TY - GEN

T1 - Intervals in Linear and Nonlinear Problems of Image Reconstruction

AU - Censor, Yair

PY - 1981

Y1 - 1981

N2 - In the “series expansion” approach (see, e.g., [15]) to image reconstruction, the mathematical formulation takes the form of a system of equations, linear or nonlinear, (1)$${\rm{\{ }}{{\rm{f}}_{\rm{i}}}{\rm{\} (x) = \{ }}{{\rm{p}}_{\rm{i}}}{\rm{\} ,i = 1,2,}}...{\rm{,m,}}$$where the functions fi:ℝn → ℝ describe the physics and geometry of the model, x ∈ ℝn (the Euclidean n-space) represents the unknown image in digitized form, and pi are the values of the measurements taken. Typically, the Jacobian matrix of the system is huge, sparse, and lacks structure in its sparsity pattern. Moreover, due to various limitations, practical restrictions, and features which are inherent in the real problem and/or in the “pixel by pixel” model, the system at hand is inevitably inconsistent. It is highly over- or under-determined, mostly ill-conditioned and describes the real problem only approximately because discretization takes place at the very beginning of the series expansion approach.

AB - In the “series expansion” approach (see, e.g., [15]) to image reconstruction, the mathematical formulation takes the form of a system of equations, linear or nonlinear, (1)$${\rm{\{ }}{{\rm{f}}_{\rm{i}}}{\rm{\} (x) = \{ }}{{\rm{p}}_{\rm{i}}}{\rm{\} ,i = 1,2,}}...{\rm{,m,}}$$where the functions fi:ℝn → ℝ describe the physics and geometry of the model, x ∈ ℝn (the Euclidean n-space) represents the unknown image in digitized form, and pi are the values of the measurements taken. Typically, the Jacobian matrix of the system is huge, sparse, and lacks structure in its sparsity pattern. Moreover, due to various limitations, practical restrictions, and features which are inherent in the real problem and/or in the “pixel by pixel” model, the system at hand is inevitably inconsistent. It is highly over- or under-determined, mostly ill-conditioned and describes the real problem only approximately because discretization takes place at the very beginning of the series expansion approach.

U2 - 10.1007/978-3-642-93157-4_13

DO - 10.1007/978-3-642-93157-4_13

M3 - Conference contribution

SN - 978-3-540-10277-9

T3 - Lecture Notes in Medical Informatics

SP - 152

EP - 159

BT - Mathematical Aspects of Computerized Tomography

A2 - T. Herman, Gabor

A2 - Natterer, Frank

PB - Springer Heidelberg

CY - Berlin

ER -