  The enhancement and detection of elongated structures in noisy image data is
relevant for many biomedical imaging applications. To handle complex crossing
structures in 2D images, 2D orientation scores $U: \mathbb{R} ^ 2\times S ^ 1
\rightarrow \mathbb{C}$ were introduced, which already showed their use in a
variety of applications. Here we extend this work to 3D orientation scores $U:
\mathbb{R} ^ 3 \times S ^ 2\rightarrow \mathbb{C}$. First, we construct the
orientation score from a given dataset, which is achieved by an invertible
coherent state type of transform. For this transformation we introduce 3D
versions of the 2D cake-wavelets, which are complex wavelets that can
simultaneously detect oriented structures and oriented edges. Here we introduce
two types of cake-wavelets, the first uses a discrete Fourier transform, the
second is designed in the 3D generalized Zernike basis, allowing us to
calculate analytical expressions for the spatial filters. Finally, we show two
applications of the orientation score transformation. In the first application
we propose an extension of crossing-preserving coherence enhancing diffusion
via our invertible orientation scores of 3D images which we apply to real
medical image data. In the second one we develop a new tubularity measure using
3D orientation scores and apply the tubularity measure to both artificial and
real medical data.
