Synthesizing extrapolated views remains a difficult task, especially in urban driving scenes, where the only reliable sources of data are limited RGB captures and sparse LiDAR points. To address this problem, we present PointmapDiff, a framework for novel view synthesis that utilizes pre-trained 2D diffusion models. Our method leverages point maps (i.e., rasterized 3D scene coordinates) as a conditioning signal, capturing geometric and photometric priors from the reference images to guide the image generation process. With the proposed reference attention layers and ControlNet for point map features, PointmapDiff can generate accurate and consistent results across varying viewpoints while respecting geometric fidelity. Experiments on real-life driving data demonstrate that our method achieves high-quality generation with flexibility over point map conditioning signals (e.g., dense depth map or even sparse LiDAR points) and can be used to distill to 3D representations such as 3D Gaussian Splatting for improving view extrapolation.

Vehicles suddenly exiting road-side parking constitute a hazardous situation for vehicle drivers as well as for Connected and Autonomous Vehicles (CAV). In order to improve the awareness of road users, we propose an original cooperative information system based on image processing to monitor vehicles parked on the road-side and on communication for sending early warning to vehicles on the road about vehicles leaving their parking space. We have implemented and tested this system in two places in France with parallel and perpendicular parking slots for several camera positions, and we report on its efficiency and its limits.

From an analysis of the priors used in previous algorithms for single image defogging, a new prior is proposed to obtain a better atmospheric veil removal. The Naka-Rushton function is used to modulate the atmospheric veil according to empirical observations on synthetic foggy images. The parameters of this function are set from features of the input image. The algorithm is able to take into account different kinds of airborne particles and different illumination conditions. The proposed method is extended to nighttime and underwater images by computing the atmospheric veil on each color channel. Qualitative and quantitative evaluations show the benefit of the proposed algorithm.

For the last decade, single image defogging has been a subject of interest in image processing. In the visible spectrum, fog and haze decrease the visibility of distant objects. Thus, the objective of the visibility restoration is to remove as much as possible the effect of the fog within the image. Infrared sensors are more and more used in automotive and aviation industries but the effect of fog and haze is not restricted to the visible spectrum and also applies in the infrared band. After recalling the effects of fog in the common sub-bands of the infrared spectrum, we tested if the approach used for single image defogging in the visible spectrum might also work for infrared. This led us to propose a new approach of single image defogging for Long-Wavelength Infra-Red (LWIR) or Thermal Infra-Red. Several experiments are presented showing that the proposed algorithm offers interesting results not only for fog and haze but for bad weather conditions in general, during day and night.

We propose a new way of performing pixel by pixel comparison between two images, taking advantage of interesting invariance properties with respect to illumination conditions and camera settings. Moreover, we show that the proposed operator is relatively robust to strong noise on one of the compared images. The new operator can be used for background subtraction which inherits its invariance properties. The useful properties of the proposed operators are illustrated in the experiments.

Stereoscopic reconstruction is important to automatic vision systems. As an intermediate step, estimating this reconstruction is not enough for good performance of the whole system, and its uncertainty must be characterized. Several methods propose uncertainty indexes based on specific data features, thus incomplete, while others are based on learning. We propose a simple index, named ambiguity index, taking into account both data and regularization, and derived directly from the optimization process. Exploiting properties of dynamic programming, this index is related to the posterior variance of the solution when the Semi-Global Matching (SGM) algorithm is used for stereo reconstruction. To illustrate its interest, improvements in refining stereo reconstruction are shown on the KITTI datasets when the index is used.

Many problems of image processing lead to the minimization of an energy, which is a function of one or several given images, with respect to a binary or multi-label image. When this energy is made of unary data terms and of pairwise regularization terms, and when the pairwise regularization term is a metric, the multi-label energy can be minimized quite rapidly, using the so-called α-expansion algorithm. α-expansion consists in decomposing the multi-label optimization into a series of binary sub-problems called move. Depending on the chosen decomposition, a different condition on the regularization term applies. The metric condition for α-expansion move is rather restrictive. In many cases, the statistical model of the problem leads to an energy which is not a metric. Based on the enlightening article [1], we derive another condition for β-jump move. Finally, we propose an alternated scheme which can be used even if the energy fulfills neither the α-expansion nor β-jump condition. The proposed scheme applies to a much larger class of regularization functions, compared to α-expansion. This opens many possibilities of improvements on diverse image processing problems. We illustrate the advantages of the proposed optimization scheme on the image noise reduction problem.

The bilateral filter and its variants, such as the joint/cross bilateral filter, are well-known edge-preserving image smoothing tools used in many applications. The reason of this success is its simple definition and the possibility of many adaptations. The bilateral filter is known to be related to robust estimation. This link is lost by the ad hoc introduction of the guide image in the joint/cross bilateral filter. We here propose a new way to derive the joint/cross bilateral filter as a particular case of a more generic filter, which we name the guided bilateral filter. This new filter is iterative, generic, inherits the robustness properties of the robust bilateral filter, and uses a guide image. The link with robust estimation allows us to relate the filter parameters with the statistics of input images. A scheme based on graduated nonconvexity is proposed, which allows converging to an interesting local minimum even when the cost function is nonconvex. With this scheme, the guided bilateral filter can handle non-Gaussian noise on the image to be filtered. A complementary scheme is also proposed to handle non-Gaussian noise on the guide image even if both are strongly correlated. This allows the guided bilateral filter to handle situations with more noise than the joint/cross bilateral filter can work with and leads to high peak signal-to-noise ratio values as shown experimentally.

Fog reduces contrast and thus the visibility of vehicles and obstacles for drivers. Each year, this causes traffic accidents. Fog is caused by a high concentration of very fine water droplets in the air. When light hits these droplets, it is scattered and this results in a dense white background, called the atmospheric veil. As pointed in [1], Advanced Driver Assistance Systems (ADAS) based on the display of defogged images from a camera may help the driver by improving objects visibility in the image and thus may lead to a decrease of fatality and injury rates. In the last few years, the problem of single image defogging has attracted attention in the image processing community. Being an ill-posed problem, several methods have been proposed. However, a few among of these methods are dedicated to the processing of road images. One of the first exception is the method in [2], [1] where a planar constraint is introduced to improve the restoration of the road area, assuming an approximately flat road. The single image defogging problem being ill-posed, the choice of the Bayesian approach seems adequate to set this problem as an inference problem. A first Markov Random Field (MRF) approach of the problem has been proposed recently in [3]. However, this method is not dedicated to road images. In this paper, we propose a novel MRF model of the single image defogging problem which applies to all kinds of images but can also easily be refined to obtain better results on road images using the planar constraint. A comparative study and quantitative evaluation with several state-of-the-art algorithms is presented. This evaluation demonstrates that the proposed MRF model allows to derive a new algorithm which produces better quality results, in particular in case of a noisy input image.

Sight distance along the pathway plays a significant role in road safety and in particular, has a clear impact on the choice of speed limits. Mapping visibility distance is thus of importance for road engineers and authorities. While visibility distance criteria are routinely taken into account in road design, few systems exist for evaluating them on existing road networks. Most available systems comprise a target vehicle followed at a constant distance by an observer vehicle. This only allows to check if a given, fixed visibility distance is available: estimating the maximum visibility distance requires several passages, with increasing inter-vehicle intervals. We propose two alternative approaches for estimating the maximum available visibility distance, that exploit 3D models of the road and its close environment. These methods involve only one acquisition vehicle and use either active vision, more specifically 3D range sensing (LIDAR), or passive vision, namely, stereovision. The first approach is based on a Terrestrial LIDAR Mobile Mapping System. The triangulated 3D model of the road and its surroundings provided by the system is used to simulate targets at different distances, which allows for estimation of the maximum geometric visibility distance along the pathway in a quite flexible way. The second approach involves the processing of two views taken by digital cameras on-board an inspection vehicle. After road segmentation, the 3D road model is reconstructed which allows maximum roadway visibility distance estimation. Both approaches are described, evaluated and compared. Their pros and cons with respect to vehicle-following systems are also discussed.

Various interest point and corner definitions were proposed in the past with associated detection algorithms. We propose an intuitive and novel detection algorithm for finding the location of such features in an image. The detection is based on the edges in the original image. Interest points are detected as accumulation points where several edge tangent lines in a neighborhood are crossing. Edge connectivity is not used and thus detected interest points are robust to partial edges, outliers and edge extraction failures at junctions. One advantage of the approach is that detected interest points are not shifted in location when the original image is smoothed compared with other approaches. Experiments performed on Oxford and Cambridge reference databases allow us to show that the proposed detection algorithm performs better than 9 existing interest point detectors in terms of repeatability from multiple camera views.

We investigate the scenario of a vehicle equipped with a camera and a GPS driving on a road whose 3D map is known. We focus on the case of a road under fog or/and snow conditions. The GPS is used to estimate the vehicle pose and yaw and then the 3D road map is projected onto the camera image. The vehicle pitch and roll angles are then refined by fitting the projected road to detected road markings. Finally, we discuss the pros and cons of the obtained road registrations in the images and of the vehicle pitch-roll estimates, with respect to the vehicle dynamics and the driving environment, in adverse weather conditions.

Road sign identification in images is an important issue, in particular for vehicle safety applications. It is usually tackled in three stages: detection, recognition and tracking, and evaluated as a whole. To progress towards better algorithms, we focus in this paper on the first stage of the process, namely road sign detection. More specifically, we compare, on the same ground-truth image database, results obtained by three algorithms that sample different state-of-the-art approaches. The three tested algorithms: Contour Fitting, Radial Symmetry Transform, and pair-wise voting scheme, all use color and edge information and are based on geometrical models of road signs. The test dataset is made of 847 images 960×1080 of complex urban scenes (available at www.itowns.fr/benchmarking.html). They feature 251 road signs of different shapes (circular, rectangular, triangular), sizes and types. The pros and cons of the three algorithms are discussed, allowing to draw new research perspectives.

Most road marking detection systems use image processing to extract potential marking elements in their first stage. Hence, the performances of extraction algorithms clearly impact the result of the whole process. In this paper, we address the problem of extracting road markings in high resolution environment images taken by inspection vehicles in a urban context. This situation is challenging since large special markings, such as crosswalks, zebras or pictographs must be detected as well as lane markings. Moreover, urban images feature many white elements that might lure the extraction process. In prior work an efficient extraction process, called Median Local Threshold algorithm, was proposed that can handle all kinds of road markings. This extraction algorithm is here improved and compared to other extraction algorithms. An experimental study performed on a database of images with ground-truth shows that the stereovision strategy reduces the number of false alarms without significant loss of true detection.

One source of accidents when driving a vehicle is the presence of homogeneous and heterogeneous fog. Fog fades the colors and reduces the contrast of the observed objects with respect to their distances. Various camera-based Advanced Driver Assistance Systems (ADAS) can be improved if efficient algorithms are designed for visibility enhancement of road images. The visibility enhancement algorithm proposed in is not dedicated to road images and thus it leads to limited quality results on images of this kind. In this paper, we interpret the algorithm in as the inference of the local atmospheric veil subject to two constraints. From this interpretation, we propose an extended algorithm which better handles road images by taking into account that a large part of the image can be assumed to be a planar road. The advantages of the proposed local algorithm are its speed, the possibility to handle both color images or gray-level images, and its small number of parameters. A comparative study and quantitative evaluation with other state-of-the-art algorithms is proposed on synthetic images with several types of generated fog. This evaluation demonstrates that the new algorithm produces similar quality results with homogeneous fog and that it is able to better deal with the presence of heterogeneous fog.

In adverse weather conditions, in particular, in daylight fog, the contrast of images grabbed by in-vehicle cameras in the visible light range is drastically degraded, which makes current driver assistance that relies on cameras very sensitive to weather conditions. An onboard vision system should take weather effects into account. The effects of daylight fog vary across the scene and are exponential with respect to the depth of scene points. Because it is not possible in this context to compute the road scene structure beforehand, contrary to fixed camera surveillance, a new scheme is proposed. Fog density is first estimated and then used to restore the contrast using a flat-world assumption on the segmented free space in front of a moving vehicle. A scene structure is estimated and used to refine the restoration process. Results are presented using sample road scenes under foggy weather and assessed by computing the visibility level enhancement that is gained by the method. Finally, we show applications to the enhancement in daylight fog of low-level algorithms that are used in advanced camera-based driver assistance.

Central to many problems in scene understanding based on using a network of tens, hundreds or even thousands of randomly distributed cameras with on-board processing and wireless communication capability is the “efficient” reconstruction of the 3D geometry structure in the scene. What is meant by “efficient” reconstruction? In this paper we investigate this from different aspects in the context of visual sensor networks and offer a distributed reconstruction algorithm roughly meeting the following goals: 1. Close to achievable 3D reconstruction accuracy and robustness; 2. Minimization of the processing time by adaptive computing-job distribution among all the cameras in the network and asynchronous parallel processing; 3. Communication Optimization and minimization of the (battery-stored) energy, by reducing and localizing the communications between cameras. A volumetric representation of the scene is reconstructed with a shape from apparent contour algorithm, which is suitable for distributed processing because it is essentially a local operation in terms of the involved cameras, and apparent contours are robust to ourdoor illumination conditions. Each camera processes its own image and performs the computation for a small subset of voxels, and updates the voxels through collaborating with its neighbor cameras. By exploring the structure of the reconstruction algorithm, we design the minimum-spanning-tree (MST) message passing protocol in order to minimize the communication. Of interest is that the resulting system is an example of “swarm behavior”. 3D reconstruction is illustrated using two real image sets, running on a single computer. The iterative computations used in the single processor experiment are exactly the same as are those used in the network computations. Distributed concepts and algorithms for network control and communication performance are theoretical designs and estimates.

One source of difficulties when processing outdoor images is the presence of haze, fog or smoke which fades the colors and reduces the contrast of the observed objects. We introduce a novel algorithm and variants for visibility restoration from a single image. The main advantage of the proposed algorithm compared with other is its speed: its complexity is a linear function of the number of image pixels only. This speed allows visibility restoration to be applied for the first time within real-time processing applications such as sign, lane-marking and obstacle detection from an in-vehicle camera. Another advantage is the possibility to handle both color images or gray level images since the ambiguity between the presence of fog and the objects with low color saturation is solved by assuming only small objects can have colors with low saturation. The algorithm is controlled only by a few parameters and consists in: atmospheric veil inference, image restoration and smoothing, tone mapping. A comparative study and quantitative evaluation is proposed with a few other state of the art algorithms which demonstrates that similar or better quality results are obtained. Finally, an application is presented to lane-marking extraction in gray level images, illustrating the interest of the approach.

We present a new transformation for angle vertex and bisector detection. The vertex and bisector transformation (VBT) takes the image gradient as input and outputs two arrays, accumulating evidence of respectively angle vertex and angle bisector. A geometric model of the gradient orientation is implemented using a pair-wise voting scheme: normal vectors of two adjacent sides of a triangle have a specific relationship depending on the corresponding vertex angle. Our approach is able to accurately detect vertices and bisectors of a triangular road sign in a 360 × 270 image in about 50 ms with no particular optimization. We tested our approach on a 48 images database containing 40 triangular signs with different colors and orientations (red intersection and give way warnings, blue pedestrian crossing): 33 are correctly detected (82%) and 7 are missed with 2 false positives.

In this paper, the duality in differential form is developed between a 3D primal surface and its dual manifold formed by the surface's tangent planes, i.e., each tangent plane of the primal surface is represented as a four-dimensional vector which constitutes a point on the dual manifold. The iterated dual theorem shows that each tangent plane of the dual manifold corresponds to a point on the original 3D surface, i.e., the dual of the dual goes back to the primal. This theorem can be directly used to reconstruct 3D surface from image edges by estimating the dual manifold from these edges. In this paper we further develop the work in our original conference papers resulting in the robust differential dual operator. We argue that the operator makes good use of the information available in the image data, by using both points of intensity discontinuity and their edge directions; we provide a simple physical interpretation of what the abstract algorithm is actually estimating and why it makes sense in terms of estimation accuracy; our algorithm operates on all edges in the images, including silhouette edges, self occlusion edges, and texture edges, without distinguishing their types (thus resulting in improved accuracy and handling locally concave surface estimation if texture edges are present); the algorithm automatically handles various degeneracies; and the algorithm incorporates new methodologies for implementing the required operations such as appropriately relating edges in pairs of images, evaluating and using the algorithm's sensitivity to noise to determine the accuracy of an estimated 3D point. Experiments with both synthetic and real images demonstrate that the operator is accurate, robust to degeneracies and noise, and general for reconstructing free-form objects from occluding edges and texture edges detected in calibrated images or video sequences.

In foggy weather, the contrast of images grabbed by in-vehicle cameras in the visible light range is drastically degraded, which makes the current applications very sensitive to weather conditions. An onboard vision system should take fog effects into account. The effects of fog varies across the scene and are exponential with respect to the depth of scene points. Because it is not possible in this context to compute the road scene structure beforehand contrary to fixed camera surveillance, a new scheme is proposed. Weather conditions are first estimated and then used to restore the contrast according to a scene structure which is inferred a priori and refined during the restoration process. Based on the aimed application, different algorithms with increasing complexities are proposed. Results are presented using sample road scenes under foggy weather and assessed by computing the contrast before and after restoration.

In this paper we propose a novel approach of the two-image alignment problem based on a functional representation of images. This allows us to derive a one-to-several correspondence, multi-scale algorithm. At the same time, it also formalizes the problem as a robust estimation problem between possible matches. We then derive an accurate, robust and faster version for the alignment of edge images. The proposed algorithm is developed and tested in the context of off-line longitudinal road profile reconstruction from stereo images.

Histogram intersection (HI) kernel has been recently introduced for image recognition tasks. The HI kernel is proved to be positive definite and thus can be used in support vector machine (SVM) based recognition. Experimentally, it also leads to good recognition performances. However, its derivation applies only for binary strings such as color histograms computed on equally sized images. In this paper, we propose a new kernel, which we named generalized histogram intersection (GHI) kernel, since it applies in a much larger variety of contexts. First, an original derivation of the positive definiteness of the GHI kernel is proposed in the general case. As a consequence, vectors of real values can be used, and the images no longer need to have the same size. Second, a hyper-parameter is added, compared to the HI kernel, which allows us to better tune the kernel model to particular databases. We present experiments which prove that the GHI kernel outperforms the simple HI kernel in a simple recognition task. Comparisons with other well-known kernels are also provided.

Kernel based methods such as support vector machine (SVM) has provided successful tools for solving many recognition problems. One of the reasons of this success is the use of kernels. Positive definiteness has to be checked for kernels to be suitable for most of these methods. For instance for SVM, the use of a positive definite kernel insures that the optimized problem is convex and thus the obtained solution is unique. Alternative class of kernels called conditionally positive definite have been studied for a long time from the theoretical point of view and have drawn attention from the community only in the last decade. We propose a new kernel, named log kernel, which seems particularly interesting for images. Moreover, we prove that this new kernel is a conditionally positive definite kernel as well as the power kernel. Finally, we show from experimentations that using conditionally positive definite kernels allows us to outperform classical positive definite kernels

We introduce the intermediate matching (IM) kernel for SVM-hased object recognition. The IM kernel operates on a feature space of vector sets where each image is represented by a set of local features. Matching algorithms have proved to be efficient for such types of features. Nevertheless, kernelizing the matching for SVM does not lead to positive definite kernels. The IM kernel overcomes this drawback, as it mimics matching algorithms while being positive definite. The IM kernel introduces an intermediary set of so-called virtual local features. These select the pairs of local features to be matched. Comparisons with the matching kernel shows that the IM kernels leads to similar performances.

On the one hand, Support Vector Machines have met with significant success in solving difficult pattern recognition problems with global features representation. On the other hand, local features in images have shown to be suitable representations for efficient object recognition. Therefore, it is natural to try to combine SVM approach with local features representation to gain advantages on both sides. We study in this paper the Mercer property of matching kernels which mimic classical matching algorithms used in techniques based on points of interest. We introduce a new statistical approach of kernel positiveness. We show that despite the absence of an analytical proof of the Mercer property, we can provide bounds on the probability that the Gram matrix is actually positive definite for kernels in large class of functions, under reasonable assumptions. A few experiments validate those on object recognition tasks.

In image retrieval systems, a variety of simple similarity measures are used. The choice for one similarity measure or another is generally driven by an experimental comparison on a labeled database. The drawback of such an approach is that, while a large number of possible similarity measures can be tested, we do not know how to extend from the obtained results. However, the choice of a good similarity measure leads to noticeable better results. It is known that this choice is related to the variability of the images within the same class. Therefore, we propose a model of image retrieval systems and deduce a scheme for deriving the best similarity measure in a set of similarity measures, assuming a parametric model of the variability of feature vectors within the same class. An experimental validation of the model and the derived similarity measures is performed on synthetic ground-truth databases. Finally, from our experiments, we give several rules to follow for the design of ground-truth databases allowing reliable conclusions on the search of better similarity measures.

We present an algorithm that extracts the largest shape within a specific class, starting from a set of image edgels. The algorithm inherits the best-first segmentation approach. However, instead of being applicable only to shapes defined within a given class of curves, we have extended our approach to tackle more general-and complex-shapes. For example, we can now process shapes obtained from sets defined over different kinds of curves and related to one another by estimated parameters. Therefore, we go from a segmentation problem to a recognition problem. In order to reduce the complexity of the searching algorithm, we work with a linearly parameterized class of shapes. This allows us, first, to use a recursive least-squares fitting, second, to cast the problem as the search of a largest edgel subset in a directed acyclic graph, and, third, to easily introduce a priori information on the location of the edgels of the searched subset. This leads us to propose a unified approach where recognition and tracking are combined. Experiments on recognizing and tracking both left and right road boundaries demonstrate that real-time processing is achievable.

Representations are introduced for handling 2D algebraic curves (implicit polynomial curves) of arbitrary degree in the scope of computer vision applications. These representations permit fast, accurate pose-independent shape recognition under Euclidean transformations with a complete set of invariants, and fast accurate pose-estimation based on all the polynomial coefficients. The latter is accomplished by a centering of a polynomial based on its coefficients, followed by rotation estimation by decomposing polynomial coefficient space into a union of orthogonal subspaces for which rotations within two-dimensional subspaces or identity transformations within one-dimensional subspaces result from rotations in x, y measured-data space. Angles of these rotations in the two-dimensional coefficient subspaces are proportional to each other and are integer multiples of the rotation angle in the x, y data space. By recasting this approach in terms of a complex variable, i.e., x+iy=z, and complex polynomial-coefficients, further conceptual and computational simplification results. Application to shape-based indexing into databases is presented to illustrate the usefulness and the robustness of the complex representation of algebraic curves.

An algebraic curve is defined as the zero set of a polynomial in two variables. Algebraic curves are practical for modeling shapes much more complicated than conics or superquadrics. The main drawback in representing shapes by algebraic curves has been the lack of repeatability in fitting algebraic curves to data. Usually, arguments against using algebraic curves involve references to mathematicians Wilkinson (and Runge). The first goal of this article is to understand the stability issue of algebraic curve fitting. Then a fitting method based on ridge regression and restricting the representation to well behaved subsets of polynomials is proposed, and its properties are investigated. The fitting algorithm is of sufficient stability for very fast position-invariant shape recognition, position estimation, and shape tracking, based on invariants and new representations. Among appropriate applications are shape-based indexing into image databases.

An algebraic curve is defined as the zero set of a polynomial in two variables. Algebraic curves are practical for modeling shapes much more complicated than conics or superquadrics. The main drawback in representing shapes by algebraic curves has been the lack of repeatability in fitting algebraic curves to data. A regularized fast linear fitting method based on ridge regression and restricting the representation to well behaved subsets of polynomials is proposed, and its properties are investigated. The fitting algorithm is of sufficient stability for very fast position-invariant shape recognition, position estimation, and shape tracking, based on new invariants and representations, and is appropriate to open as well as closed curves of unorganized data. Among appropriate applications are shape-based indexing into image databases.

We present an algorithm that extracts curves from a set of edgels within a specific class in a decreasing order of their "length". The algorithm inherits the perceptual grouping approaches. But, instead of using only local cues, a global constraint is imposed on each extracted subset of edgels, that the underlying curve belongs to a specific class. In order to reduce the complexity of the solution, we work with a linearly parameterized class of curves, a function of one image coordinate. This first allows one to use recursive Kalman based fitting and, second, to cast the problem as an optimal path search in a directed graph. Experiments on finding lane-markings on roads demonstrate that real-time processing is achievable.

New representations are developed for 2D IP (implicit polynomial) curves of arbitrary degree. These representations permit shape recognition and pose estimation with essentially single, rather than iterative, computation, and extract and use all the information in the polynomial coefficients. This is accomplished by decomposing polynomial coefficient space into a union of orthogonal subspaces for which rotations within two dimensional subspaces or identity transformations within one dimensional subspaces result from rotations in x, y measured-data space. These rotations in the two dimensional coefficient subspaces are related in simple ways to each other and to rotation in the x, y data space. By recasting this approach in terms of complex polynomials, i.e., z=x+iy and complex coefficients, further simplification occurs for rotations and some simplification occurs for translation.

This paper outlines a geometric parameterization of 2D curves where the parameterization is in terms of geometric invariants and terms that determine an intrinsic coordinate system. Thus, we present a new approach to handle two fundamental problems: single-computation alignment and recognition of 2D shapes under affine transformations. The approach is model-based, and every shape is first fit by an implicit fourth degree (quartic) polynomial. Based on the decomposition of this equation into three covariant conics, we are able to define a unique intrinsic reference system that incorporates usable alignment information contained in the implicit polynomial representation, a complete set of geometric invariants, and thus an associated canonical form for a quartic. This representation permits shape recognition based on 8 affine invariants. This is illustrated in experiments with real data sets.

This paper presents a global scheme for 3D face reconstruction and face segmentation into a limited number of analytical patches from stereo images. From a depth map, we generate a 3D model of the face which is iteratively deformed under stereo and shape-from-shading constraints as well as differential features. This model enables us to improve the quality of the depth map, from which we perform the segmentation and the approximation of the surface.

We describe a general-purpose method for the accurate and robust interpretation of a data set of p-dimensional points by several deformable prototypes. This method is based on the fusion of two algorithms: a generalization of the iterative closest point (GICP) to different types of deformations for registration purposes, and a fuzzy clustering algorithm (FCM). Our method always converges monotonically to the nearest focal minimum of a mean-square distance metric, and experiments show that the convergence is fast during the first few iterations. Therefore, we propose a scheme for choosing the initial solution to converge to an "interesting" local minimum. The method presented is very generic and can be applied: (a) to shapes or objects in a p-dimensional space, (b) to many shape patterns such as polyhedra, quadrics, splines, (c) to many possible shape deformations such as rigid displacements, similitudes, affine and homographic transforms. Consequently, our method has important applications in registration with an ideal model prior to shape inspection, i.e. to interpret 2D or 3D sensed data obtained from calibrated or uncalibrated sensors. Experimental results illustrate some capabilities of our method.