## Sinc Interpolation Without Gibbs Ringing

Our 3-D local structure filtering methods described above assume that volume data with isotropic voxels are used as input data. However, voxels in medical volume data are usually anisotropic since they generally have lower resolution along the third direction, i.e., the direction orthogonal to the slice plane, than within slices. Rotational invariant feature extraction becomes more intuitive in a space where the sample distances are uniform. That is, structures of a particular size can be detected on the same scale independent of the direction when the signal sampling is isotropic. We therefore introduce a preprocessing procedure for 3-D local structure filtering in which we perform interpolation to make each voxel isotropic. Linear and spline-based interpolation methods are often used, but blurring is inherently involved in these approaches. Because, as noted above, the original volume data is inherently blurrier in the third direction, further degradation of the data in that direction should be avoided. For this reason, we opted to employ sine interpolation so as not to introduce any additional blurring. After Gaussian-shaped slopes are added at the beginning and end of each profile in the third direction to avoid unwanted Gibbs ringing, sinc interpolation is performed by zero-filled expansion in the frequency domain [24,25].

The method for sinc interpolation without Gibbs ringing is described below. The sinc interpolation along the third (z-axis) direction is performed by zero-filled expansion in the frequency domain. Let f (i) (i = 0, 1,...,n — 1) be the profile in the third direction. In the discrete Fourier transform of f (i), f (i) should be regarded as cyclic and then f (n — 1) and f (0) are essentially adjacent. Unwanted Gibbs ringing occurs in the interpolated profile due to the discontinuity between f (n — 1) and f (0). Thus, Gaussian-shaped slopes are added at the beginning and end of f (i) to avoid the occurrence of unwanted ringing before the sinc interpolation. Let f '(i) (i =—3 ■ a, ...,0,1,...,n — 1, n,...,3 ■ a + n) be the modified profile, which is given by f'(i) =

exp(-(i-0^) • f (n - 1) i = n,..., 3 ■ a where the variation is sufficiently smooth everywhere, including between f (3 ■ a + n) and f (—3 ■ a). The discrete Fourier transform of f (i) is performed (we used a = 4). After the sinc interpolation of f '(i), the added Gaussian-shaped slopes are removed.

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