Laurent D. Cohen

Chong Di, Li Liu, Jinglin Zhang et al.

Curvature-penalized geodesic models have proven their effectiveness in image segmentation by computing globally optimal curves. Unfortunately, these models remain susceptible to shortcuts when delineating objects with complex shapes and image intensity distributions, as they lack mechanisms to enforce shape-aware tangent constraints. To address this limitation, we propose a unified geodesic framework that integrates tangent-constrained priors with curvature penalization. The key idea is to formulate tangent admissibility directly within the orientation-lifted space, where path tangents are restricted to spatially varying angular sectors derived from intrinsic shape representatives (ISR) such as skeletons or interior landmarks. This formulation gives rise to a family of tangent-constrained Finslerian metrics, extending the classical curvature-penalized geodesic models while enforcing mandatory tangent constraints. The resulting Hamilton-Jacobi-Bellman (HJB) partial differential equations (PDEs) admit efficient numerical solutions via variants of the fast marching method, preserving the single-pass computational complexity. Experiments on synthetic, natural, and medical images demonstrate that the proposed geodesic framework indeed improves robustness against weak boundaries and topological shortcuts, yielding segmentation results with enhanced shape fidelity compared to existing geodesic models.

Thomas Dagès, Laurent D. Cohen, Alfred M. Bruckstein

Traditional signal processing methods relying on mathematical data generation models have been cast aside in favour of deep neural networks, which require vast amounts of data. Since the theoretical sample complexity is nearly impossible to evaluate, these amounts of examples are usually estimated with crude rules of thumb. However, these rules only suggest when the networks should work, but do not relate to the traditional methods. In particular, an interesting question is: how much data is required for neural networks to be on par or outperform, if possible, the traditional model-based methods? In this work, we empirically investigate this question in two simple examples, where the data is generated according to precisely defined mathematical models, and where well-understood optimal or state-of-the-art mathematical data-agnostic solutions are known. A first problem is deconvolving one-dimensional Gaussian signals and a second one is estimating a circle's radius and location in random grayscale images of disks. By training various networks, either naive custom designed or well-established ones, with various amounts of training data, we find that networks require tens of thousands of examples in comparison to the traditional methods, whether the networks are trained from scratch or even with transfer-learning or finetuning.

Nicolas Makaroff, Laurent D. Cohen

When studying the results of a segmentation algorithm using convolutional neural networks, one wonders about the reliability and consistency of the results. This leads to questioning the possibility of using such an algorithm in applications where there is little room for doubt. We propose in this paper a new attention gate based on the use of Chan-Vese energy minimization to control more precisely the segmentation masks given by a standard CNN architecture such as the U-Net model. This mechanism allows to obtain a constraint on the segmentation based on the resolution of a PDE. The study of the results allows us to observe the spatial information retained by the neural network on the region of interest and obtains competitive results on the binary segmentation. We illustrate the efficiency of this approach for medical image segmentation on a database of MRI brain images.

Nicolas Makaroff, Théo Bertrand, Laurent D. Cohen

Leveraging geodesic distances and the geometrical information they convey is key for many data-oriented applications in imaging. Geodesic distance computation has been used for long for image segmentation using Image based metrics. We introduce a new method by generating isotropic Riemannian metrics adapted to a problem using CNN and give as illustrations an example of application. We then apply this idea to the segmentation of brain tumours as unit balls for the geodesic distance computed with the metric potential output by a CNN, thus imposing geometrical and topological constraints on the output mask. We show that geodesic distance modules work well in machine learning frameworks and can be used to achieve state-of-the-art performances while ensuring geometrical and/or topological properties.

Raphaël Groscot, Laurent D. Cohen

We present Deformable Voxel Grids (DVGs) for 3D shapes comparison and processing. It consists of a voxel grid which is deformed to approximate the silhouette of a shape, via energy-minimization. By interpreting the DVG as a local coordinates system, it provides a better embedding space than a regular voxel grid, since it is adapted to the geometry of the shape. It also allows to deform the shape by moving the control points of the DVG, in a similar manner to the Free Form Deformation, but with easier interpretability of the control points positions. After proposing a computation scheme of the energies compatible with meshes and pointclouds, we demonstrate the use of DVGs in a variety of applications: correspondences via cubification, style transfer, shape retrieval and PCA deformations. The first two require no learning and can be readily run on any shapes in a matter of minutes on modest hardware. As for the last two, they require to first optimize DVGs on a collection of shapes, which amounts to a pre-processing step. Then, determining PCA coordinates is straightforward and brings a few parameters to deform a shape.

Li Liu, Da Chen, Minglei Shu et al.

Geodesic models are known as an efficient tool for solving various image segmentation problems. Most of existing approaches only exploit local pointwise image features to track geodesic paths for delineating the objective boundaries. However, such a segmentation strategy cannot take into account the connectivity of the image edge features, increasing the risk of shortcut problem, especially in the case of complicated scenario. In this work, we introduce a new image segmentation model based on the minimal geodesic framework in conjunction with an adaptive cut-based circular optimal path computation scheme and a graph-based boundary proposals grouping scheme. Specifically, the adaptive cut can disconnect the image domain such that the target contours are imposed to pass through this cut only once. The boundary proposals are comprised of precomputed image edge segments, providing the connectivity information for our segmentation model. These boundary proposals are then incorporated into the proposed image segmentation model, such that the target segmentation contours are made up of a set of selected boundary proposals and the corresponding geodesic paths linking them. Experimental results show that the proposed model indeed outperforms state-of-the-art minimal paths-based image segmentation approaches.

Théo Bertrand, Laurent D. Cohen

Segmentation of tubular structures in vascular imaging is a well studied task, although it is rare that we try to infuse knowledge of the tree-like structure of the regions to be detected. Our work focuses on detecting the important landmarks in the vascular network (via CNN performing both localization and classification of the points of interest) and representing vessels as the edges in some minimal distance tree graph. We leverage geodesic methods relevant to the detection of vessels and their geometry, making use of the space of positions and orientations so that 2D vessels can be accurately represented as trees. We build our model to carry tracking on Ultrasound Localization Microscopy (ULM) data, proposing to build a good cost function for tracking on this type of data. We also test our framework on synthetic and eye fundus data. Results show that scarcity of well annotated ULM data is an obstacle to localization of vascular landmarks but the Orientation Score built from ULM data yields good geodesics for tracking blood vessels.

Tao Chen, Dan Zhang, Da Chen et al.

Choroidal neovascularization (CNV), a primary characteristic of wet age-related macular degeneration (wet AMD), represents a leading cause of blindness worldwide. In clinical practice, optical coherence tomography angiography (OCTA) is commonly used for studying CNV-related pathological changes, due to its micron-level resolution and non-invasive nature. Thus, accurate segmentation of CNV regions and vessels in OCTA images is crucial for clinical assessment of wet AMD. However, challenges existed due to irregular CNV shapes and imaging limitations like projection artifacts, noises and boundary blurring. Moreover, the lack of publicly available datasets constraints the CNV analysis. To address these challenges, this paper constructs the first publicly accessible CNV dataset (CNVSeg), and proposes a novel multilateral graph convolutional interaction-enhanced CNV segmentation network (MTG-Net). This network integrates both region and vessel morphological information, exploring semantic and geometric duality constraints within the graph domain. Specifically, MTG-Net consists of a multi-task framework and two graph-based cross-task modules: Multilateral Interaction Graph Reasoning (MIGR) and Multilateral Reinforcement Graph Reasoning (MRGR). The multi-task framework encodes rich geometric features of lesion shapes and surfaces, decoupling the image into three task-specific feature maps. MIGR and MRGR iteratively reason about higher-order relationships across tasks through a graph mechanism, enabling complementary optimization for task-specific objectives. Additionally, an uncertainty-weighted loss is proposed to mitigate the impact of artifacts and noise on segmentation accuracy. Experimental results demonstrate that MTG-Net outperforms existing methods, achieving a Dice socre of 87.21\% for region segmentation and 88.12\% for vessel segmentation.

Chong Di, Shuwang Zhou, Da Chen et al.

The computation of minimal paths for the applications in tracking tubular structures such as blood vessels and roads is challenged by complex morphologies and environmental variations. Existing approaches can be roughly categorized into two research lines: the point-wise based models and the segment-wise based models. Although segment-wise approaches have obtained promising results in many scenarios, they often suffer from computational inefficiency and heavily rely on a prescribed prior to fit the target elongated shapes. We propose a novel framework that casts segment-wise tracking as a Markov Decision Process (MDP), enabling a reinforcement learning approach. Our method leverages Q-Learning to dynamically explore a graph of segments, computing edge weights on-demand and adaptively expanding the search space. This strategy avoids the high cost of a pre-computed graph and proves robust to incomplete initial information. Experimental reuslts on typical tubular structure datasets demonstrate that our method significantly outperforms state-of-the-art point-wise and segment-wise approaches. The proposed method effectively handles complex topologies and maintains global path coherence without depending on extensive prior structural knowledge.

Lin Zhang, Chenggang Lu, Xin-yang Shi et al.

Atherosclerosis is a chronic, progressive disease that primarily affects the arterial walls. It is one of the major causes of cardiovascular disease. Magnetic Resonance (MR) black-blood vessel wall imaging (BB-VWI) offers crucial insights into vascular disease diagnosis by clearly visualizing vascular structures. However, the complex anatomy of the neck poses challenges in distinguishing the carotid artery (CA) from surrounding structures, especially with changes like atherosclerosis. In order to address these issues, we propose GAPNet, which is a consisting of a novel geometric prior deduced from.

Da Chen, Jean-Marie Mirebeau, Minglei Shu et al.

The minimal geodesic models based on the Eikonal equations are capable of finding suitable solutions in various image segmentation scenarios. Existing geodesic-based segmentation approaches usually exploit image features in conjunction with geometric regularization terms, such as Euclidean curve length or curvature-penalized length, for computing geodesic curves. In this paper, we take into account a more complicated problem: finding curvature-penalized geodesic paths with a convexity shape prior. We establish new geodesic models relying on the strategy of orientation-lifting, by which a planar curve can be mapped to an high-dimensional orientation-dependent space. The convexity shape prior serves as a constraint for the construction of local geodesic metrics encoding a particular curvature constraint. Then the geodesic distances and the corresponding closed geodesic paths in the orientation-lifted space can be efficiently computed through state-of-the-art Hamiltonian fast marching method. In addition, we apply the proposed geodesic models to the active contours, leading to efficient interactive image segmentation algorithms that preserve the advantages of convexity shape prior and curvature penalization.

Da Chen, Jian Zhu, Xinxin Zhang et al.

Minimal paths are regarded as a powerful and efficient tool for boundary detection and image segmentation due to its global optimality and the well-established numerical solutions such as fast marching method. In this paper, we introduce a flexible interactive image segmentation model based on the Eikonal partial differential equation (PDE) framework in conjunction with region-based homogeneity enhancement. A key ingredient in the introduced model is the construction of local geodesic metrics, which are capable of integrating anisotropic and asymmetric edge features, implicit region-based homogeneity features and/or curvature regularization. The incorporation of the region-based homogeneity features into the metrics considered relies on an implicit representation of these features, which is one of the contributions of this work. Moreover, we also introduce a way to build simple closed contours as the concatenation of two disjoint open curves. Experimental results prove that the proposed model indeed outperforms state-of-the-art minimal paths-based image segmentation approaches.

Da Chen, Jack Spencer, Jean-Marie Mirebeau et al.

The Voronoi diagram-based dual-front active contour models are known as a powerful and efficient way for addressing the image segmentation and domain partitioning problems. In the basic formulation of the dual-front models, the evolving contours can be considered as the interfaces of adjacent Voronoi regions. Among these dual-front models, a crucial ingredient is regarded as the geodesic metrics by which the geodesic distances and the corresponding Voronoi diagram can be estimated. In this paper, we introduce a type of asymmetric quadratic metrics dual-front model. The metrics considered are built by the integration of the image features and a vector field derived from the evolving contours. The use of the asymmetry enhancement can reduce the risk of contour shortcut or leakage problems especially when the initial contours are far away from the target boundaries or the images have complicated intensity distributions. Moreover, the proposed dual-front model can be applied for image segmentation in conjunction with various region-based homogeneity terms. The numerical experiments on both synthetic and real images show that the proposed dual-front model indeed achieves encouraging results.

Li Liu, Da Chen, Minglei Shu et al.

Tubular structure tracking is a crucial task in the fields of computer vision and medical image analysis. The minimal paths-based approaches have exhibited their strong ability in tracing tubular structures, by which a tubular structure can be naturally modeled as a minimal geodesic path computed with a suitable geodesic metric. However, existing minimal paths-based tracing approaches still suffer from difficulties such as the shortcuts and short branches combination problems, especially when dealing with the images involving complicated tubular tree structures or background. In this paper, we introduce a new minimal paths-based model for minimally interactive tubular structure centerline extraction in conjunction with a perceptual grouping scheme. Basically, we take into account the prescribed tubular trajectories and curvature-penalized geodesic paths to seek suitable shortest paths. The proposed approach can benefit from the local smoothness prior on tubular structures and the global optimality of the used graph-based path searching scheme. Experimental results on both synthetic and real images prove that the proposed model indeed obtains outperformance comparing with the state-of-the-art minimal paths-based tubular structure tracing algorithms.

Da Chen, Jean-Marie Mirebeau, Huazhong Shu et al.

The geodesic model based on the eikonal partial differential equation (PDE) has served as a fundamental tool for the applications of image segmentation and boundary detection in the past two decades. However, the existing approaches commonly only exploit the image edge-based features for computing minimal geodesic paths, potentially limiting their performance in complicated segmentation situations. In this paper, we introduce a new variational image segmentation model based on the minimal geodesic path framework and the eikonal PDE, where the region-based appearance term that defines then regional homogeneity features can be taken into account for estimating the associated minimal geodesic paths. This is done by constructing a Randers geodesic metric interpretation of the region-based active contour energy functional. As a result, the minimization of the active contour energy functional is transformed into finding the solution to the Randers eikonal PDE. We also suggest a practical interactive image segmentation strategy, where the target boundary can be delineated by the concatenation of several piecewise geodesic paths. We invoke the Finsler variant of the fast marching method to estimate the geodesic distance map, yielding an efficient implementation of the proposed region-based Randers geodesic model for image segmentation. Experimental results on both synthetic and real images exhibit that our model indeed achieves encouraging segmentation performance.

Da Chen, Laurent D. Cohen

In this chapter, we give an overview of part of our previous work based on the minimal path framework and the Eikonal partial differential equation (PDE). We show that by designing adequate Riemannian and Randers geodesic metrics the minimal paths can be utilized to search for solutions to almost all of the active contour problems and to the Euler-Mumford elastica problem, which allows to blend the advantages from minimal geodesic paths and those original approaches, i.e. the active contours and elastica curves. The proposed minimal path-based models can be applied to deal with a broad variety of image analysis tasks such as boundary detection, image segmentation and tubular structure extraction. The numerical implementations for the computation of minimal paths are known to be quite efficient thanks to the Eikonal solvers such as the Finsler variant of the fast marching method.

Da Chen, Jiong Zhang, Laurent D. Cohen

The minimal path method has proven to be particularly useful and efficient in tubular structure segmentation applications. In this paper, we propose a new minimal path model associated with a dynamic Riemannian metric embedded with an appearance feature coherence penalty and an adaptive anisotropy enhancement term. The features that characterize the appearance and anisotropy properties of a tubular structure are extracted through the associated orientation score. The proposed dynamic Riemannian metric is updated in the course of the geodesic distance computation carried out by the efficient single-pass fast marching method. Compared to state-of-the-art minimal path models, the proposed minimal path model is able to extract the desired tubular structures from a complicated vessel tree structure. In addition, we propose an efficient prior path-based method to search for vessel radius value at each centerline position of the target. Finally, we perform the numerical experiments on both synthetic and real images. The quantitive validation is carried out on retinal vessel images. The results indicate that the proposed model indeed achieves a promising performance.

Da Chen, Laurent D. Cohen

In this paper, we propose a new minimal path model for minimally interactive retinal vessel centerline extraction. The main contribution lies at the construction of a novel coherence-penalized Riemannian metric in a lifted space, dependently of the local geometry of tubularity and an external scalar-valued reference feature map. The globally minimizing curves associated to the proposed metric favour to pass through a set of retinal vessel segments with low variations of the feature map, thus can avoid the short branches combination problem and shortcut problem, commonly suffered by the existing minimal path models in the application of retinal imaging. We validate our model on a series of retinal vessel patches obtained from the DRIVE and IOSTAR datasets, showing that our model indeed get promising results.

Da Chen, Jean-Marie Mirebeau, Laurent D. Cohen

In this paper, we propose a novel curvature-penalized minimal path model via an orientation-lifted Finsler metric and the Euler elastica curve. The original minimal path model computes the globally minimal geodesic by solving an Eikonal partial differential equation (PDE). Essentially, this first-order model is unable to penalize curvature which is related to the path rigidity property in the classical active contour models. To solve this problem, we present an Eikonal PDE-based Finsler elastica minimal path approach to address the curvature-penalized geodesic energy minimization problem. We were successful at adding the curvature penalization to the classical geodesic energy. The basic idea of this work is to interpret the Euler elastica bending energy via a novel Finsler elastica metric that embeds a curvature penalty. This metric is non-Riemannian, anisotropic and asymmetric, and is defined over an orientation-lifted space by adding to the image domain the orientation as an extra space dimension. Based on this orientation lifting, the proposed minimal path model can benefit from both the curvature and orientation of the paths. Thanks to the fast marching method, the global minimum of the curvature-penalized geodesic energy can be computed efficiently. We introduce two anisotropic image data-driven speed functions that are computed by steerable filters. Based on these orientation-dependent speed functions, we can apply the proposed Finsler elastica minimal path model to the applications of closed contour detection, perceptual grouping and tubular structure extraction. Numerical experiments on both synthetic and real images show that these applications of the proposed model indeed obtain promising results.