Abstract. This article introduces a novel approach to learning monotone neural networks (NNs) through a newly defined penalization loss. The proposed method is particularly effective in solving classes of variational problems, specifically monotone inclusion problems, commonly encountered in image processing tasks. The forward-backward-forward (FBF) algorithm is employed to address these problems, offering a solution even when the Lipschitz constant of the NN is unknown. Notably, the FBF algorithm provides convergence guarantees under the condition that the learned operator is monotone. Building on plug-and-play methodologies, our objective is to apply these newly learned operators to solving nonlinear inverse problems. To achieve this, we initially formulate the problem as a variational inclusion problem. Subsequently, we train a monotone NN to approximate an operator that may not inherently be monotone. Leveraging the FBF algorithm, we then show simulation examples where the nonlinear inverse problem is successfully solved.

Infertility affects around 15% of couples worldwide. Male fertility problems include poor sperm quality and low sperm count. The advanced fertility treatment methods like ICSI are nowadays supported by vision systems to assist embryologists in selecting good quality sperm. Computer-Assisted Semen Analysis (CASA) provides quantitative and qualitative sperm analysis concerning concentration, motility, morphology, vitality, and fragmentation. However, fertility assessment algorithms often neglect individual spermatozoon tail and its beating patterns because recognizing the tails in blurry microscopic images reliably is challenging. In this article, we propose that models trained with head and tail part classes can better localize parts and segment the whole spermatozoon objects. Usually, the training of segmentation sperm models is supported by image-level augmentation. We argue that models guided by class-specific data augmentation attend to less discriminative sperm parts. To demonstrate this, we decouple the augmentation into object-level and background augmentation for the sperm part segmentation problem. Our proposed method outperforms state-of-the-art methods on the SegSperm dataset. Moreover, our ablation studies confirm the effectiveness of the proposed part-based object representation and augmentation.

For best medical imaging application results, learning-based approaches such as deep learning necessitate specific, extensive and precise annotations. Outside well-curated public benchmarks, these are rarely available in practice, and so it becomes necessary to use less-than-perfect annotations. One way of compensating for this is the embedding of anatomical knowledge. Complementing this, there is the incremental semi-supervised learning technique, whereby a small amount of annotations can be used to derive more and superior labels.In this article, we illustrate this approach on a deep learning system to help radiologists and rheumatologists finely and interactively assess MRI scans of the sacro-iliac joint in order to correctly diagnose Axial Spondyloarthritis. Our model is trained initially on a relatively small set of images with promising results, on par with expert opinion and generalizable to new datasets.

In the last ten years, Convolutional Neural Networks (CNNs) have formed the basis of deep-learning architectures for most computer vision tasks. However, they are not necessarily optimal. For example, mathematical morphology is known to be better suited to deal with binary images. In this work, we create a morphological neural network that handles binary inputs and outputs. We propose their construction inspired by CNNs to formulate layers adapted to such images by replacing convolutions with erosions and dilations. We give explainable theoretical results on whether or not the resulting learned networks are indeed morphological operators. We present promising experimental results designed to learn basic binary operators, and we have made our code publicly available online.

Coronary computed tomography angiography (CCTA) provides a non-invasive imaging solution that reliably depicts the anatomy of coronary arteries. Diagnosing coronary artery diseases (CAD) entails a clinical evaluation of stenosis and plaques, which is in turn essential for obtaining a reliable coronary-artery centerline from CCTA 3D imaging. This work proposes a centerline extraction algorithm by combining local semantic segmentation and recursive tracking. To this end we propose a Morphological Skeleton Loss (MS_Loss) suited for 3D centerline segmentation based on an improved morphological skeleton algorithm coupled with a resource-efficient back-propagation scheme. This work employs 225 CCTA examinations paired with manually annotated coronary-artery centerlines. This method is compared against the deep-learning state of the art in the literature using a standardized evaluation method for coronary-artery tracking.

Both theoretical and practical problems in deep learning classification require solutions for assessing uncertainty prediction but current state-of-the-art methods in this area are computationally expensive. In this paper, we propose a new confidence measure dubbed Over-MAP that utilizes a measure of overlap between structural attention mechanisms and segmentation methods, that is of particular interest in accurate fine-grained contexts. We show that this classification confidence increases with the degree of overlap. The associated confidence and identification tools are conceptually simple, efficient, and of high practical interest as they allow for weeding out misleading examples in training data. Our measure is currently deployed in the real-world on widely used platforms to annotate large-scale data efficiently.

Both theoretical and practical problems in deep learning classification benefit from assessing uncertainty prediction. In addition, current state-of-the-art methods in this area are computationally expensive: for example,~\cite{loquercio2020general} is a general method for uncertainty estimation in deep learning that relies on Monte-Carlo sampling. We propose a new, efficient confidence measure later dubbed Over-MAP that utilizes a measure of overlap between structural attention mechanisms and segmentation methods. It does not rely on sampling or retraining. We show that the classification confidence increases with the degree of overlap. The associated confidence and identification tools are conceptually simple, efficient and of high practical interest as they allow for weeding out misleading examples in training data. Our measure is currently deployed in the real-world on widely used platforms to annotate large-scale data efficiently.

Component-graphs provide powerful and complex structures for multi-band image processing. We propose a multiband astronomical source detection framework with the component-graphs relying on a new set of component attributes. We propose two modules to differentiate nodes belong to distinct objects and to detect partial object nodes. Experiments demonstrate an improved capacity at detecting faint objects on a multi-band astronomical dataset.

Recently, many deep learning methods have been used to handle single image super-resolution (SISR) tasks and often achieve state-of-the-art performance. From a visual point of view, the results look convincing. Yet, does it mean that those techniques are reliable and robust enough to be implemented in real business cases to enhance the performance of other computer vision tasks? In this article, we investigate the use of SISR to construct higher-resolution images of real receipt photos sent by a company's customers and evaluate its impact on the performance of an OCR task (receipt information retrieval). Using built-in task-based performance evaluation methods, we show that the use of SISR can significantly improve OCR performance in the case where recognition was poor in low-resolution, but can also deteriorate the performance for receipts that were already successfully recognized. As a conclusion, we provide recommendations on how to best use SISR in a production environment.

Curvilinear structure restoration in image processing procedures is a difficult task, which can be compounded when these structures are thin, i.e., when their smallest dimension is close to the resolution of the sensor. Many recent restoration methods involve considering a local gradient-based regularization term as prior, assuming gradient sparsity. An isotropic gradient operator is typically not suitable for thin curvilinear structures, since gradients are not sparse for these. In this paper, we propose a mixed gradient operator that combines a standard gradient in the isotropic image regions, and a directional gradient in the regions where specific orientations are likely. In particular, such information can be provided by curvilinear structure detectors (e.g., RORPO or Frangi filters). Our proposed mixed gradient operator, that can be viewed as a companion tool of such detectors, is proposed in a discrete framework and its formulation/computation holds in any dimension; in other words, it is valid in [Formula: see text], n ≥ 1 . We show how this mixed gradient can be used to construct image priors that take edge orientation, as well as intensity, into account, and then involved in various image processing tasks while preserving curvilinear structures. The experiments carried out on 2D, 3D, real, and synthetic images illustrate the relevance of the proposed gradient, and its use in variational frameworks for both denoising and segmentation tasks.

In this paper, we present a new distributed algorithm for minimizing a sum of non-necessarily differentiable convex functions composed with arbitrary linear operators. The overall cost function is assumed strongly convex. Each involved function is associated with a node of a hypergraph having the ability to communicate with neighboring nodes sharing the same hyperedge. Our algorithm relies on a primal-dual splitting strategy with established convergence guarantees. We show how it can be efficiently implemented to take full advantage of a multicore architecture. The good numerical performance of the proposed approach is illustrated in a problem of video sequence denoising, where a significant speedup is achieved.

The analysis of thin curvilinear objects in 3D images is a complex and challenging task. In this article, we introduce a new, non-linear operator, called RORPO (Ranking the Orientation Responses of Path Operators). Inspired by the multidirectional paradigm currently used in linear filtering for thin structure analysis, RORPO is built upon the notion of path operator from mathematical morphology. This operator, unlike most operators commonly used for 3D curvilinear structure analysis, is discrete, non-linear and non-local. From this new operator, two main curvilinear structure characteristics can be estimated: an intensity feature, that can be assimilated to a quantitative measure of curvilinearity; and a directional feature, providing a quantitative measure of the structure's orientation. We provide a full description of the structural and algorithmic details for computing these two features from RORPO, and we discuss computational issues. We experimentally assess RORPO by comparison with three of the most popular curvilinear structure analysis filters, namely Frangi Vesselness, Optimally Oriented Flux, and Hybrid Diffusion with Continuous Switch. In particular, we show that our method provides up to 8 percent more true positive and 50 percent less false positives than the next best method, on synthetic and real 3D images.

Flicker removal consists of filtering out rapid, artefactual changes of luminosity and colorimetry from image sequences in order to improve colorimetry consistency between video frames. It is a necessary and fundamental task in multiple applications, for instance in archived film sequences, image/video compression and time-lapse videos. In recent years, the wider availability of fast video acquisition technology has renewed the interest in the flicker removal problem, in particular periodic flicker. In this context, rapid, undesirable intensity and chroma variations are due to acquisition hardware performing faster than the frequency of alternating current powering artificial light sources. This paper proposes both theoretical and efficient experimental solutions for flicker removal from image sequences by performing simultaneous motion tracking and color correction using a superpixel image representation.

Tubular structure segmentation is an important task, with many applications in medical image analysis such as vessel segmentation both in 2D and 3D. However, this task is challenging due to the spatial sparsity of these objects, implying a high sensitivity to noise. An important cue in this context is the local orientation of the tubular structures. Using this information, it is possible to regularize the structures without destroying its integrity. In this article, we take advantage of recent advances in orientation estimation to propose a directional regularization prior for tubular structures, suitable for use in a variational framework. We illustrate on both synthetic and 2D real data.

As image processing and analysis techniques improve, an increasing number of procedures in bio-medical analyses can be automated. This brings many benefits, e.g improved speed and accuracy, leading to more reliable diagnoses and follow-up, ultimately improving patients outcome. Many automated procedures in bio-medical imaging are well established and typically consist of detecting and counting various types of cells (e.g. blood cells, abnormal cells in Pap smears, and so on). In this article we propose to automate a different and difficult set of measurements, which is conducted on the cilia of people suffering from a variety of respiratory tract diseases. Cilia are slender, microscopic, hair-like structures or organelles that extend from the surface of nearly all mammalian cells. Motile cilia, such as those found in the lungs and respiratory tract, present a periodic beating motion that keep the airways clear of mucus and dirt. In this paper, we propose a fully automated method that computes various measurements regarding the motion of cilia, taken with high-speed video-microscopy. The advantage of our approach is its capacity to automatically compute robust, adaptive and regionalized measurements, i.e. associated with different regions in the image. We validate the robustness of our approach, and illustrate its performance in comparison to the state-of-the-art.

Essential image processing and analysis tasks, such as image segmentation, simplification and denoising, can be conducted in a unified way by minimizing the Mumford-Shah (MS) functional. Although seductive, this minimization is in practice difficult because it requires to jointly define a sharp set of contours and a smooth version of the initial image. For this reason, various relaxations of the original formulations have been proposed, together with optimisation methods. Among these, the Ambrosio-Tortorelli (AT) parametric functional is of particular interest, because minimizers of AT can be shown to converge to a minimizer of MS. However this convergence is difficult to achieve numerically using standard finite difference schemes. Indeed, with AT, discontinuities need to be represented explicitly rather than implicitly. In this work, we propose to formulate AT using the full framework of Discrete Calculus (DC), which is able to sharply represent discontinuities thanks to a more sophisticated topological framework. We present our proposed formulation, its resolution, and results on synthetic and real images. We show that we are indeed able to represent sharp discontinuities and as a result significantly better stability to noise, compared with finite difference schemes.

Angiographic imaging is a crucial domain of medical imaging. In particular, Magnetic Resonance Angiography (MRA) is used for both clinical and research purposes. This article presents the first framework geared toward the design of virtual MRA images from real MRA images. It relies on a pipeline that involves image processing, vascular modeling, computational fluid dynamics and MR image simulation, with several purposes. It aims to provide to the whole scientific community (1) software tools for MRA analysis and blood flow simulation; and (2) data (computational meshes, virtual MRAs with associated ground truth), in an open-source/open-data paradigm. Beyond these purposes, it constitutes a versatile tool for progressing in the understanding of vascular networks, especially in the brain, and the associated imaging technologies. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

When acquiring high-speed video (more than 100 frames per second), artificial lighting can cause severe non-uniform luminosity and chroma variation between frames, commonly labeled periodic flicker. Non-uniform periodic flicker is not easy to correct in the presence of general motion, since its estimation requires background and object tracking, and most tracking techniques assume consistent illumination. In this paper, we propose a joint tracking/color correction scheme using a block matching technique paired with color variation estimation. We introduce a robust method for stabilizing brightness variations in image sequences. A post-processing step is also proposed in order to deal with blocking artifacts. We demonstrate the efficacy of our method both on simulated and real data.

Connected operators provide well-established solutions for digital image processing, typically in conjunction with hierarchical schemes. In graph-based frameworks, such operators basically rely on symmetric adjacency relations between pixels. In this article, we introduce a notion of directed connected operators for hierarchical image processing, by also considering non-symmetric adjacency relations. The induced image representation models are no longer partition hierarchies (i.e., trees), but directed acyclic graphs that generalize standard morphological tree structures such as component trees, binary partition trees or hierarchical watersheds. We describe how to efficiently build and handle these richer data structures, and we illustrate the versatility of the proposed framework in image filtering and image segmentation.

The Poisson--Gaussian model can accurately describe the noise present in a number of imaging systems. However most existing restoration methods rely on approximations of the Poisson--Gaussian noise statistics. We propose a convex optimization strategy for the reconstruction of images degraded by a linear operator and corrupted with a mixed Poisson--Gaussian noise. The originality of our approach consists of considering the exact, mixed continuous-discrete model corresponding to the data statistics. After establishing the Lipschitz differentiability and convexity of the Poisson--Gaussian neg-log-likelihood, we derive a primal-dual iterative scheme for minimizing the associated penalized criterion. The proposed method is applicable to a large choice of convex penalty terms. The robustness of our scheme allows us to handle computational difficulties due to infinite sums arising from the computation of the gradient of the criterion. We propose finite bounds for these sums, that are dependent on the current image estimate, and thus adapted to each iteration of our algorithm. The proposed approach is validated on image restoration examples. Then, the exact data fidelity term is used as a reference for studying some of its various approximations. We show that in a variational framework the shifted Poisson and exponential approximations lead to very good restoration results.

Path operators are powerful tools for the enhancement of thin and elongated objects in an image. In order to cope with noisy acquisition a variant of the path operators was recently proposed. However, both approaches cannot properly handle thin objects with tortuous shapes since strong variations of an object curvature produce disconnections in the paths. In order to address this issue, we propose a novel operator able to properly handle paths in tortuous shapes. It relies on the coupling of attribute filters based on the geodesic tortuosity and conventional path operators. Analogously to the complete version of the path operators, by allowing disconnections within paths it is possible also to define a path operator that is both robust and selective. The effectiveness of the proposed operators in filtering thin and tortuous image objects is proved on a 2D and 3D biomedical image.

This introductory paper aims at summarizing some problems and state-of-the-art techniques encountered in image processing for material analysis and design. Developing generic methods for this purpose is a complex task given the variability of the different image acquisition modalities (optical, scanning or transmission electron microscopy; surface analysis instrumentation, electron tomography, micro-tomography ...), and material composition (porous, fibrous, granular, hard materials, membranes, surfaces and interfaces ...). This paper presents an overview of techniques that have been and are currently developed to address this diversity of problems, such as segmentation, texture analysis, multiscale and directional features extraction, stochastic models and rendering, among others. Finally, it provides references to enter the issues, challenges and opportunities in materials characterization.

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This paper deals with noise parameter estimation from a single image under Poisson-Gaussian noise statistics. The problem is formulated within a mixed discrete-continuous optimization framework. The proposed approach jointly estimates the signal of interest and the noise parameters. This is achieved by introducing an adjustable regularization term inside an optimized criterion, together with a data fidelity error measure. The optimal solution is sought iteratively by alternating the minimization of a label field and of a noise parameter vector. Noise parameters are updated at each iteration using an Expectation-Maximization approach. The proposed algorithm is inspired from a spatial regularization approach for vector quantization. We illustrate the usefulness of our approach on macroconfocal images. The identified noise parameters are applied to a denoising algorithm, so yielding a complete denoising scheme.

Interior tomography, also called local or region-of-interest tomography is a special case of computed tomography, in which the object under study is larger than the detector. In this modality, reconstructing the tomography image is an even more ill-posed problem than standard tomography and characteristic artefacts are typically observed, even when a large number of measurements are taken. In this work we propose a reconstruction algorithm designed for interior tomography. We also investigate the case of under-sampled measurements in the case of gradient-sparse images. Our algorithm optimizes the sum of a spatial regularization terms for the image inside the region of interest, and a sinogram regularization term for the projection of the non-reconstructed part of the sample, using convex optimization techniques. We present results on simulated and real data.

The automatic detection of hedgerows in very high resolution remote sensing images is addressed in this paper. In particular, the use of advanced morphological filters, such as path operators, is proposed. Conventional path openings have been already proposed in the literature to discriminate between forest objects and hedge objects in very high resolution optical images. They have shown greater flexibility with respect to geodesic openings. However, path operators are sensitive to noise and in practical situations they are likely to produce missed detections. In particular, path operators are unable to extract long hedgerows as a single object. In order to tackle this limitation, robust path openings are investigated in this work. In the experimental results, robust path opening shows superior performances for the detection of hedgerows.

We provide conditions under which 2D digital images preserve their topological properties under rigid transformations. We consider the two most common digital topology models, namely dual adjacency and well-composedness. This paper leads to the proposal of optimal preprocessing strategies that ensure the topological invariance of images under arbitrary rigid transformations. These results and methods are proved to be valid for various kinds of images (binary, gray-level, label), thus providing generic and efficient tools, which can be used in particular in the context of image registration and warping.

Tomography is an image reconstruction task that may be viewed as a linear inverse problem akin to deconvolution. Recent progresses in optimization methods have made it possible to formulate this task so that fewer projections and higher amounts of noise can be dealt with, making use of a-priori information and domain constraints. In this article, we investigate 3MG, a new optimization method that is highly flexible and effective. In particular, we propose and compare convex and non-convex regularization potentials on both synthetic and real images. We further investigate the possibility to deal with continuous angular integration, i.e. where projections rays are no longer straight lines, but cones. This is encountered in a variety of real-life situations, but is difficult or impossible to deal with exactly using traditional reconstruction algorithms. We show that in this situation it may be beneficial to acquire fewer projections than would be required using classical methods.

We study the conditions under which the topological properties of a 2D well-composed binary image are preserved under arbitrary rigid transformations. This work initiates a more global study of digital image topological properties under such transformations, which is a crucial but under-considered problem in the context of image processing, e.g., for image registration and warping.

Thin structure filtering is an important preprocessing task for the analysis of 2D and 3D bio-medical images in various contexts. We propose a filtering framework that relies on three approaches that are distinct and infrequently used together: linear, non-linear and non-local. This strategy, based on recent progress both in algorithmic/computational and methodological points of view, provides results that benefit from the advantages of each approach, while reducing their respective weaknesses. Its relevance is demonstrated by validations on 2D and 3D images.

In this work, we consider a class of differentiable criteria for sparse image computing problems, where a nonconvex regularization is applied to an arbitrary linear transform of the target image. As special cases, it includes edge-preserving measures or frame-analysis potentials commonly used in image processing. As shown by our asymptotic results, the $\ell_2-\ell_0$ penalties we consider may be employed to provide approximate solutions to $\ell_0$-penalized optimization problems. One of the advantages of the proposed approach is that it allows us to derive an efficient majorize-minimize subspace algorithm. The convergence of the algorithm is investigated by using recent results in nonconvex optimization. The fast convergence properties of the proposed optimization method are illustrated through image processing examples. In particular, its effectiveness is demonstrated on several data recovery problems.

Algorithms based on total variation (TV) minimization are prevalent in image processing. They play a key role in a variety of applications such as image denoising, compressive sensing, and inverse problems in general. In this work, we extend the TV dual framework that includes Chambolle's and Gilboa and Osher's projection algorithms for TV minimization. We use a flexible graph data representation that allows us to generalize the constraint on the projection variable. We show how this new formulation of the TV problem may be solved by means of fast parallel proximal algorithms. In denoising and deblurring examples, the proposed approach is shown not only to perform better than recent TV-based approaches, but also to perform well on arbitrary graphs instead of regular grids. The proposed method consequently applies to a variety of other inverse problems including image fusion and mesh filtering.

A Poisson-Gaussian model accurately describes the noise present in many imaging systems such as CCD cameras or fluorescence microscopy. However most existing restoration strategies rely on approximations of the Poisson-Gaussian noise statistics. We propose a convex optimization algorithm for the reconstruction of signals degraded by a linear operator and corrupted with mixed Poisson-Gaussian noise. The originality of our approach consists of considering the exact continuous-discrete model corresponding to the data statistics. After establishing the Lipschitz differentiability of the Poisson-Gaussian log-likelihood, we derive a primal-dual iterative scheme for minimizing the associated penalized criterion. The proposed method is applicable to a large choice of penalty terms. The robustness of our scheme allows us to handle computational difficulties due to infinite sums arising from the computation of the gradient of the criterion. The proposed approach is validated on image restoration examples.

In this work, we present a hybrid algorithm to automatically delineate the heart volume in 3D cardiac computed tomography (CT) datasets for the visualization of coronary arteries. Our work eliminates the tedious and time consuming step of manually removing obscuring structures around the heart (ribs, sternum, liver...). It quickly provides a clear and well defined view of the coronaries. So far, works related to heart segmentation have mainly focused on heart cavities delineation, which is not suited for coronaries visualization. In contrast, our algorithm extracts the heart cavities, the myocardium and coronaries as a single object. The proposed approach is based on the fitting of a geometric model of the heart to a set of automatically extracted 3D points lying on the heart shell. A novel two-stage fitting scheme is used to improve the robustness to the outliers. The fitting result is further refined using a Random Walker (RW) segmentation approach. Qualitative analysis of results obtained on a 70 exam database shows the efficiency and the accuracy of our approach.

Algorithms based on the minimization of the Total Variation are prevalent in computer vision. They are used in a variety of applications such as image denoising, compressive sensing and inverse problems in general. In this work, we extend the TV dual framework that includes Chambolle's and Gilboa Osher's projection algorithms for TV minimization in a flexible graph data representation by generalizing the constraint on the projection variable. We show how this new formulation of the TV problem may be solved by means of a fast parallel proximal algorithm, which performs better than the classical TV approach for denoising, and is also applicable to inverse problems such as image deblurring.

In this paper, we consider a class of differentiable criteria for sparse image recovery problems. The regularization is applied to a linear transform of the target image. As special cases, it includes edge preserving measures or frame analysis potentials. As shown by our asymptotic results, the considered ℓ2- ℓ0penalties may be employed to approximate solutions to ℓ0penalized optimization problems. One of the advantages of the approach is that it allows us to derive an efficient Majorize-Minimize Memory Gradient algorithm. The fast convergence properties of the proposed optimization algorithm are illustrated through image restoration examples.

In this work, we extend a common framework for graph-based image segmentation that includes the graph cuts, random walker, and shortest path optimization algorithms. Viewing an image as a weighted graph, these algorithms can be expressed by means of a common energy function with differing choices of a parameter q acting as an exponent on the differences between neighboring nodes. Introducing a new parameter p that fixes a power for the edge weights allows us to also include the optimal spanning forest algorithm for watershed in this same framework. We then propose a new family of segmentation algorithms that fixes p to produce an optimal spanning forest but varies the power q beyond the usual watershed algorithm, which we term the power watershed. In particular, when q=2, the power watershed leads to a multilabel, scale and contrast invariant, unique global optimum obtained in practice in quasi-linear time. Placing the watershed algorithm in this energy minimization framework also opens new possibilities for using unary terms in traditional watershed segmentation and using watershed to optimize more general models of use in applications beyond image segmentation.

Maximum flow (and minimum cut) algorithms have had a strong impact on computer vision. In particular, graph cut algorithms provide a mechanism for the discrete optimization of an energy functional which has been used in a variety of applications such as image segmentation, stereo, image stitching, and texture synthesis. Algorithms based on the classical formulation of max-flow defined on a graph are known to exhibit metrication artifacts in the solution. Therefore, a recent trend has been to instead employ a spatially continuous maximum flow (or the dual min-cut problem) in these same applications to produce solutions with no metrication errors. However, known fast continuous max-flow algorithms have no stopping criteria or have not been proved to converge. In this work, we revisit the continuous max-flow problem and show that the analogous discrete formulation is different from the classical max-flow problem. We then apply an appropriate combinatorial optimization technique to this combinatorial continuous max-flow (CCMF) problem to find a null-divergence solution that exhibits no metrication artifacts and may be solved exactly by a fast, efficient algorithm with provable convergence. Finally, by exhibiting the dual problem of our CCMF formulation, we clarify the fact, already proved by Nozawa in the continuous setting, that the max-flow and the total variation problems are not always equivalent.

We present a simple, fast, and effective method to detect defects on textured surfaces. Our method is unsupervised and contains no learning stage or information on the texture being inspected. The new method is based on the Phase Only Transform (PHOT) which correspond to the Discrete Fourier Transform (DFT), normalized by the magnitude. The PHOT removes any regularities, at arbitrary scales, from the image while preserving only irregular patterns considered to represent defects. The localization is obtained by the inverse transform followed by adaptive thresholding using a simple standard statistical method. The main computational requirement is thus to apply the DFT on the input image. The new method is also easy to implement in a few lines of code. Despite its simplicity, the methods is shown to be effective and generic as tested on various inputs, requiring only one parameter for sensitivity. We provide theoretical justification based on a simple model and show results on various kinds of patterns. We also discuss some limitations.

Many computer vision applications such as image filtering, segmentation and stereo-vision can be formulated as optimization problems. Whereas in previous decades continuous-domain, iterative procedures were common, recently discrete, convex, globally optimal methods have received a lot of attention. However not all problems in computer vision are convex, for instance L0norm optimization such as seen in compressive sensing. Recently, a novel discrete framework encompassing many known segmentation methods was proposed: power watershed. We are interested to explore the possibilities of this minimizer to solve other problems than segmentation, in particular with respect to unusual norms optimization. In this article we reformulate the problem of anisotropic diffusion as an L0optimization problem, and we show that power watersheds are able to optimize this energy quickly and effectively. This study paves the way for using the power watershed as a useful general-purpose minimizer in many different computer vision contexts.

Quantization, defined as the act of attributing a finite number of grey-levels to an image, is an essential task in image acquisition and coding. It is also intricately linked to various image analysis tasks, such as denoising and segmentation. In this paper, we investigate quantization combined with regularity constraints, a little-studied area which is of interest, in particular, when quantizing in the presence of noise or other acquisition artifacts. We present an optimization approach to the problem involving a novel two-step, iterative, flexible, joint quantizing-regularization method featuring both convex and combinatorial optimization techniques. We show that when using a small number of grey-levels, our approach can yield better quality images in terms of SNR, with lower entropy, than conventional optimal quantization methods.

Given a set of discrete points in a 2D digital image containing noise, we formulate our problem as robust digital line fitting. More precisely, we seek the maximum subset whose points are included in a digital line, called the optimal consensus. The paper presents an efficient method for exactly computing the optimal consensus by using the topological sweep, which provides us with the quadratic time complexity and the linear space complexity with respect to the number of input points.

In this work, we extend a common framework for seeded image segmentation that includes the graph cuts, random walker, and shortest path optimization algorithms. Viewing an image as a weighted graph, these algorithms can be expressed by means of a common energy function with differing choices of a parameter q acting as an exponent on the differences between neighboring nodes. Introducing a new parameter p that fixes a power for the edge weights allows us to also include the optimal spanning forest algorithm for watersheds in this same framework. We then propose a new family of segmentation algorithms that fixes p to produce an optimal spanning forest but varies the power q beyond the usual watershed algorithm, which we term power watersheds. Placing the watershed algorithm in this energy minimization framework also opens new possibilities for using unary terms in traditional watershed segmentation and using watersheds to optimize more general models of use in application beyond image segmentation.

In general, image sensor noise is dominated by Poisson statistics, even at high illumination level, yet most standard denoising procedures often assume a simpler additive Gaussian noise, which is in fact a poor approximation. Fortunately, Poisson noise can under some circumstances be simplified via variance stabilizing methods, such as the Anscombe transform, which is well known to statisticians, medical imaging specialists and astronomers. However, in order to use such a procedure effectively, the actual photon count needs to be known and not simply an illumination intensity, which is the main reason why such procedures are not frequently used in the image processing community. In this article, we propose to use Poisson distribution characteristics to estimate the photon count from relative illumination data, under simple hypotheses. This allows us to use variance-stabilizing methods on standard digital photographs. Thanks to this, the noise becomes close to additive Gaussian and standard filtering methods become significantly more effective. As an example we exhibit the level of improvement that can be achieved using the bilateral filter.

Segmentation and analysis of blood vessels is an important issue in medical imaging. In 3D cerebral angiographic data, the vascular signal is however hard to accurately detect and can, in particular, be disconnected. In this article, we present a procedure utilising both linear, Hessian-based and morphological methods for blood vessel edge enhancement and reconnection. More specifically, multi-scale second-order derivative analysis is performed to detect candidate vessels as well as their orientation. This information is then fed to a spatially-variant morphological filter for reconnection and reconstruction. The result is a fast and effective vessel-reconnecting method.

This paper presents a new method for extracting roads in Very High Resolution remotely sensed images based on advanced directional morphological operators. The proposed approach introduces the use of Path Openings and Closings in order to extract structural pixel information. These morphological operators remain flexible enough to fit rectilinear and slightly curved structures since they do not depend on the choice of a structural element shape and hence outperform standard approaches using rotating rectangular structuring elements. The method consists in building a granulometry chain using Path Openings and Closing to perform Morphological Profiles. For each pixel, the Morphological Profile constitutes the feature vector on which our road extraction is based.

In this paper, we address the computation of globally minimal curves and surfaces for image segmentation and stereo reconstruction. We present a solution, simulating a continuous maximal flow by a novel system of partial differential equations. Existing methods are either grid-biased (graph-based methods) or suboptimal (active contours and surfaces). The solution simulates the flow of an ideal fluid with isotropic velocity constraints. Velocity constraints are defined by a metric derived from image data. An auxiliary potential function is introduced to create a system of partial differential equations. It is proven that the algorithm produces a globally maximal continuous flow at convergence, and that the globally minimal surface may be obtained trivially from the auxiliary potential. The bias of minimal surface methods toward small objects is also addressed. An efficient implementation is given for the flow simulation. The globally minimal surface algorithm is applied to segmentation in 2D and 3D as well as to stereo matching. Results in 2D agree with an existing minimal contour algorithm for planar images. Results in 3D segmentation and stereo matching demonstrate that the new algorithm is robust and free from grid bias.

A well-known problem in image analysis is the extraction of thin and elongated features such as edges, fractures, fibres, vessels, etc. In many cases, such features are not linear but curved rather than straight. A well-known morphological tool for the extraction of linear features is the opening by straight line segments of a certain length. In this paper we extend this class of morphological filters with openings with narrow structuring elements which are not necessarily straight line segments, but which can be connected paths defined by an adjacency relation.

We show that a translation invariant implementation of min/max filters along a line segment of slope in the form of an irreducible fraction dy/dx can be achieved at the cost of 2+k min/max comparisons per image pixel, where k=max(|dx|,|dy|). Therefore, for a given slope, the computation time is constant and independent of the length of the line segment. We then present the notion of periodic moving histogram algorithm. This allows for a similar performance to be achieved in the more general case of rank filters and rank-based morphological filters. Applications to the filtering of thin nets and computation of both granulometries and orientation fields are detailed. Finally, two extensions are developed. The first deals with the decomposition of discrete disks and arbitrarily oriented discrete rectangles, while the second concerns min/max filters along gray tone periodic line segments.

New morphological tools for investigating local and global image structure orientations in grey-scale images are proposed. These tools are based on recent advances in the computation of erosions and dilations with line segments in arbitrary directions.

This paper presents efficient algorithms to perform standard morphological operations on gray-level images such as erosions and dilations with any structuring elements . In a first section the general case is studied. A solution taking advantage of overlapping areas of the structuring elements and involving either a hiearchical approach or a simple sort of the pixels of the image is described, along with a comparison of this algorithm with existing methods. The proposed method shows significant improvement over these methods. Some mean of accelerating the algorithm further are also indicated. In a second section, the particular case of the line segment as structuring element is studied and a new method for dealing with these structuring elements, oriented in any direction, is proposed, which features a constant, optimal computing time with respect to the length of these structuring elements. Extensions to other types of structuring elements using the described technique are also proposed.