Design and processing of invertible orientation scores of 3D images

\u3cp\u3eThe enhancement and detection of elongated structures in noisy image data are relevant for many biomedical imaging applications. To handle complex crossing structures in 2D images, 2D orientation scores U: R \u3csup\u3e2\u3c/sup\u3e× S \u3csup\u3e1\u3c/sup\u3e→ C were introduced, which already showed their use in a variety of applications. Here we extend this work to 3D orientation scores U: R \u3csup\u3e3\u3c/sup\u3e× S \u3csup\u3e2\u3c/sup\u3e→ 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, and the second is designed in the 3D generalized Zernike basis, allowing us to calculate analytical expressions for the spatial filters. Second, we propose a nonlinear diffusion flow on the 3D roto-translation group: crossing-preserving coherence-enhancing diffusion via orientation scores (CEDOS). Finally, we show two applications of the orientation score transformation. In the first application we apply our CEDOS algorithm 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. \u3c/p\u3

Locally adaptive frames in the roto-translation group and their applications in medical imaging

Locally adaptive differential frames (gauge frames) are a well-known effective tool in image analysis, used in differential invariants and PDE-flows. However, at complex structures such as crossings or junctions, these frames are not well defined. Therefore, we generalize the notion of gauge frames on images to gauge frames on data representations U:R d ⋊S d−1 →R \u3cbr/\u3eU:Rd⋊Sd−1→R\u3cbr/\u3e defined on the extended space of positions and orientations, which we relate to data on the roto-translation group SE(d), d=2,3 \u3cbr/\u3ed=2,3\u3cbr/\u3e. This allows to define multiple frames per position, one per orientation. We compute these frames via exponential curve fits in the extended data representations in SE(d). These curve fits minimize first- or second-order variational problems which are solved by spectral decomposition of, respectively, a structure tensor or Hessian of data on SE(d). We include these gauge frames in differential invariants and crossing-preserving PDE-flows acting on extended data representation U and we show their advantage compared to the standard left-invariant frame on SE(d). Applications include crossing-preserving filtering and improved segmentations of the vascular tree in retinal images, and new 3D extensions of coherence-enhancing diffusion via invertible orientation scores

The hessian of axially symmetric functions on SE(3) and application in 3D image analysis

\u3cp\u3eWe propose a method for computation of the Hessian of axially symmetric functions on the roto-translation group SE(3). Eigen decomposition of the resulting Hessian is then used for curvature estimation of tubular structures, similar to how the Hessian matrix of 2D or 3D image data can be used for orientation estimation. This paper focuses on a new implementation of a Gaussian regularized Hessian on the roto-translation group. Furthermore we show how eigenanalysis of this Hessian gives rise to exponential curve fits on data on position and orientation (e.g. orientation scores), whose spatial projections provide local fits in 3D data. We quantitatively validate our exponential curve fits by comparing the curvature of the spatially projected fitted curve to ground truth curvature of artificial 3D data. We also show first results on real MRA data.\u3c/p\u3