Issues Related to 3D Texture Estimation and Representation

There are three major issues related to the calculation and visualisation of a 3D orientation histogram:

• Tesselation of the unit sphere

• Visualization of the 3D histogram

• Anisotropic sampling

The first two topics have been extensively discussed in relation to 3D object representation (e.g., [11]; the third is intrinsic to the medical topographic data acquisition protocols.

The issue of tesselating the unit sphere arises from the need to quantize the directions in three dimensions. Clearly, one needs to consider equal solid angles. As a solid angle is measured by the area over which it extends on the surface of the unit sphere, this requirement is equivalent to requiring the tesselation of the surface of the unit sphere in patches of equal area. There are various ways by which this can be achieved, particularly with the help of regular or semiregular polyhedra [9, 11, 16, 27]. However, in all such approaches the number of quantized directions created is very limited, leading to representations of poor orientational resolution. Besides, the use of such representations leads to orientational cells with complicated defining boundaries, which in turn leads to expensive and cumbersome ways of testing to which cell a certain point on the unit sphere belongs. The two most straightforward ways of defining a point on the unit sphere (and by extension a certain orientation) is to define it in terms of two angles, ^ and corresponding to longitude and latitude on the sphere, respectively, or to define it in terms of the longitude ^ measured along the equator of the sphere, and the height z above the equatorial plane. For a quick quantization test, we should have bins that are of equal size either in ^ and or in ^ and z. It turns out that cells defined by dividing ^ and ^ in equally sized sections are not of equal area. On the other hand, cells defined by dividing ^ and z in equally sized sections are of equal area. Thus, we choose this tesselation of the unit sphere. The division of 0 < ^ < 360° into M equal intervals and the division of — 1 < z < 1in N equal segments results in (N — 2) x M spherical quadrangles and 2M spherical triangles, all sustaining the same solid angle of 4n/(NM). Then, an arbitrary direction defined by vector (a, b, c) belongs to bin (i, j) if the following two conditions are met:

The issue of visualization is a little more difficult, as one has the medium of a 2D page to visualize a volume descriptor. In the context of 3D shape representation the extended Gaussian image has been used to unfold the Gaussian sphere [10]. This is a 2D representation where the two angles ^ and ^ are measured along the two axes and the gray value of each cell is proportional to the number density of that cell. Figure 1a shows a 3D section of a liver scan, and Fig. 1b its orientation histogram representation that corresponds to the extended Gaussian image. An alternative way is to present the latter as a landscape seen in perspective. This is shown in Fig. 1c. Finally, one may try to represent the orientation histogram as a 3D structure, with coordinates ^ and z of the center of each cell defining the orientation, and the accumulated value in each bin measured along the radius [6]. Again this 3D structure has to be viewed in projection on the plane. This representation is shown in Fig. 1d. We find this the most expressive of the three representations, and we use it throughout this chapter. It should be clarified here that for each region or window in the image, one has to have such a representation constructed separately, as it will not be possible to visualize the orientation histograms referring to two different images, one on the top of the other. This is contrary to the 2D case, where the histograms of two regions can be superimposed to visualize differences. To facilitate comparisons between two such structures, all 3D orientation histograms produced are projected on the 2D page in the same way.

The problem of anisotropic sampling is relevant when metric 3D calculations are performed with the image. As the creation from the orientation histogram necessarily involves metric calculations, the issue of anisotropic sampling is important. It is a common practice when acquiring tomographic data to choose slice separation much larger than the pixel size on each slice. Thus, the voxels of the acquired image in reality are not cubes but elongated rectangular parallelepipeds with the longest side along the z axes, i.e., the axis of slice separation. The metric used for the various calculations, then, is the Euclidean metric with a scaling factor multiplying the z value frinfrfa' ■i

FIGURE 1 Ways of representing the 3D orientation histogram, (a) Original CT liver image volume. Its orientation histogram was constructed with 24 bins in z and 11 in (b) Histogram displayed as a 24 x 11 gray image. (c) Histogram displayed as a landscape. (d) Histogram displayed as a 3D orientation indicatrix.

FIGURE 1 Ways of representing the 3D orientation histogram, (a) Original CT liver image volume. Its orientation histogram was constructed with 24 bins in z and 11 in (b) Histogram displayed as a 24 x 11 gray image. (c) Histogram displayed as a landscape. (d) Histogram displayed as a 3D orientation indicatrix.

to balance this difference. Figure 2 demonstrates the effect on the orientation histogram, if the rate of sampling along each axis is not taken into consideration. In Fig. 2a a section of an especially constructed test image is shown. An original image consisting of 151x151x151 voxels and with intensity increasing uniformly and isotropically from its center toward its outer bounds was first constructed. Then this image was subsampled with rate 1:2:3 along the x y and z axes, respectively, to emulate anisotropic sampling. The orientation histogram of the 151 x 76 x 50 image that resulted is shown in Fig. 2b without any scale correction and in Fig. 2c with the correct scaling used. As expected, the effect of ignoring the scaling factor is crucial to such a representation. In all the discussion that follows, the scaling factor in the metric is used, without actually appearing anywhere explicitly, in order to preserve the simplicity of the presentation.

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