## 4 Modification of Probability Distributions of 3-D Object Models  
The object models have the different essential parameters in number and type. For example, the nodule model has only one essential parameter \(r^{N}\), whereas the bifurcated blood vessel model has six other essential parameters \(r_{1}^{B_{b}}\), \(\delta_{23}^{B_{b}}\), \(\delta_{123}^{B_{b}}\), \(\psi_{12}^{B_{b}}\), \(\psi_{13}^{B_{b}}\) and \(\psi_{23}^{B_{b}}\). The differences in number cause a problem that generally, the probabilities of the object models that have more essential parameters are relatively underestimated. Let us consider an example case where all the standard deviations are 1 and all the set appearance probabilities \(P^{\tau}\) are \(1/3\). The radius of a nodule model \(r^{N}\) is supposed to be \(\mu+3\) and all the essential parameters of a bifurcated blood vessel model are supposed to be \(\mu\)'s. Their probabilities are calculated as follows:  
\[p(o^{N})=1/3\cdot\ g(\mu+3;\mu,1) \approx 0.00148 \tag{6}\] \[p(o^{B_{b}})=1/3\cdot\left(g(\mu\quad;\mu,1)\right)^{6} \approx 0.00134 \tag{7}\]  
Although the bifurcated blood vessel model has much more likely parameters, its probability is smaller than that of the nodule model. Here, let us give a more generalized expression to the model probabilities as follows:  
\[p\left(o^{\tau}(\mathbf{\omega}^{\tau})\right)=P^{\tau}\cdot p(\mathbf{\omega}^{\tau}) \tag{8}\]  
where \(o^{\tau}(\mathbf{\omega}^{\tau})\) represents an object model that has an essential parameter vector:  
\[\mathbf{\omega}^{\tau} = \Big{(}\,\omega_{1}^{\tau}\in\Omega_{1}^{\tau},\ \omega_{2}^{\tau}\in\Omega_{2}^{\tau},\ \cdots \tag{9}\] \[\omega_{d}^{\tau}\in\Omega_{d}^{\tau},\ \cdots,\ \omega_{D^{\tau}}^{\tau}\in\Omega_{D^{ \tau}}^{\tau}\ \Big{)}\!\in\mathbf{\Omega}^{\tau}\]The notation \(\mathbf{\Omega}^{\tau}\) represents the parameter space and \(D^{\tau}\) its dimension. For example, \(r_{1}^{B_{c}}\), \(\delta^{B_{c}}\) and \(\psi^{B_{c}}\) in Equation (2) correspond to \(\omega_{1}^{B_{c}}\), \(\omega_{2}^{B_{c}}\) and \(\omega_{3}^{B_{c}}\), respectively, and the dimension \(D^{B_{c}}\) is \(3\). In Equation (8), the auxiliary parameters are omitted for simplicity. The differences in the dimension \(D^{\tau}\) between the model classes \(\tau\) cause underestimation in \(p(\mathbf{\omega}^{\tau})\). One solution to correct the underestimation is to use the geometric average of the parameter probability: \(\sqrt[p\left(\mathbf{\omega}^{\tau}\right)\), that is adopted in, for example, recognition of language [31] and speech [32]. Because the dimension \(D^{\tau}\) is normalized, the underestimation is no longer caused. However, the geometrically averaged probability causes another problem that generally, the integral of its distribution over the parameter space is not one: \(\int_{\mathbf{\omega}\in\mathbf{\Omega}^{\tau}}\sqrt[p\left(\mathbf{\omega}\right)d\bm {\omega}\
eq 1\). We desire a probability distribution that does not cause underestimation and that integrates to one. In this paper, we realize the desired probability distribution on a parameter space that is inhomogeneously divided so that the desired distribution may approximate to the geometrically averaged distribution. First, each \(\Omega_{d}^{\tau}\) is divided into \(K\) isometric intervals \(\Delta\Omega_{d}^{\tau}(k)\)  (\(k=1,2,...,K\))  as shown in Figure 3. The interval size \(|\Delta\Omega_{d}^{\tau}(k)|\) is equivalent to \(\frac{|\Omega_{d}^{\tau}|}{K}\). By combining \(\Delta\Omega_{d}^{\tau}(k)\), the following subspaces are generated:  
\[\mathbf{\Delta}\mathbf{\Omega}^{\tau}(l)=\Delta\Omega_{1}^{\tau}(k_{ 1})\times\Delta\Omega_{2}^{\tau}(k_{2})\times\cdots\times\Delta\Omega_{D^{ \tau}}^{\tau}(k_{D^{\tau}})\]  
where \(l=1,2,\cdots,K^{D^{\tau}}\). The operator \(A\times B\) yields a product space of \(A\) and \(B\). If \(K\) is enough large, then the probability distribution \(p(\mathbf{\omega})\) can be regarded as being constant in \(\mathbf{\Delta}\mathbf{\Omega}^{\tau}(l)\).