Tikhonov Deconvolution Research Articles

Deep learning-based galaxy image deconvolution

With the onset of large-scale astronomical surveys capturing millions of images, there is an increasing need to develop fast and accurate deconvolution algorithms that generalize well to different images. A powerful and accessible deconvolution method would allow for the reconstruction of a cleaner estimation of the sky. The deconvolved images would be helpful to perform photometric measurements to help make progress in the fields of galaxy formation and evolution. We propose a new deconvolution method based on the Learnlet transform. Eventually, we investigate and compare the performance of different Unet architectures and Learnlet for image deconvolution in the astrophysical domain by following a two-step approach: a Tikhonov deconvolution with a closed-form solution, followed by post-processing with a neural network. To generate our training dataset, we extract HST cutouts from the CANDELS survey in the F606W filter (V-band) and corrupt these images to simulate their blurred-noisy versions. Our numerical results based on these simulations show a detailed comparison between the considered methods for different noise levels.

Quantitative Magnetic Force Microscopy: Transfer-Function Method Revisited

Magnetic force microscopy (MFM) is a characterization method used to obtain images of the stray field emanating from the surface of magnetic samples with high spatial resolution. The MFM signal arises from a convolution of the magnetic moment distribution of the MFM tip with the stray field of the sample. Therefore, deconvolution of the stray field of a magnetic sample requires a well-defined micromagnetic structure of a tip. While the magnetic moment distribution of the tip remains challenging to assess, it is convenient to calibrate the tip-equivalent magnetic charge in a plane parallel to the sample surface running through the tip apex using a suitable tip-calibration method. Once the tip-equivalent magnetic charge is calibrated, MFM data can be deconvolved to obtain the stray field of the sample or the equivalent magnetic surface charge, for example, at the sample surface. Here we use low-noise MFM data obtained with high-quality-factor cantilevers and a modified L-curve method for the optimization of a Tikhonov deconvolution procedure used for the calibration of the MFM tip, and---once the tip is calibrated---for obtaining the stray field or magnetic surface charge of the sample.

Deep learning for a space-variant deconvolution in galaxy surveys

The deconvolution of large survey images with millions of galaxies requires developing a new generation of methods that can take a space-variant point spread function into account. These methods have also to be accurate and fast. We investigate how deep learning might be used to perform this task. We employed a U-net deep neural network architecture to learn parameters that were adapted for galaxy image processing in a supervised setting and studied two deconvolution strategies. The first approach is a post-processing of a mere Tikhonov deconvolution with closed-form solution, and the second approach is an iterative deconvolution framework based on the alternating direction method of multipliers (ADMM). Our numerical results based on GREAT3 simulations with realistic galaxy images and point spread functions show that our two approaches outperform standard techniques that are based on convex optimization, whether assessed in galaxy image reconstruction or shape recovery. The approach based on a Tikhonov deconvolution leads to the most accurate results, except for ellipticity errors at high signal-to-noise ratio. The ADMM approach performs slightly better in this case. Considering that the Tikhonov approach is also more computation-time efficient in processing a large number of galaxies, we recommend this approach in this scenario.

Spatially varying regularization of deconvolution in 3D microscopy.

Confocal microscopy has become an essential tool to explore biospecimens in 3D. Confocal microcopy images are still degraded by out-of-focus blur and Poisson noise. Many deconvolution methods including the Richardson-Lucy (RL) method, Tikhonov method and split-gradient (SG) method have been well received. The RL deconvolution method results in enhanced image quality, especially for Poisson noise. Tikhonov deconvolution method improves the RL method by imposing a prior model of spatial regularization, which encourages adjacent voxels to appear similar. The SG method also contains spatial regularization and is capable of incorporating many edge-preserving priors resulting in improved image quality. The strength of spatial regularization is fixed regardless of spatial location for the Tikhonov and SG method. The Tikhonov and the SG deconvolution methods are improved upon in this study by allowing the strength of spatial regularization to differ for different spatial locations in a given image. The novel method shows improved image quality. The method was tested on phantom data for which ground truth and the point spread function are known. A Kullback-Leibler (KL) divergence value of 0.097 is obtained with applying spatially variable regularization to the SG method, whereas KL value of 0.409 is obtained with the Tikhonov method. In tests on a real data, for which the ground truth is unknown, the reconstructed data show improved noise characteristics while maintaining the important image features such as edges.

A proposal for regularized inversion for an ill-conditioned deconvolution operator

From the inverse problem theory aspect, deconvolution can be understood as the linear inversion of an ill-posed and ill-conditioned problem. The ill-conditioned property of the deconvolution operator make the solution of inverse problem sensitive to errors in the data.  Tikhonov regularization is the most commonly used method for stability and uniqueness of the solution. However, results from Tikhonov method do not provide sufficient quality when the noise in the data is strong. This work uses the conjugate gradient method applied to the Tikhonov deconvolution scheme, including a regularization parameter calculated iteratively and based on the improvement criterion of Morozov discrepancy applied on the objective function. Using seismic synthetic data and real stacked seismic data, we carried out a deconvolution process with regularization and without regularization based on a conjugated gradient algorithm. A comparison of results is also presented. Applying regularized deconvolution on synthetic data shows improved stability on the solution. Additionally, real post-stack seismic data shows a direct application for increasing the vertical resolution even with noisy data.

Numerical analysis of Leray–Tikhonov deconvolution models of fluid motion

This report develops and studies a new family of Navier–Stokes equation regularizations: Leray–Tikhonov regularizations with time relaxation models. This new family of turbulence models is based on a modification (consistent with the large scales) of Tikhonov–Lavrentiev regularization. With this approach, we obtain an approximation of the unfiltered solution by one filtering step. We introduce the modified Tikhonov deconvolution operator and study its mathematical properties. We also perform rigorous numerical analysis of a computationally attractive algorithm for this family of models and present numerical experiments using it.

Deconvolution of axisymmetric flame properties using Tikhonov regularization

We present a method based on Tikhonov regularization for solving one-dimensional inverse tomography problems that arise in combustion applications. In this technique, Tikhonov regularization transforms the ill-conditioned set of equations generated by onion-peeling deconvolution into a well-conditioned set that is less susceptible to measurement errors that arise in experimental settings. The performance of this method is compared to that of onion-peeling and Abel three-point deconvolution by solving for a known field variable distribution from projected data contaminated with an artificially generated error. The results show that Tikhonov deconvolution provides a more accurate field distribution than onion-peeling and Abel three-point deconvolution and is more stable than the other two methods as the distance between projected data points decreases.

The use of deconvolution and iterative optimization for spectrogram interpretation

The problem of spectrogram interpretation is considered under the assumption that the parameters of spectral peaks-their positions and magnitudes-contain the information essential for spectrometric analysis. The subsequent use of Tikhonov deconvolution and iterative correction of the estimates of those parameters is proposed. Deconvolution is used for transforming the processed spectrogram in such a way as to facilitate finding initial estimates of its parameters. The advantages of the proposed approach, i.e., gains in accuracy of estimating the parameters of peaks, are demonstrated using both synthetic and real-world spectrophotometric data.

Combined Use of Tikhonov Deconvolution and Curve Fitting for Spectrogram Interpretation

Abstract The problem of numerical correction of spectrograms is addressed. A new method of correction is developed which consists of sequential use of the Tikhonov deconvolution algorithm, for estimating the positions of spectral peaks, and a curve-fitting algorithm, for estimating their magnitudes. The metrological and numerical properties of the proposed method for spectrogram interpretation are assessed by means of spectrometry-based criteria, using synthetic and real-world spectrograms. Conclusions are drawn concerning computational complexity and accuracy of the proposed method and its metrological applicability.