Mila Nikolova

Fast Two-Phase Image Deblurring Under Impulse Noise

In this paper, we propose a two-phase approach to restore images corrupted by blur and impulse noise. In the first phase, we identify the outlier candidates—the pixels that are likely to be corrupted by impulse noise. We consider that the remaining data pixels are essentially free of outliers. Then in the second phase, the image is deblurred and denoised simultaneously by a variational method by using the essentially outlier-free data. The experiments show several dB’s improvement in PSNR with respect to the typical variational methods.

Inversion of large-support ill-posed linear operators using a piecewise Gaussian MRF

We propose a method for the reconstruction of signals and images observed partially through a linear operator with a large support (e.g., a Fourier transform on a sparse set). This inverse problem is ill-posed and we resolve it by incorporating the prior information that the reconstructed objects are composed of smooth regions separated by sharp transitions. This feature is modeled by a piecewise Gaussian (PG) Markov random field (MRF), known also as the weak-string in one dimension and the weak-membrane in two dimensions. The reconstruction is defined as the maximum a posteriori estimate. The prerequisite for the use of such a prior is the success of the optimization stage. The posterior energy corresponding to a PG MRF is generally multimodal and its minimization is particularly problematic. In this context, general forms of simulated annealing rapidly become intractable when the observation operator extends over a large support. In this paper, global optimization is dealt with by extending the graduated nonconvexity (GNC) algorithm to ill-posed linear inverse problems. GNC has been pioneered by Blake and Zisserman in the field of image segmentation. The resulting algorithm is mathematically suboptimal but it is seen to be very efficient in practice. We show that the original GNC does not correctly apply to ill-posed problems. Our extension is based on a proper theoretical analysis, which provides further insight into the GNC. The performance of the proposed algorithm is corroborated by a synthetic example in the area of diffraction tomography.

Fast Hue and Range Preserving Histogram: Specification: Theory and New Algorithms for Color Image Enhancement.

Color image enhancement is a complex and challenging task in digital imaging with abundant applications. Preserving the hue of the input image is crucial in a wide range of situations. We propose simple image enhancement algorithms, which conserve the hue and preserve the range (gamut) of the R, G, B channels in an optimal way. In our setup, the intensity input image is transformed into a target intensity image whose histogram matches a specified, well-behaved histogram. We derive a new color assignment methodology where the resulting enhanced image fits the target intensity image. We analyze the obtained algorithms in terms of chromaticity improvement and compare them with the unique and quite popular histogram-based hue and range preserving algorithm of Naik and Murthy. Numerical tests confirm our theoretical results and show that our algorithms perform much better than the Naik-Murthy algorithm. In spite of their simplicity, they compete with well-established alternative methods for images where hue-preservation is desired.

Denoising of Frame Coefficients Using

We consider the denoising of a function (an image or a signal) containing smooth regions and edges. Classical ways to solve this problem are variational methods and shrinkage of a representation of the data in a basis or a frame. We propose a method which combines the advantages of both approaches. Following the wavelet shrinkage method of Donoho and Johnstone, we set to zero all frame coefficients with respect to a reasonable threshold. The shrunk frame representation involves both large coefficients corresponding to noise (outliers) and some coefficients, erroneously set to zero, leading to Gibbs-like oscillations in the estimate. We design a specialized (nonsmooth) objective function allowing all these coefficients to be selectively restored, without modifying the other coefficients which are nearly faithful, using regularization in the domain of the restored function. We analyze the well-posedness and the main properties of this objective function. We also propose an approximation of this method which...

\ell^1

We characterise the features of a regularised ML estimate, or equivalently a MAP estimate, of an image (or a signal) in relation with the form of the regularisation. The unknown image (signal) is observed through a linear operator and the data are corrupted by white Gaussian noise. Its reconstruction is regularised by the energy of a first-order Markov random field where the contributions of the transitions between adjacent neighbours are weighted using general potential functions (PFs). We exhibit the relationship between several features of the estimate and the form of the PF. Points of interest are the edge recovery, the stability of the estimator, the estimation of locally constant regions, the bias over large transitions, the resolution. The exposed theoretical considerations are corroborated by numerical simulations.

Data-Fidelity Term and Edge-Preserving Regularization

We propose a method for the reconstruction of an image, only partially observed through a linear integral operator. As such an inverse problem is ill-posed, prior information must be introduced. We consider the case of a compound Markov random field with a non-interacting line process. In order to maximise the posterior likelihood function, we propose an extension of the graduated non convexity principle pioneered by Blake and Zisserman (1987) which allows its use for ill-posed linear inverse problems. We discuss the role of the observation scale and some aspects of the implemented algorithm. Finally, we present an application of the method to a diffraction tomography imaging problem.<<ETX>>

Regularisation functions and estimators

This paper provides a fast algorithm to order in a meaningful, strict way the integer gray values in digital (quantized) images. It can be used in any exact histogram specification-based application. Our algorithm relies on the ordering procedure based on the specialized variational approach. This variational method was shown to be superior to all other state-of-the art ordering algorithms in terms of faithful total strict ordering but not in speed. Indeed, the relevant functionals are in general difficult to minimize because their gradient is nearly flat over vast regions. In this paper, we propose a simple and fast fixed point algorithm to minimize these functionals. The fast convergence of our algorithm results from known analytical properties of the model. Our algorithm is equivalent to an iterative nonlinear filtering. Furthermore, we show that a particular form of the variational model gives rise to much faster convergence than other alternative forms. We demonstrate that only a few iterations of this filter yield almost the same pixel ordering as the minimizer. Thus, we apply only few iteration steps to obtain images, whose pixels can be ordered in a strict and faithful way. Numerical experiments confirm that our algorithm outperforms by far its main competitors.

Inversion of large-support ill-conditioned linear operators using a Markov model with a line process

A (semi)deterministic maximum likelihood (DML) approach is presented to solve the joint blind channel identification and blind symbol estimation problem for single-input multiple-output systems. A partial prior on the symbols is incorporated into the criterion which improves the estimation accuracy and brings robustness toward poor channel diversity conditions. At the same time, this method introduces fewer local minima than the use of a full prior (statistical) ML. In the absence of noise, the proposed batch algorithm estimates perfectly the channel and symbols with a finite number of samples. Based on these considerations, an adaptive implementation of this algorithm is proposed. It presents some desirable properties including low complexity, robustness to channel overestimation, and high convergence rate.

Fast Ordering Algorithm for Exact Histogram Specification

We present an original method to reconstruct 3D magnetic resonance images of high resolution in the 3 directions of space from two anisotropic volumes. The resolution of each volume is degraded in a different direction. The reconstruction method is based on an optimization technique, the constraints being fidelity to the acquired data on the one hand, smoothness and edge preservation on the other hand. The interest of such a method is to significantly decrease the acquisition time of MR images, without degrading the spatial resolution.

Blind identification/equalization using deterministic maximum likelihood and a partial prior on the input

Abstract The non-destructive evaluation (NDE) problem we treat is the testing of a globally homogeneous conductive medium for anomalies such as cracks and notches. The medium is illuminated with a monochromatic electric field; the anomalies induce eddy currents and they modify the total field which can be measured. The tomographic approach, aimed to draw up an image of the medium, is recent in this area. It corresponds to an extremely difficult ill-posed inverse problem and its resolution needs the use of pertinent prior information. The considered anomalies can be represented using images whose pixels can only take the values 0 and 1. Our main contribution lies in the regularization of a large-support ill-posed observation operator using a locally constant binary image Markov random field. The resulting high-dimensional combinatorial optimization problem is tedious: neither exact resolution nor simulated annealing are feasible. Instead, we establish an equivalent continuous-valued optimization problem. A nearly optimal solution is then calculated using a graduated non-convexity algorithm adapted for this purpose. The proposed inversion technique surpasses the particular NDE problem and can be applied whenever a binary image is observed using a linear system and corrupted by Gaussian noise.