There has been a tremendous increase in applications of the inverse problem framework to parameter estimation in magnetic resonance. Attempting to capture both the basics of this formalism and modern developments would require an article of inordinate length. Therefore, in the following, we provide basic material as a practical introduction to the topic and an entree to the literature. First, we describe the formulation of linear and nonlinear inverse problems, with an emphasis on signal equations arising in magnetic resonance. We then describe the Fredholm equation of the first kind as a paradigm for these problems. This is followed by much more detailed considerations for determining solutions in the linear case, including central concepts such as condition number, regularization, and stability. Solution methods for nonlinear inverse problems are described next, followed by a treatment of their stability and regularization. Finally, we provide an introduction to compressed sensing, with signal reconstruction formulated as the solution to an inverse problem, making use of much of the previous material. Throughout, the emphasis is on outlines of the theory and on numerical examples, rather than on mathematical rigor and completeness.
On the basis of nearfield acoustic holography (NAH) based on the equivalent source method (ESM), patch NAH based on the ESM is proposed. The method overcomes the shortcoming in the conventional NAH that the hologram surface should be larger than the source surface. It need not to discretize the whole source and its measurement need not to cover the whole source. The measurement may be performed over the region of interest, and the reconstruction can be done in the region directly. The method is flexible in applications, stable in computation, and very easy to implement. It has good potential applications in engineering. The numerical simulations show the invalidity of the conventional NAH based on the ESM and prove the validities of the proposed method for reconstructing a partial source and the regularization for reducing the error effect of the pressure measured on the hologram surface.acoustic holography, near field, equivalent source, regularization Nearfield acoustic holography (NAH) is a powerful technique for identifying noise source and visualizing acoustic field. By measuring the pressure on the hologram surface, the acoustic quantities on the source surface and in the three-dimension acoustic field can all be reconstructed. Because the measurement in the method is performed in the near field of a source, the evanescent waves that decay exponentially with distance are included, the restriction of the Rayleigh wavelength can be overcome, and a high-resolution reconstruction image can be obtained. It helps engineers to treat noise and vibration problems [1][2][3] .Patch NAH is a new idea on holographic reconstruction, which was first proposed by Williams [4,5] in 2003 on the basis of his new understanding on the "near field" in NAH. The idea is firstly introduced into the NAH based on the spatial FFT method successfully. Patch NAH is a method for holographic reconstruction when the hologram surface is smaller than the source surface. In the conventional NAH, the hologram surface is required to be larger than the source surface. For instance, in the NAH based on the spatial FFT method, the hologram surface should be
Analysis of multiexponential decay has remained a topic of active research for over 200 years. This attests to the widespread importance of this problem and to the profound difficulties in characterizing the underlying monoexponential decays. Here, we demonstrate the fundamental improvement in stability and conditioning of this classic problem through extension to a second dimension; we present statistical analysis, Monte-Carlo simulations, and experimental magnetic resonance relaxometry data to support this remarkable fact. Our results are readily generalizable to higher dimensions and provide a potential means of circumventing conventional limits on multiexponential parameter estimation.
Many methods have been developed for estimating the parameters of biexponential decay signals, which arise throughout magnetic resonance relaxometry (MRR) and the physical sciences. This is an intrinsically ill‐posed problem so that estimates can depend strongly on noise and underlying parameter values. Regularization has proven to be a remarkably efficient procedure for providing more reliable solutions to ill‐posed problems, while, more recently, neural networks have been used for parameter estimation. We re‐address the problem of parameter estimation in biexponential models by introducing a novel form of neural network regularization which we call input layer regularization (ILR). Here, inputs to the neural network are composed of a biexponential decay signal augmented by signals constructed from parameters obtained from a regularized nonlinear least‐squares estimate of the two decay time constants. We find that ILR results in a reduction in the error of time constant estimates on the order of 15%–50% or more, depending on the metric used and signal‐to‐noise level, with greater improvement seen for the time constant of the more rapidly decaying component. ILR is compatible with existing regularization techniques and should be applicable to a wide range of parameter estimation problems.
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