Baseband Detection of Bistatic Electron Spin Signals in Magnetic Resonance Force Microscopy (MRFM)

In single spin Magnetic Resonance Force Microscopy (MRFM), the objective is to detect the presence of an electron (or nuclear) spin in a sample volume by measuring spin-induced attonewton forces using a micromachined cantilever. In the OSCAR method of single spin MRFM, the spins are manipulated by an external rf field to produce small periodic deviations in the resonant frequency of the cantilever. These deviations can be detected by frequency demodulation followed by conventional amplitude or energy detection. In this paper, we present an alternative to these detection methods, based on optimal detection theory and Gibbs sampling. On the basis of simulations, we show that our detector outperforms the conventional amplitude and energy detectors for realistic MRFM operating conditions. For example, to achieve a 10% false alarm rate and an 80% correct detection rate our detector has an 8 dB SNR advantage as compared with the conventional amplitude or energy detectors. Furthermore, at these detection rates it comes within 4 dB of the omniscient matched-filter lower bound.Comment: 8 pages, 9 figures, revision of paper contains correction to a typo on the first page (introduction section

A Recursive Algorithm for Computing Cramer-Rao-Type Bouads on Estimator Covariance

We give a recursive algorithm to calculate submatrices of the Cramer-Rao (CR) matrix bound on the covariance of any unbiased estimator of a vector parameter ?_. Our algorithm computes a sequence of lower bounds that converges monotonically to the CR bound with exponential speed of convergence. The recursive algorithm uses an invertible “splitting matrix” to successively approximate the inverse Fisher information matrix. We present a statistical approach to selecting the splitting matrix based on a “complete-data-incomplete-data” formulation similar to that of the well-known EM parameter estimation algorithm. As a concrete illustration we consider image reconstruction from projections for emission computed tomography.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85950/1/Fessler104.pd

All degree six local unitary invariants of k qudits

We give explicit index-free formulae for all the degree six (and also degree four and two) algebraically independent local unitary invariant polynomials for finite dimensional k-partite pure and mixed quantum states. We carry out this by the use of graph-technical methods, which provides illustrations for this abstract topic.Comment: 18 pages, 6 figures, extended version. Comments are welcom

Penalized Maximum-Likelihood Image Reconstruction Using Space-Alternating Generalized EM Algorithms

Most expectation-maximization (EM) type algorithms for penalized maximum-likelihood image reconstruction converge slowly, particularly when one incorporates additive background effects such as scatter, random coincidences, dark current, or cosmic radiation. In addition, regularizing smoothness penalties (or priors) introduce parameter coupling, rendering intractable the M-steps of most EM-type algorithms. This paper presents space-alternating generalized EM (SAGE) algorithms for image reconstruction, which update the parameters sequentially using a sequence of small “hidden” data spaces, rather than simultaneously using one large complete-data space. The sequential update decouples the M-step, so the maximization can typically be performed analytically. We introduce new hidden-data spaces that are less informative than the conventional complete-data space for Poisson data and that yield significant improvements in convergence rate. This acceleration is due to statistical considerations, not numerical overrelaxation methods, so monotonic increases in the objective function are guaranteed. We provide a general global convergence proof for SAGE methods with nonnegativity constraints.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85850/1/Fessler102.pd

Complete-Data Spaces and Generalized EM Algorithms

Expectation-maximization (EM) algorithms have been applied extensively for computing maximum-likelihood and penalized-likelihood parameter estimates in signal processing applications. Intrinsic to each EM algorithm is a complete-data space (CDS)-a hypothetical set of random variables that is related to the parameters more naturally than the measurements are. The authors describe two generalizations of the EM paradigm: (i) allowing the relationship between the CDS and the measured data to be nondeterministic, and (ii) using a sequence of alternating complete-data spaces. These generalizations are motivated in part by the influence of the CDS on the convergence rate, a relationship that is formalized through a data-processing inequality for Fisher information. These concepts are applied to the problem of estimating superimposed signals in Gaussian noise, and it is shown that the new space alternating generalized EM algorithm converges significantly faster than the ordinary EM algorithm.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85977/1/Fessler123.pd