BINARY IMAGE RECONSTRUCTION WITH DIAGONAL AND ANTI-DIAGONAL PROJECTIONS

Abstract

Discrete tomography refers to reconstruction of discrete functions from its projections and the function has discrete finite range and the domain of discrete function is bounded which may be continuous or discrete. In particular, when the range of discrete function consists only two values 0 and 1 and the domain consists the finite discrete set, the problem of discrete tomography is changed to determination of discrete sets by their projections in few directions, here projection represents the number of lattice points on each parallel line in few directions. Thus problem of discrete tomography can be referred as reconstruction of binary images from its line sum, whereas in classical problem of Discrete Tomography the lines are rows and columns. In present thesis, the projections are considered from two directions as diagonal (450) and anti-diagonal (1350) only. Hence the problem of discrete tomography is to reconstruct the function 􀝂 on finite lattice set 􀜺 from its projections in diagonal and anti-diagonal direction. Mathematically, this problem is to get the binary matrix from its diagonal and anti-diagonal sums. The compatibility and consistency of the projection data and the existence of solution of unique reconstruction is a challenging task of discrete tomography. In case of diagonal and anti-diagonal projection set it has not yet been reported. Thus in present thesis it has been achieved to get the consistency of the projection set then the algorithms proposed to get the unique reconstruction of two orthogonal projections without using any constraints. Mathematical formulation and characterization of projection set in diagonal and anti-diagonal direction has been determined for the reconstruction of binary images and to achieve the goal of research work. viii The analysis of all proposed reconstruction algorithms has been performed on the basis of misclassification percentage of pixels between the reconstructed and original binary images. The reconstruction algorithms proposed by Chang, without using any constraints or a priori information about the object, thus algorithms in present thesis have been compared to verify the quality of proposed reconstruction algorithms. Binary images have been Reconstructed and the error occurred in reconstruction process has been reported as of percentage of misclassification. The outcome of the proposed algorithms reveals that the reconstruction of binary images is possible from orthogonal projection (Diagonal & anti-diagonal) only without using any constraints. Maximum average misclassification percentage is achieved 7.3 % and minimum 2.43% which is much significant as compare to Chang’s (max.32.4% & min 10.77%). In case of hv-convex binary images, it has been noted that no proposed algorithms has crossed this upper and lower limit of misclassification, although for non convex binary images this limit is 47% and 0%, where zero percent misclassification signifies the uniqueness of binary image.