Smoothing and sharpening are two fundamental image processing operations. The latter is usually related to the former through the unsharp masking algorithm. In this paper, we develop a new type of filter which performs smoothing or sharpening via a tuning parameter. The development of the new filter is based on (1) a new Laplacian-based filter formulation which unifies the smoothing and sharpening operations, (2) a patch interpolation model similar to that used in the guided filter which provides edge-awareness capability, and (3) the generalized Gamma distribution which is used as the prior for parameter estimation. We have conducted detailed studies on the properties of two versions of the proposed filter (self-guidance and external guidance). We have also conducted experiments to demonstrate applications of the proposed filter. In the self-guidance case, we have developed adaptive smoothing and sharpening algorithms based on texture, depth and blurriness information extracted from an image. Applications include enhancing human face images, producing shallow depth of field effects, focus-based image enhancement, and seam carving. In the external guidance case, we have developed new algorithms for combining flash and no-flash images and for enhancing multi-spectral images using a panchromatic image.INDEX TERMS Edge-aware filter, image sharpening, image smoothing, maximum a posteriori estimate.

A technique for denoising signals defined over graphs was recently proposed by Chen et al. (2014). The technique is based on a regularisation framework and denoising is achieved by solving an optimisation problem. Matrix inversion is required and an approximate solution that avoids directly calculating the inverse, by using a graph filter, was proposed by Chen et al. (2014). The technique, however, requires an eigendecomposition and the resulting filter degree is high. In this study, the authors propose a computationally efficient technique that is based on a least squares approximation of the eigenvalues of the inverse. They show that a good approximation can be achieved with a low degree graph polynomial filter without the need for any eigendecomposition. Low degree filters also have the desirable property of vertex localisation (analogous to time localisation). The filter gives denoising results that are very similar to that using the exact solution and can be implemented using distributed processing.

Fractional calculus has increased in popularity in recent years, as the number of its applications in different fields has increased. Compared to the traditional operations in calculus (integration and differentiation) which are uniquely defined, the fractional-order operators have numerous definitions. Furthermore, a consensus on the most suitable definition for a given task is yet to be reached. Fractional operators are defined as continuous operators and their implementation requires a discretization step. In this article, we propose a discrete fractional Laplacian as a matrix operator. The proposed operator is real (non-complex) which makes it computationally efficient. The construction of the proposed fractional Laplacian utilizes the DCT transform avoiding the complexity associated with the discretization step which is typical in the constructions based on signal processing. We demonstrate the utility of the proposed operator on a number of data modeling and image processing tasks. INDEX TERMS Fractional-Laplacian, discrete operator, image-processing, trend-filtering, fractional calculus.

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Edge-aware smoothing is an essential tool for computer vision, graphics and photography. In this paper, we develop a new and efficient local weighted average filter for edge-aware smoothing. The proposed filter can use guidance information which permits an iterative filtering process. Since the weights of the proposed filter depend on the local variance, the implementation requires linear filters only, leading to O(N pix ) computational complexity. We also present statistical analysis and simulations which provides new insights into its computational efficiency and its relationship with the bilateral filter. The performance of the proposed filter is comparable to those state-of-the-art filters in many applications including: edge-preserving smoothing, compression artifact removal, structure separation, edge extraction, non-photo realistic image rendering, salience detection, detail magnification and multi-focus image fusion.INDEX TERMS Edge-aware smoothing, Bilateral filter, Guided filter, Detail magnification, Multi-focus image fusion.