Invertible Orientation Scores of 3D Images

The enhancement and detection of elongated structures in noisy image data is relevant for many biomedical applications. To handle complex crossing structures in 2D images, 2D orientation scores \(U: \mathbb {R} ^ 2\times S ^ 1 \rightarrow \mathbb {R}\) were introduced, which already showed their use in a variety of applications. Here we extend this work to 3D orientation scores \(U: \mathbb {R} ^ 3 \times S ^ 2\rightarrow \mathbb {R}\). 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. For efficient implementation of the different steps in the wavelet creation we use a spherical harmonic transform. Finally, we show some first results of practical applications of 3D orientation scores.