Error analysis using ellipsoids

One can analyze the limitations of the spherical harmonic shape analysis method, as developed in this paper, by applying it to shapes for which all the properties of interest are known analytically. A sphere is a known shape, but its analysis is trivial, since only the coefficient a₀₀ is non-zero when a spherical harmonic expansion like eq. (9) is carried out for a sphere. An ellipsoid, however, has many non-zero values for its coefficients, and has known analytical results for its properties. Ellipsoids of revolution in particular were used, as there exists closed-form solutions for the properties of these shapes in terms of elementary functions [40,41,42] (see Appendix C). The shapes used were a 5:1 prolate and a 1:5 oblate ellipsoid of revolution. By this is meant ellipsoids with axes in the ratio of 1:1:5, and 5:5:1. Various digital versions, using different digital resolutions, were made of these shapes and the numerical procedures tested on them. Only results for the prolate ellipsoid will be shown, as the results were similar for the oblate ellipsoid. Each ellipsoid was centered on a voxel center, and a voxel was counted to be a part of the ellipsoid if its center were inside the analytical limits of the ellipsoid.

Figure 8 shows the volume and surface area for a 5:1 prolate ellipsoid as a function of the number of spherical harmonic coefficients used (in terms of N). The solid lines are the exact analytical result for an ellipsoid with axis dimensions, in terms of voxel length, of (21,21,105). The expansion results become very close to the exact analytical results by about N = 12. Figure 9 shows, for the same ellipsoid, the values of I₁₁ and I₃₃. The graph also shows h^-1, which is a combined measure of the average size and shape of the object. Again, the results basically become equal to the exact analytical results by about N = 12. Although not shown, the value of k stays very close to 1 (much less than 1 % difference) for all values of N up to 24.

Figure 8: The volume and surface area of a 5:1 prolate ellipsoid of revolution as a function of the number of spherical harmonic functions, N, used in the expansion. The expansion was made from a digital image of the ellipsoid that was 21 x 21 x 105 pixels in extent. The solid lines are the exact result for a continuum ellipsoid.

Figure 9: The two independent components of the moment of inertia tensor (described in text) and the reciprocal of the surface-area-weighted integrated mean curvature for a 5:1 prolate ellipsoid of revolution, plotted as a function of the number of spherical harmonic functions, N, used in the expansion. The expansion was made from a digital image of the ellipsoid that was 21 x 21 x 105 pixels in extent. The solid lines are the exact result for a continuum ellipsoid.

It is important to note that the spherical harmonic coefficients are being generated from the digital realization of the ellipsoid, not the exact formula. By using the exact formula for an ellipsoid centered on the origin and aligned with the coordinate axes,

$$\frac{x^2}{a^2}+\frac{y^2}{a^2}+\frac{z^2}{c^2} = 1$$ (28)

where again 2a = 21 and 2c = 105, one can easily show that the exact value of r(, $r(\theta,\phi)$) is given by

$$r(\theta,\phi)=a^2c \: [a^2 c^2 \sin^2(\theta)+a^4 \cos^2(\theta)]^{-1/2}.$$ (29)

Since r(, $r(\theta,\phi)$) is a function of $\theta$ only, this implies that the only spherical harmonic coefficients that are non-zero have m = 0. And since r(, $r(\theta,\phi)$) is even in $\theta$, only a_n0 with n even are non-zero. This is found to be approximately true for the numerical results, which are generated from the digital image of the (21,21,105) ellipsoid, not the exact value of r$r(\theta)$.

Table 1 shows the values of a(n,0) vs. n for the numerical and the exact analytical results (generated using the exact value of r). One can see, as N gets into double digits, an increasing amount of disagreement between the exact and numerical results. The disagreement should increase for a coarser resolution ellipsoid image, and decrease for a finer resolution image.

Table 1: The exact and numerical values of the a_n0 coefficients in the spherical harmonic expansion for an ellipsoid of revolution with axes in the ratio 21:21:105.

n	Coefficient	Exact	Numerical	% diff
0	a0 0	52.0237	52.1800	0.3
2	a2 0	19.7127	19.6905	-0.1
4	a4 0	9.9545	9.9199	-0.35
6	a6	5.5475	5.5526	0.1
8	a8 0	3.2408	3.2673	0.8
10	a10 0	1.9461	1.9885	2.2
12	a12 0	1.1899	1.1228	-5.6

The above was an analysis of the error incurred when using a varying number of coefficients on a fixed digital resolution image. One also has to address the question of changing digital resolution on the computed shape parameters of particles, such as volume and surface area. What is the resolution needed, for a given number of spherical harmonic coefficients, for a digital object to give accurate results in the spherical harmonic expansion? A similar analysis is performed, but with coarser resolution digital ellipsoids, comparing the spherical harmonic-computed quantities with their exact analytical counterparts.

Figure 10 shows the volume and surface area for a 5:1 prolate ellipsoid as a function of the number of spherical harmonic coefficients used (in terms of N), for an ellipsoid that now has axis dimensions, in terms of voxels, of 11:11:55. The expansion results become very close to the exact analytical results by about N = 12, although there is a small amount of "drift" upwards past the exact quantities for higher values of N. Figure 11 shows the same quantities, but for an ellipsoid with axis dimensions of 5:5:25. It is interesting to note that there is a flat part of the graph starting again at about N = 12. The spherical harmonic results go significantly above the exact lines as N increases beyond about 20. If one were to stop at N = 12, the volume would be off by 5 % and the surface area would be too high by about 8 %.

Figure 10: The volume and surface area of a 5:1 prolate ellipsoid of revolution as a function of the number of spherical harmonic functions, N, used in the expansion. The expansion was made from a digital image of the ellipsoid that was 11 x 11 x 55 pixels in extent. The solid lines are the exact result for a continuum ellipsoid.

Figure 11: The volume and surface area of a 5:1 prolate ellipsoid of revolution as a function of the number of spherical harmonic functions, N, used in the expansion. The expansion was made from a digital image of the ellipsoid that was 5 x 5 x 25 pixels in extent. The solid and dashed lines are the exact result for a continuum ellipsoid.

One can analyze the lowest resolution ellipsoid data in several ways. First, the digital volume is itself inaccurate at this resolution. For this ellipsoid, which has axis dimensions of 5:5:25, the exact volume is 327.25. The digital volume is 349. For N = 12, the volume calculated using the spherical harmonic expansion, which was itself taken from the digital image, is only off from the digital volume by -1.4 %. Second, one can examine the integrated Gaussian curvature of eq. (25) to see at how high a value of N should the spherical harmonic expansion be trusted. By N $N \geq 14$ 14, the value of k is more than 2 % different from 1. So the "best" results for this shape are at N = 12, which agrees with the flat part of the graph. Also, it is clearly seen that the spherical harmonic expansion faithfully follows the digital image shape, not the analytical shape. Since in the case of images of aggregates derived from x-ray tomographs, only the digital image is available, this fact is comforting.

When considering the surface area of digital images in the past [45], it was found that the digital surface area, obtained by counting voxel faces, was too high. For a sphere, it is too high by a value of about 3/2. For many random pore structures, not based on a sphere, this value also holds true [46]. However, by examining Table 2, one can see that the ratio of the digital surface area to the true surface area does depend on the shape of the ellipsoid, oblate or prolate, but only weakly on the resolution used, at least down to the lowest resolution shown in the table. But, as could be seen in Fig. 8, the spherical harmonic expansion gave a surface area that was quite close to the exact value for the prolate ellipsoid. So in this case, there is higher accuracy in using the expansion than in correcting the digital surface area, since the correction is not universal, but depends on the ellipsoid shape. For example, if the aspect ratio was 1 for both prolate and oblate shapes, so that they were spheres, the equivalent ratio would be 1.5. For an aspect ratio of 20, the oblate ratio is 1.085, and the prolate ratio is 1.371.

Table 2: The digital and exact surface areas and their ratio for three oblate and three prolate digital ellipsoids of different resolutions.

Shape	Ellipsoid axes	Digital SA	Exact SA	Ratio of digital SA to exact SA
Prolate	5:5:25	446	313.7	1.422
Prolate	11:11:55	2118	1518.3	1.395
Prolate	21:21:105	7694	5533.7	1.390
Oblate	25:25:5	1382	1073.6	1.287
Oblate	55:55:11	6678	5196.4	1.285
Oblate	105:105:21	24366	18938.8	1.287

How the prolate ratio changes with shape is hard to see intuitively. The range of variation for the oblate shapes is easier to see. As an ellipsoid of revolution becomes very oblate, its surface area approaches the surfaces of two circles (see the oblate surface area formula in Appendix C). The digital area of a circle is very accurate [35], and so the ratio should go to unity in this limit, and vary between 3/2 and 1 in between the sphere and extreme oblate limit.

Now that the accuracy of the spherical harmonic technique has been analyzed, one can go on to illustrate its use on images of real aggregates.