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An image super resolution reconstruction method was used to improve the spatial resolution of the thick pinhole imaging system and to mitigate the limitations of the image spatial resolution of the hardware of the image diagnostic system. The thick pinhole is usually applied into the diagnostics of the high energy neutron radiation image. Due to the impacts among its energy flux, spatial resolution and effective field of view, in dealing with the large area radiation source, the spatial resolution of the thick pinhole neutron image cannot meet the requirements for high precision modeling of the radiation source image. In this paper, the Lucy-Richardson image super resolution reconstruction method was used to simulate the thick pinhole imaging and super resolution image reconstruction. And the spatial resolution of the image could be increased by over three times after the image super resolution reconstruction. Besides, in dealing with the pseudo-noise, plum blossom shape appeared in the image super resolution reconstruction. The analysis of the source of the pseudo-noise was made based on the simulation of the image reconstruction under various conditions according to the characteristics of the thick pinhole image configuration.

As for the large-area high energy neutron radiation source image diagnostics [

As for the image super resolution processing method, it refers to an image data processing technique of singleframe or multi-frame images which can provide a spatial resolution far beyond the performance of the hard wares of the image diagnostic system. Due to its high effectiveness, it has become an important discipline in the image processing field. The image super resolution technique is consisting of the image super resolution restoration and the image super resolution reconstruction [

The image super resolution processing method could be divided into frequency domain method and spatial domain method. The frequency domain method is used to eliminate the spectrum aliasing in the frequency domain and to improve the spatial resolution. Principally, this method is applied into the image processing with spaceinvariant point spread function, typically including Winner filter method that is widely used in the penumbral imaging. Meanwhile, the spatial domain method is used at the image pixel scale. It is used to vary or constrain the pixels to improve the image quality. The spatial domain method could be widely used due to its strong inclusion of spatial domain apriori constraints. The major processing methods are including: the Iterative Back Projection (IBP) method [

In the image super resolution processing, the point spread function (PSF) is a very important parameter for the image diagnostic system. Most of the studies on the super resolution imaging processing method were made based on the characteristics of the space-invariant point spread function. Comparatively, few studies were made on the super resolution processing of the images with space-dependent point spread function (PSF) [10-12]. As for the super resolution image processing with spacevariant point spread function (PSF), such methods are commonly used, including the spatial coordinate transformation method, sectioned restoration algorithm and direct restoration algorithm. However, in the thick pinhole imaging system, only the direct restoration algorithm could be used for the image processing due to the continuous variation of the space-dependent point spread function (PSF).

In this paper, the image super resolution processing method of the thick pinhole imaging system will be presented, which is characterized in the continuous variation of the space-dependent PSF and relatively big image size.

The image diagnostic system for the radiation image diagnostics mainly consists of two parts: the radiation imaging system and image detecting and recording system. The radiation imaging system works according to the pinhole imaging principle. For low-energy photon image diagnostics such as the visible lights and X-rays, the thin pinhole imaging system is used, while for neutrons and high energy γ rays, the thick pinhole imaging system is used. As for the image detecting, the film, imaging plates and CCD cameras could be used to directly record the radiation images. Alternatively, the scintillator could be used to convert neutrons or γ rays into fluorescence image which will be taken with the visible light recording system [

Generally, the radiation source image is obtained in the image plane through the thick pinhole. The intensity of the image plane could be given as:

where: is the corresponding intensity distribution at the image plane point (x, y) of the luminescence intensity f(x¢, y¢) at the object plane point (x¢, y¢) in accordance with the characteristics of the imaging system. Suppose the image plane intensity abides by the linear superposition, and according to the linear propagation principle, the image plane intensity could be obtained as the following:

where: is the response function of the object plane point (x¢, y¢) at the point (x, y) in the image plane with a supposed linear response between f(x¢, y¢) and h. Again, a certain hypothesis should be introduced to simplify the issue. First of all, suppose the thick pinhole imaging should be made in an ideal condition, with a static object plane and a static image plane, then Equation (2) could be expressed as follow:

In the radiation imaging, the effects of the noise of the system have to be taken into consideration. Thus, the image plane intensity could be obtained in the following equation:

where, n is the noise of the recording system.

The point spread function (PSF) is a major characteristic parameter of the imaging system, standing for the intensity distribution of the infinitively small point light source after the pinhole imaging. However, since the point spread function (PSF) is very hard to measure in the actual tests, the simulation and computation results then could be used as an alternative. The computation of the point spread function (PSF) for the thick pinhole imaging system could be made with the geometry analytical method, and Monte Carlo N-Particle Transport Code (MCNP) neutron transportation program, etc. In the radiation imaging, due to the uniform spatial distribution of the secondary particles caused by various reactions between the neutrons and the thick pinhole shielding materials, the contribution of the secondary particles to the image detector could be negligible. Thus, in this paper, the geometry analytical method would be used for the computation of the point spread function (PSF) of the thick pinhole imaging system.

The basic principle of the pinhole imaging is given in

where: J(i, j) is the ray intensity of the point (i, j), L is the distance between the object plane and the image plane, s is

In the computation, the total thickness of the pinhole was 600 mm, the length of the straight aperture was 30 mm, the aperture size was 0.1 mm, the free path of the material was 7.5 mm, the image distance and the object

distance were both 2000 mm, and the top points of dualcone structures at the both ends were at the very center of the straight aperture to provide an optimized design for the thick pinhole. The simulated computational results of the distribution of the point spread function (PSF) of the thick pinhole imaging system are given in

In order to clarify the evolution of the intensity of the point spread function (PSF), the intensity distribution of the point spread function (PSF) nearby the axis was calculated, whose results are given in

In the thick pinhole imaging system, the point spread function (PSF) of the system is variable with the change

of the space. In dealing with the super resolution of the images with a determined point spread function (PSF) or single-frame image, Lucy-Richardson image super resolution processing method should be a very effective option. This method is characterized in a relatively big noise tolerance, which is deduced from iterative algorithm principle based on Bayesian posterior probability theory [

Besides, for the first estimation, the equation could then be expressed as:

where: a = max(1, m − K + 1); b = min(m, I), c = max(1, n − L +1 ); d = min(n, J); e = I + K − 1; f = j + L − 1); I = 1, 2,∙∙∙I, j = 1, 2,∙∙∙J. Equation (6) requires a large quantity of computation. Thus, for simplification, only the point spread functions (PSF) within a limited area should be applied into the computation. In the case of the pinhole imaging, the point spread function (PSF) has a very sharp boundary, and the values of the pixels out of the boundary should be very small. In this sense, the selection of the radius of the point spread functions (PSF) should be made to provide the function values at the end of the radius far smaller than the peak values.

Again, Equation (6) requires a large quantity of iterative computation, and the required computation time is proportional to I^{2}N^{4} (where: I is the image size and N is the size of the point spread function). As for the selection of I and N, both the image quality and the computation time should be carefully evaluated. In our computation, the pinhole diameter was 0.1 mm, the magnification was 1, the spatial resolution of the thick pinhole imaging system was 0.2 mm, and the pixel size in the image plane was 0.0088 mm/pixel. Finally, the computation of the ringtype pinhole imaging and image reconstruction were made, whose results are given in

times after the image super resolution processing.

The thick pinhole is mainly used for the radiation source image diagnostics. Based on the above mentioned computational results, the simulation of the image distribution of the stepped shape was made, and a strip-type image was overlapped in the center of the image. The simulation and the image reconstruction were made to provide results in

Throughout the above mentioned simulated thick pinhole imaging and image reconstruction, noise signals in plum blossom shape always occurred in the image reconstruction as shown in

In order to identify the source of such noise, the simulated imaging with the ideal thin pinhole was introduced before image reconstruction to provide an ideal radiation source image as shown in

For further insight, the thick pinhole imaging systems with magnifications of 1 and 10 respectively were used for the simulation of imaging and image reconstruction

as shown in

Then, two problems were found in the reconstruction of the gradient image. One is the reconstruction results is dependent on the magnification. The bigger the magnification is the better reconstruction results could be obtained. Another one is that the strongest noise signals occur nearby the axis of the pinhole. As indicated in

the pinhole axis lies in the abrupt change of the point spread function nearby the straight aperture.

In the radiation image diagnostics, the thick pinhole imaging system is a major bottleneck to improve the spatial resolution of the image diagnostic system. Due to the continuous space-dependent variation of the spatial point spread function, the spatial domain method is the only option for the image super resolution reconstruction. With the introduction of the Lucy-Richardson method into the image reconstruction, the spatial resolution of the thick pinhole imaging system could be increased by over three times after the image super resolution processing.

In the image reconstruction, relatively strong noise would occur nearby the pinhole axis due to the image distortion nearby the straight aperture. Within view of the straight aperture, the point spread function is determined by the back end of the thick pinhole, while the expansion angel of the field of view of the object plane corresponding to the image plane is determined by the front end. As for the whole range of the field of view, the point spread function of the system is determined both by the front end and the back end of the thick pinhole. Besides, abrupt change of the point spread function takes place nearby the straight aperture.

Due to fact that the point spread function of the thick pinhole imaging system is determined both by the front end and the back end of the thick pinhole, with a relatively small image distance, the length of the thick pinhole could not be negligible. Comparatively, with a relatively big image distance, the length of the thick pinhole could be negligible and to provide reconstruction results approximately as good as those with the thin pinhole imaging system.

The study presented in this paper is jointly sponsored by both the Sci. & Tech. Development Foundation of China Academy of Engineering Physics (CAEP 2010B0103006 and CAEP 2011B0103017) and the National Natural Science Foundation of China (Grant No. 11005095). And the authors are also very thankful to associate professor Zhang Jianhua, associate professor Chen Jinchuan, Mr. Zhou Lin and other colleagues for their help during this study.