Maximum entropy and NMR—A new approach

Daniell, G.J.; Hore, P. J.

doi:10.1016/0022-2364(89)90117-0

Cited by 35 publications

(33 citation statements)

References 11 publications

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“…Several numerical algorithms have been pubexpected error for each point. For convenience, we will gen-lished (4,17,18,31). They all work by iteratively improverally refer to aim instead of C 0 ; aim is defined as C 0 /2N ing a trial guess of the spectrum and simultaneously adand is comparable to the expected error in each component justing l so as to arrive at a final solution satisfying the (real and imaginary) of each point in the data set.…”

Section: Overview Of Maxent Reconstructionmentioning

confidence: 99%

Quantification of Maximum-Entropy Spectrum Reconstructions

Schmieder

Stern²,

Wagner

et al. 1997

Journal of Magnetic Resonance

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Section: Overview Of Maxent Reconstructionmentioning

confidence: 99%

Quantification of Maximum-Entropy Spectrum Reconstructions

Schmieder

Stern²,

Wagner

et al. 1997

Journal of Magnetic Resonance

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“…The mathematical problem of maximum entropy reconstruction consists of finding the spectrum f exhibiting the maximum value of the entropy S, subject to the constraint that d, the inverse Fourier transform off, is consistent with the measured data d. Now, the entropy of a positive real spectrum is given by the equation: N.,,, S = -I ft logf, W=1 [1] expressed in suitable units (4). [Note that this analysis can easily be generalized to handle complex spectra (5,8,12,13 [3] where dt = 1v/<, NIN fIv e27ritw/N. The solution we seek corresponds to a critical point of Q, with the value ofA chosen so that C = CO.…”

mentioning

confidence: 99%

Does the maximum entropy method improve sensitivity?

Donoho

Johnstone

Stern

et al. 1990

Proc. Natl. Acad. Sci. U.S.A.

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Maximum entropy reconstruction has been used in several fields to produce visually striking reconstructions of positive objects (images, densities, spectra) from noisy, indirect measurements. In magnetic resonance spectroscopy, this technique is notable for its apparent noise suppression and its avoidance of the artifacts that affect discrete Fourier transform spectra of short (zero-extended) data records. In the general case where the length of the reconstructed spectrum exceeds that of the data record or where a convolution kernel is incorporated in the reconstruction, no known analytical solution to the reconstruction problem exists. Consequently, knowledge of the properties of maximum entropy reconstruction has been mainly anecdotal, based on a small selection of published reconstructions. However, in the limiting case where the lengths of the reconstructed spectrum and the data record are the same and a convolution kernel is not applied, the problem can be solved analytically. The solution has a simple structure that helps explain several commonly observed features of maximum entropy reconstructions-for example, the biases in the recovered intensities and the fact that noise near the baseline is more successfully suppressed than is noise superimposed on broad features in the spectrum. The solution also shows that the noise suppression offered by maximum entropy reconstruction could (in this special case) be equally well obtained by a "cosmetic" device: simply displaying the conventional Fourier transform reconstruction using a certain nonlinear plotting scale for the vertical (y) coordinate.Maximum entropy reconstruction has been applied to inverse problems in a variety of fields (1-3). In NMR spectroscopy (3), for example, maximum entropy reconstruction has been used to obtain apparent improvements in signal-to-noise ratio over conventional discrete Fourier transform spectrum estimates. Unfortunately, just how maximum entropy reconstruction achieves these impressive results is not clear. This situation is largely because in the general case no analytical expression exists for the maximum entropy reconstruction, which must therefore be obtained by numerical methods. To gain some insight into the maximum entropy method, we consider maximum entropy reconstruction applied to a special class of problems for which a formal analytical solution can be found. We show that the reconstructions take the form of a single nonlinear transformation applied point-by-point to the discrete Fourier transform of the data. This result explains many published examples of maximum entropy reconstructions and shows that the maximum entropy method does not, in this special case, improve the discrimination of signal from noise. emphasized is the suppression of noise near the baseline; however, close examination reveals another characteristic feature: while deemphasized relative to strong signals, the structure of the noise near the baseline is mostly preserved. A third characteristic feature is evident in Fig. 2: noise near t...

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“…It removes NUS artifacts by modeling each transform coefficient in the spatial-spectral domain as the byproduct of an ensemble of 1 H spins whose phase dispersion determines the signal amplitude; in-phase spin ensembles have high signal amplitude and low phase entropy, while high entropy ensembles have low phase coherence and low signal amplitude. Because Shannon entropy is derived for discrete processes that result from a Poisson-based distribution, it cannot be used for reconstructing NMR data, which do not follow a Poisson distribution [26,27].…”

Section: Cs Tv and Maxent Based Mrsi Reconstructionmentioning

confidence: 99%

“…Because the mixed-domain k y − t 1 plane of a 4D EP-COSI dataset is self-sparse, no sparsifying transforms are applied to u for the CS and TV reconstruction so Ψ = I [16]. The MaxEnt signal model uses φ(u) = −S1 /2 , which is the spin-1 /2 entropy derived from the statistical distributions defined by the density matrices of spin-1 /2 nuclei, such as 1 H [26]. It removes NUS artifacts by modeling each transform coefficient in the spatial-spectral domain as the byproduct of an ensemble of 1 H spins whose phase dispersion determines the signal amplitude; in-phase spin ensembles have high signal amplitude and low phase entropy, while high entropy ensembles have low phase coherence and low signal amplitude.…”

Section: Cs Tv and Maxent Based Mrsi Reconstructionmentioning

confidence: 99%

Group Sparse Reconstruction of Multi-Dimensional Spectroscopic Imaging in Human Brain in vivo

Burns

Wilson²,

Thomas

2014

Algorithms

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Four-dimensional (4D) Magnetic Resonance Spectroscopic Imaging (MRSI) data combining 2 spatial and 2 spectral dimensions provides valuable biochemical information in vivo; however, its 20-40 min acquisition time is too long to be used for a clinical protocol. Data acquisition can be accelerated by non-uniformly under-sampling (NUS) the k y − t 1 plane, but this causes artifacts in the spatial-spectral domain that must be removed by non-linear, iterative reconstruction. Previous work has demonstrated the feasibility of accelerating 4D MRSI data acquisition through NUS and iterative reconstruction using Compressed Sensing (CS), Total Variation (TV), and Maximum Entropy (MaxEnt) reconstruction. Group Sparse (GS) reconstruction is a variant of CS that exploits the structural sparsity of transform coefficients to achieve higher acceleration factors than traditional CS. In this article, we derive a solution to the GS reconstruction problem within the Split Bregman iterative framework that uses arbitrary transform grouping patterns of overlapping or non-overlapping groups. The 4D Echo-Planar Correlated Spectroscopic Imaging (EP-COSI) gray matter brain phantom and in vivo brain data are retrospectively under-sampled 2×, 4×, 6×, 8×, and 10× and reconstructed using CS, TV, MaxEnt, and GS with overlapping or non-overlapping groups. Results show that GS reconstruction with overlapping groups outperformed the other reconstruction methods at each NUS rate for both phantom and in vivo data. These results can potentially reduce the scan time of a 4D EP-COSI brain scan from 40 min to under 5 min in vivo.Algorithms 2014, 7 277

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Maximum entropy and NMR—A new approach

Cited by 35 publications

References 11 publications

Quantification of Maximum-Entropy Spectrum Reconstructions

Quantification of Maximum-Entropy Spectrum Reconstructions

Does the maximum entropy method improve sensitivity?

Group Sparse Reconstruction of Multi-Dimensional Spectroscopic Imaging in Human Brain in vivo

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