Tikhonov Regularization in Image Reconstruction with Kaczmarz Extended Algorithm 1 Andrei B˘ autu abautu@anmb.ro “Mircea cel B˘ atrˆ an” Naval Academy 1 st Fulgerului St, Constantza, 900218 Romania Elena B˘ autu, Constantin Popa {erogojina, cpopa}@univ-ovidius.ro “Ovidius” University 124 th Mamaia Blvd., Constantza, 900527 Romania Abstract In a previous paper we proposed a simple and natural extension of Kaczmarz’s projection algorithm (KE, for short) to inconsistent least-squares problems arising in ART image re- construction in computerized tomography. In the present one we describe two versions of this extension for a Tikhonov regularization of the original inconsistent least-squares problem. The first version deals directly with an (augmented) equivalent formulation of the Tikhonov regularization problem, whereas the second one uses the gradient of the Tikhonov regularized functional. For both new versions of the KE algorithm we present some theo- retical considerations together with numerical experiments and comparisons with the initial KE method. 1 Introduction Many problems in the field of tomographic image reconstruction are modeled by the linear least-squares problem: find x ∈ IR n such that ‖Ax − ˜ b‖ = min!, (1) where A is an m × n real matrix and ˜ b ∈ IR m a given vector ( ‖·‖ and 〈·, ·〉 will denote the Euclidean norm and scalar product on some space IR q ). Although from a theoretical view point the problem (1) is consistent, i.e ˜ b ∈ R(A), in real world applications, usually due to measurements errors, the right hand side of (1) is perturbed as b = ˜ b + δb, δb = δb A + δb ∗ A ∈ R(A) ⊕ N (A t ) (2) and the problem becomes inconsistent (R(A),N (A),A t will denote the range, null space and transpose of A). In this case, the classical Kaczmarz projection algorithm (see [6]) can no longer be used, thus the extended version KE from [5] has to be applied. If a i ∈ 1 Paper supported by the PNCDI INFOSOC Grant 131/2004