Nilpotent Approximations of Sub-Riemannian Distances for Fast Perceptual Grouping of Blood Vessels in 2D and 3D

We propose an efficient approach for the grouping of local orientations (points on vessels) via nilpotent approximations of sub-Riemannian distances in the 2D and 3D roto-translation groups $SE(2)$ and $SE(3)$. In our distance approximations we consider homogeneous norms on nilpotent groups that locally approximate $SE(n)$, and which are obtained via the exponential and logarithmic map on $SE(n)$. In a qualitative validation we show that the norms provide accurate approximations of the true sub-Riemannian distances, and we discuss their relations to the fundamental solution of the sub-Laplacian on $SE(n)$. The quantitative experiments further confirm the accuracy of the approximations. Quantitative results are obtained by evaluating perceptual grouping performance of retinal blood vessels in 2D images and curves in challenging 3D synthetic volumes. The results show that 1) sub-Riemannian geometry is essential in achieving top performance and 2) that grouping via the fast analytic approximations performs almost equally, or better, than data-adaptive fast marching approaches on $\mathbb{R}^n$ and $SE(n)$.Comment: 18 pages, 9 figures, 3 tables, in review at JMI

The optimal connection model for blood vessels segmentation and the MEA-Net

Vascular diseases have long been regarded as a significant health concern. Accurately detecting the location, shape, and afflicted regions of blood vessels from a diverse range of medical images has proven to be a major challenge. Obtaining blood vessels that retain their correct topological structures is currently a crucial research issue. Numerous efforts have sought to reinforce neural networks' learning of vascular geometric features, including measures to ensure the correct topological structure of the segmentation result's vessel centerline. Typically, these methods extract topological features from the network's segmentation result and then apply regular constraints to reinforce the accuracy of critical components and the overall topological structure. However, as blood vessels are three-dimensional structures, it is essential to achieve complete local vessel segmentation, which necessitates enhancing the segmentation of vessel boundaries. Furthermore, current methods are limited to handling 2D blood vessel fragmentation cases. Our proposed boundary attention module directly extracts boundary voxels from the network's segmentation result. Additionally, we have established an optimal connection model based on minimal surfaces to determine the connection order between blood vessels. Our method achieves state-of-the-art performance in 3D multi-class vascular segmentation tasks, as evidenced by the high values of Dice Similarity Coefficient (DSC) and Normalized Surface Dice (NSD) metrics. Furthermore, our approach improves the Betti error, LR error, and BR error indicators of vessel richness and structural integrity by more than 10% compared to other methods, and effectively addresses vessel fragmentation and yields blood vessels with a more precise topological structure.Comment: 19 page

Minimal Path Methods for Segmentation and Analysis of 2D and 3D Line Structures

Image segmentation plays a vital role in many applications of computer vision. Segmentation is not only an important task in its own right, but also a prerequisite for many further image analysis steps. Consequently, segmentation is one of the most active research areas of computer vision. In this thesis, line structures are considered, which have quite different characteristics compared to common objects in natural 2D images: Line structures are much thinner and longer, and often they have little color or texture information such as blood vessels in medical images. To cope with these challenges, minimal path methods are commonly used. In this thesis, two new methods are introduced which are extensions of existing minimal path methods. The first method is a novel hybrid approach for automatic 3D segmentation and quantification of high-resolution 7 Tesla magnetic resonance angiography (MRA) images of the human cerebral vasculature. Our approach consists of two main steps. First, a 3D model-based approach is used to segment and quantify thick vessels and most parts of thin vessels. Second, remaining vessel gaps of the first step in low-contrast and noisy regions are completed using a 3D minimal path approach, which exploits directional information. We present two novel minimal path approaches: The first is an explicit approach based on energy minimization using probabilistic sampling, and the second is an implicit approach based on fast marching with anisotropic directional prior. The second method we introduce is a novel minimal path method for the segmentation of 2D and 3D line structures. Minimal path methods perform propagation of a wavefront emanating from a start point at a speed derived from image features, followed by path extraction using backtracing. Usually, the computation of the speed and the propagation of the wave are two separate steps, and point features are used to compute a static speed. We introduce a new continuous minimal path method which steers the wave propagation progressively using dynamic speed based on path features. We present three instances of our method, using an appearance feature of the path, a geometric feature based on the curvature of the path, and a joint appearance and geometric feature based on the tangent of the wavefront. Such features have not been used in previous continuous minimal path methods. We compute the features dynamically during the wave propagation, and also efficiently using a fast numerical scheme and a low-dimensional parameter space. Our method does not suffer from discretization or metrication errors. We conducted quantitative and qualitative experimental evaluations of our methods using 2D and 3D images from different application areas, including synthetic images, retinal images, satellite images of streets, rivers, and bridges, and 3D 7T MRA images of human brain vessels

Study of Image Local Scale Structure Using Nonlinear Diffusion

Multi-scale representation and local scale extraction of images are important in computer vision research, as in general , structures within images are unknown. Traditionally, the multi-scale analysis is based on the linear diusion (i.e. heat diusion) with known limitation in edge distortions. In addition, the term scale which is used widely in multi-scale and local scale analysis does not have a consistent denition and it can pose potential diculties in real image analysis, especially for the proper interpretation of scale as a geometric measure. In this study, in order to overcome limitations of linear diusion, we focus on the multi-scale analysis based on total variation minimization model. This model has been used in image denoising with the power that it can preserve edge structures. Based on the total variation model, we construct the multi-scale space and propose a denition for image local scale. The new denition of local scale incorporates both pixel-wise and orientation information. This denition can be interpreted with a clear geometrical meaning and applied in general image analysis. The potential applications of total variation model in retinal fundus image analysis is explored. The existence of blood vessel and drusen structures within a single fundus image makes the image analysis a challenging problem. A multi-scale model based on total variation is used, showing the capabilities in both drusen and blood vessel detections. The performance of vessel detection is compared with publicly available methods, showing the improvements both quantitatively and qualitatively. This study provides a better insight into local scale study and shows the potentials of total variation model in medical image analysis

Mathematical Methods for the Quantification of Actin-Filaments in Microscopic Images

In cell biology confocal laser scanning microscopic images of the actin filament of human osteoblasts are produced to assess the cell development. This thesis aims at an advanced approach for accurate quantitative measurements about the morphology of the bright-ridge set of these microscopic images and thus about the actin filament. Therefore automatic preprocessing, tagging and quantification interplay to approximate the capabilities of the human observer to intuitively recognize the filaments correctly. Numerical experiments with random models confirm the accuracy of this approach

Automatic detection of fluorescein tear breakup sequence

Dry Eye Syndrome is a common disease in the western world, with effects from uncomfortable itchiness to permanent damage to the ocular surface. Almost 5 million Americans over 50 years old suffer from dry eye. A conservative estimate shows that approximately 17 million Americans have contact lens related dry eye -one of the main factors to contact lens discontinuation. In addition, the incidence of the disease is on the rise. Nevertheless, there is still no gold standard test that can reliably detect dry eye. One of the most commonly used tests by clinicians to detect dry eye is the Fluorescein Break Up Time (FBUT). However, results vary a lot between clinicians. Other tests such as observing the tear meniscus height are also performed regularly by the clinicians but not necessarily in conjunction with the FBUT test. Therefore there is a real need for a reliable, robust and operator-dependent method to evaluate dry eye. To our knowledge, no previous research has been conducted on automatic evaluation of dry eye in fluorescein images. In this thesis, we present new algorithms to automatically detect various dryness signs and make a number of original contributions. The first problem we address is how to detect the dry areas in fluorescein videos of the anterior of the eye, which are captured using a portable camera. We present a new multi-step algorithm which first locates the iris in each image in the video, then aligns the images according to the location of the iris and finally analyzes the aligned video to find the regions of dryness. We produce a novel segmentation result called dryness image, which depicts the various degrees of tear film thinning over the corneal surface. Then, we demonstrate through experiments that there is a large variation in the estimated Break Up Time (BUT) between clinicians and no ground-truth can be defined. To overcome that, we define a new value based on the clinical definitions of the BUT. These definitions are converted to image processing properties and an estimate of the BUT is computed using temporal analysis of the aligned video. We demonstrate that our new value is in the accepted range of the BUT values provided by the clinicians. We present an extension to the dryness algorithm, which is based on transforming the video to a volume by considering each video frame as a slice in a 3D volume. On a volume, a temporal monotonic constraint can be applied between pixels in consecutive slices. The constraint enforces the clinical definition of tear film thinning over time -the amount of fluid cannot increase while not blinking. The constraint is applied directly into the cost function and the whole volume is segmented simultaneously using graph-cuts. As a consequence, the approach is more robust and less sensitive to alignment errors. Finally, we generalize the idea and explain how monotonic constraints can be applied to other imaging modalities. In the last part of the thesis, we develop a new algorithm to evaluate the tear meniscus height and shape using graph-cuts. We formulate the segmentation problem using asymmetric cost functions and demonstrate its power and usefulness for the task. The asymmetry induces which directional moves are permitted in the minimization process and thus produces a result that adheres to the known shape properties of the tear meniscus. The iterative algorithm provides simultaneously the best segmentation result and shape prior of the meniscus