We propose an approximated L_1 curvature model to reconstruct implicit surfaces from unstructured point cloud data. The L_1 curvature |κ | keeps sharp features better than a typical κ ^2 based curvature models, yet it is computationally more challenging. We approximate the L_1 curvature by a smooth function, consider the reconstructed implicit surfaces as a denoised signed distance function (SDF), and propose a simple gradient descent scheme for fast and efficient computation. We demonstrate not only the effectiveness of the model but also the advantages of considering the L_1 curvature.
Multi-object segmentation algorithms are of great interest in a very large range of fields. Deep learning brought major improvements in terms of processing speed or prediction accuracy. Nevertheless, some traditional methods such as active surfaces have features that conventional deep learning methods cannot provide, especially representing the object in a continuous geometrical way and encoding prior information on the shapes to segment. Those features are of particular interest in biology to efficiently segment noisy and poorly resolved data, and then understand the interactions between segmented cells. We introduce NAGINI-3D (N-Active shapes for seGmentINg 3D biological Images), a new hybrid segmentation method dedicated to multi-object segmentation of 3D images that combines the efficiency of deep learning and the powerful representation of active surfaces. We evaluate our method on real and synthetic 3D datasets of fluorescence microscopy.
Using classical smoothers in restoration processes, such as total variation regularisation, while capturing well discontinuities, is known to induce an estimation bias in the final result, materialised by a loss of contrast. If the literature is prolific when dealing with standard modalities of images (grayscale or RGB images), it is more tenuous when the involved modality encodes some intrinsic geometrical properties, requiring the design of specific purpose-built algorithms. In this work, focused on such a specific modality, namely polarimetric imaging, we address the joint restoration and contrast re-enhancement (equivalently referred to as debiasing or refitting) question within an extension of the CLEAR framework (Covariant LEAst-square Refitting, [5]), emphasising the importance of preserving the Jacobian (with respect to the observed signal) of the original estimator.
We propose a functional for extracting curves between a list of possible endpoints, based on a discretization of a variational energy and Smirnov's decomposition theorem for vector fields. It is then used to design a bi-level minimization approach to automatically extract curves and 1D structures from an image, which is mostly unsupervised.
We present a proof-of-concept study demonstrating the application of the linear osmotic flow to unstructured domains in ℝ^3 , such as irregular meshes. This study aims to extend the osmotic filter to surfaces, thereby simplifying some geometry processing tasks, such as surface inpainting and completion (cloning), that typically require costly and complex algorithms. This will include two new challenges: the numerical solution of the drift-diffusion osmotic model on meshes, and the use of surface geometric descriptors, which play the role of reference function for the osmotic flow.
The clustering of data points in n-dimensional Euclidean space, i. e., assigning each data point to exactly one group (cluster) to detect previously unseen relations within the data set, has become a standard task for (unsupervised) machine learning. In this paper, this concept is generalized to shape data consisting of three-dimensional volumetric objects. As the underlying distance on the space of such objects, the optimal transport-based Wasserstein distance is considered and different variants of such a clustering approach are presented and compared. A variational autoencoder representing these cluster centers is incorporated to counteract an over-smoothing of objects representing the cluster centers. Numerical experiments for three distinct, volumetric data sets are presented to validate the performance of the proposed methods.
An atomic autoencoder is a neural network architecture that decomposes an image as a sum of low dimensional atoms. While it is efficient for image datasets which are well represented by this summation model, it is more limited for the representation of more generic images. In this article, we propose a new atomic model, the max-sparsity model to better represent images. We study some theoretical properties of this model and implement the corresponding atomic autoencoder. We show experimentally that it leads to a sparse decomposition of input images with interpretable low-level visual features. With this new architecture, we solve a super resolution inverse problem via a projected gradient descent that uses the trained network as a projection operator. The resulting estimation shows improved robustness compared to previous architectures.
We introduce a class of processing architectures for node features on a graph, that are equivariant with respect to local actions of a general symmetry group G, i.e. G may act on each feature φ _v at some node v by an individual transformation g_v∈ G . Our method is based on modelling node feature data in terms of sections (E) of associated vector bundles, which ensures preservation of local symmetries by construction. Processing architectures are then derived from mappings of bundle sections ℱ : (E) → (E) . We focus on mappings ℱ induced by diffusion PDEs on vector bundles associated to a generalized Laplacian. We define a bundle scale space, which contains non-trivial fixed points corresponding to harmonic sections of the associated vector bundle, in contrast to the basic Gaussian scale space. We utilize vector diffusion maps and lattice gauge theory in order to discretize the geometric PDEs such that local symmetries are still respected. We outline parametrizations suited for supervised machine learning scenarios and provide a proof-of-concept numerical experiment.
Chromatic degradation is often caused by multiple factors that are difficult to retrieve after images have been acquired, making scene recovery a highly ill-posed inverse problem. In this paper, we leverage the nature of human vision and propose an effective algorithm to achieve robust and visually appealing reconstructions. Specifically, we develop a reliable method for estimating chromatic bias by averaging colors from large, blurred regions identified through scale-space analysis. We then utilize CIELAB perceptual hue similarity to determine the degree of degradation caused by transmission and backscattering. By combining these estimators, we recover the scene by adaptively discounting the bias across the image domain. We conduct numerical experiments and comparative studies to demonstrate the effectiveness of our method.
We enumerate a framework for optimal transportation on Lie Groups using costs that depend on geodesic distances derived from spatially varying data-driven metrics. We build on the entropic regularized formulation which can be efficiently solved using Sinkhorn iterations. We estimate local distances on the Lie group using logarithmic distance approximations and formulate their extension to a more general setting of data-driven metric tensors. Our formulation leads to a data-driven approximation of the Gibbs kernel which is essential to the Sinkhorn framework. We demonstrate our method with two experiments: Tractography with Diffusion-Weighted MRI and crossing-preserving interpolations of measures in SE(2).
We investigate a variational method for ill-posed problems, which embeds the fractional power of the standard graph Laplacian operator in the regularization term. We explore the dependence of the regularizer on a preliminary approximation of the solution, which is obtained using various existing reconstruction methods from the literature. As a result, the regularization term is both dependent on and adaptive to the observed data, noise, and the choice of the fractional exponent. We present a selected numerical example problem on 2D computerized tomography, for which we consider various reconstruction techniques , including Filtered Back Projection, Total Variation, and a trained deep neural network. Incorporating the fractional power of the graph Laplacian operator into the regularization term significantly enhances the quality of the approximated solutions for each method . Additionally, we show that our proposal behaves as a regularization method and is also stable with respect to variations in the noise level.
Direct reconstruction of macromolecular structures from cryogenic electron microscopy (Cryo-EM) data has shown to be a challenge both in the homogeneous and heterogeneous setting. In this work we propose a new direct reconstruction method based on a combination of recent developments on protein geometry and orientation estimation. Even though this method is set up for the homogeneous setting, we aim to gain insight into challenges that atomistic methods have been facing for the heterogeneous case. In numerical experiments we observe that the method is able to recover the structure to almost inter-atomic resolution from as few as 100 2D Cryo-EM images due to the strong bias the regularizer gives. We conclude this work with a discussion on how the obtained results indicate possibilities and challenges for the generalization to the heterogeneous case.
Diffusion echoes are a fundamental concept for understanding the behaviour of nonlinear diffusion filters. They describe the accumulated data exchange during a diffusion process and are given by the columns or rows of the corresponding state transition matrix. Unfortunately, they involve a prohibitively large amount of data. Therefore, we propose the first compression strategy to efficiently represent and reconstruct diffusion echoes. Using a truncated singular value decomposition (SVD) we can reduce the storage requirements substantially, while still obtaining very accurate reconstructions. The SVD works on all echoes jointly and captures the redundancy between them. To approximate the singular value decomposition, we use a powerful probabilistic approach: the randomised subspace iteration. We show on a test case with a difficult, rapidly decaying diffusivity that we can reduce the storage requirements by a factor of 20, without creating visually noticeable errors. Furthermore, our compressed data representation enables an efficient reconstruction, which allows a fast and detailed echo investigation. This paves the way for various future applications of the diffusion echo that have been prevented by its huge amount of data so far.
In this paper, we investigate the computation of anisotropic metrics for geodesic distances using the heat equation and its application to image segmentation, particularly for tubular structures. Building upon the work of Bertrand et al. [2], we extend this approach to anisotropic media by incorporating spatially varying and direction-dependent diffusion tensors derived from the image's structure tensor field. Our contributions are twofold: first, we formulate anisotropic geodesic distance computation using the heat equation and integrate it within deep learning models for image segmentation; second, we propose two methods that learn the anisotropic metric directly from image data-one requiring explicit seed point selection and another eliminating the need for seed points by predicting a probability map that serves as the initial condition for heat diffusion. Experiments on synthetic and medical image datasets demonstrate the effectiveness of our methods in accurately segmenting vascular tree structures by leveraging the anisotropic properties inherent in the images without relying on manual seed point selection.
We present a novel variational model for the additive decomposition of 1D noisy signals. The model relies on sparsifying fractional-order derivatives of the sought-for components to capture intricate signal structures. To efficiently solve the resulting optimization problem, an alternating direction method of multipliers-based algorithm is developed. Furthermore, a bilevel optimization framework is proposed to automatically select "optimal" values of all the free parameters in the model, including the orders of the sparsified fractional derivatives. Preliminary results validate the effectiveness of the proposed approach in accurately decomposing noisy signals, even in the presence of abrupt changes.
We propose an image registration model for computing a dense, piecewise diffeomorphic deformation map between 3D thorax images which incorporates sliding motion as often occurring in the pleural cavity. Our approach is based on stationary velocity fields and neural implicit representations, with a particular focus on facilitating motion interpolation. This allows for generating a time-continuous model of the respiratory cycle based on end-inspiration and end-expiration images. We investigate the effect of composing deformations in motion interpolation, using a hybrid technique to enforce domain alignment. We experimentally validate our approach for registration and motion interpolation between thoracic end-inspiration and end-expiration images.
We introduce a time-fractional telegraph diffusion model to remove high level multiplicative noise while preserving critical features such as texture and edges. The model utilizes α ^th and β ^ th order time-fractional Caputo derivatives to capture sub-diffusive and super-diffusive behaviors, enhancing noise removal in various images such as synthetic aperture radar (SAR), texture, and natural images. To accurately capture the non-local effects of the Caputo derivatives, we apply first-order and second-order discretization schemes. Extensive experiments demonstrate superior performance of the proposed model over state-of-the-art methods in terms of PSNR, SI, MAE, and MSSIM.
Image registration is a demanding task that is required in many different areas of application, in particular in medical imaging. Due to the ill-posedness of image registration problems, regularization is unavoidable. This paper focuses on a family of so-called vector-field (VF) regularizers which consist of second-order energies based on a convex combination of gradients of divergence and rotation. Following a discretize-then-optimize approach, this paper proposes a staggered-grid discretization of the VF regularizers and applies a quasi-Newton type minimization to the image registration problem. Here the most costly part is solving linear systems, which can be regarded as a discretization of a linearization of a partial differential equation of fourth order. This paper proposes a highly efficient multigrid (MG) solver. In particular, the paper presents a local Fourier analysis to show that the suggested discretization is well suited for MG. More specifically, the paper provides an explicit number for the h-ellipticity measure and the local smoothing factor for a collective ω -relaxed Jacobi-type iteration. Our numerical results, including 3D image registration tasks, underline that the MG solver has in fact a complexity of 𝒪(n) , where n is the number of unknowns.
Image classification has gained significant attention in recent years. However, the quantification of uncertainty in classification outcomes has only been more intensively studied in recent times. In this paper, we propose an equivariant bootstrap method, combined with an autoencoder, to quantify the inherent uncertainty of classification predictions. Specifically, the exact method involves applying group actions, such as rotations or translations, to the image. This process includes two applications of encoding, a single decoding step, and the application of the inverse of the group action, which generates a single bootstrap sample. Additionally, we introduce an inexact method that requires only a single pass through the encoder, trading off computational efficiency for slightly lower accuracy. We validate our methods through several numerical experiments, evaluating their performance using various metrics and benchmarking against other recent uncertainty quantification approaches. In particular, we focus on ablation studies that examine the impact of different group actions on the results.
Image segmentation and registration are pivotal preliminary steps in image analysis exhibiting a dual nature, especially in a multimodal context where salient features are to be matched. As such, intertwining them in a unified framework reduces uncertainty propagation and yields positive mutual influence. Registration compensates for weak boundary definition and encodes intrinsic topological requirements like preserving contextual relations between objects. In return, accurate segmented structures foster relevant registration, breeding reliable estimations of the deformation pairing the encoded structures. These observations underpin the proposed contribution blending variational techniques (versatility/adaptability/interpretability) and deep-learning-based approaches (more proficient at handling computationally intensive tasks) through coordinate Multi-Layer Perceptron (MLP) with periodic sinusoidal activation functions. More precisely, in a hybrid nonlinear-elasticity-grounded framework, an unsupervised pairwise joint segmentation/registration 2D model is introduced. Working in the class of mappings with finite distortion enables one to guarantee that the engendered deformation is a homeomorphism. Also, combining a directional-total-variation-based term promoting gradient alignment with the segmentation task, viewed as a substitute for classical intensity-based data terms in registration, enlarges the scope of applications to multimodal images. The existence of a minimiser for the optimisation problem constitutes the core of the paper. Eventually, we test and evaluate our model on both synthetic and medical images to exhibit the accuracy and relevance of our model.
