Structural health monitoring (SHM) systems are critical for monitoring aging infrastructure (such as buildings or bridges) in a cost-effective manner. Wireless sensor networks that sample vibration data over time are particularly appealing for SHM applications due to their flexibility and low cost. However, in order to extend the battery life of wireless sensor nodes, it is essential to minimize the amount of vibration data these sensors must collect and transmit. In recent work, we have studied the performance of the Singular Value Decomposition (SVD) applied to the collection of data and provided new finite sample analysis characterizing conditions under which this simple technique{also known as the Proper Orthogonal Decomposition (POD){can correctly estimate the mode shapes of the structure. Specifically, we provided theoretical guarantees on the number and duration of samples required in order to estimate a structure's mode shapes to a desired level of accuracy. In that previous work, however, we considered simplified Multiple-Degree-Of-Freedom (MDOF) systems with no damping. In this paper we consider MDOF systems with proportional damping and show that, with sufficiently light damping, the POD can continue to provide accurate estimates of a structure's mode shapes. We support our discussion with new analytical insight and experimental demonstrations. In particular, we study the tradeoffs between the level of damping, the sampling rate and duration, and the accuracy to which the structure's mode shapes can be estimated.
We propose a multiscale, iterative algorithm for reconstructing video signals from streaming compressive measurements. Our algorithm is based on the observation that, at the imaging sensor, many videos should have limited temporal bandwidth due to the spatial lowpass filtering that is inherent in typical imaging systems. Under modest assumptions about the motion of objects in the scene, this spatial filtering prevents the temporal complexity of the video from being arbitrarily high. Thus, even though streaming measurement systems may measure a video thousands of times per second, we propose an algorithm that only involves reconstructing a much lower rate stream of “anchor frames.” Our analysis of the temporal complexity of videos reveals an interesting tradeoff between the spatial resolution of the camera, the speed of any moving objects, and the temporal bandwidth of the video. We leverage this tradeoff in proposing a multiscale reconstruction algorithm that alternates between video reconstruction and motion estimation as it produces finer resolution estimates of the video.
KEYWORDS: Modulation, Statistical analysis, Detection theory, Compressed sensing, Cesium, Monte Carlo methods, Signal to noise ratio, Interference (communication), System on a chip, Associative arrays
The framework of computing Higher Order Cyclostationary Statistics (HOCS) from an incoming signal has
proven useful in a variety of applications over the past half century, from Automatic Modulation Recognition
(AMR) to Time Dierence of Arrival (TDOA) estimation. Much more recently, a theory known as Compressive
Sensing (CS) has emerged that enables the ecient acquisition of high-bandwidth (but sparse) signals via nonuni-
form low-rate sampling protocols. While most work in CS has focused on reconstructing the high-bandwidth
signals from nonuniform low-rate samples, in this work, we consider the task of inferring the modulation of a
communications signal directly in the compressed domain, without requiring signal reconstruction. We show
that the HOCS features used for AMR are compressible in the Fourier domain, and hence, that AMR of various
linearly modulated signals is possible by estimating the same HOCS features from nonuniform compressive sam-
ples. We provide analytical support for the accurate approximation of HOCS features from nonuniform samples
and derive practical rules for classication of modulation type using these samples based on simulated data.
The theory of compressive sensing (CS) enables the reconstruction of a sparse or compressible
image or signal from a small set of linear, non-adaptive (even random) projections. However, in
many applications, including object and target recognition, we are ultimately interested in making
a decision about an image rather than computing a reconstruction. We propose here a framework
for compressive classification that operates directly on the compressive measurements without first
reconstructing the image. We dub the resulting dimensionally reduced matched filter the smashed
filter. The first part of the theory maps traditional maximum likelihood hypothesis testing into the
compressive domain; we find that the number of measurements required for a given classification
performance level does not depend on the sparsity or compressibility of the images but only on
the noise level. The second part of the theory applies the generalized maximum likelihood method
to deal with unknown transformations such as the translation, scale, or viewing angle of a target
object. We exploit the fact the set of transformed images forms a low-dimensional, nonlinear
manifold in the high-dimensional image space. We find that the number of measurements required
for a given classification performance level grows linearly in the dimensionality of the manifold but
only logarithmically in the number of pixels/samples and image classes. Using both simulations
and measurements from a new single-pixel compressive camera, we demonstrate the effectiveness
of the smashed filter for target classification using very few measurements.
Compressive Sensing is an emerging field based on the revelation that a small number of linear projections of a compressible signal contain enough information for reconstruction and processing. It has many promising implications and enables the design of new kinds of Compressive Imaging systems and cameras. In this paper, we develop a new camera architecture that employs a digital micromirror array to perform optical calculations of linear projections of an image onto pseudorandom binary patterns. Its hallmarks include the ability to obtain an image with a single detection element while sampling the image fewer times than the number of pixels. Other attractive properties include its universality, robustness, scalability, progressivity, and computational asymmetry. The most intriguing feature of the system is that, since it relies on a single photon detector, it can be adapted to image at wavelengths that are currently impossible with conventional CCD and CMOS imagers.
In this paper, we study families of images generated by varying a
parameter that controls the appearance of the object/scene in each image. Each image is viewed as a point in high-dimensional space; the family of images forms a low-dimensional submanifold that we call an image appearance manifold (IAM). We conduct a detailed study of some representative IAMs generated by translations/rotations of simple objects in the plane and by rotations of objects in 3-D space. Our central, somewhat surprising, finding is that IAMs generated by images with sharp edges are nowhere differentiable. Moreover, IAMs have an inherent multiscale structure in that approximate tangent planes fitted to ε-neighborhoods continually twist off into new dimensions as the scale parameter ε varies. We explore and explain this phenomenon. An additional, more exotic kind of local non-differentiability happens at some exceptional parameter points where occlusions cause image edges to disappear. These non-differentiabilities help to understand some key phenomena in image processing. They imply that Newton's method will not work in general for image registration, but that a multiscale Newton's method will work. Such a multiscale Newton's method is similar to existing coarse-to-fine differential estimation algorithms for image registration; the manifold perspective offers a well-founded theoretical motivation for the multiscale approach and allows quantitative study of convergence and approximation. The manifold viewpoint is also generalizable to other image understanding problems.
Natural images can be viewed as combinations of smooth regions,
textures, and geometry. Wavelet-based image coders, such as the
space-frequency quantization (SFQ) algorithm, provide reasonably
efficient representations for smooth regions (using zerotrees, for
example) and textures (using scalar quantization) but do not properly
exploit the geometric regularity imposed on wavelet coefficients by
features such as edges. In this paper, we develop a representation for
wavelet coefficients in geometric regions based on the wedgelet
dictionary, a collection of geometric atoms that construct
piecewise-linear approximations to contours. Our wedgeprint
representation implicitly models the coherency among geometric
wavelet coefficients. We demonstrate that a simple compression
algorithm combining wedgeprints with zerotrees and scalar quantization
can achieve near-optimal rate-distortion performance D(R) ~ (log R)2/R2 for the class of piecewise-smooth images containing smooth C2 regions separated by smooth C2 discontinuities. Finally, we extend this simple algorithm and propose a complete compression framework for natural images using a rate-distortion criterion to balance the three representations. Our Wedgelet-SFQ (WSFQ) coder outperforms SFQ in terms of visual quality and mean-square error.
In the last few years, it has become apparent that traditional wavelet-based image processing algorithms and models have significant shortcomings in their treatment of edge contours. The standard modeling paradigm exploits the fact that wavelet coefficients
representing smooth regions in images tend to have small magnitude, and that the multiscale nature of the wavelet transform implies that these small coefficients will persist across scale (the canonical
example is the venerable zero-tree coder). The edge contours in the image, however, cause more and more large magnitude wavelet coefficients as we move down through scale to finer resolutions. But if the contours are smooth, they become simple as we zoom in on them, and are well approximated by straight lines at fine scales. Standard wavelet models exploit the grayscale regularity of the smooth regions of the image, but not the geometric regularity of the contours.
In this paper, we build a model that accounts for this geometric regularity by capturing the dependencies between complex wavelet coefficients along a contour. The Geometric Hidden Markov Tree (GHMT) assigns each wavelet coefficient (or spatial cluster of wavelet
coefficients) a hidden state corresponding to a linear approximation of the local contour structure. The shift and rotational-invariance properties of the complex wavelet transform allow the GHMT to model
the behavior of each coefficient given the presence of a linear edge at a specified orientation --- the behavior of the wavelet coefficient given the state. By connecting the states together in a quadtree, the GHMT ties together wavelet coefficients along a contour, and also models how the contour itself behaves across scale.
We demonstrate the effectiveness of the model by applying it to feature extraction.
Since their introduction a little more than 10 years ago, wavelets
have revolutionized image processing. Wavelet based
algorithms define the state-of-the-art for applications
including image coding (JPEG2000), restoration, and segmentation.
Despite their success, wavelets have significant shortcomings in their
treatment of edges. Wavelets do not parsimoniously capture even the
simplest geometrical structure in images, and wavelet based processing
algorithms often produce images with ringing around the edges.
As a first step towards accounting for this structure, we will show
how to explicitly capture the geometric regularity of contours in
cartoon images using the wedgelet representation and a multiscale geometry model. The wedgelet representation builds up an image out of simple piecewise constant functions with linear discontinuities. We will show how the geometry model, by putting a joint distribution on the orientations of the linear discontinuities, allows us to weigh several factors when choosing the wedgelet representation: the error between the representation and the original image, the parsimony of the representation, and whether the wedgelets in the representation form "natural" geometrical structures. Finally, we will analyze a simple wedgelet coder based on these principles, and show that it has optimal asymptotic performance for simple cartoon images.
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