# Three-Dimensional Object Recognition Using Orthogonal Polynomials: An Embedded Kernel Approach
## 2 Preliminaries
### Computation of 3D Moments  
In the literature, there are different methods proposed for computing 3D moments. Examples of these methods are the direct method, cubes method, and matrix product method, as well as Kronecker tensor and matrix product method [50]. The primary goal of these methods is to minimize computation time while maintaining accuracy. However, the computational complexity of these methods varies. For example, the direct method has a computational complexity of \(\mathcal{O}(n^{6})\), while the cubes method has a complexity of \(\mathcal{O}((\frac{n}{C\text{thos}})^{6})\), the matrix product method has a complexity of \(\mathcal{O}((\frac{n}{8})^{6})\), and finally, the Kronecker tensor and matrix product method has a complexity of \(\mathcal{O}(n^{3.81})\)[50]. As illustrated above, the computational complexity burden could be very high, especially with 3D signals. To overcome this issue, this paper proposes an efficient method of using the Kronecker tensor as well as the matrix multiplication method to compute the moments in an efficient way with lower computational complexity. To this end, for a 3D signal \(f(x,y,z)\), the moments \(\mathbf{M}\) can be computed as follows [50]:  
\[\begin{split}\mathbf{M}_{:,k}=\sum_{z}\mathbf{R}  
In the spatially separable convolution process, the first kernel, i.e., the \(A\times 1\times 1\) kernel, is convolved with \(f(x,y,z)\), followed by convolving the result with the \(1\times B\times 1\) kernel and then convolving the output of the previous step with the \(1\times 1\times C\) kernel. This technique has been explained in detail in [51]. As a result of this decomposition, the original convolution equation, represented by (7), can be rewritten as follows:  
\[g(x,y,z)=\sum_{u=-a}^{a}\left[\sum_{v=-b}^{b}\left[\sum_{w=-c}^{c}f(x,y,z)h(w-z )\right]h(v-y)\right]h(u-x). \tag{9}\]  
To calculate the moments of the function \(g(x,y,z)\), we can use Equation (6). This equation will yield the moments (\(\mathbf{MC}_{n,m,k}\)) of the original function \(f(x,y,z)\) after it has been convolved with the kernel and can be expressed, in the form of a matrix, as follows:  
\[\mathbf{MC}_{:,k}=\sum_{z}\mathbf{R}_{k,z}\otimes\left[\mathbf{R }_{n,x}\:\mathbf{G}\:\mathbf{R}_{n,y}^{T}\right] \tag{10}\] \[n,m,k=0,1,\ldots,Ord,\]  
where \(\mathbf{G}\) denotes the matrix form of the function \(g(x,y,z)\) and \(\otimes\) represents the mathematical kronecker product. In what follows, we provide proof of the above expression. **Proposition 1**. : _Suppose we have a function \(\mathbf{p}\) that is convolved with a kernel \(\mathbf{h}\). The outcome of this convolution operation is a new function \(\mathbf{q}\), which can be expressed as follows:_  
\[\mathbf{q}=\mathbf{p}*\mathbf{h}. \tag{11}\]  
_Typically, the convolution operation presented above can be reformulated using the properties of matrix multiplication. The resulting matrix can be expressed as follows:_  
\[\mathbf{Q}=\mathbf{H}\,\mathbf{P}, \tag{12}\]  
_In the equation mentioned above, the matrix \(H\) represents the Toeplitz matrix of the kernel \(\mathbf{h}\). Using this equation, we can compute the moments \(\mathbf{\Phi}\) as a function of \(\mathbf{Q}\) as follows:_  
\[\mathbf{\Phi}=\mathbf{R}\,\mathbf{Q}. \tag{13}\]  
_To this end, we can substitute matrix \(\mathbf{P}\) given above into the moment function \(\mathbf{\Phi}\). The resulting moment function \(\mathbf{\Phi}\) can be written as follows:_  
\[\mathbf{\Phi}=\mathbf{R}\,\mathbf{H}\,\mathbf{P} \tag{14}\]  
_Let us assume that \(\mathbf{R_{e}}=\mathbf{R}\,\mathbf{H}\), which represents the orthogonal polynomials embedded with the kernel \(\mathbf{H}\) so that we have the following:_  
\[\mathbf{\Phi}=\mathbf{R_{e}}\,\mathbf{P} \tag{15}\]  
_The concept we discussed above can also be extended to multidimensional signals. In other words, we can use the same approach to compute moments for signals with more than two dimensions.