In this section we investigate the properties of the two approaches to texture mentioned earlier with the help of simulated images. In particular we examine the effect of the various parameter values on the behavior of the anisotropy measures, and the influence of varied degrees of noise on them.

The simulated 3D image used is shown in the top left of Fig. 4. The image shows a fourfold symmetry, and although it is isotropic along the three axes, it is in general anisotropic. The dark areas have gray value 100 and the bright planes are 1 voxel thick, have gray value 180, and are 10 voxels apart.

The INV method relies on the displacement parameter d. One usually does not have a priori knowledge of its correct value, so it is necessary for a range of values to be used. The first and second column of plots in Fig. 4 show the indicatrices computed with values of d equal to 2, 5, 10, and 15. A 3D version of the indicatrix is shown as well as its projection on one of its principal planes. All projections on the three principal planes must be identical because of the symmetry of the image. However, some differences in appearance are expected because the sampling points are not homogeneously arranged on the surface of the unit sphere and the triangulation created from them, as a consequence, introduces some anisotropy in the appearance of the structure. This apparent anisotropy becomes unnoticeable when the number of sam pling points used to tessellate the orientations is increased. The consequence, of course, is increase in the computational cost.

It is interesting to note that the indicatrix is relatively smooth when the choice of d is less than the texture periodicity, whereas it shows very high anisotropy when d has been chosen to be equal to the texture characteristic length. We can monitor this behavior best with the help of feature F1: Its value from a modest 2.43 for d = 2 and 2.91 for d = 5 shoots up to 17.34 for d = 10, only to drop again to 3.71 for d = 15. The other two features also increase in value, but not in such a dramatic way. This behavior of INV is preserved for moderate levels of noise: The second column ofresults shown in Fig. 4 and those shown in Fig. 5 have been computed from the same image with 20, 50, and 80% noise added to it. The noise is zero mean uniformly distributed and 20% means that its range is 16 units, while the contrast in the image is 80 units. From these results we may say that the value of F1 remains a good indicator of whether the value of d coincides with the basic periodicity of the texture or not, for quite high levels of noise (up to 50%). We used uniformly distributed noise as one of the worse types of noise. In reality the distribution of noise will probably be more centrally concentrated and its effect will be even less prominent.

This behavior of the INV method is not surprising, as this approach is based on integration and therefore it is expected to be robust to noise. The behavior of the gradient-based method, however, is not expected to be as robust. The gradient-based method does not rely on any parameter, but as it estimates the local gradient at each voxel by using masks that only take into consideration the immediate neighbors of a voxel, it is expected to produce results that are different for different characteristic lengths of the texture. We do not present results here for zero noise situations because the indicatrix in that case consists of four line segments perpendicular to each other (the delta functions that represent the derivatives of perfect step edges), except for a few extra short lines arising from the misbehavior of the gradient masks at the intersection places of the bright planes. Figures 6, 7, and 8 show the results of applying the method to images where the characteristic length of the texture is 4, 6, and 10 respectively. In all cases, results obtained in the presence of noise are also shown. In particular, examples with 20, 50, and 80% added noise are included. It can be seen that the indicatrix becomes rounder and rounder as noise increases, while the values of the features change very quickly so that they cannot be used to identify a texture irrespective of the level of noise. This method, however, is very fast and for setups where the level of noise is expected to be constant, it can be used to characterize texture locally.

The INV method also can be used for local calculations, if we choose d = 1. However, the two methods produce very different results: INV projects the gradients of all the voxels along certain directions (the sampling directions on the unit sphere) and adds up all the square magnitudes of those projections. The GD method, on the other hand, simply counts how many voxels have gradient in a certain cone of directions.

20% nobie rtÃ» noise

FIGURE 4 Experiments with a synthetic texture without noise and with 20% additive uniform noise. The indicatrices calculated by the INV method are presented as 3D structures and in projection on the plane z = 0.

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