Anisotropic adaptive diffusion filter

As we saw in the previous sections, improvement of the SNR seems to be an inevitable trade off between increasing the acquisition time (i.e., time averaging) and decreasing the spatial resolution (i.e., spatial averaging). In this section, the adaptive anisotropic diffusion filter is discussed, which is able to increase the image SNR significantly while leaving the spatial resolution almost intact.
Suppose $f(\vec r)$ is a 3D image, where $\vec{r}=(x_1,x_2,x_3)$ is a three-dimensional position vector. The filtering process consists of convolving $f(\vec r)$ with a Gauss kernel $h(\vec r)$ of which the shape is pointwise adapted to the local structure within a neighbourhood $\Omega$. De resulting filtered function $g(\vec{r})$ can be written as follows:

$$g(\vec{r_0})= {\int\!\!\int\limits_\Omega\!\!\int h(\vec r_... ...t\limits_\Omega\!\!\int h(\vec r_0 - \vec r) d \vec r}\quad ,$$ (8.10)

where

$$h(\vec{r} - \vec r_0) = \exp \left[-{1 \over 2}\sum_{i=1}^3... ...c n_i \right)^2 \over \sigma_i^2(\vec r_0)} \right] \quad .$$ (8.11)

In Eq. (8.11), the vectors $\vec n$ are the eigenvectors of the 3 x 3 second moment matrix R of the Fourier spectrum |F(u)|². The coefficients of R can as well be calculated in the spatial domain:

$$\int\!\!\int\limits_\Omega\!\!\int\ u_{i} u_{j} \vert F(u_1,u_2,u_3)\vert^2 du_1 du_2 du_3$$ R_ij = (8.12)

$${1 \over {(2\pi)^3 }} \int\!\!\int\limits_\Omega\!\!\int\ \left({... ...ht) \left({{\partial f} \over {\partial x_j}}\right) \; d x_1 dx_2 dx_3 \quad ,$$ = (8.13)

where i,j = 1,2,3. R is positive semi-definite and Hermitian, hence having only positive eigenvalues. The direction of the eigenvector $\vec n_1$, corresponding to the smallest eigenvalue, say $\lambda_1$, determines the main direction of the pattern in the neighbourhood $\Omega$ of the spatial domain. Alternatively, the three eigenvalues $\lambda_1$, $\lambda_2$, and $\lambda_3$, with $\lambda_1 \le \lambda_2 \le \lambda_3$, determine the relative orientation of the pattern in the respective directions.
The shape of the kernel h [see Eq. (8.11)] is controlled by the standard deviations $\sigma_1$, $\sigma_2$, and $\sigma_3$, which are functions of the local gradient strength in the respective directions. From the eigenvalues, anisotropy measures are derived, which are used to design the standard deviations:

$$a_{12} = {{\lambda}_2 - {\lambda}_1 \over \sum\limits_i {\l... ...bda}_3 - {\lambda}_1 \over \sum\limits_i {\lambda}_i} \quad .$$ (8.14)

The standard deviations should be large along the main direction(s) of the pattern, such that the data are only smoothed in homogeneous regions and along instead of across edge surfaces.
In addition, corners should be preserved during filtering. A corner is identified as a condition, where the pattern is relative isotropic ( $a_{12} \approx 0$ ; $a_{13} \approx 0$), while the local gradient strength $\vert\nabla f(\vec r)\vert^2$ is large. Therefore, a spatial dependent corner strength C is defined as:

$$C( \vec r) = (1-a_{12}-a_{13}) \vert\nabla f(\vec r)\vert^2 \quad .$$ (8.15)

From Eq. (8.14) and Eq. (8.15), the standard deviations are designed as follows:

$$\sigma_1 (\vec r) = {\sigma \over {1 + C(\vec r)}} \quad ... ...r) = {\sigma (1-a_{12}-a_{13}) \over {1 + C(\vec r)}} \quad ,$$ (8.16)

where $\sigma$ denotes the standard deviation of the image noise. As we saw in Chapter 4, $\sigma$ can be estimated directly from a large, uniform signal region as the standard deviation of the voxel values in that region [40]. However, the noise is commonly estimated from the amplitude values of non-signal regions [40], as these are often easier to find than large homogeneous signal regions. The noise standard deviation is then given by 1.53 times the measured standard deviation of the background voxel values. The multiplication constant results from the fact that background noise obeys a Rayleigh distribution [19,49]. We used the background for noise estimation.
The effect of the adaptive anisotropic diffusion filter can be appreciated from Fig. (8.2), where the filter was applied to the cucumber image described in the previous section. Although it is acknowledged that the SNR improvement strongly depends on the image content, the SNR of this example was 2.4, while the general resolution loss as was defined above was observed to be 8 %.
In the next chapter, a segmentation scheme will be discussed, where the anisotropic adaptive diffusion filter, described here, is first applied to the image data. It will be shown that a priori processing of the MR data with this filter has a positive influence on the segmentation time.

Figure 8.2: Diffusion filter: perceptual comparison of the SNR improvement.

[Original]

[Adaptive anisotropic diffusion filter]