Kenneth Kreutz-Delgado

Sparse solutions to linear inverse problems with multiple measurement vectors

We address the problem of finding sparse solutions to an underdetermined system of equations when there are multiple measurement vectors having the same, but unknown, sparsity structure. The single measurement sparse solution problem has been extensively studied in the past. Although known to be NP-hard, many single-measurement suboptimal algorithms have been formulated that have found utility in many different applications. Here, we consider in depth the extension of two classes of algorithms-Matching Pursuit (MP) and FOCal Underdetermined System Solver (FOCUSS)-to the multiple measurement case so that they may be used in applications such as neuromagnetic imaging, where multiple measurement vectors are available, and solutions with a common sparsity structure must be computed. Cost functions appropriate to the multiple measurement problem are developed, and algorithms are derived based on their minimization. A simulation study is conducted on a test-case dictionary to show how the utilization of more than one measurement vector improves the performance of the MP and FOCUSS classes of algorithm, and their performances are compared.

The attitude control problem

A general framework for the analysis of the attitude tracking control problem for a rigid body is presented. A large family of globally stable control laws is obtained by using the globally nonsingular unit quaternion representation in a Lyapunov function candidate whose form is motivated by the consideration of the total energy of the rigid body. The controllers share the common structure of a proportional-derivative feedback plus some feedforward which can be zero (the model-independent case), the Coriolis torque compensation, or an adaptive compensation. These controller structures are compared in terms of the requirement on the a priori model information, guaranteed transient performance, and robustness. The global stability of the Luh-Walker-Paul robot end-effector controller is also analyzed in this framework. >

Dictionary learning algorithms for sparse representation

Algorithms for data-driven learning of domain-specific overcomplete dictionaries are developed to obtain maximum likelihood and maximum a posteriori dictionary estimates based on the use of Bayesian models with concave/Schur-concave (CSC) negative log priors. Such priors are appropriate for obtaining sparse representations of environmental signals within an appropriately chosen (environmentally matched) dictionary. The elements of the dictionary can be interpreted as concepts, features, or words capable of succinct expression of events encountered in the environment (the source of the measured signals). This is a generalization of vector quantization in that one is interested in a description involving a few dictionary entries (the proverbial 25 words or less), but not necessarily as succinct as one entry. To learn an environmentally adapted dictionary capable of concise expression of signals generated by the environment, we develop algorithms that iterate between a representative set of sparse representations found by variants of FOCUSS and an update of the dictionary using these sparse representations. Experiments were performed using synthetic data and natural images. For complete dictionaries, we demonstrate that our algorithms have improved performance over other independent component analysis (ICA) methods, measured in terms of signal-to-noise ratios of separated sources. In the overcomplete case, we show that the true underlying dictionary and sparse sources can be accurately recovered. In tests with natural images, learned overcomplete dictionaries are shown to have higher coding efficiency than complete dictionaries; that is, images encoded with an overcomplete dictionary have both higher compression (fewer bits per pixel) and higher accuracy (lower mean square error).

An affine scaling methodology for best basis selection

A methodology is developed to derive algorithms for optimal basis selection by minimizing diversity measures proposed by Wickerhauser (1994) and Donoho (1994). These measures include the p-norm-like (l/sub (p/spl les/1)/) diversity measures and the Gaussian and Shannon entropies. The algorithm development methodology uses a factored representation for the gradient and involves successive relaxation of the Lagrangian necessary condition. This yields algorithms that are intimately related to the affine scaling transformation (AST) based methods commonly employed by the interior point approach to nonlinear optimization. The algorithms minimizing the (l/sub (p/spl les/1)/) diversity measures are equivalent to a previously developed class of algorithms called focal underdetermined system solver (FOCUSS). The general nature of the methodology provides a systematic approach for deriving this class of algorithms and a natural mechanism for extending them. It also facilitates a better understanding of the convergence behavior and a strengthening of the convergence results. The Gaussian entropy minimization algorithm is shown to be equivalent to a well-behaved p=0 norm-like optimization algorithm. Computer experiments demonstrate that the p-norm-like and the Gaussian entropy algorithms perform well, converging to sparse solutions. The Shannon entropy algorithm produces solutions that are concentrated but are shown to not converge to a fully sparse solution.

Subset selection in noise based on diversity measure minimization

We develop robust methods for subset selection based on the minimization of diversity measures. A Bayesian framework is used to account for noise in the data and a maximum a posteriori (MAP) estimation procedure leads to an iterative procedure which is a regularized version of the focal underdetermined system solver (FOCUSS) algorithm. The convergence of the regularized FOCUSS algorithm is established and it is shown that the stable fixed points of the algorithm are sparse. We investigate three different criteria for choosing the regularization parameter: quality of fit; sparsity criterion; L-curve. The L-curve method, as applied to the problem of subset selection, is found not to be robust, and we propose a novel modified L-curve procedure that solves this problem. Each of the regularized FOCUSS algorithms is evaluated through simulation of a detection problem, and the results are compared with those obtained using a sequential forward selection algorithm termed orthogonal matching pursuit (OMP). In each case, the regularized FOCUSS algorithm is shown to be superior to the OMP in noisy environments.

A spatial operator algebra for manipulator modeling and control

A recently developed spatial operator algebra for manipu lator modeling, control, and trajectory design is dis cussed. The elements of this algebra are linear operators whose domain and range spaces consist of forces, moments, velocities, and accelerations. The effect of these operators is equivalent to a spatial recursion along the span of a manipulator. Inversion of operators can be efficiently obtained via techniques of recursive filtering and smoothing. The operator algebra provides a high- level framework for describing the dynamic and kinematic behavior of a manipulator and for control and trajectory design algorithms. The interpretation of expressions within the algebraic framework leads to enhanced concep tual and physical understanding of manipulator dynamics and kinematics. Furthermore, implementable recursive algorithms can be immediately derived from the abstract operator expressions by inspection. Thus the transition from an abstract problem formulation and solution to the detailed mechanization of specific algorithms is greatly simplified.

Improved disk-drive failure warnings

Improved methods are proposed for disk-drive failure prediction. The SMART (self monitoring and reporting technology) failure prediction system is currently implemented in disk-drives. Its purpose is to predict the near-term failure of an individual hard disk-drive, and issue a backup warning to prevent data loss. Two experimental tests of SMART show only moderate accuracy at low false-alarm rates. (A rate of 0.2% of total drives per year implies that 20% of drive returns would be good drives, relative to /spl ap/1% annual failure rate of drives). This requirement for very low false-alarm rates is well known in medical diagnostic tests for rare diseases, and methodology used there suggests ways to improve SMART. Two improved SMART algorithms are proposed. They use the SMART internal drive attribute measurements in present drives. The present warning-algorithm based on maximum error thresholds is replaced by distribution-free statistical hypothesis tests. These improved algorithms are computationally simple enough to be implemented in drive microprocessor firmware code. They require only integer sort operations to put several hundred attribute values in rank order. Some tens of these ranks are added up and the SMART warning is issued if the sum exceeds a prestored limit. These new algorithms were tested on 3744 drives of 2 models. They gave 3-4 times higher correct prediction accuracy than error thresholds on will-fail drives, at 0.2% false-alarm rate. The highest accuracies achievable are modest (40%-60%). Care was taken to test will-fail drive prediction accuracy on data independent of the algorithm design data. Additional work is needed to verify and apply these algorithms in actual drive design. They can also be useful in drive failure analysis engineering. It might be possible to screen drives in manufacturing using SMART attributes. Marginal drives might be detected before substantial final test time is invested in them, thereby decreasing manufacturing cost, and possibly decreasing overall field failure rates.

Motion and force control of multiple robotic manipulators

This paper addresses the motion and force control problem of multiple robot arms manipulating a cooperatively held object. A general control paradigm is introduced which decouples the motion and force control problems. For motion control, different control strategies are constructed based on the variables used as the control input in the controller design. There are three natural choices; acceleration of a generalized coordinate, arm tip force vectors, and the joint torques. The first two choices require full model information but produce simple models for the control design problem. The last choice results in a class of relatively model independent control laws by exploiting the Hamiltonian structure of the open loop system. The motion control only determines the joint torque to within a manifold, due to the multiple-arm kinematic constraint. To resolve the nonuniqueness of the joint torques, two methods are introduced. If the arm and object models are available, an optimization can be performed to best allocate the desired and effector control force to the joint actuators. The other possibility is to control the internal force about some set point. It is shown that effective force regulation can be achieved even if little model information is available.

A grid algorithm for autonomous star identification

An autonomous star identification algorithm is described that is simple and requires less computer resources than other such algorithms. In simulations using an 8/spl times/8 degree field of view (FOV), the algorithm identifies the correct section of sky on 99.7% of the sensor orientations where spatial accuracy of the imaged star is 1 pixel (56.25 arc seconds) in standard deviation and the apparent brightness deviates by 0.4 units stellar magnitude. This compares very favorably with other algorithms in the literature.

Kinematic analysis of 7-DOF manipulators

This article presents a kinematic analysis of seven-degree-of- freedom serial link spatial manipulators with revolute joints. To uniquely determine the joint angles for a given end-effector position and orientation, the redundancy is parameterized by a scalar variable that defines the angle between the arm plane and a reference plane. The forward kinematic mappings from joint space to end-effector coordinates and arm angle and the augmented Jacobian matrix that gives end-effector and arm angle rates as functions of joint rates are presented. Conditions under which the augmented Jacobian becomes singular are also given and are shown to correspond to the arm being either at a kinematically singular configuration or at a nonsingular configuration for which the arm angle ceases to parameterize the redundancy.