Data-fusion using small intensity deviations

Volume $h(\vec{x})$ will have a global mean, which we take to be the sample mean

$$\begin{eqnarray} \bar{h} &=& \frac{1}{V(S)} \int_{S} \: {h(\vec{x}) \: d\vec{x}} , \ V(S) &=& \int_{S} \: {d\vec{x}} ,\end{eqnarray}$$ (1) (2)

where S represents the data space and the integrals are taken over all S. The value $\bar{h}$ represents the intensity of the gray background of $h(\vec{x})$. If we define a quantity $d(\vec{x}) = h(\vec{x}) - \bar{h}$ for all $\vec{x}$, we have a measure of the local intensity deviation from the intensity of the gray background. These deviations produce the local structures observed in $h(\vec{x})$. The scarcity of such deviations in $l(\vec{x})$ corresponds to the near-absence of local structures there. It becomes evident that a new volume $m_1(\vec{x}) = l(\vec{x}) + d(\vec{x})$ should be created. This approach adjusts the slowly changing background intensities of $l(\vec{x})$ by the rapidly changing intensity deviations from $h(\vec{x})$.

Since volume $d(\vec{x})$ maps to $D(\vec{w}) = H(\vec{w}) - H(\vec{0})/V(S)$, with V(S) defined by Eq. 2, the DC component of $H(\vec{w})$ corresponding to the gray background is mostly removed. Then, $M_1(\vec{w}) = L(\vec{w}) + D(\vec{w})$. The algorithm adds the primarily low-frequency $L(\vec{w})$ to the almost DC-less $D(\vec{w})$ to synthesize $M_1(\vec{w})$.

In practice, a scaled $\alpha \cdot d(\vec{x})$ is added to $l(\vec{x})$ to avoid significant alteration of local means when going from $l(\vec{x})$ to $m_1(\vec{x})$. The exact value of alpha depends on the relative signal levels in $h(\vec{x})$ and $l(\vec{x})$. Relating back to the framework in Fig. 3, the upper branch's extraction of high-frequency components from $h(\vec{x})$ and scaling are contained in the term $\alpha \cdot d(\vec{x})$. The lower branch's behavior is simpler: all frequency components are extracted from $l(\vec{x})$ and the scaling factor is unity.

The synthesized result of the slices from Fig. 1 is shown in Fig. 4. It can be seen that the first algorithm performs well in presenting the fine details of $h(\vec{x})$. Because we avoided significant alteration of the local means of $l(\vec{x})$ through the scaling factor $\alpha$, the textural smoothness and regional boundaries of $l(\vec{x})$ are well preserved.

 

chen-velview-merge1
Figure 4 Result using the first data-fusion algorithm.