Convolution filter

The linear operator of PDE (7) $\Lambda$ is defined as:

$$\Lambda :=\mu \nabla ^{2}+(\mu +\lambda )\nabla (\nabla \cdot)$$ (23)

and its eigenvalues are (5):

$$\begin{array}{l} \kappa _{1,i,j,k}=-\pi ^{2}(2\mu +\lambda )(... ...\kappa _{3,i,j,k}=-\pi ^{2}\mu (i^{2}+j^{2}+k^{2}), \end{array}$$ (24)

with associated eigenvectors:

$$\begin{array}{c} \phi _{1,i,j,k}(\vec{x})=\sqrt{\frac{8}{i^{2... ...+j^{2})\: ccs_{i,j,k}(\vec{x}) \end{array}\right) , \end{array}$$ (25)

where $\vec{x}\in \Omega$,

$$\begin{array}{c} scc_{i,j,k}(\vec{x})=\sin (i\pi x)\cos (j\pi... ...}(\vec{x})=\cos (i\pi x)\cos (j\pi y)\sin (k\pi z), \end{array}$$ (26)

and

$$\Gamma _{i,j,k}=2^{\text{sign}(i)+\text{sign}(j)+\text{sign}(k)}.$$ (27)

By introducing a filter width parameter $w>0$, $w\in \mathbf{N}$, which spawns a filter of size $2w+1$, and with the shortcut:

$$\alpha _{i,j,k}=\frac{8}{\pi ^{2}\mu (2\mu +\lambda )(i^{2}+j^{2}+k^{2})^{2}\Gamma _{i,j,k}}$$ (28)

the components of the impulse response $\Theta \in {\mathbb{R}}^{3× 3}$ of the linear operator $\Lambda$ can be written as (3):

$$\begin{array}{c} \Theta ^{x}(\mathbf{y})=\sum^{2w}_{i,j,k=0}\... ...,k}(\mathbf{y}+\mathbf{y}_{c}) \end{array}\right) , \end{array}$$ (29)

with $\mathbf{y}_{c}=(0.5,0.5,0.5)^{T}$ and $\mathbf{y}\in \{y_{r,s,t}=(\frac{r}{d},\frac{s}{d},\frac{t}{d})^{T}\vert\: r,s,t\in [-d,d]\cap \mathbf{Z}\}$.