Point spread function (PSF) deconvolution is an attractive software-based technique for resolution improvement in the scanning electron microscope (SEM) because it can restore information which has been blurred by challenging operating conditions. In Part 1, we studied a modern PSF determination method for SEM and explored how various parameters affected the method's ability to accurately estimate the PSF. In Part 2, we extend this exploration to PSF deconvolution for image restoration. The parameters include reference particle size, PSF smoothing (K), background correction, and restoration denoising (λ). Image quality was assessed by visual inspection and Fourier analysis. Overall, PSF deconvolution improved image quality. Low λ enhanced image sharpness at the cost of noise, while high λ created smoother restorations with less detail. λ should be chosen to balance feature preservation and denoising based on the application. Reference particle size within ±0.9 nm and K within a reasonable range had little effect on restoration quality. Restorations using background-corrected PSFs had superior quality compared with using no background correction, but if the correction was too high, the PSF was cut off causing blurrier restorations. Future efforts to automatically determine parameters would remove user guesswork, improve this method's consistency, and maximize interpretability of outputs.

A method is presented to determine the spatial distribution of electrons in the focused beam of a scanning electron microscope (SEM). Knowledge of the electron distribution is valuable for characterizing and monitoring SEM performance, as well as for modeling and simulation in computational scanning electron microscopy. Specifically, it can be used to characterize astigmatism as well as study the relationship between beam energy, beam current, working distance, and beam shape and size. In addition, knowledge of the distribution of electrons in the beam can be utilized with deconvolution methods to improve the resolution and quality of backscattered, secondary, and transmitted electron images obtained with thermionic, FEG, or Schottky source instruments. The proposed method represents an improvement over previous methods for determining the spatial distribution of electrons in an SEM beam. Several practical applications are presented.

This paper presents a heuristic Learning-based Non-Negativity Constrained Variation (L-NNCV) aiming to search the coefficients of variational model automatically and make the variation adapt different images and problems by supervised-learning strategy. The model includes two terms: a problem-based term that is derived from the prior knowledge, and an image-driven regularization which is learned by some training samples. The model can be solved by classical ε-constraint method. Experimental results show that: the experimental effectiveness of each term in the regularization accords with the corresponding theoretical proof; the proposed method outperforms other PDE-based methods on image denoising and deblurring.

We consider the total curvature of graphs of curves in high-codimension Euclidean space. We introduce the corresponding relaxed energy functional and prove an explicit representation formula. In the case of continuous Cartesian curves, i.e. of graphs cu of continuous functions u on an interval, we show that the relaxed energy is finite if and only if the curve cu has bounded variation and finite total curvature. In this case, moreover, the total curvature does not depend on the Cantor part of the derivative of u. We treat the wider class of graphs of one-dimensional functions of bounded variation, and we prove that the relaxed energy is given by the sum of the length and total curvature of the new curve obtained by closing the holes in cu generated by jumps of u with vertical segments.

The accuracy of attitude determination using a star sensor tends to be degraded if the host vehicle's manoeuvring smears the star image. In this paper, a new restoration algorithm with the aid of a Strap-down Inertial Navigation System (SINS) is proposed to reduce the effect of the smeared image. The smeared trace length is estimated with aid of the SINS angular rate. The restoration algorithm based on a Wiener filter is designed after the smeared zone is derived with the aid of the SINS coarse attitude. A tracking method is proposed to reject the stars of low centroid extraction accuracy. Simulations demonstrate that the success rate and the accuracy of attitude determination are improved significantly by the restoration algorithm even under a very fast rotation. The impact of the SINS error on the restoration is evaluated in the simulations.

In this paper, we propose a fast proximity point algorithm and apply it to total variation (TV) based image restoration. The novel method is derived from the idea of establishing a general proximity point operator framework based on which new first-order schemes for total variation (TV) based image restoration have been proposed. Many current algorithms for TV-based image restoration, such as Chambolle’s projection algorithm, the split Bregman algorithm, the Bermúdez-Moreno algorithm, the Jia-Zhao denoising algorithm, and the fixed point algorithm, can be viewed as special cases of the new first-order schemes. Moreover, the convergence of the new algorithm has been analyzed at length. Finally, we make comparisons with the split Bregman algorithm which is one of the best algorithms for solving TV-based image restoration at present. Numerical experiments illustrate the efficiency of the proposed algorithms.

A method is presented for determining the point spread function (PSF) of an electron beam in a scanning electron microscope for the examination of near planar samples. Once measured, PSFs can be used with two or more low-resolution images of a selected area to create a high-resolution reconstructed image of that area. As an example, a 4× improvement in resolution for images is demonstrated for a fine gold particle sample. Since thermionic source instruments have high beam currents associated with large probe sizes, use of this approach implies that high-resolution images can be produced rapidly if the probe diameter is less of a limiting factor. Additionally, very accurate determination of the PSFs can lead to a better understanding of instrument performance as exemplified by very accurate measurement of the beam shape and therefore the degree of astigmatism.

Image deconvolution problems with a symmetric point-spread function arise in many areas of science and engineering. These problems often are solved by the Richardson-Lucy method, a nonlinear iterative method. We first show a convergence result for the Richardson-Lucy method. The proof sheds light on why the method may converge slowly. Subsequently, we describe an iterative active set method that imposes the same constraints on the computed solution as the Richardson-Lucy method. Computed examples show the latter method to yield better restorations than the Richardson-Lucy method and typically require less computational effort.

This paper addresses the problem of how to restore degraded images where the pixels have been partly lost during transmission or damaged by impulsive noise. A wide range of image restoration tasks is covered in the mathematical model considered in this paper - e.g. image deblurring, image inpainting and super-resolution imaging. Based on the assumption that natural images are likely to have a sparse representation in a wavelet tight frame domain, we propose a regularization-based approach to recover degraded images, by enforcing the analysis-based sparsity prior of images in a tight frame domain. The resulting minimization problem can be solved efficiently by the split Bregman method. Numerical experiments on various image restoration tasks - simultaneously image deblurring and inpainting, super-resolution imaging and image deblurring under impulsive noise - demonstrated the effectiveness of our proposed algorithm. It proved robust to mis-detection errors of missing or damaged pixels, and compared favorably to existing algorithms.

A method based on the minimization of variation is presented for the identification of a completely unknown blur operator. We assume the knowledge of a blurred image and its original version. The class of blurring operators is identified in the class of compact operators. A variational method with negative norms is then used for the restoration of a blurred and noised image. The restoration method works for a wide class of blurring operators and we do not assume that the blur operator commutes with the Laplacian.

We study a time-delay regularization of the anisotropic diffusion model for image denoising of Perona and Malik [IEEE Trans. Pattern Anal. Mach. Intell12 (1990) 629–639], which has been proposed by Nitzberg and Shiota [IEEE Trans. Pattern Anal. Mach. Intell14 (1998) 826–835]. In the two-dimensional case, we show the convergence of a numerical approximation and the existence of a weak solution. Finally, we show some experiments on images.

This paper deals with two complementary methods in noisy image deblurring: a nonlinear shrinkage of wavelet-packets coefficients called FCNR and Rudin-Osher-Fatemi's variational method. The FCNR has for objective to obtain a restored image with a white noise. It will prove to be very efficient to restore an image after an invertible blur but limited in the opposite situation. Whereas the Total Variation based method, with its ability to reconstruct the lost frequencies by interpolation, is very well adapted to non-invertible blur, but that it tends to erase low contrast textures. This complementarity is highlighted when the methods are applied to the restoration of satellite SPOT images.

The spherical aberration in the primary mirror of the Hubble Space Telescope causes more than 80% of the light from a point source to be spread into a halo of radius of 2–3 arcsec. The point spread function (PSF) is both time variant (resulting from spacecraft jitter and desorption of the secondary mirror support structure) and space variant (owing to the Cassegrain repeater optics in the Wide Field / Planetary Camera). A variety of image restoration algorithms have been utilized on HST data with some success, although optimal restorations require better modeling of the PSF and the development of efficient restoration algorithms that accommodate a spacevariant PSF. The first HST servicing mission (December 1993) will deploy a corrective optics system for the Faint Object Camera and the two spectrographs and a second generation WF/PC with internal corrective optics. As simulations demonstrate, however, the restoration algorithms developed now for aberrated images will be very useful for removing the remaining diffraction features and optimizing dynamic range in post-servicing mission data.