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Dentin is a hierarchically structured biomineralized composite material, and dentin’s tubules are difficult to study in situ. Nano-CT provides the requisite resolution, but the field of view typically contains only a few tubules. Using a plate-like specimen allows reconstruction of a volume containing specific tubules from a number of truncated projections typically collected over an angular range of about 140°, which is practically accessible. Classical computed tomography (CT) theory cannot exactly reconstruct an object only from truncated projections, needless to say a limited angular range. Recently, interior tomography was developed to reconstruct a region-of-interest (ROI) from truncated data in a theoretically exact fashion via the total variation (TV) minimization under the condition that the ROI is piecewise constant. In this paper, we employ a TV minimization interior tomography algorithm to reconstruct interior microstructures in dentin from truncated projections over a limited angular range. Compared to the filtered backprojection (FBP) reconstruction, our reconstruction method reduces noise and suppresses artifacts. Volume rendering confirms the merits of our method in terms of preserving the interior microstructure of the dentin specimen.

Teeth are important and interesting biomineralized tissues with remarkable mechanical properties through their hierarchy of structures [

Dentin tubules and their surroundings remain of interest not just to microanatomists but also to those studying the efficacy of prostheses’ attachment. The small dimensions of dentin tubules make them difficult to evaluate and have motivated major research efforts. Up to date, tubules and their surroundings have been characterized with microradiography [

Of particular interest, nano-CT provides the requisite 3D spatial resolution for studying dentin’s tubules and canaliculi. However, exact nano-CT reconstruction typically requires that the specimen remains within the field-of-view (FOV) during a 180° scan. Suitable cross sections, say 25

In classic CT theory, an interior ROI cannot be reconstructed exactly from truncated projections. As a result, features outside the ROI may seriously disturb the features inside the ROI, often hiding or distorting vital information. A recent progress demonstrated that the interior problem can be exactly and stably solved if a subregion in the ROI is known [

For dentin image reconstruction, we employed an ordered-subset simultaneous algebraic reconstruction technique (OS-SART) to reconstruct the dentin slice images [

This paper is organized as follows. Section

A thin wafer of bovine dentin was cut from a molar using a diamond wafering saw (Isomet 1000, Buehler, Lake Bluff, IL) to a thickness of about 150

Compared to medical and micro-CT, nano-CT uses an X-ray lens to bring spatial resolution into the nanometer domain. To date, better than 20 nm has been achieved for routine use with multi-keV hard X-ray radiation which is able to penetrate hundreds of microns of dental tissue [

The dentin specimen was imaged by the transmission X-ray microscope at Sector 32-ID of the Advanced Photon Source, Argonne National Laboratory, USA. The synchrotron nano-CT system can be viewed as in a typical parallel-beam geometry and employs monochromatic 8 keV X-radiation. The X-ray detector contained

One truncated projection of the dentin specimen.

Sinogram consisting of 561 truncated projections for the slice marked by the red line in Figure

The conventional CT approach cannot exactly reconstruct an internal ROI only from truncated projections through the ROI because this interior problem does not have a unique solution in an unconstrained setting. Interestingly, recent results show that the interior problem is solvable if appropriate yet practical prior information is available. In particular, if the attenuation coefficient distribution on a small sub-region in an ROI is known, or the attenuation coefficient distribution over the ROI is piecewise constant, the interior problem has a unique solution. Theoretically, a function can be well approximated by piecewise constant functions, so the present dentin specimen is modeled as being piecewise constant. In this project, the piecewise-constant-model-based interior tomography algorithm was used to reconstruct dentin images from truncated projections over a limited angular range. The interior tomography algorithm is robust against noise by minimizing the image TV. Specifically, we employed the ordered-subset simultaneous algebraic reconstruction technique (OS-SART) for interior reconstruction of the dentin specimen.

The imaging process can be modeled as a linear system in terms of the popular pixel basis functions:

While the ART method is the first iterative algorithm used for CT reconstruction [

Then, a possible version of the OS-SART formulation can be expressed as

The above OS-SART reconstruction method can be empowered by the CS technique to improve the image quality under less favorable measurement conditions. As mentioned earlier, the discrete gradient transform (DGT) is a valid sparse transform for dental images. Hence, a dentin image can be reconstructed from truncated projections data via the

Equation (

The whole iteration process can be summarized in the following steps.

Input measured data

Update the current image using OS-SART by (

Minimize the TV of the current image using the gradient descent method by (

Go to Step

In our implementation, the gradient descent control coefficient was

To evaluate the performance of interior tomography for studying the dentin specimen, we designed a dentin phantom as shown in Figure ^{−1} for an X-ray energy 8 keV according to the X-ray Attenuation Databases reported by the National Institute of Standards and Technology (NIST). The scanning range was −70° to +70° (0° is for the normal to the plate-like specimen) with either a 0.25° or 1° angular increment and captured two groups of truncated projection data (a total of 561 or 141 projections, resp.). We then used FBP and CS-based interior tomography methods, respectively, to reconstruct the ROI from the two datasets for comparison.

Dentin phantom. The circular region indicates a ROI, with the line labeled “X” for subsequent profiling.

The reconstructed results are in Figure

Reconstruction results. (a) The reconstructed ROI using FBP from 561 projections, (b) the reconstructed ROI using interior tomography from 561 projections, (c) the reconstructed ROI using FBP from 141 projections, and (d) the reconstructed ROI using interior tomography from 141 projections. The display window is [0, 585] cm^{−1}.

Profiles corresponding to the line “X” in Figure

To test the stability of interior tomography against data noise, we repeated the reconstructions from projections contaminated with 1% Gaussian noise level. The reconstructed results from the data with 1% Gaussian noise are in Figure

Reconstruction results from the data with 1% Gaussian noise. (a) The reconstructed ROI using FBP from 561 projections, (b) the reconstructed ROI using interior tomography from 561 projections, (c) the reconstructed ROI using FBP from 141 projections, and (d) the reconstructed ROI using interior tomography from 141 projections. The display window is [0, 585] cm^{−1}.

Profiles corresponding to the line “X” in Figure

Then, we used the root mean square error (RMSE) to quantify the reconstructed results, which is expressed as

RMSE values for the reconstructed ROI images.

Reconstruction protocol | Noise-free data | Data with 1% noise |
---|---|---|

FBP (561 projections) | 117.00 | 122.85 |

FBP (141 projections) | 140.40 | 152.10 |

Interior tomography (561 projections) | 9.65 | 10.59 |

Interior tomography (141 projections) | 11.81 | 12.75 |

Supported by our encouraging numerical results, we applied the interior tomography method to study the dentin specimen. Figure

ROI reconstruction of the dentin specimen. (a) The reconstruction using FBP from 561 projections, (b) The reconstruction using interior tomography from 561 projections, and (c) the reconstruction of interior tomography from 141 projections. The display window is consistent.

To analyze the internal microstructures, two volumes of 600 high-resolution dentin slices were reconstructed using interior tomography from 561 projections and 141 projections, respectively. As a benchmark, a volume of the same 600 high-resolution dentin slices was also reconstructed using FBP from the 561 projections. All of these image volumes were rendered, as shown in Figure

Reconstructed dentin structures. (a) The volume rendering based on the FBP reconstruction from 561 projections, (b) the volume rendering based on the interior tomographic reconstruction from 561 projections, and (c) the volume rendering based on the interior tomographic reconstruction from 141 projections. The 3D visualization display window is consistent.

For high-resolution image reconstructions, the FBP algorithm is very efficient and accurate. With truncated datasets, however, the FBP method is subject to more noises and artifacts than those reconstructed by the iterative approach. In the piecewise-constant-model-based interior tomography framework, we have employed several techniques to increase convergence rate while improving image quality. First, an OS version of the Landweber scheme has been used. Second, the code has been optimized, combining the merits of C++ and multicore techniques. Third, a high-performance computer has been utilized to run our code program. Particularly, we have simultaneously reconstructed 8 slices using 8 central processing units (CPU).

The CS theory indicates that an image can be often accurately reconstructed from a rather limited amount of data when it can be sparsely represented in an appropriate domain. The internal feature of the dentin specimen is complex, and porosity is characteristic. We consider a dentin object approximately piecewise constant. Then, a dentin image is sparsified by a discrete gradient transform. Because the dentin projections are intrinsically truncated, it is inevitable that there are some artifacts in the image reconstructed using the FBP method. On the other hand, interior tomography is shown to be promising in meeting the challenge. In particular, the ability to generate a volume rendering with a high signal-to-noise ratio from a very limited number of truncated projections is quite feasible using interior tomography.

For real data study, our purpose is to reconstruct a high-quality dentin image. For 2D image reconstruction, there were some shadows (lower brightness) in the canaliculi and tubules regions reconstructed by the CS-based interior tomography method, which could reflect the attenuation characteristics of dentin interior structure. However, for 3D image reconstruction, the CS-based interior tomography method could suppress image artifacts and noises for the reconstructed images from truncated projections. Moreover, the CS-based interior tomography minimizes the TV of a reconstructed image by the steepest gradient descent method to generate a better looking 3D perspective view, which might oversmooth fine details if the number of views is too small. In the future studies we will analyze more dentin specimens to evaluate the performance of interior tomography and use the dictionary learning technique to capture more information.

In conclusion, we have developed a piecewise-constant-model-based interior tomography method to deal with truncated projections collected over a limited angular range, and investigated the feasibility and potential of the interior tomographic application in dentin characterization. It has been demonstrated that the CS-based interior tomography method is advantageous for the dentin reconstruction from incomplete nano-CT data. Further improvements are underway to facilitate dental research.

This work was partially supported by the US NIH/NIBIB Grant EB011785, US NICDR Grant DE001374, and the National Natural Science Foundation of China Grant 61171157. The use of the Advanced Photon Source was supported by the US Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract no. DE-AC02-06CH11357.