Algebraic Reconstruction Technique for Inverse Problem Solving

The Algebraic Reconstruction Technique (ART) is an iterative method used to solve inverse problems, specifically designed to find the solution vector x that satisfies the equation Ax=b. This algorithm requires three primary inputs: the system matrix A, the measurement vector b, and the number of iterations k. During implementation, ART performs sequential row-wise projections to gradually refine the x vector's values toward the solution. The method utilizes a correction term based on the residual between the current estimate and actual measurements, updating the solution vector after processing each equation row. This technique finds applications across multiple domains including signal processing, medical imaging (particularly in CT reconstruction), and data analysis. The iteration count k serves as a crucial parameter controlling the trade-off between computational accuracy and speed - higher iterations yield better convergence but require more processing time. The algorithm typically initializes x with zeros or an educated guess, then iteratively applies correction steps using the formula: x^(i+1) = x^i + λ * (b_i - A_i·x^i) / ||A_i||² * A_i^T, where λ represents the relaxation parameter. This makes ART a powerful and flexible tool for addressing various inverse problems where direct matrix inversion is computationally expensive or numerically unstable.