Reconstructions of Truncated Projections Using an Optimal Basis Expansion Derived From the Cross-Correlation of a “Knowledge Set” of a priori Cross-Sections

An algorithm was developed to obtain reconstructions from truncated projections by utilizing cross-correlation of a “knowledge set” of a priori nontruncated cross-sections with a similar structure. A cross-correlation matrix was constructed for the known set of cross-sectional images. The eigenvectors of this matrix form a set of orthogonal basis vectors for the reconstructed image. The basis set is optimal in the sense that the average of the differences between members of a given set of a priori images, and their truncated linear expansion for any basis set, is minimal for this particular basis set. A procedure for finding optimal basis vectors is fundamental for deriving the Karhunen-Loeve (K-L) transform. Therefore, one can represent an image not in the “knowledge set,” but of similar structure by a linear combination of basis vectors corresponding to the larger eigenvalues; thus, the number of basis vectors is reduced to a number less than the total number of pixels. The projection of an image represented by this linear combination of basis vectors is a linear combination of projected basis vectors which are not necessarily orthogonal. A constrained least-squares method was used to evaluate the coefficients of this expansion by minimizing the sum of squares difference between the expansion and the projection measurements taking into account the distribution of coefficients over basis vectors. The constrained least-squares estimates of the coefficients were used in an expansion of the orthogonal basis to obtain the reconstructed image. The constrained solution has a reduced noise level in this inverse problem. It is shown that the reconstruction of truncated projections can be significantly improved over that of commonly used iterative reconstruction algorithms.