In recent years, several algorithms have been developed for the segmentation of the Inferior Alveolar Canal (IAC) in Cone-Beam Computed Tomography (CBCT) scans. However, the availability of public datasets in this domain is limited, resulting in a lack of comparative evaluation studies on a common benchmark. To address this scientific gap and encourage deep learning research in the field, the ToothFairy challenge was organized within the MICCAI 2023 conference. In this context, a public dataset was released to also serve as a benchmark for future research. The dataset comprises 443 CBCT scans, with voxel-level annotations of the IAC available for 153 of them, making it the largest publicly available dataset of its kind. The participants of the challenge were tasked with developing an algorithm to accurately identify the IAC using the 2D and 3D-annotated scans. This paper presents the details of the challenge and the contributions made by the most promising methods proposed by the participants. It represents the first comprehensive comparative evaluation of IAC segmentation methods on a common benchmark dataset, providing insights into the current state-of-the-art algorithms and outlining future research directions. Furthermore, to ensure reproducibility and promote future developments, an open-source repository that collects the implementations of the best submissions was released.

In sport scenarios like football or basketball, we often deal with central views where only the central circle and some additional primitives like the central line and the central point or a touch line are visible. In this paper we first characterize, from a mathematical point of view, the set of homographies that project a given ellipse into the unit circle, next, using some extra minimal additional information like the knowledge of the position in the image of the central line and central point or a touch line we show a method to fully determine the plane homography. We present some experiments in sport scenarios to show the ability of the proposed method to properly recover the plane homography.

In this paper we study multiscale analyses for images defined on Riemannian manifolds and extend the axiomatic approach proposed by Álvarez, Guichard, Lions, and Morel to this general case. This covers the case of two- and three-dimensional images and video sequences. After obtaining the general classification, we consider the case of morphological scale spaces, which are given in terms of geometric equations, and the linear case given by the Laplace--Beltrami flow. We consider in some detail the case of image metrics given in terms of the structure tensor and compute some cases of such a tensor for video. Then we comment on the connections with variational formulations of image diffusion comparing the anisotropies that appear. Finally, we include numerical experiments illustrating some of the models. Namely, we compare some examples for still images using the Laplace--Beltrami flow and some variational models. We also consider several examples in video: the mean curvature motion and the extension of the morphological and Galilean invariant scale spaces to the video manifold case, and the Laplace--Beltrami flow. We point out that the number of models that appear is huge, and we have restricted ourselves to such cases for the sake of brevity and illustration.

We present a novel segmentation algorithm based on a hierarchical representation of images. The main contribution of this work is to explore the capabilities of the a contrario reasoning when applied to the segmentation problem and to overcome the limitations of current algorithms within that framework. This exploratory approach has three main goals. Our first goal is to extend the search space of greedy merging algorithms to the set of all partitions spanned by a certain hierarchy and to cast the segmentation as a selection problem within this space. In this way we increase the number of tested partitions, and thus we potentially improve the segmentation results. In addition, this space is considerably smaller than the space of all possible partitions, and thus we still keep the complexity controlled. Our second goal aims to improve the locality of region merging algorithms, which usually merge pairs of neighboring regions. In this work, we overcome this limitation by introducing a validation procedure for complete partitions rather than for pairs of regions. The third goal is to perform an exhaustive experimental evaluation methodology in order to provide reproducible results. Finally, we embed the selection process on a statistical a contrario framework which allows us to have only one free parameter related to the desired scale.

This paper is concerned with the relaxation of nonconvex functionals used in image processing. We review most of the recently introduced relaxation methods, and we propose a new convex one based on a probabilistic approach, which has the advantages of being intuitive, flexible, and involving an algorithm without inner loops. We investigate in detail the connections between the solutions of the relaxed functionals with minimizers of the original one. Such connection is demonstrated only for a nonconvex relaxation which turns out to be quite robust to initialization. As a case of study, we illustrate our theoretical analysis with numerical experiments, namely, for the optical flow problem.

We present a novel formulation of exemplar-based inpainting as a global energy optimization problem, written in terms of the offset map. The proposed energy function combines a data attachment term that ensures the continuity of reconstruction at the boundary of the inpainting domain with a smoothness term that ensures a visually coherent reconstruction inside the hole. This formulation is adapted to obtain a global minimum using the graph cuts algorithm. To reduce the computational complexity, we propose an efficient multiscale graph cuts algorithm. To compensate the loss of information at low resolution levels, we use a feature representation computed at the original image resolution. This permits alleviation of the ambiguity induced by comparing only color information when the image is represented at low resolution levels. Our experiments show how well the proposed algorithm performs compared with other recent algorithms.

In this work we propose a non-local gradient-based energy for interpolating incomplete disparity maps. It represents an extension of the bilateral filter adapted to reconstruct locally planar disparity maps. We assume that we have at our disposal a reference image from which similarity weights can be computed. When the spatial extend of the weights tends to zero, the proposed model can be shown to converge to an energy involving second order derivatives, explaining thus its ability to obtain higher order interpolations. The proposed energy can be minimized by solving its Euler-Lagrange equation via an iteration of second order Poisson equations. By including an edge map our model permits also to recover depth discontinuities.

Using a classical result on algebraic invariants of the unimodular group, we present in this paper some basic geometric affine invariant quantities, and we use them to construct some distinctive descriptors for object detection. Although full affine invariance cannot be guaranteed due to noncommutativity of camera blur with affine maps and the domain problem (that is, the difficulty of finding an affine covariant domain), the proposed descriptors behave more robustly than SIFT with respect to affine deformations. This is supported by our comparisons both with the version of SIFT computed on an affine normalized neighborhood, and with ASIFT, which solves both the previously mentioned camera blur and domain problems by cleverly sampling the orbit of affine transformations of the images.

We present a method to compute the relative depth of moving objects in video sequences. The method relies on the fact that the boundary between two moving objects follows the movement of the object which is closest to the camera. Thus, the input of the method is a segmentation (to know the boundaries of objects) and an optical flow (to know the movement of the objects). The output of the method is a relative ordering of the neighboring segments. In fact, this output only provides a cue of the desired relative ordering, just like T-junctions typically provide a cue of the relative ordering of the objects around them. These cues can be used later as heuristics or as starting points for higher-level algorithms for image and video-processing.

We present a method for detecting appearances of logos in low-resolution video sequences. The method is based on matching of SIFT descriptors, plus several heuristics. The logos must come from a small database of possible logos. The emphasis is not on speed but on reliability, although the method can be executed in real time using a parallel computer.

In this paper we propose an algorithm for stereoscopic image inpainting, given the inpainting mask in both images. We also assume that depth map is known in one of the images of the stereo pair, taken as reference. This image is clustered in homogeneous color regions using a mean-shift procedure. In each clustered region, depths are fitted by planes and then extended into the mask. Then we inpaint the visible parts of each extended region using a modified exemplar-based inpainting algorithm. Finally, we extend the algorithm to stereoscopic image inpainting. We display some experiments showing the performance of the proposed algorithm.

Tone Mapping is the problem of compressing the range of a High-Dynamic Range image so that it can be displayed in a Low-Dynamic Range screen, without losing or introducing novel details: The final image should produce in the observer a sensation as close as possible to the perception produced by the real-world scene. We propose a tone mapping operator with two stages. The first stage is a global method that implements visual adaptation, based on experiments on human perception, in particular we point out the importance of cone saturation. The second stage performs local contrast enhancement, based on a variational model inspired by color vision phenomenology. We evaluate this method with a metric validated by psychophysical experiments and, in terms of this metric, our method compares very well with the state of the art.

We evaluate the performance of different optimization techniques developed in the context of optical flow computation with different variational models. In particular, based on truncated Newton (TN) methods that have been an effective approach for large-scale unconstrained optimization, we develop the use of efficient multilevel schemes for computing the optical flow. More precisely, we compare the performance of a standard unidirectional multilevel algorithm—called multiresolution optimization (MR/Opt)—with that of a bidirectional multilevel algorithm—called full multigrid optimization (FMG/Opt). The FMG/Opt algorithm treats the coarse grid correction as an optimization search direction and eventually scales it using a line search. Experimental results on three image sequences using four models of optical flow with different computational efforts show that the FMG/Opt algorithm outperforms both the TN and MR/Opt algorithms in terms of the computational work and the quality of the optical flow estimation.

In this paper, we propose a variational formulation for histogram transfer of two or more color images. We study an energy functional composed by three terms: one tends to approach the cumulative histograms of the transformed images, the other two tend to maintain the colors and geometry of the original images. By minimizing this energy, we obtain an algorithm that balances equalization and the conservation of features of the original images. As a result, they evolve while approaching an intermediate histogram between them. This intermediate histogram does not need to be specified in advance, but it is a natural result of the model. Finally, we provide experiments showing that the proposed method compares well with the state of the art.

In this paper the problem of optical flow and occlusion mask estimation is aborded. To that end, we consider a multi-label representation of the optical flow and we define an energy that models the problem. The convexification of the energy and its minimization with an iterative algorithm are studied. Our algorithm is implemented in GPU, since each pixel can be processed in parallel. From our experiments, the relation between the quality of the results obtained and computing time seems to be very promising.

In this paper we propose an algorithm for in painting of stereo images. The issue is to reconstruct the holes in a pair of stereo image as if they were the projection of a 3D scene. Hence, the reconstruction of the missing information has to produce a consistent visual perception of depth. Thus, first step of the algorithm consists in the computation and in painting of disparity maps in the given holes. The second step of the algorithm is to fill-in missing regions using the complete disparity maps in a way that avoids the creation of 3D artifacts. We present some experiments on several pairs of stereo images.

This paper presents a spatial string matching method to incorporate spatial information into the bag-of-words model, which represents an image as an unordered distribution of local features. Spatial constraints among neighboring features are explored in order to achieve better discrimination power for image classification. The features from neighboring points are combined together and taken as a spatial string, and then our method matches the images according to the similarity of string pairs. The categorization problem can be formulated using KNN or SVM classifier based on the spatial string matching kernel. The proposed method is able to capture spatial dependencies across the neighboring features. Experiment results show promising performance for image classification tasks.

Inpainting is the art of modifying an image in a form that is not detectable by an ordinary observer. There are numerous and very different approaches to tackle the inpainting problem, though as explained in this paper, the most successful algorithms are based upon one or two of the following three basic techniques: copy-and-paste texture synthesis, geometric partial differential equations (PDEs), and coherence among neighboring pixels. We combine these three building blocks in a variational model, and provide a working algorithm for image inpainting trying to approximate the minimum of the proposed energy functional. Our experiments show that the combination of all three terms of the proposed energy works better than taking each term separately, and the results obtained are within the state-of-the-art.

Hierarchies are a powerful tool for image segmentation, they produce a multiscale representation which allows to design robust algorithms and can be stored in tree-like structures which provide an efficient implementation. These hierarchies are usually constructed explicitly or implicitly by means of region merging algorithms. These algorithms obtain the segmentation from the hierarchy by either using a greedy merging order or by cutting the hierarchy at a fixed scale. Our main contribution is to enlarge the search space of these algorithms to the set of all possible partitions spanned by a certain hierarchy, and to cast the segmentation as a selection problem within this space. The importance of this is two-fold. First, we are enlarging the search space of classic greedy algorithms and thus potentially improving the segmentation results. Second, this space is considerably smaller than the space of all possible partitions, thus we are reducing the complexity. In addition, we embed the selection process on a statistical a contrario framework which allows us to reduce the number of free parameters of our algorithm to only one.

We propose an interpolation method for sparse data that incorporates the geometric information of a reference image. The idea consists in defining for each sample a geodesic neighborhood and then fit a model (affine for instance) to interpolate at the current point. In the field of remote sensing for urban areas, two widely used techniques are laser range scanning (LIDAR) and stereo photogrammetry. Both techniques have a common drawback, for a variety of reasons the information they provide is sparse or incomplete. But in both cases it is fair to assume that a high resolution image of the scene is available, and we propose in this paper a diffusion algorithm that takes into account the geometry of the image u to refine the range data. This allows us to interpolate the data set while respecting the edges of u. The core of the algorithm is a fast method for computing geodesic distances between image points, which has been successfully applied to colorization by Yatziv et al. and supervised segmentation by Bai et al. The geodesic distance is used to find the set of points that are used to interpolate a piecewise affine model in the current sample. This first interpolation result is refined by merging the obtained affine patches using a greedy Mumford-Shah like algorithm. The output is a piecewise affine interplation of the data set that respects both the given data and the radiometric information provided by u.

Basic phenomenology of human color vision has been widely taken as an inspiration to devise explicit color correction algorithms. The behavior of these models in terms of significative image features (such as, e.g., contrast and dispersion) can be difficult to characterize. To cope with this, we propose to use a variational formulation of color contrast enhancement that is inspired by the basic phenomenology of color perception. In particular, we devise a set of basic requirements to be fulfilled by an energy to be considered as 'perceptually inspired', showing that there is an explicit class of functionals satisfying all of them. We single out three explicit functionals that we consider of basic interest, showing similarities and differences with existing models. The minima of such functionals is computed using a gradient descent approach. We also present a general methodology to reduce the computational cost of the algorithms under analysis from O(N2) to O(N logN), being N the number of pixels of the input image.

The main purpose of this paper is to develop the mathematical analysis of anisotropic total variation problems with a degenerate metric and the computation of the associated Cheeger sets. We illustrate our analysis with the computation of Cheeger sets with respect to different anisotropic norms of relevance in applications to image processing. In particular, we describe the computation of global minima of geodesic active contour models, and we illustrate the use of Cheeger sets for the problem of edge linking.

Demosaicking is a particular case of interpolation problems where, from a scalar image in which each pixel has either the red, the green or the blue component, we want to interpolate the full-color image. State-of-the-art demosaicking algorithms perform interpolation along edges, but these edges are estimated locally. We propose a level-set-based geometric method to estimate image edges, inspired by the image inpainting literature. This method has a time complexity of O(S) , where S is the number of pixels in the image, and compares favorably with the state-of-the-art algorithms both visually and in most relevant image quality measures.

Video is usually acquired in interlaced format, where each image frame is composed of two image fields, each field holding same parity lines. However, many display devices require progressive video as input; also, many video processing tasks perform better on progressive material than on interlaced video. In the literature, there exist a great number of algorithms for interlaced to progressive video conversion, with a great tradeoff between the speed and quality of the results. The best algorithms in terms of image quality require motion compensation; hence, they are computationally very intensive. In this paper, we propose a novel deinterlacing algorithm based on ideas from the image inpainting arena. We view the lines to interpolate as gaps that we need to inpaint. Numerically, this is implemented using a dynamic programming procedure, which ensures a complexity of O(S), where S is the number of pixels in the image. The results obtained with our algorithm compare favorably, in terms of image quality, with state-of-the-art methods, but at a lower computational cost, since we do not need to perform motion field estimation.

Here, we present an efficient method for movie denoising that does not require any motion estimation. The method is based on the well-known fact that averaging several realizations of a random variable reduces the variance. For each pixel to be denoised, we look for close similar samples along the level surface passing through it. With these similar samples, we estimate the denoised pixel. The method to find close similar samples is done via warping lines in spatiotemporal neighborhoods. For that end, we present an algorithm based on a method for epipolar line matching in stereo pairs which has per-line complexity O (N), where N is the number of columns in the image. In this way, when applied to the image sequence, our algorithm is computationally efficient, having a complexity of the order of the total number of pixels. Furthermore, we show that the presented method is unsupervised and is adapted to denoise image sequences with an additive white noise while respecting the visual details on the movie frames. We have also experimented with other types of noise with satisfactory results.

In this paper, we present a discussion about perceptual-based color correction of digital images in the framework of variational techniques. We propose a novel image functional whose minimization produces a perceptually inspired color enhanced version of the original. The variational formulation permits a more flexible local control of contrast adjustment and attachment to data. We show that a numerical implementation of the gradient descent technique applied to this energy functional coincides with the equation of automatic color enhancement (ACE), a particular perceptual-based model of color enhancement. Moreover, we prove that a numerical approximation of the Euler-Lagrange equation reduces the computational complexity of ACE from theta(N2) to theta(N log N), where N is the total number of pixels in the image.

Minimal surface regularization has been used in several applications ranging from stereo to image segmentation, sometimes hidden as a graph-cut discrete formulation, or as a strictly convex approximation to TV minimization. In this paper we consider a modified version of minimal surface regularization coupled with a robust data fitting term for interpolation purposes, where the corresponding evolution equation is constrained to diffuse only along the isophotes of a given image u and we design a convergent numerical scheme to accomplish this. To illustrate the usefulness of our approach, we apply this framework to the digital elevation model interpolation and to constrained vector probability diffusion. 1

The tree of shapes is a powerful tool for image representation which holds many interesting properties. Many works in the literature use it for image segmentation, but most of them use only boundary information along the level lines. In many real images this is not enough to achieve a good segmentation, and region information must be introduced. In this work we present a novel region-based segmentation algorithm using the tree of shapes. The approach taken consists in the selection of relevant level-lines according to region based descriptors computed from their interior. We describe a region using the histogram of its features and we select interesting regions by identifying parts of the tree with an homogeneous histogram. The main contribution of this work is the joint use of histograms and suitable metrics between them, with the powerful representation of the tree of shapes. This allows us to handle complex region models and thus improves on previous works which were only able to deal with piecewise constant models. We validate our approach with real images and we obtain results which are favorably compared with some well known related approaches.

Two complementary geometric structures for the topographic representation of an image are developed in this work. The first one computes a description of the Morse-topological structure of the image, while the second one computes a simplified version of its drainage structure. The topographic significance of the Morse and drainage structures of digital elevation maps (DEMs) suggests that they can been used as the basis of an efficient encoding scheme. As an application, we combine this geometric representation with an interpolation algorithm and lossless data compression schemes to develop a compression scheme for DEMs. This algorithm achieves high compression while controlling the maximum error in the decoded elevation map, a property that is necessary for the majority of applications dealing with DEMs. We present the underlying theory and compression results for standard DEM data.

In this paper we study a variational approach for filling-in regions of missing data in 2D and 3D digital images. Applications of this technique include the restoration of old photographs and removal of superimposed text like dates, subtitles, or publicity, or the zooming of images. The approach presented here, initially introduced in [C. Ballester et al., 2001], is based on a joint interpolation of the image gray-levels and gradient/isophotes directions, smoothly extending the isophote lines into the holes of missing data. The process underlying this approach can be considered as an interpretation of the Gestaltist's principle of good continuation. After discussing the model, we present the numerical algorithm used to minimize it and display some numerical experiments.

We discuss possible algorithms for interpolating data given in a set of curves and/or points in a surface in /spl Ropf//sup 3/. We propose a set of basic assumptions to be satisfied by the interpolation algorithms which lead to a set of models in terms of possibly degenerate elliptic partial differential equations. The absolute minimal Lipschitz extension model (AMLE) is singled out and studied in more detail. We show experiments illustrating the interpolation of data on the sphere and the torus.

Two complementary geometric structures for the topographic representation of an image are developed in this work. The first one computes a description of the Morse structure of the image, while the second one computes a simplified version of its drainage structure. The topographic significance of the Morse and drainage structures of digital elevation maps (DEM) suggests that they can been used as the basis of an efficient encoding scheme. We combine this geometric representation with an interpolation algorithm and lossless data compression schemes to develop a compression scheme for DEM. This algorithm permits to obtain compression results while controlling the maximum error in the decoded elevation map, a property that is necessary for the majority of applications dealing with DEM.

An algorithm for filling-in surface holes is introduced in this paper. The basic idea is to represent the surface of interest in implicit form, and fill-in the holes with a system of geometric partial differential equations derived from image inpainting algorithms. The framework and examples with synthetic and real data are presented.

A variational approach for filling-in regions of missing data in digital images is introduced. The approach is based on joint interpolation of the image gray levels and gradient/isophotes directions, smoothly extending in an automatic fashion the isophote lines into the holes of missing data. This interpolation is computed by solving the variational problem via its gradient descent flow, which leads to a set of coupled second order partial differential equations, one for the gray-levels and one for the gradient orientations. The process underlying this approach can be considered as an interpretation of the Gestaltist's principle of good continuation. No limitations are imposed on the topology of the holes, and all regions of missing data can be simultaneously processed, even if they are surrounded by completely different structures. Applications of this technique include the restoration of old photographs and removal of superimposed text like dates, subtitles, or publicity. Examples of these applications are given. We conclude the paper with a number of theoretical results on the proposed variational approach and its corresponding gradient descent flow.

A novel approach for color image denoising is proposed in this paper. The algorithm is based on separating the color data into chromaticity and brightness, and then processing each one of these components with partial differential equations or diffusion flows. In the proposed algorithm, each color pixel is considered as an n-dimensional vector. The vectors' direction, a unit vector, gives the chromaticity, while the magnitude represents the pixel brightness. The chromaticity is processed with a system of coupled diffusion equations adapted from the theory of harmonic maps in liquid crystals. This theory deals with the regularization of vectorial data, while satisfying the intrinsic unit norm constraint of directional data such as chromaticity. Both isotropic and anisotropic diffusion flows are presented for this n-dimensional chromaticity diffusion flow. The brightness is processed by a scalar median filter or any of the popular and well established anisotropic diffusion flows for scalar image enhancement. We present the underlying theory, a number of examples, and briefly compare with the current literature.

A novel approach for color image denoising is proposed. The algorithm is based on separating the color data into chromaticity and brightness, and then processing each one of these components with partial differential equations or diffusion flows. In the proposed algorithm, each color pixel is considered as an n-dimensional vector. The vectors' direction, a unit vector, gives the chromaticity, while the magnitude represents the pixel brightness. The chromaticity is processed with a system of coupled diffusion equations adapted from the theory of harmonic maps in liquid crystals. This theory deals with the regularization of vectorial data, while satisfying the intrinsic unit norm constraint of directional data such as chromaticity. Both isotropic and anisotropic diffusion flows are presented for this n-dimensional chromaticity diffusion flow. The brightness is processed by a scalar median filter or any of the popular and well established anisotropic diffusion flows for scalar image enhancement. We present the underlying theory, a number of examples, and comparison with the current literature.

Inpainting, the technique of modifying an image in an undetectable form, is as ancient as art itself. The goals and applications of inpainting are numerous, from the restoration of damaged paintings and photographs to the removal/replacement of selected objects. In this paper, we introduce a novel algorithm for digital inpainting of still images that attempts to replicate the basic techniques used by professional restorators. After the user selects the regions to be restored, the algorithm automatically fills-in these regions with information surrounding them. The fill-in is done in such a way that isophote lines arriving at the regions' boundaries are completed inside. In contrast with previous approaches, the technique here introduced does not require the user to specify where the novel information comes from. This is automatically done (and in a fast way), thereby allowing to simultaneously fill-in numerous regions containing completely different structures and surrounding backgrounds. In addition, no limitations are imposed on the topology of the region to be inpainted. Applications of this technique include the restoration of old photographs and damaged film; removal of superimposed text like dates, subtitles, or publicity; and the removal of entire objects from the image like microphones or wires in special effects.

In a number of disciplines, directional data provides a fundamental source of information. A novel framework for isotropic and anisotropic diffusion of directions is presented in this paper. The framework can be applied both to regularize directional data and to obtain multiscale representations of it. The basic idea is to apply and extend results from the theory of harmonic maps in liquid crystals. This theory deals with the regularization of vectorial data, while satisfying the unit norm constraint of directional data. We show the corresponding variational and partial differential equations formulations for isotropic diffusion, obtained from an L/sub 2/ norm, and edge preserving diffusion, obtained from an L/sub 1/ norm. In contrast with previous approaches, the framework is valid for directions in any dimensions, supports non-smooth data, and gives both isotropic and anisotropic formulations. We present a number of theoretical results, open questions, and examples for gradient vectors, optical flow, and color images.

We formally connect between vector median filters, morphological operators, and geometric partial differential equations. Considering a lexicographic order, which permits us to define an order between vectors in IR/sup N/, we first show that the vector median filter of a vector valued image is equivalent to a collection of infimum-supremum morphological operations. We then proceed and study the asymptotic behavior of this filter. We also provide an interpretation of the infinitesimal iteration of this vectorial median filter in terms of systems of coupled geometric partial differential equations. The main component of the vector evolves according to curvature motion, while, intuitively, the others regularly deform their level sets toward those of this main component. These results extend to the vector case classical connections between scalar median filters, mathematical morphology, and mean curvature motion.

A novel approach for shape preserving contrast enhancement is presented in this paper. Contrast enhancement is achieved by means of a local histogram equalization algorithm which preserves the level-sets of the image. This basic property is violated by common local schemes, thereby introducing spurious objects and modifying the image information. The scheme is based on equalizing the histogram in all the connected components of the image, which are defined based both on the grey-values and spatial relations between pixels in the image, and following mathematical morphology, constitute the basic objects in the scene. We give examples for both grey-value and color images.

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We discuss possible algorithms for interpolating data given in a set of curves and/or points in the plane. We propose a set of basic assumptions to be satisfied by the interpolation algorithms which lead to a set of models in terms of possibly degenerate elliptic partial differential equations. The absolute minimal Lipschitz extension model (AMLE) is singled out and studied in more detail. We show experiments suggesting a possible application, the restoration of images with poor dynamic range.

A novel approach for shape preserving contrast enhancement is presented. Contrast enhancement is achieved by means of a local histogram equalization algorithm which preserves the level-sets of the image. This basic property is violated by common local schemes, thereby introducing spurious objects and modifying the image information. The scheme is based on equalizing the histogram in all the connected components of the image, which are defined based on the image grey-values and spatial relations between its pixels. Following mathematical morphology, these constitute the basic objects in the scene. We give examples for both grey-valued and color images.

We discuss possible algorithms for interpolating data given in a set of curves and/or points in the plane. We propose a set of basic assumptions to be satisfied by the interpolation algorithms which lead to a set of models in terms of possibly degenerate elliptic partial differential equations. The absolute minimal Lipschitz extension model (AMLE) is singled out and studied in more detail. We show experiments suggesting a possible application, the restoration of images with poor dynamic range.

A geometric approach for 3D object segmentation and representation is presented. The segmentation is obtained by deformable surfaces moving towards the objects to be detected in the 3D image. The model is based on curvature motion and the computation of surfaces with minimal areas, better known as minimal surfaces. The space where the surfaces are computed is induced from the 3D image (volumetric data) in which the objects are to be detected. The model links between classical deformable surfaces obtained via energy minimization, and intrinsic ones derived from curvature based flows. The new approach is stable, robust, and automatically handles changes in the surface topology during the deformation.

We discuss the physical generation process of images as a combination of basic operations: occlusions, transparencies and contrast changes. These operations generate the essential singularities which we call junctions. We deduce a mathematical and computational model for image analysis according to which the "atoms" of the image must be "pieces of level lines joining junctions", fitting the phenomenological description of Gaetano Kanizsa (1990). A parameter free junction detection algorithm is proposed for the computation of the previously defined "atoms". Then we propose an adequate modification of the morphological filtering algorithms so that they smooth the "atoms" without altering the junctions. Finally, we give some experiments on real and synthetic images.

A novel scheme for the detection of object boundaries is presented. The technique is based on active contours deforming according to intrinsic geometric measures of the image. The evolving contours naturally split and merge, allowing the simultaneous detection of several objects and both interior and exterior boundaries. The proposed approach is based on the relation between active contours and the computation of geodesics or minimal distance curves. The minimal distance curve lays in a Riemannian space whose metric as defined by the image content. This geodesic approach for object segmentation allows to connect classical "snakes" based on energy minimization and geometric active contours based on the theory of curve evolution. Previous models of geometric active contours are improved as showed by a number of examples. Formal results concerning existence, uniqueness, stability, and correctness of the evolution are presented as well.>

A geometric formulation of active contours for 2D, 3D boundary detection and motion tracking is presented. The technique is based on active contours evolving in time according to intrinsic geometric measures of the image. The evolving contours naturally split and merge, allowing the simultaneous detection of several objects and both interior and exterior boundaries. The proposed approach is based on the relation between active contours and the computation of minimal distance curves or minimal surfaces in a Riemannian space whose metric is derived from the image. Previous models of geometric active contours are improved allowing stable boundary detection when their gradients suffer from large variations, including gaps. Numerical experiments are also presented.

An algorithm for histogram modification via image evolution equations is presented. We show that the image histogram can be modified to achieve any given distribution as the steady state solution of this partial differential equation. We then prove that this equation corresponds to a gradient descent flow of a variational problem. That is, the proposed PDE is solving an energy minimization problem. This gives a new interpretation to histogram modification and contrast enhancement in general. This interpretation is completely formulated in the image domain, in contrast with classical techniques for histogram modification which are formulated in a probabilistic domain. From this, new algorithms for contrast enhancement, which include for example image modeling, can be derived. Based on the energy formulation and its corresponding PDE, we show that the proposed histogram modification algorithm can be combined with denoising schemes. This allows one to perform simultaneous contrast enhancement and denoising, avoiding common noise sharpening effects in classical algorithms. The approach is extended to focal contrast enhancement as well. Theoretical results regarding the existence of solutions to the proposed equations are presented.