Our method employs neighborhood matching in g-space for effective atlas construction. For each point in the x-g space, (x_{;}, q_{k}/, where x; e R^{3} is a voxel location and q_{k} e R^{3} is a wavevector, we define a spherical patch, V_{i;}k, centered at q_{k} with fixed q_{k} = |q_{k}| and subject to a neighborhood angle op. The diffusion measurements on this spherical patch are mapped to a disc using azimuthal equidistant projection (AEP) before computing the rotation invariant features via polar complex exponential transform (PCET) (Sect. 2.1) [7] for patch matching (Sect. 2.2). The similarity weights resulting from patch matching will be used in the mean shift algorithm (Sect. 2.3) to determine the most probable signal at each point in q-space.

Patch Features

Azimuthal equidistant projection (AEP) [6] maps the coordinates on a sphere to a plane where the distances and azimuths of points on the sphere are preserved with respect to a reference point [6]. This provides a good basis for subsequent computation of invariant features for matching. The reference point (ф_{0} , X_{0}/, with ф being the latitude and X being the longitude, corresponds in our case to the center of the spherical patch and will be projected to the center of a disc. Viewing the reference point as the ‘North pole’, all points along a given azimuth, в, will project along a straight line from the center of the disc. In the projection plane, this line subtends an angle в with the vertical. The distance from the center to another projected point is given as p. The projection can be described as q ! (q, p, в/. Based on [6], distance p associated with a point (ф, X/ on the sphere is computed as the great circle distance between the point and the reference (ф_{0}, X_{0}/ and is given by

The angle в is computed as the azimuth of the point in relation to the reference:

Note that, since the diffusion signals are antipodal symmetric, we map antipodally all the points on the sphere to the same hemisphere as the reference point prior to performing AEP. After projection, the q-space spherical patch P is mapped to a 2D circular patch P.

After AEP, we proceed with the computation of rotation invariant features. We choose to use the polar complex exponential transform (PCET) [7] for its computation efficiency. PCET with order n,
= 0,1,2,, 1, and repetition l, l= 0, 1,2, ..., 1, of AEP-projected signal profile S(x, q, p, в/ is defined as

where [•]* denotes the complex conjugate and H_{n;l}(p, 9) is the basis function defined as

For each patch P consisting of signal vector S(P), the associated PCET features {Mn,i(V) |} computed up to maximum order m (i.e., 0 < l, n < m) are concatenated into a feature vector M(P).