We present a novel method for aligning images in an HDR (high-dynamic-range) image stack to produce a new exposure stack where all the images are aligned and appear as if they were taken simultaneously, even in the case of highly dynamic scenes. Our method produces plausible results even where the image used as a reference is either too dark or bright to allow for an accurate registration.

We have previously reported on coded aperture snapshot spectral imagers (CASSI) that can capture a full frame spectral image in a snapshot. Here we describe the use of CASSI for spectral imaging of a dynamic scene at video rate. We describe significant advances in the design of the optical system, system calibration procedures and reconstruction method. The new optical system uses a double Amici prism to achieve an in-line, direct view configuration, resulting in a substantial improvement in image quality. We describe NeAREst, an algorithm for estimating the instantaneous three-dimensional spatio-spectral data cube from CASSI's two-dimensional array of encoded and compressed measurements. We utilize CASSI's snapshot ability to demonstrate a spectral image video of multi-colored candles with live flames captured at 30 frames per second.

An extended sequentially semiseparable (SSS) representation derived from timevarying system theory is used to capture, on the one hand, the low-rank of the off-diagonal blocks of a matrix for the purposes of efficient computations and, on the other, to provide for sufficient descriptive richness to allow for backward stability in the computations. We present (i) a fast algorithm (linear in the number of equations) to solve least squares problems in which the coefficient matrix is in SSS form, (ii) a fast algorithm to find the SSS form of X such that AX = B, where A and B are in SSS form, and (iii) a fast model reduction technique to improve the SSS form.

Abstract. In this paper we develop a new superfast solver for Toeplitz systems of linear equations. To solve Toeplitz systems many people use displacement equation methods. With displacement structures, Toeplitz matrices can be transformed into Cauchy-like matrices using the FFT or other trigonometric transformations. These Cauchy-like matrices have a special property, that is, their off-diagonal blocks have small numerical ranks. This low-rank property plays a central role in our superfast Toeplitz solver. It enables us to quickly approximate the Cauchy-like matrices by structured matrices called sequentially semiseparable (SSS) matrices. The major work of the constructions of these SSS forms can be done in precomputations (independent of the Toeplitz matrix entries). These SSS representations are compact because of the low-rank property. The SSS Cauchy-like systems can be solved in linear time with linear storage. Excluding precomputations the main operations are the FFT and SSS system solvers, which are both very efficient. Our new Toeplitz solver is stable in practice. Numerical examples are presented to illustrate the efficiency and the practical stability.

We develop an algorithmic framework for reducing the bandwidth of symmetric matrices via orthogonal similarity transformations. This framework includes the reduction of full matrices to banded or tridiagonal form and the reduction of banded matrices to narrower banded or tridiagonal form, possibly in multiple steps. Our framework leads to algorithms that require fewer floating-point operations than do standard algorithms, if only the eigenvalues are required. In addition, it allows for space-time tradeoffs and enables or increases the use of blocked transformations.

The QR factorization with column pivoting (QRP), originally suggested by Golub [Numer. Math., 7 (1965), 206-216], is a popular approach to computing rank-revealing factorizations. Using Level 1 BLAS, it was implemented in LINPACK, and, using Level 2 BLAS, in LAPACK. While the Level 2 BLAS version delivers superior performance in general, it may result in worse performance for large matrix sizes due to cache effects. We introduce a modification of the QRP algorithm which allows the use of Level 3 BLAS kernels while maintaining the numerical behavior of the LINPACK and LAPACK implementations. Experimental comparisons of this approach with the LINPACK and LAPACK implementations on IBM RS/6000, SGI R8000, and DEC AXP platforms show considerable performance improvements.

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We present a software toolbox for symmetric band reduction via orthogonal transformations, together with a testing and timing program. The toolbox contains drivers and computational routines for the reduction of full symmetric matrices to banded form and the reduction of banded matrices to narrower banded or tridiagonal form, with optional accumulation of the orthogonal transformations, as well as repacking routines for storage rearrangement. The functionality and the calling sequences of the routines are described, with a detailed discussion of the “control” parameters that allow adaptation of the codes to particular machine and matrix characteristics. We also briefly describe the testing and timing program included in the toolbox.