive white Gaussian noise environment at first. We assumed that the Gaussian noise

The computational cost is mainly consumed by hyperparameters inference. In this study, we adopted gradient method for the hyperparameter inference, so the computational cost depends on the initial state of these hyperparameters and learning coefficients

Figure

Comparison of the reconstructed tomography images derived using the Bayesian method and conventional FBP. The top row shows the Bayesian FBP methods, and the bottom one shows the conventional one. Each column corresponds to the strength of the observation Gaussian noise standard deviations. We show the magnification of a part of the reconstructed images around the edge of the phantom, whose location is indicated by white rectangle in the true image.

We used the peak signal-to-noise ratio (PSNR) to evaluate the quality of the reconstructed image. The result of this evaluation is shown in Figure

Qualities of reconstruction images measured by PSNR. The horizontal axis shows the SD of the Gaussian noise. The vertical axis shows the PSNR. The solid line shows the median of the 10 trials of our Bayesian inference results, and box plot shows quartile deviation. The dashed line shows the results of the conventional FBP method.

Figure

Reconstruction performance against hyperparameter

Gauss noise observation is the assumed model in our formulation equation (

In the computer simulation, we used R PET package for Poissonian noise sampling [

Figure

Comparison of the reconstructed tomography images derived using the Bayesian method and conventional FBP under the Poissonian noise. The top row shows the results of our method, and the bottom one shows the results of the conventional FBP method. Each column corresponds to the strength of the observation noise which can be denoted as the number of sampling in the acceptance rejection method. We show the magnification of a part of the reconstructed images around the edge of the phantom, whose location is indicated by a white rectangle in the true image.

Figure

Qualities of reconstruction images measured by PSNR. The horizontal axis shows the SD of the sampling level, which means the inverse of the Poissonian noise. The vertical axis shows the PSNR. The solid line shows the median of the 10 trials of our Bayesian inference results, and box plot shows quartile deviation. The dashed line shows the results of the conventional FBP method.

In order to evaluate the performance of our method for the CT/PET image, we applied our method to a real CT image reconstruction.

We prepare several real CT images provided by Tokushima University Hospital. The acquisition parameters of those HRCT images are as follows: Toshiba “Aquilion 16” is used for imaging device, and each slice image consists of 512 × 512 pixels, and pixel size corresponds to 0.546~0.826 mm; slice thickness is 1 mm. Thus, we set the sampling parameters as

In order to obtain noise-corrupted data

Figure

Comparison of the reconstructed images using real CT data with Gaussian noise between Bayes method and conventional FBP method. The top row shows the results of our method, and the bottom one shows the conventional FBP results. Each column corresponds to the strength of the observation noise that can be denoted as standard deviation (SD) of adding noise. We also show the magnification of a part around the bronchus, whose location is indicated by black rectangle in the true image.

Comparison of the reconstructed images using real CT data with Poissonian noise between Bayes method and conventional FBP method. The top row shows the results of our method, and the bottom one shows the conventional FBP results. Each column corresponds to the strength of the observation noise that can be denoted as the number of sampling in the acceptance-rejection method. We also show the magnification of around bronchus indicated by black rectangle in the true image.

Moreover, we evaluate the quantitative reconstruction performance by PSNR for the real CT image. Figure

Qualities of reconstruction images measured by PSNR for real CT image reconstruction under the Gaussian noise. The horizontal axis shows the observation SD of Gaussian noise, and the vertical axis shows the PSNR. The solid line shows the result of our Bayesian method, and the dashed one shows that of the conventional FBP method.

Qualities of reconstruction images measured by PSNR for real CT image reconstruction under the Poissonian noise. The horizontal axis shows the sampling level, and the vertical axis shows the PSNR. The solid line shows the result performance of Bayesian method, and the dashed line shows the results of the conventional FBP method.

We proposed a hyperparameter inference based on the Bayesian inference in order to reconstruct tomography image formulated by Radon transform. As a stochastic model, we introduced a simple MRF-like distribution

We discretized the image signals in the frequency domain expressed by the polar coordinate in order to evaluate the posterior distribution analytically, resulting in the ability to conduct posterior mean for the reconstructed image. Using the marginal-likelihood maximization method, we show that the hyperparameters introduced as

In order to evaluate the performance of our method, we simulated two observation noise cases, that is, Gaussian and Poissonian noises. We controlled noise strength by SD for Gaussian noise and sampling levels for Poissonian noise. In the phantom simulation for the Gaussian noise, we confirmed that our hyperparameter inference worked well against the PSNR, and the performance for the reconstruction was better than that of the conventional FBP. The computational cost for the hyperparameter inference depend on the initial state of them; however, about 1200~2000 times iterations made convergence to them for typical cases. In the Poissonian cases, the tendency of the reconstruction performance is similar to the Gaussian case. Our Bayesian method made better performance than the conventional FBP in any noise strength area. However, in the strong Poissonian noise case, that is, the noise could not approximate well by Gaussian noise, we confirmed that the performance of the reconstruction was not good enough for diagnosing. Moreover, we evaluated the performance by a real chest CT image. The real image has a little complex shape against the phantom image. Thus, in the low-noise strength area for both noise cases, the prior components worked too much for the smoothness effect. As a result, the PSNR was just worse than the conventional FBP in such area. However, detail structure of the organ was easy to identify in the obtained image of our model.

In this study, we demonstrate applying our algorithm to the only 2-dimensional image reconstruction. We consider the algorithm easy to extend for 3-dimensional case. Thus, we would reformulate our algorithm for applying to the 3-dimensional image reconstruction and confirm the performance in the future work.

This work is supported by Grant-in-Aids for Scientific Research (C) 21500214 and Innovative Areas 21103008, MEXT, Japan. The authors thank Professor Shoji Kido, Yamaguchi University. He provided a lot of advice about this study and made effort to prepare the CT images. They also thank Professor Junji Ueno, Tokushima University. He provided them with several high-resolution CT image for this study.