129 research outputs found
Discussion on the Time-Harmonic Elastodynamic Half-Space Green’s Function Obtained by Superposition
The time-harmonic elastodynamic half-space Green’s function derived by Banerjee and Mamoon by way of superposition is discussed and examined against another semianalytical solution and a numerical solution. It is shown that Banerjee and Mamoon’s solution gives infinite z-displacement response when the depth of the source goes to infinity, which is unreasonable and does not agree with other solutions. A possible problem in the derivation is that it is inappropriate to directly extend the results of Mindlin’s superposition method for the elastostatic half-space problem to the dynamic case. The superposition of the six full-space elastodynamic solutions does not satisfy the required boundary conditions of the half-space elastodynamic problem as in the static case and thus does not solve the dynamic half-space problem
Phase Retrieval with Background Information: Decreased References and Efficient Methods
Fourier phase retrieval(PR) is a severely ill-posed inverse problem that
arises in various applications. To guarantee a unique solution and relieve the
dependence on the initialization, background information can be exploited as a
structural priors. However, the requirement for the background information may
be challenging when moving to the high-resolution imaging. At the same time,
the previously proposed projected gradient descent(PGD) method also demands
much background information.
In this paper, we present an improved theoretical result about the demand for
the background information, along with two Douglas Rachford(DR) based methods.
Analytically, we demonstrate that the background required to ensure a unique
solution can be decreased by nearly for the 2-D signals compared to the
1-D signals. By generalizing the results into -dimension, we show that the
length of the background information more than folds of
the signal is sufficient to ensure the uniqueness. At the same time, we also
analyze the stability and robustness of the model when measurements and
background information are corrupted by the noise. Furthermore, two methods
called Background Douglas-Rachford (BDR) and Convex Background Douglas-Rachford
(CBDR) are proposed. BDR which is a kind of non-convex method is proven to have
the local R-linear convergence rate under mild assumptions. Instead, CBDR
method uses the techniques of convexification and can be proven to own a global
convergence guarantee as long as the background information is sufficient. To
support this, a new property called F-RIP is established. We test the
performance of the proposed methods through simulations as well as real
experimental measurements, and demonstrate that they achieve a higher recovery
rate with less background information compared to the PGD method
Untrained neural network embedded Fourier phase retrieval from few measurements
Fourier phase retrieval (FPR) is a challenging task widely used in various
applications. It involves recovering an unknown signal from its Fourier
phaseless measurements. FPR with few measurements is important for reducing
time and hardware costs, but it suffers from serious ill-posedness. Recently,
untrained neural networks have offered new approaches by introducing learned
priors to alleviate the ill-posedness without requiring any external data.
However, they may not be ideal for reconstructing fine details in images and
can be computationally expensive. This paper proposes an untrained neural
network (NN) embedded algorithm based on the alternating direction method of
multipliers (ADMM) framework to solve FPR with few measurements. Specifically,
we use a generative network to represent the image to be recovered, which
confines the image to the space defined by the network structure. To improve
the ability to represent high-frequency information, total variation (TV)
regularization is imposed to facilitate the recovery of local structures in the
image. Furthermore, to reduce the computational cost mainly caused by the
parameter updates of the untrained NN, we develop an accelerated algorithm that
adaptively trades off between explicit and implicit regularization.
Experimental results indicate that the proposed algorithm outperforms existing
untrained NN-based algorithms with fewer computational resources and even
performs competitively against trained NN-based algorithms
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