Geodesic Distance Function Research Articles

Temperature-driven coordination of circadian transcriptional regulation.

The circadian clock is an evolutionarily-conserved molecular oscillator that enables species to anticipate rhythmic changes in their environment. At a molecular level, the core clock genes induce circadian oscillations in thousands of genes in a tissue-specific manner, orchestrating myriad biological processes. While previous studies have investigated how the core clock circuit responds to environmental perturbations such as temperature, the downstream effects of such perturbations on circadian regulation remain poorly understood. By analyzing bulk-RNA sequencing of Drosophila fat bodies harvested from flies subjected to different environmental conditions, we demonstrate a highly condition-specific circadian transcriptome: genes are cycling in a temperature-specific manner, and the distributions of their phases also differ between the two conditions. Further employing a reference-based gene regulatory network (Reactome), we find evidence of increased gene-gene coordination at low temperatures and synchronization of rhythmic genes that are network neighbors. We report that the phase differences between cycling genes increase as a function of geodesic distance in the low temperature condition, suggesting increased coordination of cycling on the gene regulatory network. Our results suggest a potential mechanism whereby the circadian clock mediates the fly's response to seasonal changes in temperature.

Uniform convergence rates for Lipschitz learning on graphs

Abstract Lipschitz learning is a graph-based semisupervised learning method where one extends labels from a labeled to an unlabeled data set by solving the infinity Laplace equation on a weighted graph. In this work we prove uniform convergence rates for solutions of the graph infinity Laplace equation as the number of vertices grows to infinity. Their continuum limits are absolutely minimizing Lipschitz extensions (AMLEs) with respect to the geodesic metric of the domain where the graph vertices are sampled from. We work under very general assumptions on the graph weights, the set of labeled vertices and the continuum domain. Our main contribution is that we obtain quantitative convergence rates even for very sparsely connected graphs, as they typically appear in applications like semisupervised learning. In particular, our framework allows for graph bandwidths down to the connectivity radius. For proving this we first show a quantitative convergence statement for graph distance functions to geodesic distance functions in the continuum. Using the ‘comparison with distance functions’ principle, we can pass these convergence statements to infinity harmonic functions and AMLEs.

Segmentation only uses sparse annotations: Unified weakly and semi-supervised learning in medical images.

Since segmentation labeling is usually time-consuming and annotating medical images requires professional expertise, it is laborious to obtain a large-scale, high-quality annotated segmentation dataset. We propose a novel weakly- and semi-supervised framework named SOUSA (Segmentation Only Uses Sparse Annotations), aiming at learning from a small set of sparse annotated data and a large amount of unlabeled data. The proposed framework contains a teacher model and a student model. The student model is weakly supervised by scribbles and a Geodesic distance map derived from scribbles. Meanwhile, a large amount of unlabeled data with various perturbations are fed to student and teacher models. The consistency of their output predictions is imposed by Mean Square Error (MSE) loss and a carefully designed Multi-angle Projection Reconstruction (MPR) loss. Extensive experiments are conducted to demonstrate the robustness and generalization ability of our proposed method. Results show that our method outperforms weakly- and semi-supervised state-of-the-art methods on multiple datasets. Furthermore, our method achieves a competitive performance with some fully supervised methods with dense annotation when the size of the dataset is limited.

Continuum Limit of Lipschitz Learning on Graphs

Tackling semi-supervised learning problems with graph-based methods has become a trend in recent years since graphs can represent all kinds of data and provide a suitable framework for studying continuum limits, for example, of differential operators. A popular strategy here is p-Laplacian learning, which poses a smoothness condition on the sought inference function on the set of unlabeled data. For p<infty continuum limits of this approach were studied using tools from varGamma -convergence. For the case p=infty , which is referred to as Lipschitz learning, continuum limits of the related infinity Laplacian equation were studied using the concept of viscosity solutions. In this work, we prove continuum limits of Lipschitz learning using varGamma -convergence. In particular, we define a sequence of functionals which approximate the largest local Lipschitz constant of a graph function and prove varGamma -convergence in the L^{infty }-topology to the supremum norm of the gradient as the graph becomes denser. Furthermore, we show compactness of the functionals which implies convergence of minimizers. In our analysis we allow a varying set of labeled data which converges to a general closed set in the Hausdorff distance. We apply our results to nonlinear ground states, i.e., minimizers with constrained L^p-norm, and, as a by-product, prove convergence of graph distance functions to geodesic distance functions.

Initial Condition Assessment for Reaction-Diffusion Glioma Growth Models: A Translational MRI-Histology (In)Validation Study.

Reaction-diffusion models have been proposed for decades to capture the growth of gliomas. Nevertheless, these models require an initial condition: the tumor cell density distribution over the whole brain at diagnosis time. Several works have proposed to relate this distribution to abnormalities visible on magnetic resonance imaging (MRI). In this work, we verify these hypotheses by stereotactic histological analysis of a non-operated brain with glioblastoma using a 3D-printed slicer. Cell density maps are computed from histological slides using a deep learning approach. The density maps are then registered to a postmortem MR image and related to an MR-derived geodesic distance map to the tumor core. The relation between the edema outlines visible on T2-FLAIR MRI and the distance to the core is also investigated. Our results suggest that (i) the previously proposed exponential decrease of the tumor cell density with the distance to the core is reasonable but (ii) the edema outlines would not correspond to a cell density iso-contour and (iii) the suggested tumor cell density at these outlines is likely overestimated. These findings highlight the limitations of conventional MRI to derive glioma cell density maps and the need for other initialization methods for reaction-diffusion models to be used in clinical practice.

Non-invasive local pulse wave velocity using 4D-flow MRI

Pulse Wave Velocity (PWV) corresponds to the velocity at which pressure waves, generated by the systolic contraction in the heart, propagate along the arterial tree. Due to the complex interplay between blood flow and the artery wall, PWV is related to inherent mechanical properties and arterial morphology. PWV has been widely accepted as a biomarker and early predictor to evaluate global arterial distensibility. Still, several local abnormalities often remain hidden or difficult to detect using non-invasive techniques. Here, we introduce a novel method to efficiently construct a local estimate of PWV along the aorta using 4D-Flow MRI data. A geodesic distance map was used to track advancing pulses for efficient flow calculations, based on the observation that the propagation of velocity wavefronts strongly depends on the arterial morphology. This procedure allows us a robust evaluation of the local transit time due to the pulse wave at each position in the aorta. Moreover, the estimation of the local PWV map did not require centerlines, and the final result is projected back to 3D using the same geodesic map. We evaluated PWV values in healthy young and adult volunteers and patients with univentricular physiology after a Fontan procedure. Our method is fast, semi-automatic, and depicts differences between young versus adult volunteers and young volunteers versus Fontan patients, showing consistent results compared to global methods. Remarkably, the technique could detect local differences of PWV on the aortic arch for all subjects, being consistent with previous findings of reduced PWV in the aortic arch.

Attention-Guided Multi-Scale Segmentation Neural Network for Interactive Extraction of Region Objects from High-Resolution Satellite Imagery

Automatic extraction of region objects from high-resolution satellite imagery presents a great challenge, because there may be very large variations of the objects in terms of their size, texture, shape, and contextual complexity in the image. To handle these issues, we present a novel, deep-learning-based approach to interactively extract non-artificial region objects, such as water bodies, woodland, farmland, etc., from high-resolution satellite imagery. First, our algorithm transforms user-provided positive and negative clicks or scribbles into guidance maps, which consist of a relevance map modified from Euclidean distance maps, two geodesic distance maps (for positive and negative, respectively), and a sampling map. Then, feature maps are extracted by applying a VGG convolutional neural network pre-trained on the ImageNet dataset to the image X, and they are then upsampled to the resolution of X. Image X, guidance maps, and feature maps are integrated as the input tensor. We feed the proposed attention-guided, multi-scale segmentation neural network (AGMSSeg-Net) with the input tensor above to obtain the mask that assigns a binary label to each pixel. After a post-processing operation based on a fully connected Conditional Random Field (CRF), we extract the selected object boundary from the segmentation result. Experiments were conducted on two typical datasets with diverse region object types from complex scenes. The results demonstrate the effectiveness of the proposed method, and our approach outperforms existing methods for interactive image segmentation.

A Riemannian approach for free-space extraction and path planning using catadioptric omnidirectional vision

This paper presents a Riemannian approach for free-space extraction and path planning using color catadioptric vision. The problem is formulated considering color catadioptric images as Riemannian manifolds and solved using the Riemannian Eikonal equation with an anisotropic fast marching numerical scheme. This formulation allows the integration of adapted color and spatial metrics in an incremental process. First, the traversable ground (namely free-space) is delimited using a color structure tensor built on the multi-dimensional components of the catadioptric image. Then, the Eikonal equation is solved in the image plane incorporating a generic metric tensor for central catadioptric systems. This built Riemannian metric copes with the geometric distortions in the catadioptric image plane introduced by the curved mirror in order to compute the geodesic distance map and the shortest path between image points. We present comparative results using Euclidean and Riemannian distance transforms and show the effectiveness of the Riemannian approach to produce safest path planning.

Robust Pulmonary Nodule Segmentation in CT Image for Juxta-pleural and Juxta-vascular Case

Background: Lung cancer is a greatest threat to people's health and life. CT image leads to unclear boundary segmentation. Segmentation of irregular nodules and complex structure, boundary information is not well considered and lung nodules have always been a hot topic. Objective: In this study, the pulmonary nodule segmentation is accomplished with the new graph cut algorithm. The problem of segmenting the juxta-pleural and juxta-vascular nodules was investigated which is based on graph cut algorithm. Methods: Firstly, the inflection points by the curvature was decided. Secondly, we used kernel graph cut to segment the nodules for the initial edge. Thirdly, the seeds points based on cast raying method is performed; lastly, a novel geodesic distance function is proposed to improve the graph cut algorithm and applied in lung nodules segmentation. Results: The new algorithm has been tested on total 258 nodules. Table 1 summarizes the morphologic features of all the nodules and given the results between the successful segmentation group and the poor/failed segmentation group. Figure 1 to Fig. (12) shows segmentation effect of Juxta-vascular nodules, Juxta-pleural nodules, and comparted with the other interactive segmentation methods. Conclusion: The experimental verification shows better results with our algorithm, the results will measure the volume numerical approach to nodule volume. The results of lung nodules segmentation in this study are as good as the results obtained by the other methods.

Fast Asymmetric Fronts Propagation for Image Segmentation

In this paper, we introduce a generalized asymmetric fronts propagation model based on the geodesic distance maps and the Eikonal partial differential equations. One of the key ingredients for the computation of the geodesic distance map is the geodesic metric, which can govern the action of the geodesic distance level set propagation. We consider a Finsler metric with the Randers form, through which the asymmetry and anisotropy enhancements can be taken into account to prevent the fronts leaking problem during the fronts propagation. These enhancements can be derived from the image edge-dependent vector field such as the gradient vector flow. The numerical implementations are carried out by the Finsler variant of the fast marching method, leading to very efficient interactive segmentation schemes. We apply the proposed Finsler fronts propagation model to image segmentation applications. Specifically, the foreground and background segmentation is implemented by the Voronoi index map. In addition, for the application of tubularity segmentation, we exploit the level set lines of the geodesic distance map associated to the proposed Finsler metric providing that a thresholding value is given.

Bounded Plurisubharmonic Exhaustion Functions for Lipschitz Pseudoconvex Domains in $${\mathbb {C}}{\mathbb {P}}^n$$ C P n

In this paper, we use Takeuchi’s Theorem to show that for every Lipschitz pseudoconvex domain \(\Omega \) in \({\mathbb {C}}{\mathbb {P}}^n\) there exist a Lipschitz defining function \(\rho \) and an exponent \(0<\eta <1\) such that \(-(-\rho )^\eta \) is strictly plurisubharmonic on \(\Omega \). This generalizes a result of Ohsawa and Sibony for \(C^2\) domains. In contrast to the Ohsawa–Sibony result, we provide a counterexample demonstrating that we may not assume \(\rho =-\delta \), where \(\delta \) is the geodesic distance function for the boundary of \(\Omega \).

Spanners for Geodesic Graphs and Visibility Graphs

Let \(\mathcal{P}\) be a set of n points inside a polygonal domain \(\mathcal{D}\). A polygonal domain with h holes (or obstacles) consists of h disjoint polygonal obstacles surrounded by a simple polygon which itself acts as an obstacle. We first study t-spanners for the set \(\mathcal{P}\) with respect to the geodesic distance function \(\pi \) where for any two points p and q, \(\pi (p,q)\) is equal to the Euclidean length of the shortest path from p to q that avoids the obstacles interiors. For a case where the polygonal domain is a simple polygon (i.e., \(h=0\)), we construct a (\(\sqrt{10}+\epsilon \))-spanner that has \(O(n \log ^2 n)\) edges. For a case where there are h holes, our construction gives a (\(5+\epsilon \))-spanner with the size of \(O(n\sqrt{h}\log ^2 n)\). Moreover, we study t-spanners for the visibility graph of \(\mathcal{P}\) (\(VG(\mathcal{P})\), for short) with respect to a hole-free polygonal domain \(\mathcal{D}\). The graph \(VG(\mathcal{P})\) is not necessarily a complete graph or even connected. In this case, we propose an algorithm that constructs a (\(3+\epsilon \))-spanner of size \(O(n^{4/3+\delta })\) for some \(\delta >0\). In addition, we show that there is a set \(\mathcal{P}\) of n points such that any \((3-\epsilon )\)-spanner of \(VG(\mathcal{P})\) must contain \(\varOmega (n^2)\) edges.

Abnormal Profiles of Local Functional Connectivity Proximal to Focal Cortical Dysplasias.

IntroductionFocal cortical dysplasia (FCD) is a congenital malformation of cortical development that often leads to medically refractory epilepsy. Focal resection can be an effective treatment, but is challenging as the surgically relevant abnormality may exceed the MR-visible lesion. The aim of the current study is to develop methodology to characterize the profile of functional connectivity around FCDs using resting-state functional MRI and in the individual patient. The detection of aberrant connectivity may provide a means to more completely delineate the clinically relevant lesion.Materials and MethodsFifteen FCD patients (age, mean±SD: 31±11 years; 11 males) and 16 matched healthy controls (35±9 years; 7 males) underwent structural and functional imaging at 3 Tesla. The cortical surface was reconstructed from the T1-weighted scan and the registered functional MRI data was spatially normalized to a common anatomical standard space employing the gyral pattern. Seed-based functional connectivity was determined in all subjects for all dysplasia locations. A single patient was excluded based on an aberrant FCD seed time series. Functional connectivity as a function of geodesic distance (along the cortical surface) was compared between the individual patients and the homotopic normative connectivity profiles derived from the controls.ResultsIn 12/14 patients, aberrant profiles of functional connectivity were found, which demonstrated both hyper- and hypoconnectivity as well as combinations. Abnormal functional connectivity was typically found (also) beyond the lesion visible on structural MRI, while functional connectivity profiles not related to a lesion appeared normal in patients.ConclusionThis novel functional MRI technique has potential for delineating functionally aberrant from normal cortex beyond the structural lesion in FCD, which remains to be confirmed in future research.

Measurement of repeat effects in Chicago’s criminal social network

The “near-repeat” effect is a well-known criminological phenomenon in which the occurrence of a crime incident gives rise to a temporary elevation of crime risk within close physical proximity to an initial incident. Adopting a social network perspective, we instead define a near repeat in terms of geodesic distance within a criminal social network, rather than spatial distance. Specifically, we report a statistical analysis of repeat effects in arrest data for Chicago during the years 2003–2012. We divide the arrest data into two sets (violent crimes and other crimes) and, for each set, we compare the distributions of time intervals between repeat incidents to theoretical distributions in which repeat incidents occur only by chance. We first consider the case of the same arrestee participating in repeat incidents (“exact repeats”) and then extend the analysis to evaluate repeat risks of those arrestees near one another in the social network. We observe repeat effects that diminish as a function of geodesic distance and time interval, and we estimate typical time scales for repeat crimes in Chicago.

Hybrid detection of lung nodules on CT scan images.

The diversity of lung nodules poses difficulty for the current computer-aided diagnostic (CAD) schemes for lung nodule detection on computed tomography (CT) scan images, especially in large-scale CT screening studies. We proposed a novel CAD scheme based on a hybrid method to address the challenges of detection in diverse lung nodules. The hybrid method proposed in this paper integrates several existing and widely used algorithms in the field of nodule detection, including morphological operation, dot-enhancement based on Hessian matrix, fuzzy connectedness segmentation, local density maximum algorithm, geodesic distance map, and regression tree classification. All of the adopted algorithms were organized into tree structures with multi-nodes. Each node in the tree structure aimed to deal with one type of lung nodule. The method has been evaluated on 294 CT scans from the Lung Image Database Consortium (LIDC) dataset. The CT scans were randomly divided into two independent subsets: a training set (196 scans) and a test set (98 scans). In total, the 294 CT scans contained 631 lung nodules, which were annotated by at least two radiologists participating in the LIDC project. The sensitivity and false positive per scan for the training set were 87% and 2.61%. The sensitivity and false positive per scan for the testing set were 85.2% and 3.13%. The proposed hybrid method yielded high performance on the evaluation dataset and exhibits advantages over existing CAD schemes. We believe that the present method would be useful for a wide variety of CT imaging protocols used in both routine diagnosis and screening studies.

Surface shape recognition method for crack detection

Computer vision-based crack detection methods for large-scale civil structures are mainly developed within the framework of image enhancement and segmentation. In contrast, we propose an idea that converts the images into parametric surfaces and then detects the crack surfaces using shape recognition techniques. The shape variations among noncrack surfaces are caused by approximately isometric deformations, but the dissimilarities between crack and noncrack surfaces are produced by nonisometric deformations. Therefore, the two classes of surfaces can be discriminated by their geodesic distance maps. To tackle the disturbances caused by the gaps in cracks, we develop a dedicated method, steady marching method, for the computation of the distance maps. In the subsequent quantitative comparison, we first construct distance difference matrices from the distance maps. Next, sub-block contrast ratios of these matrices are calculated and used as shape descriptors, which can be directly compared for the surface classification. Experimental results demonstrate that our method achieves a better performance than some typical methods. The extension of our method for crack localization is also presented.

Inverse kinematics solution of a new circumferential drilling machine for aircraft assembly

SUMMARYInverse kinematics solutions are the basis for position and orientation control of automated machines in their Cartesian workspace. This paper presents an efficient and robust inverse kinematics algorithm for a new circumferential drilling machine for aircraft fuselage assembly. After a brief introduction to the circumferential drilling machine and its forward kinematics, the paper discusses the nonlinear optimization method for solving inverse kinematics problems. The objective function is defined as a weighted combination of a position error function and an orientation error function. By representing orientation error as the geodesic distance between two points on a unit sphere, the paper proposes to define the orientation error function by using faithful geodesic distance functions, which are accurate approximations to the geodesic distance when it is small. For increased efficiency, robustness, and easy setting of initial values, the inverse kinematics problem is decomposed into two subproblems. The revolute joint coordinates are obtained by nonlinear optimization, and the prismatic joint coordinates are calculated with closed-form formulas. Numerical experiments show that the objective function defined with faithful geodesic distance functions is effective, and the proposed algorithm is efficient, robust, and accurate. The algorithm has been successfully integrated into the control system of the circumferential drilling machine. Preliminary drilling experiments show that the position accuracy of drilled holes is within ±0.5 mm, which is acceptable for the assembly of large aircrafts.

Efficient algorithm for finding the exact minimum barrier distance

The minimum barrier distance, MBD, introduced recently in [1], is a pseudo-metric defined on a compact subset D of the Euclidean space Rn and whose values depend on a fixed map (an image) f from D into R. The MBD is defined as the minimal value of the barrier strength of a path between the points, which constitutes the length of the smallest interval containing all values of f along the path.In this paper we present a polynomial time algorithm, that provably calculates the exact values of MBD for the digital images. We compare this new algorithm, theoretically and experimentally, with the algorithm presented in [1], which computes the approximate values of the MBD. Moreover, we notice that every generalized distance function can be naturally translated to an image segmentation algorithm. The algorithms that fall under such category include: Relative Fuzzy Connectedness, and those associated with the minimum barrier, fuzzy distance, and geodesic distance functions. In particular, we compare experimentally these four algorithms on the 2D and 3D natural and medical images with known ground truth and at varying level of noise, blur, and inhomogeneity.

Gray matter contamination in arterial spin labeling white matter perfusion measurements in patients with dementia

IntroductionWhite matter (WM) perfusion measurements with arterial spin labeling can be severely contaminated by gray matter (GM) perfusion signal, especially in the elderly. The current study investigates the spatial extent of GM contamination by comparing perfusion signal measured in the WM with signal measured outside the brain.Material and methodsFour minute 3T pseudo-continuous arterial spin labeling scans were performed in 41 elderly subjects with cognitive impairment. Outward and inward geodesic distance maps were created, based on dilations and erosions of GM and WM masks. For all outward and inward geodesic distances, the mean CBF was calculated and compared.ResultsGM contamination was mainly found in the first 3 subcortical WM voxels and had only minor influence on the deep WM signal (distances 4 to 7 voxels). Perfusion signal in the WM was significantly higher than perfusion signal outside the brain, indicating the presence of WM signal.ConclusionThese findings indicate that WM perfusion signal can be measured unaffected by GM contamination in elderly patients with cognitive impairment. GM contamination can be avoided by the erosion of WM masks, removing subcortical WM voxels from the analysis. These results should be taken into account when exploring the use of WM perfusion as micro-vascular biomarker.

ESTIMATION OF TORTUOSITY AND RECONSTRUCTION OF GEODESIC PATHS IN 3D

The morphological tortuosity of a geodesic path in a medium can be defined as the ratio between its geodesic length and the Euclidean distance between its two extremities. Thus, the minimum tortuosity of all the geodesic paths into a medium in 2D or in 3D can be estimated by image processing methodsusing mathematical morphology. Considering a medium, the morphological tortuosities of its internal paths are estimated according to one direction, which is perpendicular to both starting and ending opposite extremities of the geodesicpaths. The used algorithm estimates the morphological tortuosities from geodesic distance maps, which are obtained from geodesic propagations. The shape of the propagated structuring element used to estimate the geodesic distance maps on a discrete grid has a direct influence on the morphological tortuosity and has to be chosen very carefully. The results of our algorithm is an image with pixels p having a value equal to the length of the shortest path containing p and connected to two considered opposite boundaries A and B of the image. The analysis of the histogram of the morphological tortuosities gives access to their statistical distribution. Moreover, for each tortuosity the paths can be extracted from the original image, which highlights the location of them into the sample. However, these geodesic paths have to be reconstructed for further processing. The extraction, because applying a threshold on the tortuosities, results in disconnected components, especially for highly tortuous paths. This reconstruction consists in reconnecting these components to the geodesic path linking the two opposite faces, by means of a backtracking algorithm.