Left-invariant evolutions of wavelet transforms on the Similitude Group

Enhancement of multiple-scale elongated structures in noisy image data is relevant for many biomedical applications but commonly used PDE-based enhancement techniques often fail at crossings in an image. To get an overview of how an image is composed of local multiple-scale elongated structures we construct a multiple scale orientation score, which is a continuous wavelet transform on the similitude group, SIM(2). Our unitary transform maps the space of images onto a reproducing kernel space defined on SIM(2), allowing us to robustly relate Euclidean (and scaling) invariant operators on images to left-invariant operators on the corresponding continuous wavelet transform. Rather than often used wavelet (soft-)thresholding techniques, we employ the group structure in the wavelet domain to arrive at left-invariant evolutions and flows (diffusion), for contextual crossing preserving enhancement of multiple scale elongated structures in noisy images. We present experiments that display benefits of our work compared to recent PDE techniques acting directly on the images and to our previous work on left-invariant diffusions on orientation scores defined on Euclidean motion group.Comment: 40 page

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 $U:\mathbb{R}^{d} \rtimes S^{d-1} \to \mathbb{R}$ defined on the extended space of positions and orientations, which we relate to data on the roto-translation group $SE(d)$, $d=2,3$. 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

Vesselness via multiple scale orientation scores

The multi-scale Frangi vesselness filter is an established tool in (retinal) vascular imaging. However, it cannot cope with crossings or bifurcations, since it only looks for elongated structures. Therefore, we disentangle crossing structures in the image via (multiple scale) invertible orientation scores. The described vesselness filter via scale-orientation scores performs considerably better at enhancing vessels throughout crossings and bifurcations than the Frangi version. Both methods are evaluated on a public dataset. Performance is measured by comparing ground truth data to the segmentation results obtained by basic thresholding and morphological component analysis of the filtered images

Geodesic Tracking of Retinal Vascular Trees with Optical and TV-Flow Enhancement in SE(2)

Retinal images are often used to examine the vascular system in a non-invasive way. Studying the behavior of the vasculature on the retina allows for noninvasive diagnosis of several diseases as these vessels and their behavior are representative of the behavior of vessels throughout the human body. For early diagnosis and analysis of diseases, it is important to compare and analyze the complex vasculature in retinal images automatically. In previous work, PDE-based geometric tracking and PDE-based enhancements in the homogeneous space of positions and orientations have been studied and turned out to be useful when dealing with complex structures (crossing of blood vessels in particular). In this article, we propose a single new, more effective, Finsler function that integrates the strength of these two PDE-based approaches and additionally accounts for a number of optical effects (dehazing and illumination in particular). The results greatly improve both the previous left-invariant models and a recent data-driven model, when applied to real clinical and highly challenging images. Moreover, we show clear advantages of each module in our new single Finsler geometrical method

Rivulet: 3D Neuron Morphology Tracing with Iterative Back-Tracking

The digital reconstruction of single neurons from 3D confocal microscopic images is an important tool for understanding the neuron morphology and function. However the accurate automatic neuron reconstruction remains a challenging task due to the varying image quality and the complexity in the neuronal arborisation. Targeting the common challenges of neuron tracing, we propose a novel automatic 3D neuron reconstruction algorithm, named Rivulet, which is based on the multi-stencils fast-marching and iterative backtracking. The proposed Rivulet algorithm is capable of tracing discontinuous areas without being interrupted by densely distributed noises. By evaluating the proposed pipeline with the data provided by the Diadem challenge and the recent BigNeuron project, Rivulet is shown to be robust to challenging microscopic imagestacks. We discussed the algorithm design in technical details regarding the relationships between the proposed algorithm and the other state-of-the-art neuron tracing algorithms

Geodesic Tracking via New Data-driven Connections of Cartan Type for Vascular Tree Tracking

We introduce a data-driven version of the plus Cartan connection on the homogeneous space $\mathbb{M}_2$ of 2D positions and orientations. We formulate a theorem that describes all shortest and straight curves (parallel velocity and parallel momentum, respectively) with respect to this new data-driven connection and corresponding Riemannian manifold. Then we use these shortest curves for geodesic tracking of complex vasculature in multi-orientation image representations defined on $\mathbb{M}_{2}$. The data-driven Cartan connection characterizes the Hamiltonian flow of all geodesics. It also allows for improved adaptation to curvature and misalignment of the (lifted) vessel structure that we track via globally optimal geodesics. We compute these geodesics numerically via steepest descent on distance maps on $\mathbb{M}_2$ that we compute by a new modified anisotropic fast-marching method. Our experiments range from tracking single blood vessels with fixed endpoints to tracking complete vascular trees in retinal images. Single vessel tracking is performed in a single run in the multi-orientation image representation, where we project the resulting geodesics back onto the underlying image. The complete vascular tree tracking requires only two runs and avoids prior segmentation, placement of extra anchor points, and dynamic switching between geodesic models. Altogether we provide a geodesic tracking method using a single, flexible, transparent, data-driven geodesic model providing globally optimal curves which correctly follow highly complex vascular structures in retinal images. All experiments in this article can be reproduced via documented Mathematica notebooks available at GitHub (https://github.com/NickyvdBerg/DataDrivenTracking)

Retinal vessel segmentation:An efficient graph cut approach with Retinex and local phase

Our application concerns the automated detection of vessels in retinal images to improve understanding of the disease mechanism, diagnosis and treatment of retinal and a number of systemic diseases. We propose a new framework for segmenting retinal vasculatures with much improved accuracy and efficiency. The proposed framework consists of three technical components: Retinex-based image inhomogeneity correction, local phase-based vessel enhancement and graph cut-based active contour segmentation. These procedures are applied in the following order. Underpinned by the Retinex theory, the inhomogeneity correction step aims to address challenges presented by the image intensity inhomogeneities, and the relatively low contrast of thin vessels compared to the background. The local phase enhancement technique is employed to enhance vessels for its superiority in preserving the vessel edges. The graph cut-based active contour method is used for its efficiency and effectiveness in segmenting the vessels from the enhanced images using the local phase filter. We have demonstrated its performance by applying it to four public retinal image datasets (3 datasets of color fundus photography and 1 of fluorescein angiography). Statistical analysis demonstrates that each component of the framework can provide the level of performance expected. The proposed framework is compared with widely used unsupervised and supervised methods, showing that the overall framework outperforms its competitors. For example, the achieved sensitivity (0:744), specificity (0:978) and accuracy (0:953) for the DRIVE dataset are very close to those of the manual annotations obtained by the second observer