## 3 New Methodology for Ultrasonic CT
### A Theoretical Justification of [PERSON]'s Idea  
We give a justification why [PERSON]'s idea worked well in practice. His iteration method is justified in terms of the least square solutions. If a linear system Equation (17) has a solution \(\mathbf{x}\), then it holds \(A\mathbf{x}-\mathbf{s}=\mathbf{0}\), in view of which the following the idea of the least square solution is defined. **Definition 1**. : _A vector \(\mathbf{x}\in\mathbb{R}^{n}\) is called a least square solution of Equation (17) (or Equation (18)) if and only if it minimizes the norm as Equation (23)_  
\[\|A\mathbf{x}-\mathbf{s}\|, \tag{23}\]  
_where \(\|\mathbf{y}\|:=\sqrt{y_{1}^{2}+\cdots+y_{m}^{2}}\) for \(\mathbf{y}\in\mathbb{R}^{m}\)._  
We note that there are many researches to give least square solutions to the overdetermined system Equation (17), consult [15; 16] for example, our theory is different from them in the following points. * We a priori know that the least square solution to the overdetermined system Equation (17) is unique (Theorem 1 below). Figure 2: Image of algebraic reconstruction technique (ART). * We shall discuss not only the solution to Equation (17) but also the limit where the size of the all pixels tends to 0. * We know by experience that, even for fixed number of pixels (even if the number \(n\) is fixed), the more observation X-ray data we have (the larger the number \(m\) is), the better resolution we can obtain, which shall be discussed in a stochastic way (Theorem 2 below). As for the least square solutions, there hold the following propositions. Proposition 1: _For \(A\in M_{mn}(\mathbb{R})\), the following conditions are equivalent_. * _There exists at least one least square solution of Equation_ (17) _for any_ \(\mathbf{s}\in\mathbb{R}^{m}\)_._
* _Furthermore, if_ \(\text{rank }(A)=n\) _then the solution of Equation_ (17) _is unique._  
Proposition 2: _For \(A\in M_{mn}(\mathbb{R})\), the following conditions are equivalent_. * \(\mathbf{x}\in\mathbb{R}^{n}\) _is a least square solution of Equation_ (17)_._
* _There holds the following Equation_ (24)_._ \[{}^{t}A\mathbf{x}-{}^{t}A\mathbf{s}=\mathbf{0}.\] (24)  
The proofs of Propositions 1 and 2 being exercises of linear algebras, we would not mention them. In the practical CT, the rank of the coefficient matrix \(A\) in Equation (17) must be \(n\). Therefore, we have the following theorem. Theorem 1: _In the practical CT, there always exists the unique least square solution to the system Equation_ (17) _of the linear equations._  
In the following, we assume the following conditions. * \(f(x,y)\in L^{1}\cap L^{2}(\mathbb{R}^{2})\). * For any line \(l\subset\mathbb{R}^{2}\), \(f|_{l}\in L^{1}(l)\) and is piecewisely continuous. Note that these conditions, (I) and (II), are satisfied by the density distribution of the human body. We also assume that  
* \(m>>n\). With these assumptions, (I), (II) and (III), we shall give a justification of [PERSON]'s idea applying a stochastic approach. In the system Equation (17), we divide the right hand side into there components as Equation (25);  
\[\mathbf{s}=\mathbf{s}^{0}+\mathbf{s}_{\varepsilon}^{app}+\mathbf{s}_{\varepsilon}^{obs}, \tag{25}\]  
where \(\mathbf{s}^{0}:=A\mathbf{x}^{0}\) with Equation (26)  
\[\mathbf{x}^{0}=(x_{1}^{0},\cdots,x_{n}^{0}),\  
system \(A\mathbf{x}^{0}=\mathbf{s}^{0}\) assigning the integral mean value of the function \(x_{j}\) in each pixel \(c_{j}\), that is, we take the pixel function as Equation (27)  
\[g_{n}^{0}(x,y):=\sum_{j=1}^{n}x_{j}^{0}\chi_{c_{j}}(x,y). \tag{27}\]  
The least square solution \(\mathbf{x}^{app}\) of the system \(A\mathbf{x}=\mathbf{s}_{\epsilon}^{app}\) gives the effect of the approximation of the function \(f\) by the pixel function Equation (27) to the least square solution of the system \(A\mathbf{x}=\mathbf{s}\). The least square solution \(\mathbf{x}^{obs}\) of the system \(A\mathbf{x}=\mathbf{s}_{\epsilon}^{obs}\) give the effect of the observation errors, numerical rounding errors by computers and so on.