The Weighted Least Squares Problem Sample Clauses

The Weighted Least Squares Problem. ‌ In Chapter 4, we have discussed energy-windowed spectral CT model and introduced a new preconditioner based on the corresponding nonlinear least squares problem and the ▇▇▇▇▇-▇▇▇▇▇▇ approximation of the Hessian. A two-step optimization method that includes projected line search and trust region method is implemented to solve this problem. However, there are two main concerns related to the preconditioner and optimization. At first, the preconditioner requires the information of the current iteration and even if it is cheap to compute, we still need to repeat the computational process in each ▇▇▇▇▇▇ iteration. Secondly, since the nonlinear optimization is based on the ▇▇▇▇▇-▇▇▇▇▇▇ approximation of the Hessian, it raises a question if we can come up with a new preconditioner and use a first order method to solve it. In this chapter, we still focus on the energy-windowed spectral CT model and we present a linearization technique to transform the nonlinear equation into an optimization problem that is based on a weighted least squares term and a bound constraint. Recall that the basic energy-windowed spectral CT model is expressed by ∫ y(k) = S(k)(e) exp .− ∫ t∈l µ (˙r (t) , e) d tΣ d e + η(k).  i = 1, 2, · · · , Nd × Np,  k = ▇, ▇, · · · , ▇▇, ▇ Using the material decomposition, µ (˙r (t) , e) = Nm um,ewm (˙r), discretizing Equation