Tomographic Image Reconstruction Using Training Images with Matrix and Tensor Formulations

Reducing X-ray exposure while maintaining the image quality is a major challenge in computed tomography (CT); since the imperfect data produced from the few view and/or low intensity projections results in low-quality images that are suffering from severe artifacts when using conventional reconstruction methods.

Incorporating a priori information about the solution is a necessity to improve the reconstruction. For example, Total Variation (TV) regularization method –assuming a piecewise constant image model – has been shown to allow reducing X-ray exposure significantly, while maintaining the image resolution compared to a classical reconstruction method such as Filtered Back Projection (FBP).

Some priors for the tomographic reconstruction take the form of cross-section images of similar objects, providing a set of the so-called training images, that hold the key to the structural information about the solution. The training images must be reliable and application-specific. This PhD project aims at providing a mathematical and computational framework for the use of training sets as non-parametric priors for the solution in tomographic image reconstruction.

Through an unsupervised machine learning technique (here, the dictionary learning), prototype elements from the training images are extracted and then incorporated in the tomographic reconstruction problem both with matrix and tensor representations of the training images. First, an algorithm for the tomographic image reconstruction using training images, where the training images are represented as vectors in a training matrix, is described.

The dictionary learning problem is formulated as a regularized non-negative matrix factorization in order to compute a nonnegative dictionary. Then a tomographic solution with a sparse representation in the dictionary is obtained through a convex optimization formulation. Computational experiments clarify the choice and interplay of the model parameters and the regularization parameters.

Furthermore, the assumptions in the tomographic problem formulation are analyzed. The sensitivity and robustness of the reconstruction to variations of the scale and rotation in the training images is investigated and algorithms to estimate the correct relative scale and orientation of the unknown image to the training images are suggested.

Then, a third-order tensor representation for the training images images is used. The dictionary and image reconstruction problem are reformulated using the tensor representation. The dictionary learning problem is presented as a nonnegative tensor factorization problem with sparsity constraints and the reconstruction problem is formulated in a convex optimization framework by looking for a solution with a sparse representation in the tensor dictionary.

Numerical results show considering a tensor formulation over a matrix formulation significantly reduces the approximation error by the dictionary as well as leads to very sparse representations of both the training images and the reconstructions. Further computational experiments show that in few-projection and low-dose settings our algorithm is while (not surprisingly) being superior to the classical reconstruction methods, is competitive with (or even better of) the TV regularization and tends to include more texture and sharper edges in the reconstructed images.

The focus of the thesis is the study of mathematical and algorithmic prospectives and thus the training images and tomographic scenarios are mostly simulation based. More studies are however needed for implementing the proposed algorithm in a routine use for clinical applications and materials testing.