Simultaneous Localization and Recognition of Dynamic Hand Gestures

A framework for the simultaneous localization and recognition of dynamic hand gestures is proposed. At the core of this framework is a dynamic space-time warping (DSTW) algorithm, that aligns a pair of query and model gestures in both space and time. For every frame of the query sequence, feature detectors generate multiple hand region candidates. Dynamic programming is then used to compute both a global matching cost, which is used to recognize the query gesture, and a warping path, which aligns the query and model sequences in time, and also finds the best hand candidate region in every query frame. The proposed framework includes translation invariant recognition of gestures, a desirable property for many HCI systems. The performance of the approach is evaluated on a dataset of hand signed digits gestured by people wearing short sleeve shirts, in front of a background containing other non-hand skin-colored objects. The algorithm simultaneously localizes the gesturing hand and recognizes the hand-signed digit. Although DSTW is illustrated in a gesture recognition setting, the proposed algorithm is a general method for matching time series, that allows for multiple candidate feature vectors to be extracted at each time step.National Science Foundation (CNS-0202067, IIS-0308213, IIS-0329009); Office of Naval Research (N00014-03-1-0108

Piercing convex sets

A family of sets has the $(p,q)$ property if among any $p$ members of the family some $q$ have a nonempty intersection. It is shown that for every $p\ge q\ge d+1$ there is a $c=c(p,q,d)<\infty$ such that for every family \scr F of compact, convex sets in $R^d$ that has the $(p,q)$ property there is a set of at most $c$ points in $R^d$ that intersects each member of \scr F. This extends Helly's Theorem and settles an old problem of Hadwiger and Debrunner.Comment: 5 page

Compressibility and probabilistic proofs

We consider several examples of probabilistic existence proofs using compressibility arguments, including some results that involve Lov\'asz local lemma.Comment: Invited talk for CiE 2017 (full version

Distortion-Rate Function of Sub-Nyquist Sampled Gaussian Sources

The amount of information lost in sub-Nyquist sampling of a continuous-time Gaussian stationary process is quantified. We consider a combined source coding and sub-Nyquist reconstruction problem in which the input to the encoder is a noisy sub-Nyquist sampled version of the analog source. We first derive an expression for the mean squared error in the reconstruction of the process from a noisy and information rate-limited version of its samples. This expression is a function of the sampling frequency and the average number of bits describing each sample. It is given as the sum of two terms: Minimum mean square error in estimating the source from its noisy but otherwise fully observed sub-Nyquist samples, and a second term obtained by reverse waterfilling over an average of spectral densities associated with the polyphase components of the source. We extend this result to multi-branch uniform sampling, where the samples are available through a set of parallel channels with a uniform sampler and a pre-sampling filter in each branch. Further optimization to reduce distortion is then performed over the pre-sampling filters, and an optimal set of pre-sampling filters associated with the statistics of the input signal and the sampling frequency is found. This results in an expression for the minimal possible distortion achievable under any analog to digital conversion scheme involving uniform sampling and linear filtering. These results thus unify the Shannon-Whittaker-Kotelnikov sampling theorem and Shannon rate-distortion theory for Gaussian sources.Comment: Accepted for publication at the IEEE transactions on information theor

The number of independent sets in a graph with small maximum degree

Let ${\rm ind}(G)$ be the number of independent sets in a graph $G$. We show that if $G$ has maximum degree at most $5$ then $${\rm ind}(G) \leq 2^{{\rm iso}(G)} \prod_{uv \in E(G)} {\rm ind}(K_{d(u),d(v)})^{\frac{1}{d(u)d(v)}}$$ (where $d(\cdot)$ is vertex degree, ${\rm iso}(G)$ is the number of isolated vertices in $G$ and $K_{a,b}$ is the complete bipartite graph with $a$ vertices in one partition class and $b$ in the other), with equality if and only if each connected component of $G$ is either a complete bipartite graph or a single vertex. This bound (for all $G$) was conjectured by Kahn. A corollary of our result is that if $G$ is $d$-regular with $1 \leq d \leq 5$ then $${\rm ind}(G) \leq \left(2^{d+1}-1\right)^\frac{|V(G)|}{2d},$$ with equality if and only if $G$ is a disjoint union of $V(G)/2d$ copies of $K_{d,d}$. This bound (for all $d$) was conjectured by Alon and Kahn and recently proved for all $d$ by the second author, without the characterization of the extreme cases. Our proof involves a reduction to a finite search. For graphs with maximum degree at most $3$ the search could be done by hand, but for the case of maximum degree $4$ or $5$, a computer is needed.Comment: Article will appear in {\em Graphs and Combinatorics

Lossy Compression of Decimated Gaussian Random Walks

We consider the problem of estimating a Gaussian random walk from a lossy compression of its decimated version. Hence, the encoder operates on the decimated random walk, and the decoder estimates the original random walk from its encoded version under a mean squared error (MSE) criterion. It is well-known that the minimal distortion in this problem is attained by an estimate-and-compress (EC) source coding strategy, in which the encoder first estimates the original random walk and then compresses this estimate subject to the bit constraint. In this work, we derive a closed-form expression for this minimal distortion as a function of the bitrate and the decimation factor. Next, we consider a compress-and-estimate (CE) source coding scheme, in which the encoder first compresses the decimated sequence subject to an MSE criterion (with respect to the decimated sequence), and the original random walk is estimated only at the decoder. We evaluate the distortion under CE in a closed form and show that there exists a nonzero gap between the distortion under the two schemes. This difference in performance illustrates the importance of having the decimation factor at the encoder