We need to know the vector formula_1. formula_1 and formula_11 are the original image, while formula_12 and formula_13 are measurements of responses. Vector formula_1 is inside the region of interest formula_3, (formula_18). Vector formula_11 is outside the region formula_3. The outside region is called formula_4, (formula_21) and formula_12 is inside a region in the measurement corresponding to formula_3. This region is denoted formula_25, (formula_26). The region of vector formula_13 (outside the region formula_25) also corresponds to formula_4 and is denoted as formula_30, (formula_31). In CT image reconstruction, it has  To simplify the concept of interior reconstruction, the matrices formula_6, formula_7, formula_8, formula_9 are applied to image reconstruction instead of a complex operator. The response in the outside region can be a guess formula_30; for example, assume it is formula_73  A solution of formula_1 is written as formula_76, and is known as the extrapolation method. The result depends on how good the extrapolation function formula_73 is. A frequent choice is  at the boundary of the two regions. The extrapolation method is often combined with "a priori" knowledge, and an extrapolation method which reduces calculation time is shown below. Assume a rough solution, formula_76 and formula_80, is obtained from the extrapolation method described above. The response in the outside region formula_81 can be calculated as follows:  The reconstructed image can be calculated as follows:  It is assumed that  at the boundary of the interior region; formula_85 solves the problem, and is known as the adaptive extrapolation method. formula_86 is the adaptive extrapolation function. It is assumed that a rough solution, formula_76 and formula_80, is obtained from the extrapolation method described below:  or  The reconstruction can be obtained as  Here formula_86 is an extrapolation function, and it is assumed that formula_85 is one solution of this problem. Local tomography, with a very short filter, is also known as lambda tomography. The local inverse method extends the concept of local tomography. The response in the outside region can be calculated as follows:  Consider the generalized inverse formula_96 satisfying  Define   so that Hence,  The above equation can be solved as  considering that formula_104 is the generalized inverse of formula_104, i.e.