Recovering fine details from under-resolved electron tomography data using higher order total variation<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si0004.gif" overflow="scroll"><mml:msub subscriptshift="65%"><mml:mrow><mml:mo>ℓ</mml:mo></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math>regularization

Applied Numerical Mathematics

2018

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Popular methods for finding regularized solutions to inverse problems include sparsity promoting 1 regularization techniques, one in particular which is the well known total variation (TV) regularization. More recently, several higher order (HO) methods similar to TV have been proposed, which we generally refer to as HOTV methods. In this letter, we investigate the problem of the often debated selection of λ, the parameter used to carefully balance the interplay between data fitting and regularization terms. We theoretically for a scaling of the operators for a uniform parameter selection for all orders of HOTV regularization. In particular, parameter selection for all orders of HOTV may be determined by scaling an initial parameter for TV, which the imaging community may be more familiar with. We also provide several numerical results which justify our theoretical findings.

Section: Introductionmentioning

confidence: 99%

Parameter selection for HOTV regularization

Applied Numerical Mathematics

2018

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“…|γ j (C)| 2 + λ4 r sin 2r (πj/n) −1 4 r sin 2r (πj/n) (25) An algorithm summarizing these ideas and their implementation into our general strategy for finding λ are provided in Algorithm 2, where for simplicity we present the denoising case. In Figure 6, we have demonstrated the time saved as well as improved convergence when using these exact formulas, when compared with the iterative procedures for finding the solutions and Define H k = I + λ k T T r T r .…”

Section: Accelerated Iterations For Denoising and Deconvolutionmentioning

confidence: 99%

Effective new methods for automated parameter selection in regularized inverse problems

Applied Numerical Mathematics

Platte

Skeel

2020

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The choice of the parameter value for regularized inverse problems is critical to the results and remains a topic of interest. This article explores a criterion for selecting the right parameter value by maximizing the probability of the data, with no prior knowledge of the noise variance. These concepts are developed for 2 and consequently 1 regularization models by way of their Bayesian interpretations. Based on these concepts, an iterative scheme is proposed and demonstrated to converge accurately, and analytical convergence results are provided that substantiate these empirical observations. The computational concerns associated with the algorithm are carefully addressed, and methods for significant acceleration of the algorithm are provided for denoising and deconvolution problems. A robust set of 1D and 2D numerical simulations confirm the effectiveness of the proposed approach.

“…As an alternative to TV regularization, general order TV methods have been shown to be effective for regularization [ 8 , 14 , 16 , 20 ]. The TV transform can be thought of as a finite difference approximation of the first derivative, thus annihilating a function in locations where the function is a constant, i.e., a polynomial of degree zero.…”

Section: Hotv and Multiscale Generalizationsmentioning

confidence: 99%

“…For the next experiment with this data set, we reconstruct the 3D volume of the object from the available tilt series. To see how this problem is formulated as a regularized reconstruction in the form of ( 1 ), see for instance [ 8 ]. The tilt series is “full”, in the sense that the range of angles is over taken at every .…”

Section: Application To Multidimensional Electron Microscopy and Tomomentioning

confidence: 99%

Multiscale higher-order TV operators for L1 regularization

Platte

2018

Adv Struct Chem Imag

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In the realm of signal and image denoising and reconstruction, regularization techniques have generated a great deal of attention with a multitude of variants. In this work, we demonstrate that the formulation can sometimes result in undesirable artifacts that are inconsistent with desired sparsity promoting properties that the formulation is intended to approximate. With this as our motivation, we develop a multiscale higher-order total variation (MHOTV) approach, which we show is related to the use of multiscale Daubechies wavelets. The relationship of higher-order regularization methods with wavelets, which we believe has generally gone unrecognized, is shown to hold in several numerical results, although notable improvements are seen with our approach over both wavelets and classical HOTV. These results are presented for 1D signals and 2D images, and we include several examples that highlight the potential of our approach for improving two- and three-dimensional electron microscopy imaging. In the development approach, we construct the tools necessary for MHOTV computations to be performed efficiently, via operator decomposition and alternatively converting the problem into Fourier space.