Volume rendering modes or computer visualization methods are, for example, x-ray projection (X-Ray) or maximum intensity projection (MIP). Applying these modes the volume cells (regular or irregular) can be processed (projected) in any order, since the volume rendering integral degenerates to a commutative function, i.e. the rays have not to be processed in a front-to-back manner. In contrast, for iso-surface or full volume rendering a depth ordering of the cells is required to satisfy the compositing order, thus rays have to be processed in a front-to-back manner, since the generalized volume rendering integral is not commutative.
However, each computer visualization method can be combined with any rending ap- proach (cf. Sec. 5), e.g. ray-casting or shear-warp. Although a straight forward imple- mentation of a volume rendering algorithm would apply ray-casting, the choice of the rendering method has great influence on the speed of the final algorithm as well as the resulting image quality. Similarly, for performance reasons or the requirements of the image quality, one can utilize different strategies to find the values along rays for the different modes:
• Analytical solution: Depending on the data reconstruction model, for each cube intersected by the ray a local polynomial of an appropriate degree has to be deter- mined. According to the mode used, one has to compute the roots (for iso-surface visualizations), the maxima (for MIP) or the integral (for full volume rendering) of the polynomial pieces to determine the final result. Such an analytical compu- tation and evaluation of the local polynomials is the most accurate but also the most expensive method.
• Sampling and interpolation: In this less expensive approach data values are sam- pled with a distance d along a ray using a data reconstruction model. No ex- plicit polynomial pieces are considered (determined) within cubes, thus the cost of this approach depends only on the data reconstruction scheme and the sampling
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distance d. However, the method can be greatly accelerated by using longer re- sampling distances d along rays, which, on the other side, produces lower quality images.
• Nearest neighbor interpolation: No real interpolation is performed at all, i.e. the data reconstruction model generates piecewise constant polynomials. Values on the data grid closest to the current considered samples along the ray are only considered for estimation. Since no interpolation is applied, the data grid structure becomes visible as aliasing or as staircase artifacts in the resulting images.
Figure 4.7: Volume rendering modes reveal different information about the volume data set, e.g. maximum intensity projection (left), full volume rendering (middle), and iso-surface rendering with a gradient-based color transfer function (right) where high gradient magnitudes are mapped to red colors.
4.6.1 Maximum Intensity Projection
Maximum intensity projection (MIP) [MGK99] [CKG99] is a computer visualization method that projects sample values into the projection plane with maximum intensity [MHG00] [MKG00] obtained along a ray from the three-dimensional data set [PHK+03] [ME05]. Applying a parallel (orthographic) projection method using MIP implies that two images obtained from opposite viewpoints are symmetrical. More over, the viewer cannot distinguish between left or right, front or back and even not if the object is rotated clockwise or not. On the one hand, within angiography data sets usually the data values of vascular structures are brighter than the values of the surrounding tissue, the structure of vessels contained in these data sets can be captured easily. On the other hand, an application of early-ray termination as acceleration technique (cf. Sec. 6) is not possible, making the standard MIP often even more expensive than full volume rendering or iso-surface rendering. Further, an image obtained by MIP does not contain any shading information, thus there is no depth and occlusion information, i.e. objects with brighter image data values appear to be in front – even though lying behind of objects corresponding to darker image values. A common way to simplify the interpretation of such images is to rotate (animate) the data. That means, the relative three-dimensional positions of objects are revealed by rendering the object’s data sets from different viewing positions or directions. However, this improves the viewer’s perception. In other words, it is easier for the viewer to reveal the relative three-dimensional positions of objects.
4.6.2 X-Ray Projection
Volume rendering can be considered as the inverse problem of tomographic reconstruc- tion [Mal93] where the aim is to reconstruct the unknown volume density function f (k) by rotating an X-ray emitter/dedector pair at some angle and collecting a set of measured projections. However, X-ray like images [Lev92] of a volume data set can be obtained with a rendering technique that evaluates the line integral in the frequency domain. Namely, the Fourier Projection-Slice Theorem allows to compute a two-dimensional X- ray like image from a three-dimensional data set by using a two-dimensional slice of that data in the frequency domain (cf. also [Max95]).
4.6.3 Iso-Surface Rendering
The goal in iso-surface rendering is to find the smallest intersection parameter t≥ 0 of each ray r(t) = rs+ t rd going through the volume where the data has an user-defined iso-value. This global problem is usually split into smaller local problems. However, to find the correct intersection of a ray and a surface of the volume data using the trilinear reconstruction model is an essential task. Many available methods are accurate but slow or fast but for an approximate solution only. In [MKW+04] available techniques are compared, analyzed and a new intersection algorithm is presented, which is said to be three times faster and provides the same image quality and better numerical stability in opposite to previous methods.
4.6.4 Full Volume Rendering
In full volume rendering (FVR), usually the volume rendering integral is solved by appro- priate evaluation of equations (4.6), (4.7), and (4.8) as already discussed in the previous sections. However, from the above mentioned integral one can derive an absorption only model where the volume is assumed to consist of perfectly black particles that absorb all the light that goes through them. In the emission only model the volume is assumed to be of particles that emit – but do not absorb – light. Other models take scattering and shading/shadowing into account (cf. [Max95]).
4.6.5 Non-Photorealistic
Volume illustration techniques, as, for example, presented in [TC00] [CMH+01] [LME+02] [LM02] [MHIL02], can be considered as a subclass of non-photorealistic rendering (NPR) [GGSC98]. The particular characteristic of illustrations [GSG+99] [ER00] is to empha- size important features or parts of an object and to advise the information to the viewer in the most efficient way. This characterization matches the requirements of traditional visualization techniques as well. However, in volume rendering transfer functions are of- ten set such that objects are semi-transparent making spatial relationship between these objects difficult. Silhouette edge illustration (as discussed in section 4.5) can be par- ticularly useful here (further non-photorealistic techniques can be found in [GGSC98], [GSG+99] [NSW02], [WKME03b]. The manipulation of color based on distance can also improve depth perception [FvDFH97]. For depth cues typically warmer hues are used for the foreground and cooler in the background and color values tend to become lighter and less intense with distance. In addition, non-linear fading of the alpha channel along
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the view direction allows making closer material more transparent such that underlying features are more visible, but foreground material is still slightly preserved to provide context for the features of interest.
4.6.6 Focus and Context Techniques
Traditionally, objects within the volume data set are classified by optical properties like color and opacity using transfer functions. Another technique is to classify or segment the data during a pre-processing stage into different regions where each region specified by an identification number (id) represents an object of the volume. Now different optical properties can be assigned to each object such that some are enhanced and others not. An importance driven approach [VKG04] additionally assigns objects another dimension, which describes their importance. This scalar importance value encodes which objects are the most interesting ones and have the highest priority to be clearly visible. During rendering the model evaluates the visibility of each object according to its importance, i.e. such that less important objects are not occluding features that are more interesting. In other words, the less important ones are rendered more sparsely. Two approaches have been proposed using importance values, the first is called maximum importance
projection (MImP), whereas the second average importance projection. In the former
method for each ray the object with highest importance along the ray is determined. This object is displayed densely, whereas all the remaining objects along the ray are displayed with the highest level of sparseness, i.e. fully transparent. This approach can be considered as a cut-away view similar as in [SCC+04], where a cut-away technique for CT angiography of peripheral arteries in human legs is applied. The goal is to have a clear view on the vessels, which are partially segmented by their centerline.
A discussion of expressive visualization techniques, e.g. cut-away views, ghosted views, and exploded views, originating from technical illustration that expose the most impor- tant information in order to maximize the visual information of the underlying data can be found in [VG05] (cf. also [KTH+05] [BGKG05], and [BG05]).