298				Q. Zhang et al.
	dI(λ)		= c(λ)τ (λ) −I(λ)τ (λ) = c˜(λ) −I˜(λ).	(13.1)
	dλ

The solution to this equation is given below, showing the intensity of each pixel.

I(D) = I0T (D) +		D c˜(λ)T (λ)dλ,
		0
T (λ ) = exp −	0λ τ (x)d x ,		(13.2)

describes the volume transparency. The first term I0 illustrates light coming from the background, and D is the extent of the ray over which light is emitted. The last term demonstrates the behavior of the volume emitting and absorbing incoming light. The source term c() indicates the color change, and the extinction coefficient (tau)(D) defines the occlusion of light per unit length due to light scattering or extinction.

13.3.2 Color and Opacity Mapping

13.3.2.1 Voxel Classification

To display 3D medical images with DVR, the scalar values must first be mapped to optical properties such as color and opacity through transfer function (TF), a process referred to as voxel classification. Preand postclassification approaches differ with respect to the order in which the TF and sampling interpolation are applied [16, 35]. Pre-classification first maps every scalar value at the grid into color and opacity in a pre-processing step, where the color and opacity are assigned at the resampling points. However, for post-classification, we first sample the scalar value by interpolation, and then map the acquired values to colors and opacity through TFs. Both the pre-and post-classification operations introduce high frequency components via the nonlinear TFs [36–38]. Pre-classification suppresses this high-frequency information, so the rendered image appears blurry (Fig. 13.7a), while postclassification maintains all the high frequencies, but introduces “striping” artifacts in the final images (Fig. 13.7b).

To address the undersampling problem, Rottger et al. [39, 41] proposed a preintegrated classification algorithm for hardware-accelerated tetrahedra projection, which was first introduced by Max et al. [40]. Later, Engel et al. [36] applied this classification algorithm for 3D texture-mapping-based volume visualization of regular-grid volume data. Preintegrated classification separates the DVR integral into two parts, one for the continuous scalar value, and the other for the TF parameters c(colors) and tau(extinction). This algorithm renders the volume segment- by-segment, instead of point-by-point. In this manner, the Nyquist frequency for reconstructing the continuous signal is not increased by the TF nonlinear properties. Assuming there are n + 1 sampling points along the viewing ray, then the segment

13 Medical Image Volumetric Visualization: Algorithms, Pipelines...

299

Fig. 13.7 DVR of cardiac vessels via different voxel classification algorithms: (a) preclassification; (b) postclassification; (c) preintegrated classification; (d) post color-attenuated classification

length d equals D = n, where D is the maximum ray length. For the ith segment, the front and back points are sa = s(id) and sb = s((i + 1)d). The calculated opacity and color of this segment are then given by (13.3) and (13.4), respectively.

n	i−1	n	i−1
I(D) f b = ∑ αiCi ∏ Tj = ∑ αiCi ∏ (1 −α j )				(13.3)
i=0	j=0	i=0	j=0
n	n	n	n
I(D)b f = ∑ αiCi	∏ Tj = ∑ αiCi		∏ (1 −α j).	(13.4)
i=0	j=i+1	i=0	j=i+1

13.3.2.2 Transfer Function

The important step of voxel classification is implemented through a TF adjustment, which plays an important role in DVR. However, TF specification is a complex procedure and is a major obstacle for the widespread clinical use of DVR [42, 43]. In this section, we briefly review the TFs that are of crucial clinical importance. In DVR, the TF was typically used for tissue classification based on local intensities in the 3D dataset [44]. Multidimensional TF is efficient for multiple spatial feature detection, for example, Kniss et al. [45] designed such a TF and demonstrated its medical applications, while Higuera et al. [46] built a 2D TF to effectively