Operation and planning of electric grids involve the analysis of certain power flow equations, which exhibit multiple solutions. One solution generally corresponds to a preferred operating condition, while other less desirable solutions are often useful for assessing dynamic angular stability and static voltage stability margins. The number and nature of solutions to these network equations has been studied for decades, yet without revealing a complete understanding. In light of observations that the number of solutions encountered in practice is smaller than the number predicted by the usual mathematical bounds, the most recent theoretical papers on this topic have sought to produce more accurate bounds on the number of solutions, accounting for network structure. In this paper we further refine the analysis to describe distributions of solutions. We present algebraic results for a certain small system and empirical results for others.

Consider an r-dimensional subspace of ℝd, r <; d, and suppose that we are only given projections of this subspace onto small subsets of the canonical coordinates. The paper establishes necessary and sufficient deterministic conditions on the subsets for subspace identifiability. The results also shed new light on low-rank matrix completion.

Tail-biting trellises are important graphical representations of codes, which can be simpler to decode than conventional trellises. While conventional trellises are well understood, the general theory of (tail-biting) trellises is still under development. In this paper we first develop a new algebraic framework for a systematic analysis of linear trellises which enables us to address open foundational questions. In particular, we present a useful and powerful characterization of linear trellis isomorphy. We also obtain a new proof of the linear factorization theorem of Koetter/Vardy, and point out unnoticed problems for the group case. Next, we apply our framework to: 1) describe all the factorizations of linear trellises; 2) determine and classify all the minimal linear trellises for a given linear code; and 3) prove that nonmergeable, one-to-one, linear trellises are determined by the edge-label sequences of certain closed paths. We also provide new insight into mergeability and state how results on reduced linear trellises can be extended to nonreduced ones. Our classification results are relevant to iterative/linear programming decoding, as we show that minimal linear trellises can yield different pseudocodewords even if they have the same graph structure.

This paper addresses an old, yet challenging issue - curvature estimation from discrete sampling points over a curve. We introduce a novel algorithm based on performing line integrals. The proposed method is computationally more efficient than the previous integration-based methods because of the constant computation time. Qualitative tests on synthesized shapes in the presence of noise are performed, which show the robustness of our approach. Also, the discriminative capability of the estimated curvature is evaluated by conducting experiments on the FRGC (Face Recognition Grand Challenge) v2.0 dataset which contains 4007 3D facial images recorded from 466 subjects. The results show that recognition performance is significantly improved by using our curvature estimation method. This novel approach presents potential for a broad class of multimedia applications.

Applications of summation invariant features to multi-region face recognition are explored in this work. Earlier, we have demonstrated the potential benefits of this approach. In this paper, we provide a systematic, thorough comparison of all the summation invariant features derived to-date, and propose a new multi-feature fusion approach to further improve the overall performance. We also identify summation invariant features that yield superior performance for face recognition applications. Special attention is given to the implementation of 3D summation invariants. Extensive experimental results with the FRGC (Face Recognition Grand Challenge) 2.0 data set confirms the advantage of summation invariant features for 3D face recognition.

We propose a space–time block coding framework based on linear precoding. The codes for$P$transmit antennas are formed by transmitting the information vector (with$P$independent information symbols)$L$times where each time it is rotated by a distinct precoding matrix. The framework generalizes conventional spatial multiplexing techniques and facilitates tradeoff between rate and diversity. We propose a simple construction for precoding matrices whose parameters are chosen to guarantee maximal diversity using algebraic number theory. Our codes exhibit circular structure, which greatly simplifies the performance analysis and facilitates linear decoding. Theoretical analysis and numerical simulations demonstrated excellent performance of the proposed algebraic precoding framework.

Recently a novel family of geometrically invariant features, called summation invariant, has been developed and applied to object recognition. The range of this family of features is expanded here beyond the Euclidean and afline transformation groups to planar projective transformations. Whereas other methods require small changes in view, or collinear points, this method removes those limitations and allows recognition of general planar objects over wide ranges of viewpoint. The derivation of these new features requires the innovation of deriving the invariants in the homogeneous coordinate space, yet yields results formulated in terms of Cartesian coordinates. Simulations demonstrate the effectiveness of this new approach to object recognition under projective transformations like those encountered in camera networks.

We propose a handwritten digit recognition algorithm that uses 4×4 2D hidden Markov models to extract basic features from an unclassified image. The novel idea given here is that we use powerful techniques from the emerging mathematical fields of tropical geometry and algebraic statistics to determine parameters for the model. The distance between the unclassified images and prototypes is calculated in stages, where estimates of the distance become finer as obviously distant prototypes are discarded from the pool of possible K-nearest neighbors. Our algorithm achieves a 95.51 percent recognition rate with zero rejection on the MNIST database of handwritten digits.

We present a novel 3D face recognition method that incorporates summation invariant features extracted from multiple sub-regions of a facial range images, and optimal fusion of similarity scores between corresponding sub-regions. The key innovation of this paper is the development of the fusion-based face recognition algorithm that delivers significant performance enhancement while requiring very little computation. Experiments on the FRGC (Face Recognition Grand Challenge) version 2 dataset show that our algorithm improves the recognition performance significantly in the presence of facial expressions.

In this paper, we describe a fusion-based face recognition method that is able to compensate for facial expressions even when training samples contain only neutral expression. The similarity metric between two facial images are calculated by combining the similarity scores of the corresponding facial regions, e.g. the similarity between two mouths, the similarity between two noses, etc. In contrast with other approaches where equal weights are assigned on each region, a novel fusion method based on linear discriminant analysis (LDA) is developed to maximize the verification performance. We also conduct a comparative study on various face recognition schemes, including the FRGC baseline algorithm, the fusion of multiple regions by sum rule, and the fusion of multiple regions by LDA. Experiments on the FRGC (Face Recognition Grand Challenge) V2.0 dataset, containing 4007 face images recorded from 266 subjects, show that the proposed method significantly improves the verification performance in the presence of facial expressions.

In this paper, we introduce a novel 3D surface landmark detection method using a 3D integral invariant feature extended from that proposed by Manay et al. for 2D contours. We apply this new feature to detect the nose tips of 2.5D range images of human faces. Using the Face Recognition Grand Challenge 2.0 dataset, our method compares favorably with a recently proposed competing method.

This paper presents a novel 3-D multiregion face recognition algorithm that consists of new geometric summation invariant features and an optimal linear feature fusion method. A summation invariant, which captures local characteristics of a facial surface, is extracted from multiple subregions of a 3-D range image as the discriminative features. Similarity scores between two range images are calculated from the selected subregions. A novel fusion method that is based on a linear discriminant analysis is developed to maximize the verification rate by a weighted combination of these similarity scores. Experiments on the Face Recognition Grand Challenge V2.0 dataset show that this new algorithm improves the recognition performance significantly in the presence of facial expressions.

A novel family of 2D and 3D geometrically invariant features, called summation invariants is proposed for the recognition of the 3D surface of human faces. Focusing on a rectangular region surrounding the nose of a 3D facial depth map, a subset of the so called semi-local summation invariant features is extracted. Then the similarity between a pair of 3D facial depth maps is computed to determine whether they belong to the same person. Out of many possible combinations of these set of features, we select, through careful experimentation, a subset of features that yields best combined performance. Tested with the 3D facial data from the on-going Face Recognition Grand Challenge v1.0 dataset, the proposed new features exhibit significant performance improvement over the baseline algorithm distributed with the datase

A novel family of geometrically invariant features, called summation invariants are proposed for the recognition of the 3D surface of human faces. In particular, a 2D semi-local summation invariant feature is extracted from each column and each row of a rectangular region surrounding the nose of a 3D facial depth map. Through extensive experimentation, we empirically identify the most efficient 2D summation invariant features. We also investigate the proper pre-processing method for the 2D summation invariant features. Tested with the 3D facial data from the Face Recognition Grand Challenge v1.0 dataset, the proposed new features exhibit significant performance improvement over the baseline algorithm distributed with the dataset.

In this paper, we developed a family of 2D and 3D invariant features with applications to 3D human faces recognition. The main contributions of this paper are: (a) systematically deriving a family of novel features, called summation invariant that are invariant to Euclidean transformation in both 2D and 3D; (b) developing an effective method to apply summation invariant to the 3D face recognition problem. Tested with the 3D data from the face recognition grand challenge v1.0 dataset, the proposed new features exhibit achieves a performance that rivals the best 3D face recognition algorithms reported so far

Transmit beamforming with receive combining is a low-complexity solution that achieves the full diversity afforded by a multi-antenna channel. Building on our recent result which shows that even channel statistics are sufficient to achieve perfect feedback performance (in the limit of antenna dimensions) with beamforming and combining, we propose near-optimal codebook designs for correlated channels with a focus on few antennas at the transmitter and the receiver. In the process, we refine the answer to the question: When are channel statistics sufficient to achieve near perfect feedback performance? We show that the condition number of the transmit and receive covariance matrices hold the key to this question. We partition the transmit and receive covariance spaces into 4 regions based on well and ill-conditioning of the covariance matrices and show that the number of bits required for near perfect feedback performance is dependent on the condition numbers of these matrices

A novel summation invariant of curves under transformation group action is proposed. This new invariant is less sensitive to noise than the differential invariant and does not require an analytical expression for the curve as the integral invariant does. We exploit this summation invariant to define a shape descriptor called a semi-local summation invariant and use it as a new feature for shape recognition. Tested on a database of noisy shapes of fish, it was observed that the summation invariant feature exhibited superior discriminating power compared to that of wavelet-based invariant features.

Digital video watermarking is increasingly important. Video signals are very susceptible to attacks like frame averaging, dropping, swapping, collusion. This paper presents a new video watermarking method. The most important information to be watermarked such as a company's name is hidden in a sequence of statistical data, which is then embedded into each frame. Every segment of the statistical data is typically enough for the key watermark extraction. A very detailed example of this method is given, based on the cryptanalysis. The most important information is used as the key to encrypt a plaintext. The plaintext can be some less important information about the product. The ciphertext is then embedded in the video frames. With statistical knowledge of the English language, we can quickly recover the key. Mathematical analysis and simulation results are given in the paper.

Transmit beamforming and receive combining are low complexity techniques that help in achieving the full diversity afforded by the multi-antenna channel. Complete channel state information at the transmitter may be possible only under rare instances. Thus quantized beamforming with limited feedback on a reverse link has been a topic that has attracted great attention recently. But almost all of the work to date has focussed on modeling the channel with independent, identically distributed (i.i.d.) Rayleigh fading between antenna pairs. In this paper, we consider the correlated channel case, and show that Grassmannian line packing is an artificial artifact of the i.i.d. assumption. We show that there are dominant peaks in the eigen-domain when correlation is imposed and the codebook construction should be matched to the correlation in the channel, which renders the problem in this case easier than for i.i.d. channels

A novel summation invariant feature under transformation group action for 3D surface recognition is proposed, and its application to 3D face recognition is investigated. Based on a systematic mathematical procedure called moving frame, we derived the summation invariant feature that is invariant under affine transformation. Compared with classical differential invariants, such as the mean curvature or the Gaussian curvature, summation invariant feature is far less sensitive to observation noise in the data. A further enhancement leads to a new type of invariant 3D surface shape descriptor called a semi-local summation invariant. We demonstrate one important, potential application of this new feature to 3D human face recognition

The IEEE Signal Processing Society (2000) offered a prize of $1000 for proving or disproving Makhoul's conjecture, which says that, given a causal all-pass digital signal x/sub n/ of order p, with nonzero x/sub 0/, the location of the peak of x/sub n/ always lies between n = 0 and n = 2p-1. The case of p = 1 is trivial, and no further progress had been made in 25 years until Lertniphonphun, Rajagopal, and Wenzel gave counter examples for large p. In this paper, Makhoul's conjecture is proven for p = 2. It is also shown that the conjecture fails dramatically in the case of complex coefficients.

We find that the minimum distance of the binary [137,69] quadratic residue code is 21.