A new methodology for the estimation of fiber populations in the white matter of the brain with the Funk-Radon transform

Abstract

The Funk-Radon Transform (FRT) is a powerful tool for the estimation of fiber populations with High Angular Resolution Diffusion Imaging (HARDI). It is used in Q-Ball imaging (QBI), and other HARDI techniques such as the recent Orientation Probability Density Transform (OPDT), to estimate fiber populations with very few restrictions on the diffusion model. The FRT consists in the integration of the attenuation signal, sampled by the MRI scanner on the unit sphere, along equators orthogonal to the directions of interest. It is easily proved that this calculation is equivalent to the integration of the diffusion propagator along such directions, although a characteristic blurring with a Bessel kernel is introduced. Under a different point of view, the FRT can be seen as an efficient way to compute the angular part of the integral of the attenuation signal in the plane orthogonal to each direction of the diffusion propagator. In this paper, Stoke's theorem is used to prove that the FRT can in fact be used to compute accurate estimates of the true integrals defining the functions of interest in HARDI, keeping the diffusion model as little restrictive as possible. Varying the assumptions on the attenuation signal, we derive new estimators of fiber orientations, generalizing both Q-Balls and the OPDT. Extensive experiments with both synthetic and real data have been intended to show that the new techniques improve existing ones in many situations.

Fig. E.1

The real-valued exponential integral E_i(x). For negative arguments, the value of the function is not unique, so the principal (real) branch has been depicted.

Fig. E.2

The non-singular exponential integral E_in(x). For a large enough x, E_in rapidly converges to minus the logarithm of x (minus the constant γ).

Fig. 1

Auxiliary coordinates systems (left: cylindrical coordinates; right: spherical coordinates) for the computation of the integrals in the plane Π, the disk Ω, or the curve Γ. The plane Π is orthogonal to the direction of interest, r.

Fig. 2

Angular errors in the estimation of two crossing fibers, as a function of the initial angle between the ground-truth directions, for the estimators of the ODF, Ψ(r). Together with Q-Balls, the results for cQ-Balls and pQ-Balls are presented.

Fig. 3

Glyph representations of the estimators of Ψ(r) in a noise-free environment (S-4), for two fibers crossing in an angle of 80°. The glyphs are min-max normalized for visualization purposes.

Fig. 4

Angular errors in the estimation of two crossing fibers, as a function of the initial angle between the ground-truth directions, for the estimators of the OPDF Φ(r). Together with the OPDT, the cOPDT and the pOPDT are tested. Additionally, the DOT is included for comparison purposes, though it is an estimator of Υ(r) and not Φ(r).

Fig. 5

Glyph representations of the estimators of Φ(r) (and Υ(r), in the last row) in a noise-free environment (S-4), for two fibers crossing in an angle of 55°. Min-max normalization is not necessary in this case.

Fig. 6

Angular errors in the estimation of two crossing fibers, as a function of the initial angle between the ground-truth directions, for the estimators of the ODF, Ψ(r). Noisy scenario with PSNR=13.33 (top), and PSNR=5 (bottom).

Fig. 7

Glyph representations of the estimators of the ODF in a noisy scenario: S-2 with PSNR=13.3. The ground-truth angle between the fibers is 90°. The glyphs are min-max normalized for visualization purposes.

Fig. 8

Angular errors in the estimation of two crossing fibers, as a function of the initial angle between the ground-truth directions, for the estimators of the OPDF, Φ(r). Noisy scenario with PSNR=13.33 (top), and PSNR=5 (bottom).

Fig. 9

Glyph representations of the estimators of the OPDF in a noisy scenario: S-2 with PSNR=13.3. The ground-truth angle between the fibers is 70°, and no min-max normalization is performed.

Fig. 10

Relative errors in the estimation of the integral in the plane Π as the integral in the disk ω, as a function of the b-value, for crossing angles of 90° (left), 65° (center) and 50° right. Each curve represents the relative error ε for an arbitrary spatial direction r (including those yielding maximum and minimum errors) as a function of the b-value.

Fig. 11

Sample axial slices (15 and 23 of 78, respectively) of the real data set used, following usual color-coding conventions: red for the ‘x’ axis, green for the ‘y’ axis, and blue for the ‘z’ axis. For convenience, some tracts of interest have been highlighted: the cerebellar peduncle (cp), the corticopontine tract (cpt), the corticospinal tract (cst), the middle cerebellar peduncle (mcp), the medial lemnicus (ml), the pontine crossing tract (pct), and the superior cerebellar peduncle (scp).

Fig. 12

Glyphs obtained with the cOPDT for the corresponding ROI marked in Fig. 11. The ROI for slice 15 has been down-sampled for visual convenience. The gray levels in the background correspond to the Fractional Anisotropy (FA) of the voxels.

Fig. 13

Glyphs obtained for the small ROI marked in Fig. 12, for slice 15 (top) and 23 (bottom). In both cases, the OPDT (left), the pOPDT (center), and the cOPDT (right) are compared.

Fig. 14

Glyphs obtained for the small ROI marked in Fig. 12 for slice 15, without applying the denoising filter. The OPDT (left), the pOPDT (center), and the cOPDT (right) are compared.