Daniel Keren

Describing complicated objects by implicit polynomials

This paper introduces and focuses on two problems. First is the representation power of closed implicit polynomials of modest degree for curves in 2-D images and surfaces in 3-D range data. Super quadrics are a small subset of object boundaries that are well fitted by these polynomials. The second problem is the stable computationally efficient fitting of noisy data by closed implicit polynomial curves and surfaces. The attractive features of these polynomials for Vision is discussed. >

Practical reliable Bayesian recognition of 2D and 3D objects using implicit polynomials and algebraic invariants

We treat the use of more complex higher degree polynomial curves and surfaces of degree higher than 2, which have many desirable properties for object recognition and position estimation, and attack the instability problem arising in their use with partial and noisy data. The scenario discussed in this paper is one where we have a set of objects that are modeled as implicit polynomial functions, or a set of representations of classes of objects with each object in a class modeled as an implicit polynomial function, stored in the database. Then, given partial data from one of the objects, we want to recognize the object (or the object class) or collect more data in order to get better parameter estimates for more reliable recognition. Two problems arising in this scenario are discussed: 1) the problem of recognizing these polynomials by comparing them in terms of their coefficients; and 2) the problem of where to collect data so as to improve the parameter estimates as quickly as possible. We use an asymptotic Bayesian approximation for solving the two problems. The intrinsic dimensionality of polynomials and the use of the Mahalanobis distance are discussed.

A geometric approach to monitoring threshold functions over distributed data streams

Monitoring data streams in a distributed system is the focus of much research in recent years. Most of the proposed schemes, however, deal with monitoring simple aggregated values, such as the frequency of appearance of items in the streams. More involved challenges, such as the important task of feature selection (e.g., by monitoring the information gain of various features), still require very high communication overhead using naive, centralized algorithms. We present a novel geometric approach which reduces monitoring the value of a function (vis-a-vis a threshold) to a set of constraints applied locally on each of the streams. The constraints are used to locally filter out data increments that do not affect the monitoring outcome, thus avoiding unnecessary communication. As a result, our approach enables monitoring of arbitrary threshold functions over distributed data streams in an efficient manner. We present experimental results on real-world data which demonstrate that our algorithms are highly scalable, and considerably reduce communication load in comparison to centralized algorithms.Monitoring data streams in a distributed system is the focus of much research in recent years. Most of the proposed schemes, however, deal with monitoring simple aggregated values, such as the frequency of appearance of items in the streams. More involved challenges, such as the important task of feature selection (e.g., by monitoring the information gain of various features), still require very high communication overhead using naive, centralized algorithms. We present a novel geometric approach which reduces monitoring the value of a function (vis-à-vis a threshold) to a set of constraints applied locally on each of the streams. The constraints are used to locally filter out data increments that do not affect the monitoring outcome, thus avoiding unnecessary communication. As a result, our approach enables monitoring of arbitrary threshold functions over distributed data streams in an efficient manner. We present experimental results on real-world data which demonstrate that our algorithms are highly scalable, and considerably reduce communication load in comparison to centralized algorithms.

A Bayesian method for fitting parametric and nonparametric models to noisy data

We present a simple paradigm for fitting models, parametric and nonparametric, to noisy data, which resolves some of the problems associated with classical MSE algorithms. This is done by considering each point on the model as a possible source for each data point. The paradigm can be used to solve problems which are ill-posed in the classical MSE approach, such as fitting a segment (as opposed to a line). It is shown to be nonbiased and to achieve excellent results for general curves, even in the presence of strong discontinuities. Results are shown for a number of fitting problems, including lines, circles, elliptic arcs, segments, rectangles, and general curves, contaminated by Gaussian and uniform noise.

Fitting curves and surfaces with constrained implicit polynomials

A problem which often arises while fitting implicit polynomials to 2D and 3D data sets is the following: although the data set is simple, the fit exhibits undesired phenomena, such as loops, holes, extraneous components, etc. Previous work tackled these problems by optimizing heuristic cost functions, which penalize some of these topological problems in the fit. The paper suggests a different approach-to design parameterized families of polynomials whose zero-sets are guaranteed to satisfy certain topological properties. Namely, we construct families of polynomials with star-shaped zero-sets, as well as polynomials whose zero-sets are guaranteed not to intersect an ellipse circumscribing the data or to be entirely contained in such an ellipse. This is more rigorous than using heuristics which may fail and result in pathological zero-sets. The ability to parameterize these families depends heavily on the ability to parameterize positive polynomials. To achieve this, we use some powerful results from real algebraic geometry.

Restoring subsampled color images

Abstract. In some capturing devices, such as digital cameras, there is only one color sensor at each pixel. Usually, 50% of the pixels have only a green sensor, 25% only a red sensor, and 25% only a blue sensor. The problem is then to restore the two missing colors at each pixel – this is called “demosaicing”, because the original samples are usually arranged in a mosaic pattern. In this short paper, a few demosaicing algorithms are developed and compared. They all incorporate a notion of “smoothness in chroma space”, by imposing conditions not only on the behavior of each color channel separately, but also on the correlation between the three channels.

Shape Sensitive Geometric Monitoring

An important problem in distributed, dynamic databases is to continuously monitor the value of a function defined on the nodes, and check that it satisfies some threshold constraint. We introduce a monitoring method, based on a geometric interpretation of the problem, which enables to define local constraints at the nodes. It is guaranteed that as long as none of these constraints is violated, the value of the function did not cross the threshold. We generalize previous work on geometric monitoring, and solve two problems which seriously hampered its performance: as opposed to the constraints used so far, which depend only on the current values of the local data, here we incorporate their temporal behavior. Also, the new constraints are tailored to the geometric properties of the specific monitored function. In addition, we extend the concept of safe zones for the monitoring problem, and show that previous work on geometric monitoring is a special case of the proposed extension. Experimental results on real data reveal that the new approach reduces communication by up to three orders of magnitude in comparison to existing approaches, and considerably narrows the gap between achievable results and a newly defined lower bound on communication complexity.

Using symbolic computation to find algebraic invariants

Implicit polynomials have proved themselves as having excellent representation power for complicated objects, and there is growing use of them in computer vision, graphics, and CAD. A must for every system that tries to recognize objects based on their representation by implicit polynomials are invariants, which are quantities assigned to polynomials that do not change under coordinate transformations. In the recognition system developed at the Laboratory for Engineering Man-Machine Studies in Brown University (LEMS), it became necessary to use invariants which are explicit and simple functions of the polynomial coefficients. A method to find such invariants is described and the new invariants presented. This work addresses only the problem of finding the invariants; their stability is studied in another paper. >

Shape sensitive geometric monitoring

An important problem in distributed, dynamic databases is to continuously monitor the value of a function defined on the nodes, and check that it satisfies some threshold constraint. We introduce a monitoring method, based on a geometric interpretation of the problem, which enables to define local constraints at the nodes. It is guaranteed that as long as none of these constraints is violated, the value of the function did not cross the threshold. We generalize previous work on geometric monitoring, and solve two problems which seriously hampered its performance: as opposed to the constraints used so far, which depend only on the current values of the local data, here we incorporate their temporal behavior. Also, the new constraints are tailored to the geometric properties of the specific monitored function. In addition, we extend the concept of safe zones for the monitoring problem, and show that previous work on geometric monitoring is a special case of the proposed extension. Experimental results on real data reveal that the new approach reduces communication by up to three orders of magnitude in comparison to existing approaches, and considerably narrows the gap between achievable results and a newly defined lower bound on communication complexity.

Sketch-based geometric monitoring of distributed stream queries

Emerging large-scale monitoring applications rely on continuous tracking of complex data-analysis queries over collections of massive, physically-distributed data streams. Thus, in addition to the space- and time-efficiency requirements of conventional stream processing (at each remote monitor site), effective solutions also need to guarantee communication efficiency (over the underlying communication network). The complexity of the monitored query adds to the difficulty of the problem -- this is especially true for nonlinear queries (e.g., joins), where no obvious solutions exist for distributing the monitor condition across sites. The recently proposed geometric method offers a generic methodology for splitting an arbitrary (non-linear) global threshold-monitoring task into a collection of local site constraints; still, the approach relies on maintaining the complete stream(s) at each site, thus raising serious efficiency concerns for massive data streams. In this paper, we propose novel algorithms for efficiently tracking a broad class of complex aggregate queries in such distributed-streams settings. Our tracking schemes rely on a novel combination of the geometric method with compact sketch summaries of local data streams, and maintain approximate answers with provable error guarantees, while optimizing space and processing costs at each remote site and communication cost across the network. One of our key technical insights for the effective use of the geometric method lies in exploiting a much lower-dimensional space for monitoring the sketch-based estimation query. Due to the complex, highly nonlinear nature of these estimates, efficiently monitoring the local geometric constraints poses challenging algorithmic issues for which we propose novel solutions. Experimental results on real-life data streams verify the effectiveness of our approach.