Audio-visual localization with hierarchical topographic maps: Modeling the superior colliculus A key attribute of the brain is its ability to seamlessly integrate sensory information to form a multisensory representation of the world. In early perceptual processing, the superior colliculus (SC) takes a leading role in integrating visual, auditory and somatosensory stimuli in order to direct eye movements. The SC forms a representation of multisensory space through a layering of retinotopic maps which are sensitive to different types of stimuli. These eye-centered topographic maps can adapt to crossmodal stimuli so that the SC can automatically shift our gaze, moderated by cortical feedback. In this paper we describe a neural network model of the SC consisting of a hierarchy of nine topographic maps that combine to form a multisensory retinotopic representation of audio-visual space. Our motivation is to evaluate whether a biologically plausible model of the SC can localize audio-visual inputs live from a camera and two microphones. We use spatial contrast and a novel form of temporal contrast for visual sensitivity, and interaural level difference for auditory sensitivity. Results are comparable with the performance observed in cats where coincident stimuli are accurately localized, while presentation of disparate stimuli causes a significant drop in performance. The benefit of crossmodal localization is shown by adding increasing amounts of noise to the visual stimuli to the point where audiovisual localization significantly out performs visual-only localization. This work demonstrates how a novel, biologically motivated model of low level multisensory processing can be applied to practical, real-world input in real-time, while maintaining its comparability with biology. A key attribute of the brain is its ability to seamlessly integrate sensory information to form a multisensory representation of the world. Examples such as the McGurk and MacDonald [1] and rubber hand [2] effects and other perceptual phenomena [3, 4] demonstrate that the processing of one sense can influence significantly what we perceive in another. This influence occurs in both late and early stages of perceptual processing: as a result of semantic similarity in crossmodal stimuli, or because of spatial and temporal similarity [5] . Understanding how such seamless fusion of sensory information is achieved is therefore an important goal, and one which may have a significant impact on computational systems if we can replicate it.. In early perceptual processing, the SC takes a leading role in this multisensory integration [6] . The SC is a paired structure found in the midbrain of vertebrates, also known as the optic tectum in non-mammalian species [7] . As an evolutionary stable structure, the SC successfully combines sensory information in order to direct eye movements. In mammals, this involves combining visual, auditory and somatosensory stimuli through layers of aligned topographic maps [5] with a retinotopic organization [8] . The neurons which represent this multisensory space are significant in that they have a superadditive response to weaker crossmodal stimuli [9] , which makes the SC particularly important, for example, from a survival perspective. Imagine you are being stalked by a predator which is moving quietly and is camouflaged. The weak crossmodal audio-visual stimuli detected from the quiet sounds and camouflaged motion will provoke a disproportionately strong response from the SC (called multisensory enhancement), thus directing your eyes to the approaching predator [7] even if conflicting loud sounds are present elsewhere (multisensory suppression). As a pivotal structure in the saccadic system with significant influence over a large number of subcortical and cortical areas (Fig. 1) , the SC is therefore an important structure to study in neuroscience and computationally.. Computationally the SC has four key principles. First, the SC uses aligned topographic maps that develop spatial sensory feedback and movement initiation outputs. For convenience, ipsilateral and contralateral areas are shown combined. The superficial layers of the SC consist of the SZ which is cell free, uSGS and lSGS, and SO, all processing visual stimuli only. The deep layers of the SC are the uSGI and lSGI, SAI, SGP, and SAP, of which the SGI and SGP are known to process multisensory stimuli. Connections within the layers of the SC are not shown but are known to descend from the SGS, through the SO to the SGI. The ipsilateral and contralateral areas of the SGS, SO, SGI and SGP are also connected to the deep SC of their opposing areas, terminating mostly in the SGI. Constructed from [10] .. representations of each different sensory stimuli. Each map is aligned to form an eye-centered (retinotopic) representation with vision driving the alignment [7] . Second, during normal postnatal development, the SC gains the ability to integrate the senses after the first few months. Crucially, however, in abnormal development where one sense is deprived, the ability to learn how to integrate is not lost, and integration is gained after only a short period of exposure to crossmodal stimuli [11] . Here then, the SC not only demonstrates automatic development, detection and alignment of sensory information, but it also shows postdevelopmental adaptation. Third, while the localization of crossmodal stimuli in the SC is autonomous, superadditive multisensory integration is moderated by cortical feedback from unisensory areas [12] , demonstrating how different parts of the perceptual system work together through both feed forward and feedback connections. Fourth, the SC is a prime example of a real-time sensorimotor system that fuses crossmodal stimuli to react rapidly; in as little as 80ms for express saccades [13] . We might exploit these principles in numerous ways. For example, as a real-time sensorimotor system, computational principles extracted from the SC may have a number of applications, such as controlling binocular gaze in robotic systems [14] or enhancing camera surveillance to focus on small-scale anomalies or multisensory cues [15] . This is further enhanced with the ability to adapt, which could be used to make sensory systems more robust or allow them to be trained to discriminate between different types of crossmodal stimuli, such as for audio-visual speech recognition [16] . The ability to automatically align sensory representations could be applied in medical systems, such as in operating theaters to combine, say, fMRI and EEG data to aid surgery [17] in real-time. Lastly, the feedback and cooperation between subcortical and cortical areas that the SC exhibits may demonstrate how we can include top-down influences in bottom-up processing [18] using neural computation. Learning from the SC through modeling could therefore prove pivotal.. A number of different models of the SC have been developed to explore its properties. In general, models have been used to explore how sensory stimuli are integrated in order to characterize multisensory enhancement and suppression. For example, Bayes' rule and simple perceptrons have been applied to describe the computational properties of enhancement and suppression [19, 20] . Other mathematical models of integration have also been defined and compared with biological data [21] . Similar analysis has been conducted using neurobiologically motivated models with explicit cortical feedback [22, 23, 24] .. A small number of studies have modeled the development and adaptation of sensory alignment. Simple Hebbian association with rate-coded neurons has been used to align abstract audio and visual stimuli to produce a multisensory representation of space [25] . Other studies have used a more neurobiologically motivated spiking model of the inferior and superior colliculi coupled with spike-timing dependent plasticity (STDP) to learn how auditory stimuli can be re-aligned to displaced visual cues [26, 27] .. By far the most extensive models have simulated alignment, multisensory integration and cortical feedback together because of their interdependence. An early model by Grossberg et al. [28] examined the interaction between visual and multimodal cues for reactive and planned eye movement using a neurobiologically motivated model of burst and buildup neurons found in the deep SC. With this model Grossberg et al. explored how different senses can be automatically aligned and how enhancement and suppression can be achieved through cortical feedback. A similar model has also been used to explore the interaction between reactive and planned movement for antisaccades [29] . Perhaps the most capable neurobiological model to date has been applied within a robot simulation that learns to saccade to a target using head, neck and eye movements [30] . However, despite this model's complexity, there has been no exploitation of the potential for the SC to form the basis of a real-time fusion system going beyond mere simulations. For example, no model of the SC has been applied to real-time video and audio input. This is in contrast to the more established models of cortical visual processing, such as those which explore bottom-up [31] and top-down visual [18] attention. These models combine visual-only topographic maps sensitive to color, intensity, orientation and change in intensity over time to highlight salient areas in images. They therefore use very similar computational techniques to those employed in neurobiologically motivated models of the SC, while they have also been deployed to operate in real-time in, for example, robot platforms [32] , albeit using only a single modality.. While neurobiologically motivated models of the SC have been used to explore key attributes of multisensory integration, none have done this on a large-scale or in real-time in the same way that visual-only attention models have been applied. Applying a model of the SC to real-world audio-visual stimuli may therefore offer us insight into how such principles might be used for practical computational intelligence (CI) tasks. An application of a neuroscience model to a practical task would be significant in that it would help us understand if computational principles of low level sensorimotor brain structures can be utilized. The development and application of practical subcortical models is also significant for large-scale cortical models [33, 34] as they will allow us to explore how such models behave when faced with real-world stimuli, aiding the analysis of causal flow, such as in Darwin X [35] , and perhaps providing the hybrid architecture needed for us to bridge the gap between large-scale brain simulations [36] and cognitive architectures [37] .. In this paper, we describe a behavioral model of the SC that has sensory representations of visual and audio space and combines these into a multisensory representation. While on the one hand this attempts to provide a biologically plausible model of the SC by starting with its connectivity and functional specialization, on the other hand this is balanced with the desire to allow the model to operate on real-time video and audio signals. Although there are necessary abstractions made to achieve this, notably the use of over 12,000 rate-coded neurons in a fixed hierarchy, we demonstrate for the first time how a subcortical computational neuroscience model can be applied to real-world stimuli, and hence how such models may be applied to practical CI problems. The model also includes a novel method using temporal inhibition to discriminate between visual stimuli moving in different directions.. In section 2 we explore the computational properties of the SC in relation to known physiology and behavior to drive the development of the model. In section 3 we describe the hierarchical topographic map architecture we use to model the SC, including properties of the maps and the required inhibition between neurons. In section 4 we evaluate the model on real-time audio-visual localization and discuss the implications of the results. Finally in section 5 we summarize the contributions of this work and relate it to the wider context. We set out in this paper to demonstrate whether a biologically motivated, behavioral model of the SC could be applied to localize real-time audio-visual stimuli. With 12,240 neurons arranged into nine topographic maps, the model is capable of localizing in real-time (at two frames per second using a Java implementation). The localization performed is deliberately simple in order to match the computational capabilities of the SC. For visual sensitivity we focus on spatial sensitivity to detect broad patterns of high contrast, and temporal sensitivity to detect motion in preferred directions from the most intense changes extracted by frame differencing. Our auditory sensitivity is based upon locating the loudest sound. In the SC, this simple but fast sensitivity determines the focus of subsequent midbrain through to cortical sensory processing. For example, through a subcortical pathway to the amygdala, relevant multimodal stimuli can be assessed in more detail to detect potential threats [72] .. The principle we are following is therefore to construct a hierarchy of increasingly more complex functional processing. Hierarchical approaches are often used in computational neuroscience models, but few aim at achieving functionality. For example, the large-scale model developed by Izhikevich and Edelman [34] is impressive in its scale and use of biologically plausible computational units (a model thalamocortical column). While the model is not designed to be functional, for us a key objection to the approach is the smaller amount of detail in modeling low level structures in comparison with the detail applied to the cortical neurons. This imbalance reflects the focus on cortical operation, which is an understandable focus in exploring higher cognition, but at the expense of low level detail. Without low level detail or the influence of realworld stimuli, such models may not be able to demonstrate comparative cognitive properties. Darwin X [35] is a good example of how a combination of levels of processing on real-world stimuli can be utilized, and this is something our model aspires to by showing how low level processing can be used practically, despite its deliberate simplicity. A natural next step in our model development is to explore whether a spiking model of the SC could achieve similar results to our rate-coded model and then use this to enhance models of the thalamocortical column to make them functional. The potential for this has been demonstrated with a hierarchical model of auditory localization in the inferior colliculus using leaky integrate-and-fire neurons [73] .. To improve localization, we could implement different spatial maps with varying contrast sensitivity. The model of visual attention developed by Itti et al. [31] is a good example of this with a real-time implementation. Our approach is very similar to theirs in that we also use a hierarchy of topographic maps to select salient locations. Since they use cortical visual structures for motivation, their model is sensitive to color, intensity and orientation. With our focus on the SC, we model visual contrast and motion. In other respects our models are also very similar, such as the combination of hierarchical processing and competitive selection of saliency. Yet our model is distinguished by its more biologically motivated implementation of maps, plus the combination of auditory sensitivity which we use to explore how multimodal cues interact.. Our biological motivation is evident in the way we use temporal inhibition to detect motion. Sensory neurons use pre-filtering, delay filtering and non-linear interactions [62] to detect motion. We use simple frame differencing to prefilter incoming stimuli, temporal inhibition between neurons to concentrate sensitivity on stimuli moving in preferred directions, and non-linear interactions between neurons to select the highest response. This combination of simple mechanisms achieves high detection rates in simple scenes, but lower rates in scenes with conflicting motion stimuli. Further evaluation of this mechanism is needed to understand how it performs against computational, rather than biologically motivated, approaches [cf. 74] .. For auditory localization we use ILD. This intensitybased approach suffers from a number of limitations. First, in order to detect energy differences between the two fixed microphones, they need to be calibrated. We chose two identically branded microphones which reduced the need for calibration, yet both varied in their performance. Reverberation within the room also impacts on localization. Second, the energy recorded at a microphone represents all locations on the radius of a sphere centered on the microphone, where the radius is determined by the comparative energy received by the second microphone. In order to localize we have to assume that the sound source is horizontal between the microphones (to avoid the 'cone of confusion'). With more microphones, some of these assumptions can be avoided. Third, if more than one sound source is present, this can disrupt the localization because the energies compared may come from different locations. To overcome these limitations a range of techniques can be used across frequency bands, or more accurate localization via calibrated equipment such as Microsoft's Kinect [64] .. Returning back to the mammalian localization, an important aspect of the SC is its ability to automatically translate auditory spatial coordinates into visual coordinates. This means that we can move our eyes relative to our ears and still combine coincident cues. This is achieved through feedback from the SC to the IC. In our model we have assumed a fixed camera to microphone setup and a purely feedforward operation. Feedback from the multisensory space could be used to adjust the relative position of the eyes in the auditory space. An example of how this might be achieved is provided by Huo and Murray [27] . Notably their model uses spiking neurons with spike timing dependent plasticity to learn the alignment between multimodal stimuli. This type of approach, using more biologically plausible models of neurons and learning, coupled with cortical feedback and a movable camera position is the next step in our work. These results demonstrate that our model of the SC is capable of processing visual and auditory stimuli in realtime to form a localization to relevant multimodal stimuli. This localization performs well on single-modality stimuli, and is significantly enhanced for coincident multimodal stimuli, especially when the visual stimulus is weakened. However, the increase in performance is not a straightforward combination of single-modality performance since audio-visual localization achieves a maximum of 94% and a minimum of 41% when noise (variance 176 or 240) is applied. This compares to 99% for auditory stimuli alone. If we assume that each stimulus is treated independently in combined localization as per Stein et al. [67] , then the probability of correctly locating an audio-visual stimulus P av can be calculated using the independent probabilities of correctly responding to audio P a and visual P v stimuli:. Since P a = 1 for all targets, P av = 1. This holds for any level of visual noise. However, we can see from our results that the actual performance does not achieve this. Stein et al. [67] found a similar response in cats, and concluded that the combination must therefore be multiplicative, rather than additive, and this led to investigations into multisensory enhancement and suppression [9] .. In addition to this, two general models of sensory integration for spatial localization have been proposed. Knudsen and Brainard [68] reviewed audio-visual integration in the tectum and the forebrain to conclude that vision dominates whenever visual information is available. In contrast, Ernst and Banks [69] proposed a maximum-likelihood integrator in which the brain's estimate of the reliability of a given modality is used to weight the relative contribution the modality has to the integration. Both models have been evaluated by Battaglia et al. [70] who conclude that each is partially correct in that noise within a modality is taken into account in the integration, but that one sense still tends to dominate.. In our model the combination of responses is achieved through a linear weighting of the outputs from independent modality topographic maps, yet the resulting localization performance appears to be non-linear. For exam-ple, for stimuli at −15. • , we get 63% visual-only localization, 100% audio-only, but 63% combined (Table 5 ). This effect is caused by the competition between neurons in the multisensory map. With conflicting audio-visual localizations, the map forms a combined response by selecting a winning neuron which has the highest combined input. The net effect is that regions with lower responses which overlap can have higher combined responses than a single modality. It is therefore the overlap which becomes as important as a large single-modality response. This is a crude form of multisensory enhancement of coincident low intensity cues, versus suppression of high intensity singlemodality cues and is demonstrated in the way that localization performance is maintained in our model even when the visual stimulus is degraded by noise.. Although our model can successfully combine audiovisual stimuli, the combination is fixed and is not based on maximum-likelihood. We selected weightings of each map within the hierarchy ( Table 4 ) to ensure that any one map can provoke a multisensory response when presented with test stimuli consisting of noise. A limitation of our choice of weights was to bias activity towards motion detection, and hence our model is dominated by vision as per Knudsen and Brainard [68] . However, current work on the mechanisms underlying enhancement and suppression have shown that integration is more complex in that it depends upon coincident activity in sensory cortical areas feeding back into the SC [12] . This feedback may explain the findings of Battaglia et al. [70] such that the SC internally provides a visual bias, while cortical feedback performs the likelihood estimate to override the integration. In this paper we modeled integration without feedback, but we have explored how such feedback might be modeled with topographic maps representing cortical structures [24] . Alternate approaches using spiking neurons have also been explored in higher level vision [71] . So far such techniques are too crude to shed any further light on this phenomenon, yet they show promise in that they can be trained using conditioning-type feedback to respond differently to varying stimuli [54, 59] , perhaps showing how coincident cues could be recognized and enhanced cortically.

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O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O 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Strongly Consistent Model Order Selection for Estimating 2-D Sinusoids in Colored Noise Abstract-The problem of jointly estimating the number as well as the parameters of 2-D sinusoidal signals, observed in the presence of an additive colored noise field, is considered. We begin by establishing the strong consistency of the nonlinear least squares estimator of the parameters of 2-D sinusoids, when the number of sinusoidal signals assumed in the field is incorrect. Based on these results, we prove the strong consistency of a new family of model order selection rules. We have considered the problem of jointly estimating the number as well as the parameters of 2-D sinusoidal signals, observed in the presence of an additive colored noise field. We have established the strong consistency of the LSE when the number of sinusoidal signals is underestimated, or overestimated. In the case where the number of sinusoidal signals is underestimated, we have shown the almost sure convergence of the LSEs to the parameters of the dominant sinusoids. In the case where this number is overestimated, the estimated parameter vector obtained by the least squares estimator contains a subvector that converges almost surely to the correct parameters of the sinusoids. Based on these results, we proved the strong consistency of a new family of model order selection rules for the number of sinusoidal components and their parameters. The applicability and validity of the asymptotic results to finite-dimensional observations are demonstrated using Monte Carlo experiments.

O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O


Towards persistence-based reconstruction in euclidean spaces Manifold reconstruction has been extensively studied for the last decade or so, especially in two and three dimensions. Recent advances in higher dimensions have led to new methods to reconstruct large classes of compact subsets of R d . However, the complexities of these methods scale up exponentially with d, making them impractical in medium or high dimensions, even on data sets of low intrinsic dimensionality.In this paper, we introduce a novel approach that stands in-between classical reconstruction and topological estimation, and whose complexity scales up with the intrinsic dimension of the data. Our algorithm combines two paradigms: greedy refinement, and topological persistence. Given a point cloud in R d , we build a set of landmarks iteratively, while maintaining a nested pair of abstract complexes, whose images in R d lie close to the data, and whose persistent homology eventually coincides with the homology of the underlying shape. When the data points are densely sampled from a smooth m-submanifold X of R d , our method retrieves the homology of X in time at most c(m)n 5 , where n is the size of the input and c(m) is a constant depending solely on m.To prove the correctness of our algorithm, we investigate onČech, Rips, and witness complex filtrations in Euclidean spaces. More precisely, we show how previous results on unions of balls can be transposed toČech filtrations, and from there to Rips and witness complex filtrations. Finally, investigating further on witness complexes, we quantify a conjecture of Carlsson and de Silva, which states that witness complex filtrations should have cleaner persistence barcodes thanČech or Rips filtrations, at least on smooth submanifolds of Euclidean spaces. The problem of reconstructing unknown structures from finite collections of data samples is ubiquitous in the Sciences, where it has many different variants, depending on the nature of the data and on the targeted application. In the last decade or so, the computational geometry community has gained a lot of interest in manifold reconstruction, where the goal is to reconstruct submanifolds of Euclidean spaces from point clouds. Efficient solutions have been proposed in dimensions two and three, based on the use of the Delaunay triangulation -see [8] for a survey. Recently, significant steps were made towards a full understanding of the potential and limitations of the Delaunay-based approach in arbitrary dimensions [14, 30] . In parallel, new sampling theories were developped, such as the critical point theory for distance functions [9] , which provides sufficient conditions for the topology of a shape X ⊂ R d to be captured by the offsets of a point cloud L lying at small Hausdorff distance. These advances lay the foundations of a new theoretical framework for the reconstruction of smooth submanifolds [11, 29] , and more generally of large classes of compact subsets of R d [9, 10, 12] . Combined with the introduction of more lightweight data structures, such as the witness complex [16] , they have led to new provably-good algorithms [6] whose complexities can be orders of magnitude below the one of the classical Delaunay-based approach. For instance, on a data set with n points in R d , the algorithm of [6] remains too large for these methods to be practical, even when the data points lie on a low-dimensional submanifold.. A weaker yet similarly difficult version of the reconstruction paradigm is topological estimation, where the goal is to infer the topological invariants of X from an input point cloud L. This problem has received a lot of attention in the recent years, and it finds applications in a number of areas of Science, e.g. sensor networks [18] , statistical analysis [7] , or dynamical systems [28, 31] . A classical approach consists in building a nested sequence of spaces K 0 ⊆ K 1 ⊆ · · · ⊆ K m , and in studying the persistence of homology classes throughout this sequence. It has been independently proved in [12] and [15] that the persistent homology of the sequence defined by the α-offsets of a point cloud L coincides with the homology of the underlying shape X under mild sampling conditions. Specifically, if the Hausdorff distance between L and X is less than ε, for some small enough ε, then, for all sufficiently small α ≥ ε, the canonical inclusion map L α → L α+2ε induces homomorphisms between homology groups, whose images are isomorphic to the homology groups of X. Combined with the structure theorem of [33] , which states that the persistent homology of the sequence {L α } α≥0 is fully described by a finite set of intervals, called a persistence barcode or a persistence diagram -see Figure 1 , the above result means that the homology of X can be deduced from this barcode, simply by removing the intervals of length less than 2ε, which are therefore viewed as topological noise.. From an algorithmic point of view, the persistent homology of a nested sequence of simplicial complexes (called a filtration) can be efficiently computed using the persistence algorithm [21, 33] . Among the many filtrations that can be built on top of a point set L, the α-shape enables to reliably recover the homology of the underlying space X, since it is known to be a deformation retract of L α [20] . However, this property is useless in high dimensions, since computing the α-shape requires to build the full-dimensional Delaunay triangulation. It is therefore appealing to consider other filtrations that are easy to compute in arbitrary dimensions, such as the Rips and witness complex filtrations. In this paper, we produce an equivalent of the result of [12, 15] for these filtrations, and more generally for any filtration that is intertwined with theČech filtration. Recall that, for all α > 0, theČech complex C α (L) is the nerve of the union of the open balls of same radius α about the points of L. It is known to be homotopy equivalent to L α . However, combining this fact with the result of [12, 15] is not enough to prove that the persistent homology of C α (L) → C α+2ε (L) coincides with the homology of X, because it is unclear whether the homotopy equivalences. In the paper, we show that there exist homotopy equivalences that commute with canonical inclusions, at least at homology and homotopy levels. This enables us to extend the result of [12, 15] to theČech filtration, and from there to the Rips and witness complex filtrations.. Another common concern in topological data analysis is the size of the vertex set on top of which the filtration is built. Indeed, in practical situations where the input data set W samples the underlying shape very finely, it makes sense to build the filtration on top of a small subset L of landmarks to avoid a waste of computational resources. However, downsampling the vertex set may result in a significant degradation in the quality of the persistence barcode. This is true in particular with theČech and Rips filtrations, whose barcodes can have topological noise of amplitude depending directly on the density of the landmark set L. The introduction of the witness complex filtration appeared as an elengant way of solving this issue [17] . The witness complex of L relative to W , or CW (L) for short, can be viewed as a relaxed version of the Delaunay triangulation of L, in which the points of W \ L are used to drive the construction of the complex [16] . Due to its special nature, which takes advantage of the points of W \ L, the witness complex filtration is likely to give persistence barcodes whose topological noise depends on the density of W rather than on the one of L, as conjectured in [17] . We prove that this statement is only true to some extent, namely: whenever the points of W are sufficiently densely sampled from some smooth submanifold of R d , the topological noise in the barcode can be arbitrarily small compared to the density of L. Nevertheless, it cannot depend solely on the density of W . This shows that the witness complex filtration does provide cleaner persistence barcodes, though maybe not as clean as expected.. Taking advantage of the above results, we propose a novel approach to reconstruction that stands in-between the classical reconstruction and topological estimation paradigms. Our algorithm is a variant of the method of [6, 26] that combines greedy refinement and topological persistence. Given an input point cloud W , the algorithm builds a subset L of landmarks iteratively, and in the meantime it maintains a nested pair of simplicial complexes (Rips or witness complexes) and computes its persistent Betti numbers. The outcome of the algorithm is the diagram showing the evolution of these persistent Betti numbers. Using this diagram, a user or software agent can determine a relevant scale at which to process the data. It is then easy to rebuild the corresponding set of landmarks, as well as its nested pair of complexes. Although our method does not really compute an embedded complex that is close to X topologically and geometrically, it comes with theoretical guarantees, it is easily implementable, and it has reasonable complexity. Indeed, in the case where the input point cloud W is densely sampled from a smooth submanifold X of R d , we show that the complexity of our algorithm is bounded by c(m)n 5 , where c(m) is a quantity depending solely on the intrinsic dimension m of X, and n is the size of W . To the best of our knowledge, this is the first provably-good topological estimation or reconstruction method whose complexity scales up with the intrinsic dimension of the data. When X is a more general compact set in R d , our complexity bound becomes c(d)n 5 . The paper is organized as follows: after introducing thě Cech, Rips, and witness complex filtrations in Section 2, we prove our structural results in Sections 3 and 4, focusing on compact subsets of R d in Section 3, and on the particular case of smooth submanifolds in Section 4. Finally, we present our algorithm and its analysis in Section 5. This paper makes effective the approach developped in [12, 15] by providing an efficient, provably-good and easy-toimplement algorithm for the topological and geometric analysis of point cloud data in arbitrary dimensions. Addressing a weaker version of the classical reconstruction problem, the algorithm ultimately outputs a nested pair of complexes at a user-defined scale, from which the homology of the underlying shape X can be inferred. When X is a smooth submanifold of R d , the complexity of the algorithm scales up with the intrinsic dimension of X and not with the ambient dimension d, assuming that the pairwise distances between the data points have been pre-computed. Thus, a new step is made towards reconstructing (low-dimensional) manifolds in high-dimensional spaces in reasonable time with guarantees. However, there still remains the challenging problem of constructing an embedded complex that is topologically equivalent and geometrically close to the sampled shape.. The theoretical framework developed in the paper can be used for the analysis of various persistence-based methods in Euclidean spaces. It can also virtually be applied in any metric space (provided that the result of [12, 15] on unions of balls can be extended), thanks to the genericity of Lemma 3.4 and of the arguments of Section 3.2.1. A class of spaces of particular interest to us is the class of compact Riemannian manifolds, possibly with boundaries, with applications in machine learning and sensor networks.

O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O B O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O B I O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I B O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Exact Random Coding Exponents for Erasure Decoding Abstract-Random coding of channel decoding with an erasure option is studied. By analyzing the large deviations behavior of the code ensemble, we obtain exact single-letter formulas for the error exponents in lieu of Forney's lower bounds. The analysis technique we use is based on an enhancement and specialization of tools for assessing the statistical properties of certain distance enumerators. We specialize our results to the setup of the binary symmetric channel case with uniform random coding distribution and derive an explicit expression for the error exponent which, unlike Forney's bounds, does not involve optimization over two parameters. We also establish the fact that for this setup, the difference between the exact error exponent corresponding to the probability of undetected decoding error and the exponent corresponding to the erasure event is equal to the threshold parameter. Numerical calculations indicate that for this setup, as well as for a Z-channel, Forney's bound coincides with the exact random coding exponent.

O B I O O B O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O B I O


Enabling

knowledge

representation

on

the

Web

by

extending

RDF

schema

Recently,

there

has

been

a

wide

interest

in

using

ontologies

on

the

Web.

As

a

basis

for

this,

RDF

Schema

(RDFS)

provides

means

to

define

vocabulary,

structure

and

constraints

for

expressing

metadata

about

Web

resources.

However,

formal

semantics

are

not

provided,

and

the

expressivity

of

it

is

not

enough

for

full-fledged

ontological

modeling

and

reasoning.

In

this

paper,

we

will

show

how

RDFS

can

be

extended

in

such

a

way

that

a

full

knowledge

representation

(KR)

language

can

be

expressed

in

it,

thus

enriching

it

with

the

required

additional

expressivity

and

the

semantics

of

this

language.

We

do

this

by

describing

the

ontology

language

OIL

as

an

extension

of

RDFS.

An

important

benefit

of

our

approach

is

that

it

ensures

maximal

sharing

of

meta-data

on

the

Web:

even

partial

interpretation

of

an

OIL

ontology

by

less

semantically

aware

processors

will

yield

a

correct

partial

interpretation

of

the

meta-data.

We

conclude

that

our

method

of

extending

is

equally

applicable

to

other

KR

formalisms.

Currently,

computers

are

changing

from

single

isolated

devices

into

entry

points

into

a

worldwide

network

of

information

exchange

and

business

transactions

(cf.

Support

in

data,

information,

and

knowledge

exchange

is

becoming

the

key

issue

in

current

computer

technology.

Ontologies

will

play

a

major

role

in

supporting

information

exchange

processes

in

various

areas..

Many

definitions

of

ontologies

have

been

given

in

the

last

decade,

but

one

that,

in

our

opinion,

best

characterizes

the

essence

of

an

ontology

is

based

on

the

related

definitions

by

An

on-tology

is

a

formal,

explicit

specification

of

a

shared

conceptualisation.

A

conceptualisation

refers

to

an

abstract

model

of

some

phenomenon

in

the

world

which

identifies

the

relevant

concepts

of

that

phenomenon.

Explicit

means

that

the

type

of

concepts

used

and

the

constraints

on

their

use

are

explicitly

defined.

Formal

refers

to

the

fact

that

the

ontology

should

be

machine

understandable,

i.e.

the

machine

should

be

able

to

interpret

the

semantics

of

the

information

provided.

Shared

reflects

the

notion

that

an

ontology

captures

consensual

knowledge,

that

is,

it

is

not

restricted

to

some

individual,

but

accepted

by

a

group..

Ontologies

will

allow

structural

and

semantic

definitions

of

documents,

providing

completely

new

possibilities:

intelligent

search

instead

of

keyword

matching,

query

answering

instead

of

information

retrieval,

document

exchange

via

ontology

mappings,

and

definition

of

views

on

documents..

RDF

Schema

provides

means

to

define

vocabulary,

structure

and

constraints

for

expressing

metadata

about

Web

resources.

However,

formal

semantics

for

the

primitives

defined

in

RDF

Schema

are

not

provided,

and

the

expressivity

of

these

primitives

is

not

enough

for

full-fledged

ontological

modeling

and

reasoning.

To

perform

these

tasks,

an

additional

layer

on

top

of

RDF

Schema

is

needed.

Tim

Berners-Lee

calls

this

layered

architecture

the

Semantic

Web

At

the

lowest

level

of

the

Semantic

Web

a

generic

mechanism

for

expressing

machine

readable

semantics

of

data

is

required.

The

Resource

Description

Framework

(RDF)

is

this

foundation

for

processing

metadata,

providing

a

simple

data

model

and

a

standardized

syntax

for

metadata.

Basically,

it

provides

the

language

for

writing

down

factual

statements.

The

next

layer

is

the

schema

layer

(provided

by

the

RDF

Schema

specification

We

will

show

how

a

formal

knowledge

representation

language

can

be

used

as

the

third,

logical,

layer.

We

will

illustrate

this

by

defining

the

ontology

language

OIL

as

an

extension

of

RDF

Schema..

OIL

(Ontology

Inference

Layer),

a

major

spin-off

from

the

IST

project

On-To-Knowledge

is

a

Web-based

representation

and

inference

layer

for

ontologies,

which

unifies

three

important

aspects

provided

by

different

communities:

formal

semantics

and

efficient

reasoning

support

as

provided

by

Description

Logics,

epistemological

rich

modeling

primitives

as

provided

by

the

Frame

community,

and

a

standard

proposal

for

syntactical

exchange

notations

as

provided

by

the

Web

community.

The

content

of

the

paper

is

organized

as

follows.

In

section

we

provide

a

short

introduction

to

RDF

and

RDF

Schema.

Section

provides

a

very

brief

introduction

into

OIL.

Section

illustrates

in

detail

how

RDF

Schema

can

be

extended,

using

OIL

as

an

example

knowledge

representation

language.

The

result

is

an

RDF

Schema

definition

of

OIL

primitives,

which

allows

one

to

express

any

OIL

ontology

in

RDF

syntax.

In

section

we

discuss

how

our

approach

enables

the

added

benefits

of

OIL,

such

as

reasoning

support

and

formal

semantics,

to

be

used

on

the

Web,

while

retaining

maximal

compatibility

with

'pure'

RDF(S).

Finally,

we

provide

our

conclusions

in

section

The

serialization

we

propose

gives

us

enough

expressiveness

to

translate

any

possible

OIL

class

definition

to

an

RDF

syntax.

Use

of

RDF(S)

specific

constructs

is

maximized

without

sacrificing

clarity

of

the

specification,

to

enable

RDF

agents

that

are

not

OIL-aware

to

understand

as

much

of

the

specification

as

possible,

while

retaining

the

possibility

to

translate

back

to

OIL

unambiguously..

In

the

next

section,

we

will

examine

how

to

serialize

global

slot

definitions.

In

this

article,

we

have

shown

why

a

machine-accessible

representation

of

the

information

available

on

the

Web

is

both

useful

and

necessary.

We

have

also

shown

that

RDFS

is

only

a

small

step

towards

the

required

expressiveness

for

the

Semantic

Web.

Finally,

we

have

illustrated

how

RDFS

still

can

be

used

to

this

end,

by

extending

it

with

additional

modeling

primitives

as

defined

by

a

more

formal

knowledge

representation

scheme,

such

as

OIL..

An

important

advantage

of

our

approach

is

the

maximization

of

the

compatibility

with

RDFS:

not

only

is

every

RDF

Schema

document

a

valid

OIL

ontology

declaration,

but

every

OIL

ontology

can

be

partially

interpreted

by

a

semantically

poorer

processing

agent.

This

partial

interpretation

will

of

course

be

incomplete,

but

correct

under

the

intented

semantics

of

the

ontology.

We

firmly

believe

that

our

way

of

extending

is

generally

applicable

across

knowledge

representation

formalisms.

Work

on

ontology

representation

languages

dates

back

to

the

work

on

frame-languages

in

the

early

days

of

AI.

However,

efforts

of

designing

ontology-representation

languages

that

are

Webenabled

only

date

from

recent

years.

The

most

prominent

(or

even:

the

only)

efforts

in

this

area

have

been

SHOE

Ontobroker

OIL

and

DAML-ONT

and

more

recently,

as

a

replacement

for

DAML-ONT,

DAML+OIL

Of

these,

only

the

last

three

have

been

defined

on

top

of

RDF(S).

Since

DAML+OIL

is

essentially

a

merger

between

OIL

and

DAML-ONT,

we

will

focus

on

the

comparison

of

our

own

proposal

(OIL)

to

DAML-ONT..

DAML-ONT

shares

with

our

own

proposal

the

principle

that

an

ontology

language

should

maintain

maximum

backwards

compatibility

with

existing

web

standard

languages,

and

in

particular

RDF

Schema.

The

difference

between

OIL

and

DAML-ONT

lies

in

the

degree

to

which

the

languages

succeed

in

maximising

the

ontological

content

that

can

be

understood

by

an

"RDF

Schema

agent"

(i.e.

an

application

that

understands

RDF

Schema

but

does

not

recognise

the

language

specific

extensions,

OIL

or

DAML-ONT).

Unlike

OIL,

DAML-ONT

is

built

on

top

of

RDFS

in

a

way

that

allows

little

if

any

ontology

content

to

be

understood

by

an

RDFS

agent.

In

OIL,

for

example,

stating

simple

subclass

relationships

between

classes

is

done

using

the

RDFS

subClassOf

property:.

<rdfs:Class

ID="Male">

<rdfs:subClassOf

rdf:resource="#Animal"/>

</rdfs:Class>

This

part

of

OIL

ontologies

is

therefore

accessible

to

any

RDFS

agent.

In

contrast,

DAML-ONT

uses

its

own

locally

defined

"subClassOf"

property,

for

example:

The

DAML-ONT

subClassOf

property

is

then

defined

to

be

"equivalentTo"

rdfs:subClassOf,

but

the

definition

of

"daml:equivalentTo"

itself

relies

cyclicly

on

the

definition

of

daml:sub-PropertyOf.

Therefore

even

simple

subclass

relationships

in

a

DAML

ontology

are

inaccessible

to

an

RDFS

agent.

The

situation

is

even

worse

when

it

comes

to

more

complex

class

definitions.

For

example,

the

definition

of

the

class

"TallMan"

is

the

intersection

of

the

classes

"Man"

and

"TallThing"

is

expressed

in

DAML-ONT

as:.

<daml:Class

ID="TallMan">

<daml:intersectionOf

parseType="daml:collection">

<daml:Class

about="#TallThing"/>

<daml:Class

about="#Man"/>

</daml:intersectionOf>

</daml:Class>

This

is

completely

opaque

to

an

RDFS

agent

as

it

will

not

understand

the

semantics

of

"daml:intersectionOf"

In

OIL,

the

definition

of

TallMan

would

rely

on

the

fact

that

intersection

is

implicit

in

the

semantics

of

rdfs:subClassOf:.

<rdfs:Class

ID="TallMan">

<rdf:type

rdf-schema/2000/11/10-oil-standard#DefinedClass"/>

<rdfs:subClassOf

rdf:resource="#TallThing"/>

<rdfs:subClassOf

rdf:resource="#Man"/>

</rdfs:Class>

making

the

sub-class

relationships

accessible

to

any

RDF

Schema

agent.

In

conclusion,

we

argue

that:.

OIL

and

DAML-ONT

are

currently

the

only

two

web-based

ontology

languages

that

are

built

on

top

of

RDF;

of

these,

OIL

achieves

a

much

larger

degree

of

compatibility

with

RDF

processors..

As

mentioned

earlier,

DAML+OIL

is

a

proposal

for

an

ontology

language

that

merges

the

ideas

incorporated

in

DAML-ONT

with

those

in

OIL.

Specifically,

many

of

the

ideas,

presented

in

this

article,

on

how

to

represent

a

KR

language

in

RDFS

have

been

adopted

by

DAML+OIL.

In

effect,

DAML+OIL

is

"backward

compatible"

with

RDF

to

a

much

larger

degree

than

the

initial

DAML-ONT

language

is.

O B I O O O O O B O O O O O O O O O O B O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O B I O O O O O O O O B I O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O B O O O O O O B O B O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O B O O O O O O O O O B O O O O O O B O O O O O B O O O B I O O O O O B O O O B O O O O O O O B O O B O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O B O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O B O O O O O O O O O O O O O O O O O O B O O O O B O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O B O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O B O O O O O O O O O B O O O B O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O B B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O B O O O O O O O O O O O O O O O O B O O B O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O


Evaluating invariances in document layout functions With the development of variable-data-driven digital presseswhere each document printed is potentially unique there is a need for pre-press optimization to identify material that is invariant from document to document. In this way rasterisation can be confined solely to those areas which change between successive documents thereby alleviating a potential performance bottleneck. Given a template document specified in terms of layout functions, where actual data is bound at the last possible moment before printing, we look at deriving and exploiting the invariant properties of layout functions from their formal specifications. We propose future work on generic extraction of invariance from such properties for certain classes of layout functions. This research is concerned with optimising the rendering of template-based documents as exemplified in the DDF approach [1] [2] in which the placement of items in the document is defined by nested layout functions. An Example of such a function is an X Flow, which places each item in the flow to the right of the previous item.. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. This work shows very clearly the possibilities for pre-press optimisation of advanced template-based documents in a variabledata-driven workflow. If for example a retail company wanted to customise its catalogue based on its subscribers' tastes and preferences they might create a variable data document which uses the information in their database to drive a digital press. By analysing the document before printing begins it will be possible to take advantage of the bound and invariant sections (the b items in the foregoing discussion) which can be identified as fully-bound SVG and possibly pre-rasterised. Even the late-bound sections (u), with customer-specific information, may still benefit from an analysis of the properties of each unbound item with respect to ranges of values of the missing arguments.. The underlying premise of our work is that each layout function operates on axis-aligned rectangles so, in principle, the layout process only requires to know the relative x and y coordinates of any given item and the relative coordinates of its bounding box. It may be possible to determine that some properties of a given rectangle's contents (e,g, fill colour, line Under these circumstances the layout process can still go ahead for later items in that particular flow and, if desired, a PPML job specification could be drawn up to help a digital press to optimise the rendering [4] .. At the moment the code to evaluate invariances is handgenerated for each function. A formal standardisation and declaration of the interface to each function (including resources used, graphical and combinatorial properties e.g. associativity and an indication of the as-yet-unbound arguments) would enable SVG code generation to take place based simply on the values in the interface and without having to know the internal details of the function's coding. The system is implemented as an XSLT program which takes as its input a mixed namespace document containing both SVG and elements in its own layout namespace. It then processes this document, possibly with some extra data, and produces another mixed content document as the output. By successively re-evaluating the resultant document with additional data it reaches a stage where there is no more work to be done and the output is a fully grounded document which can be rendered by an SVG renderer.. Evaluation occurs in two depth-first passes over the document. The first pass replaces unbound items in the source document with any provided arguments that make these items become more fully bound. The result of this first phase of evaluation is then used as the input for the second phase. The second phase again recursively descends the document copying through, unchanged, anything in the SVG namespace and attempting to evaluate items in the layout namespace. Note that given the nature of the processing these two passes could be combined into one pass however they have been seperated out for reasons of code tidyness and ease of debugging.. The template matching system in XSLT is ideal for the above process because the code to evaluate each different layout function can be written as a matching template for that function e.g. < layout : xflow/ >. These templates are separated out from the main program and included using the 'import' instruction in XSLT which allows different files to be included and executed as though they were a single program.. The semantics of this system are such that each layout function in a document is self contained and its ultimate geometric size is in no way affected by either its siblings or ancestors. This benefits the evaluation code because it is guaranteed that once a layout function has been passed through the layout processor and partially evaluated there is nothing more that can be done unless more unbound items become bound. Also, from a performance point of view, this allows for a high degree of parallelisation.

O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


The relative value of labeled and unlabeled samples in pattern recognition with an unknown mixing parameter We observe a training set Q composed of 1 labeled samples {(X,,O1),..., (Xl, O,)} and u unlabeled samples {Xi, . . , Xg }. The labels 0, are independent random variables satisfying Pr (0, = 1) = 7, Pr (0, = Z} = 1 -p. The labeled observations X; are independently distributed with conditional density f~, (.) given 0,. Let (X0, 0,) be a new sample, independently distributed as the samples in the training set. We observe X0 and we wish to infer the classification 00. In this paper we first assume that the distributions fl (.) and fi (.) are given and that the mixing parameter 11 is unknown. We show that the relative value of labeled and unlabeled samples in reducing the risk of optimal classifiers is the ratio of the Fisher informations they carry about the parameter 7. We then assume that two densities g1 (.) and g2 (.) are given, but we do not know whether g1 (.) = fl (.) and g2 (.) = f2 (.) or if the opposite holds, nor do we know 7. Thus the learning problem consists of both estimating the optimum partition of the observation space and assigning the classifications to the decision regions. Here, we show that labeled samples are necessary to construct a classification rule and that they are exponentially more valuable than unlabeled samples. w E ADDRESS the problem of the optimal use of a training set Q composed of 1 labeled samples {(Xl, &), . . , (Xl, &)} and u unlabeled samples {Xi,... , Xl} in the construction of a classifier that discriminates between two classes of observations, called Class 1 and Class 2, and we propose a solution under two particular sets of hypotheses. We denote the prior probability of observing a sample of Class 1 by 7, and the densities of the observations of Class 1 and Class 2 by fi (.) and fa (.) , respectively. We analyze the behavior of R(Z, u), the probability of classification error of an optimal procedure based on a training set that contains I labeled samples and u unlabeled samples. Manuscript received December 16, 1994; revised October 6, 1995. This work was partially supported by NSF under Grant NCR-8914538-02, by ARPA under , and by JSEP under Contract DAAHOC 94-G-0058, and was done when both authors were with Stanford University. The material in this paper was presented in at the International Symposium on Information Theory, Trondheim, Norway, 1994 We have analyzed this problem in previous work [2] under the assumption that the density of the unlabeled samples is identifiable, and we have considered training sets composed of an infinite number of unlabeled samples and a finite number of labeled samples. Optimum classification rules exist in'such a framework. Using the notation R(Z, a) to denote the risk of such optimum rules, it is shown in [2] that R(0, u) = R(0, co) = l/2, for all u and thus labeled samples are necessary to construct a classifier.. It is also shown in [2] that when the training set contains an infinite number of unlabeled samples and one labeled sample, the probability of error of the optimum classifier is given by R(1, co) = 2R*(l -R*) / where Rx = s min{rlfl(x),Tfdx)} dx denotes the Bayes risk. Finally, it is also shown in [2] that additional labeled samples make the probability of error converge exponentially fast to the Bayes risk in the sense that R(Z, co) -R* = exp {-I D + o(Z)}.. We have! proved that the exponent D is equal to -log 2&i c s d@?%?dx > where 17 is a shorthand notation for 1 -r~. In this paper, we first assume that the densities fi (.) of the samples of Class 1 and fz(.) of the samples of Class 2 are given and that the probability n of observing a sample of Class 1 is unknown. Using a result by O'Neill [21] , we show that the risk R(Z, r~) of a Bayesian classifier with positive and continuous prior h(n) over [0, l] satisfies the asymptotic relation AR(z,u) k R(Z,u) -R* = z~ JuI. (1 + o(l)) I u where c is a function of TZ, fi (.), and fa(.) but not of the prior h( .) . We conclude that the first-order term in the expansion of AR (Z, u) is the same for all nondegenerate Bayesian classifiers and that the relative value of labeled and unlabeled samples in reducing R(Z, ZL) is equal to the ratio of their Fisher informations. Moreover, labeled samples are not necessary in this framework and one can construct a decision rule based solely on unlabeled observations, the risk of which converges to R* with rate u-l. The main result of this paper is based on the assumption that two densities g1 (.) and ga (.) are given, and that we do not know whether gl(.) = fl(.) or gl(.) = fz(.). As before, the mixing parameter n is unknown. We show that labeled samples are now necessary to construct a classifier, and we prove that, when 13+Eu-1 + 0 as I + u + co, the risk R(Z, This implies that, if uexp {-D 1) + 0 and 1 = o(u), the additional risk AR(Z, u) is essentially determined by the number of unlabeled samples in the training set, while if the number of unlabeled samples TL grows faster than exp {D Z}, then AR(Z, u) is essentially determined by the number of labeled samples.. These results should extend to the more general framework where the mixing parameter v and the the densities fi (.) of the samples of Class 1 and fa (.) of the samples of Class 2 are unknown, fl( .) and fi(.) belong to a regular parametric family F', and the class of mixtures 4 of two elements of .F is identifiable and satisfies the conditions for Laplace regularity [13] . We can 'show [l] that the main result of the present work still holds, namely, that labeled samples are exponentially more valuable than unlabeled samples in reducing the probability of error of nondegenerate Bayesian classifiers.. The general problem of learning with both labeled and unlabeled observations is of practical relevance. In fact, the construction of a training set is often performed by collecting observations first and by labeling them afterwards, and in many instances the labeling process is harder or more expensive than the sampling step required to obtain the observations. Sometimes it is impossible to label most samples, and in some cases it is desirable to keep to a minimum the number of labeled samples, as for example when labeling involves the death of the patient. Therefore, situations in which both labeled and unlabeled samples are available arise naturally, and the investigation of the simultaneous use of both kinds of observations in learning leads immediately to questions of the relative value of labeled and unlabeled samples.. Among the practical learning schemes that use training sets composed of both labeled and unlabeled samples there are neural network classifiers [20] , [23] , [lo] , [I 11, [5] , 281 and parametric methods in discriminant analysis [22] , [17] , [ 181, [26] . The problem of supervised learning in a parametric setting has been widely studied, the most commonly encountered approaches being maximum-likelihood parameter estimation and Bayesian parameter estimation [19, ch. 2.11, [4, ch. VS] , [9, ch. 51, [27, ch. 71 . Unsupervised learning is also often addressed in a parametric setting [S, ch. 61, [4, ch. V] . Many studies exist that analyze the behavior of the risk of classifiers in a supervised learning framework both for small training set size [12] , [24] and for large training set size [7] , [27, ch. 91.

O O O O B I I I O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I I O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I I O O O O O O O O O O O O B I I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O


Proving Sequential Function Chart Programs Using Timed Automata Applications described by Sequential Function Chart (SFC) often being critical, we h a ve i n vestigated the possibilities of program checking. In particular, physical time can be handled by S F C programs using temporisations, which i s w h y w e a r e interested in the quantitative temporal properties. We h a ve proposed a modeling of SFC in timed automata, a formalism which takes time into account. In this modeling, we use the physical constraints of the environment. Veri cation of properties can be carried out using the model-checker Kronos. We apply this method to SFC programs of average size like that of the control part of the production cell Korso. The size of the programs remains however a limit and we are studying the means of solving this problem. The control language in which w e a r e i n terested is Sequential Function Chart 1 (SFC). Developed since 1977, this graphical language is based on the steptransition model. Through temporisations, it makes it possible to take t i m e into account. The perfect adaptation of this intuitive and practical language to the programming of automated systems has been clearly demonstrated. It is one of the languages de ned by the IEC1133;3 for the programming of Programmable Logic Controllers. For these, safety is required it is necessary to make sure that their speci cations are respected by the program. To carry 1 In this work we s h o w that it is possible to take t i m e i n to account i n m o d eling of SFC programs and to check their qualitative o r q u a n titative temporal properties.. In modeling with the synchronous languages, we represent discrete time. During the checking, this modeling leads to a combinatorial explosion of the number of states, each instant being represented by a state.. We t h e n h a ve turned to timed automata. This formalism takes into account continuous time in its de nition, owing to real variables called clocks. With each step i referred in a temporisation t 1 =X i =t 2 , w e associate a clock. This computes the time for which the step has been active or inactive. On a timed automaton resulting from this modeling, we could check temporal properties such accessibility in a minimum or maximum time, or the durations of minimum activity and maximum. F or this veri cation, the delays are no longer a limitation.. On the other hand, the size of the automata remains a barrier to the checking. An automaton should not have more than 65000 transitions, so that Kronos can treat it. Unfortunately, some of the automata generated from the SFC programs can have more than 100000 transitions. In order to solve this problem, several solutions have b e e n i n vestigated. Taking the environment a t t h e level of the states and the transitions into account enables us to decrease the size of the automata considerably. In the same way, a proposed new modeling makes it possible to reduce the number of states and transitions from the automata generated. However to increase the size of the SFC programs that can be treated, e orts must continue in these directions as in the study of the veri cation techniques of composition and abstraction.. To ensure more safety o f S F C programs, checking only does not seem to be su cient. In parallel, we think that a methodology should be developed making it possible to avoid design errors. This methodology could perhaps also support the building of more easily veri able SFC programs. Furthermore it seems important to us to confront the models and theories already developed with the industrial applications.

O O O O O O B I O O O B I I I O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Tracking Control of a Class of Differential Inclusion Systems via Sliding Mode Technique The tracking problem for a class of differential inclusion systems is investigated. Using global sliding mode control approach, a tracking control is proposed such that the output of a differential inclusion system tracks the desired trajectory asymptotically. An extensive reaching law is proposed to achieve the chattering reduction. Finally, an example is given to illustrate the validity of the proposed design. Tracking control is always a very active research area due to its wide applications. In the past few decades, various approaches for tracking control have been presented, such as fuzzy control approach [1] , composite tracking control approach [2] , adaptive control approach [3] , sliding mode control approach [4] , etc. Because sliding mode control has attractive features such as fast response, good transient response and insensitivity to variations in system parameters and external disturbances [5−11] , it is a substantial method for the tracking control design of nonlinear systems.. Generally speaking, the differential inclusion systems are considered a generalization of differential equations, and many practical systems are described by differential inclusion systems, so the study of differential inclusion systems has been paid much attention [12−21] . In [12] , a necessary and sufficient condition for the stability of polytopic linear differential inclusion systems was derived by bilinear matrix equations. In [13] , the problem of tracking control of nonlinear uncertain dynamical systems described by differential inclusions was studied. In [14] , a nonlinear control design method for linear differential inclusion systems was presented by using quadratic Lyapunov functions of their convex hull. Using the method in [14] , Sun [15] considered the uniformly ultimately bounded tracking control of linear differential inclusions with stochastic disturbance and Huang et al. [16] considered the stabilization of linear differential inclusion system with time delay. In [17] , a frequency-domain approach was proposed to analyze the globally asymptotic stability of differential inclusion systems with discrete and distributed time-delays. In [18, 19] , the fundamental information about differential inclusions is introduced. But the research of tracking control of differential inclusion systems using sliding mode technique is not sufficient. In [20, 21] This paper applies global sliding mode control to study the tracking control problem for a class of differential inclusion systems. The main contributions of this paper lie in the following aspects: 1) the system model is a generalized model of those considered in [4, 8, 9] ; 2) an extensive reaching law is proposed to design a sliding mode controller to make the error system asymptotically stable. The tracking control for a class of differential inclusion systems has been investigated. Using global sliding mode control, a novel controller is designed to make the output of differential inclusion systems track the desired trajectory asymptotically. Finally, an example illustrates the validity of the proposed method.

B O O O O O B I I O O O O O B O O O O O B I I O O O B I I I O O B O O O O O O O O O O O O O O O O O B I I I O O O O O B I O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O B I I O O O O O O O O O O O O O O B I I O O O O B I I O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O B O O B I I O O O O O O O O O O O O O B I I I O O O B O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I I O O O O O O O O O O O O O O O O B O O O O O B I I O O O O B I I I O O O O O O O O O O B I I O O O O O O O O O O O O O O O


New Characterizations of Simple Points in 2D, 3D, and 4D Discrete Spaces Abstract-A point of a discrete object is called simple if it can be deleted from this object without altering topology. In this paper, we present new characterizations of simple points, which hold in dimensions 2, 3, and 4 and lead to efficient algorithms for detecting such points. In order to prove these characterizations, we establish two confluence properties of the collapse operation which hold in the neighborhood of a point in spaces of low dimension. We develop this work in the framework of cubical complexes, which provides a sound topological basis for image analysis and retrieves the main notions and results of digital topology, in particular the notion of simple point. T OPOLOGY-PRESERVING operators, such as homotopic skeletonization and thinning, are used in many applications of image analysis to transform an object while leaving unchanged its topological characteristics. Applications in 2D and 3D are already widely spread. In particular, skeletons are often used as a simplification of the original data, which facilitates shape recognition, registration, or animation. Fig. 1 shows examples of 3D skeletons of different kinds: surfacic (Fig. 1a) , curvilinear (Fig. 1b) , and ultimate (Fig. 1c) .. With the emergence of fast 3D image acquisition devices, such as medical X-ray and MRI scanners, there is a growing interest in considering a time sequence of 3D objects as a coherent 4D structure. For example, the segmentation of a moving heart muscle can be facilitated in this way [12] .. In discrete grids (Z Z 2 , Z Z 3 , Z Z 4 ), a topology-preserving transformation can be defined due to the notion of simple point [19] : Intuitively, a point of an object (a subset of Z Z n ) is called simple if it can be deleted from this object without altering topology. This notion, pioneered by Duda et al. [14] , Golay [16] , and Rosenfeld [25] , has since been the subject of abundant literature. In particular, local characterizations of simple points have been proposed in which efficient implementation of thinning procedures is based.. Let us informally illustrate the notion of simple point through some examples, first in 2D, then in 3D. In Fig. 2 , the points (or pixels) x, y, z, t are not simple: The removal of x from the set X of pixels would create a new connected component of the complement X of X, the removal of y would merge two connected components of X, the removal of z would split a connected component of X, and the removal of t would delete a connected component of X. On the other hand, the pixels a, b, and c are simple pixels. We see that, in 2D, the notion of connectedness (for both X and X) suffices to characterize simple pixels.. Things are more difficult in 3D. Consider the example of the set X depicted in Fig. 3 ; removing the voxel x or the voxel y from X would not split, merge, create, or suppress any component of X nor any component of X. However, neither x nor y is simple, for the deletion of x (respectively, y) causes the suppression (respectively, creation) of a tunnel. Surprisingly, it is still possible to characterize 3D simple points by local conditions, which are only based on connectedness (see [1] , [7] , [26] ), but connectedness alone is not sufficient to characterize 4D simple points. This paper is settled in the framework of cubical complexes. Abstract (cubical) complexes have been promoted in particular by Kovalevsky [21] in order to provide a sound topological basis for image analysis. For instance, in this framework, we retrieve the main notions and results of digital topology, such as the notion of simple point. Intuitively, a cubical complex may be thought of as a set of elements having various dimensions (e.g., cubes, squares, edges, vertices) glued together according to certain rules (see Fig. 4 ).. The collapse operation is an elementary topology-preserving transformation, which has been introduced by Whitehead [27] , and plays an important role in combinatorial topology; it can be seen as a discrete analog of a continuous deformation. It consists of removing two distinct elements ðf; gÞ from a complex X under the condition that g is contained in f and is not contained in any other element of X, such a pair of elements is called a free pair. This operation may be repeated several times (see X, X 1 , X 2 in Fig. 4 ).. We use a definition of simple point ( [3] , see also [6] ) based on the collapse operation. If X is a complex and x is a square in X, then we define the detachment of x from X as the complex obtained by removing from X the square x and all of the elements of X (edges, vertices), which are only contained in x. For example, the complex X 2 (respectively, X 4 ) is the detachment of x (respectively, y) from X. Schematically, a point (or element) x in an object X is said to be simple if X collapses onto the detachment of x from X. For example in Fig. 4 , the element x is simple for X, as proved by the sequence X, X 1 , X 2 . On the other hand, the element y is not simple for X: we can see that X collapses onto X 3 but not onto X 4 , which is the detachment of y from X. Notice that this definition of simple points makes sense in any dimension.. We present new characterizations of 2D, 3D, and 4D simple points based on the collapse (Theorems 14 and 15), which lead to simple greedy linear-time algorithms for simplicity checking. We also retrieve in our framework, a characterization of 4D simple points established by Kong [18] and some previously proposed characterizations of 3D simple points [18] , [1] , [7] , [26] .. In order to prove these characterizations, we establish some confluence properties of the collapse (Theorems 12 and 13). To informally illustrate such a property, let us consider a full rectangle R in the discrete plane. It is plain that R collapses onto a single point. Let S be any subset of R such that R collapses onto S, a confluence property states that S must also collapse onto a point.. Surprisingly, this property does not hold in the 3D space due to the existence of "topological monsters" such as Bing's house. In Fig. 5 , we see a classical representation of Bing's house. The house has two rooms separated by a floor; one can enter the lower room of the house by the chimney passing through the upper room and vice versa.. This object can be obtained by collapse from a full cuboid (see Section 4). Nevertheless, it does not collapse onto any of its subsets: This object has no free pair.. In this paper, we show that confluence properties of the collapse indeed hold in the neighborhood of a point when the dimension of the space is such that this neighborhood is not large enough to contain such counterexamples.. This paper is self-contained. Some of the results presented here were stated, without proofs, in a conference paper [11] . The new characterizations of simple points that we proved in this paper lead to simple and efficient algorithms for checking simplicity. In 2D and 3D, configurations of simple and nonsimple points may be stored in a lookup table, but in 4D this is clearly impossible (there are 2 80 possible configurations); thus, such algorithms may be of practical interest. On the theoretical point of view, we proved these characterizations on the basis of new confluence properties, which turn out to also be keystones of a set of new results linking minimal nonsimple sets [24] , P-simple points [2] , and critical kernels [3] , to appear in another paper [10] . We also proved (Theorem 20) that these characterizations and confluence properties do not hold beyond dimension 4.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O B O O O O O O O O O O O O O O O O O B O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O