Missing Surface Estimation Based on Modified Tikhonov Regularization: Application for Destructed Dental Tissue

Estimation of missing digital information is mostly addressed by 1- or 2-D signal processing methods; however, this problem can emerge in multi-dimensional data including 3-D images. Examples of 3-D images dealing with missing edge information are often found using dental micro-CT, where the natural contours of dental enamel and dentine are partially dissolved or lost by caries. In this paper, we present a novel sequential approach to estimate the missing surface of an object. First, an initial correct contour is determined interactively or automatically, for the starting slice. This contour information defines the local search area and provides the overall estimation pattern for the edge candidates in the next slice. The search for edge candidates in the next slice is performed in the perpendicular direction to the obtained initial edge in order to find and label the corrupted edge candidates. Subsequently, the location information of both initial and nominated edge candidates are transformed and segregated into two independent signals (X-coordinates and Y-coordinates) and the problem is changed into error concealment. In the next step, the missing samples of these signals are estimated using a modified Tikhonov regularization model with two new terms. One term contributes in the denoising of the corrupted signal by defining an estimation model for a group of mildly destructed samples, and the other term contributes in the estimation of the missing samples with the highest similarity to the samples of the obtained signals from the previous slice. Finally, the reconstructed signals are transformed inversely to edge pixel representation. The estimated edges in each slice are considered as initial edge information for the next slice, and this procedure is repeated slice by slice until the entire contour of the destructed surface is estimated. The visual results as well as quantitative results (using both contour-based and area-based metrics) for seven image data sets of tooth samples with considerable destruction of the dentin-enamel junction demonstrates that the proposed method can accurately interpolate the shape and the position of the missing surfaces in computed tomography images in both two and 3-D (e.g., 14.87 ± 3.87 $\mu \text{m}$ of mean distance (MD) error for the proposed method versus 7.33 ± 0.27 $\mu \text{m}$ of MD error between human experts and 1.25± ~ 0 % error rate (ER) of the proposed method versus 0.64± ~ 0 % of ER between human experts (~1% difference)).