A fast implementation of bilateral filtering is presented, which is based on an optimal expansion of the filter kernel into a sum of factorized terms. These terms are computed by minimizing the expansion error in the mean-square-error sense. This leads to a simple and elegant solution in terms of eigenvectors of a square matrix. In this way, the bilateral filter is applied through computing a few Gaussian convolutions, for which very efficient algorithms are readily available. Moreover, the expansion functions are optimized for the histogram of the input image, leading to improved accuracy. It is shown that this further optimization it made possible by removing the commonly deployed constrain of shiftability of the basis functions. Experimental validation is carried out in the context of digital rock imaging. Results on large 3D images of rock samples show the superiority of the proposed method with respect to other fast approximations of bilateral filtering.

This paper describes modeling and numerical computation of orthogonal bases, which are used to describe images and motion fields. Motion estimation from image data is then studied on subspaces spanned by these bases. A reduced model is obtained as the Galerkin projection on these subspaces of a physical model, based on Euler and optical flow equations. A data assimilation method is studied, which assimilates coefficients of image data in the reduced model in order to estimate motion coefficients. The approach is first quantified on synthetic data: it demonstrates the interest of model reduction as a compromise between results quality and computational cost. Results obtained on real data are then displayed so as to illustrate the method.

A new diadic family of eigenfunctions of the 2-D Fourier transform has been discovered. Specifically, new wavelets are derived by steering the elongated Hermite-Gauss filters with respect to rotations, thus obtaining a natural generalization of the Laguerre-Gauss harmonics. Interestingly, these functions are also proportional to their 2-D Fourier transform. Their analytical expression is provided in a compact and treatable form, by means of a new ad hoc matrix notation in which the cases of even and odd orders of the Hermite polynomials are unified. Moreover, these functions can be efficiently implemented by means of a recursive formula that is derived in this paper. The proposed filters are applied to the problem of gradient estimation to improve the theoretical Canny tradeoff of position accuracy versus noise rejection that occurs in edge detection. Experimental results show considerable improvements in using the new wavelets over both isotropic Gaussian derivatives and other elongated steerable filters more recently introduced. Finally, being the proposed wavelets a set of Fourier eigenfunctions, they can be of interest in other fields of science, such as optics and quantum mechanics.

We provide a closed form, both in the spatial and in the frequency domain, of a family of wavelets which arise from steering elongated Hermite-Gauss filters. These wavelets have interesting mathematical properties, as they form new dyadic families of eigenfunctions of the 2D Fourier transform, and generalize the well known Laguerre-Gauss harmonics. A special notation introduced here greatly simplifies our proof and unifies the cases of even and odd orders. Applying these wavelets to edge detection increases the performance of about 12.5% with respect to standard methods, in terms of the Pratt's figure of merit, both for noisy and noise-free input images.

In this paper, we perform approximate steering of the elongated 2D Hermite-Gauss functions with respect to rotations and provide a compact analytical expressions for the related basis functions. A special notation introduced here considerably simplifies the derivation and unifies the cases of even and odd indices. The proposed filters are applied to edge detection. Quantitative analysis shows a performance increase of about 12.5% in terms of the Pratt's figure of merit with respect to the well-established Gaussian gradient proposed by Canny.

Glass patterns have been exhaustively studied both in the vision literature and from a purely mathematical point of view. We extend the related formalism to the continuous case and we show that continuous Glass patterns can be used for artistic imaging applications. The general idea is to replace natural texture present in an input image with synthetic painterly texture that is generated by means of a continuous Glass pattern, whose geometrical structure is controlled by the gradient orientation of the input image. The behavior of the proposed algorithm is analytically interpreted in terms of the theory of dynamical systems. Experimental results on a broad range of input images validate the effectiveness of the proposed method in terms of lack of undesired artifacts, which are present with other existing methods, and easy interpretability of the input parameters.

We consider the problem of detecting object contours in natural images. In many cases, local luminance changes turn out to be stronger in textured areas than on object contours. Therefore, local edge features, which only look at a small neighborhood of each pixel, cannot be reliable indicators of the presence of a contour, and some global analysis is needed. We introduce a new morphological operator, called adaptive pseudo-dilation (APD), which uses context dependent structuring elements in order to identify long curvilinear structure in the edge map. We show that grouping edge pixels as the connected components of the output of APD results in a good agreement with the gestalt law of good continuation. The novelty of this operator is that dilation is limited to the Voronoi cell of each edge pixel. An efficient implementation of APD is presented. The grouping algorithm is then embedded in a multithreshold contour detector. At each threshold level, small groups of edges are removed, and contours are completed by means of a generalized reconstruction from markers. The use of different thresholds makes the algorithm much less sensitive to the values of the input parameters. Both qualitative and quantitative comparison with existing approaches prove the superiority of the proposed contour detector in terms of larger amount of suppressed texture and more effective detection of low-contrast contours.

Two important visual properties of paintings and painting-like images are the absence of texture details and the increased sharpness of edges as compared to photographic images. Painting-like artistic effects can be achieved from photographic images by filters that smooth out texture details, while preserving or enhancing edges and corners. However, not all edge preserving smoothers are suitable for this purpose. We present a simple nonlinear local operator that generalizes both the well known Kuwahara filter and the more general class of filters known in the literature as "criterion and value filter structure." This class of operators suffers from intrinsic theoretical limitations which give rise to a dramatic instability in presence of noise, especially on shadowed areas. Such limitations are discussed in the paper and overcome by the proposed operator. A large variety of experimental results shows that the output of the proposed operator is visually similar to a painting. Comparisons with existing techniques on a large set of natural images highlight conditions on which traditional edge preserving smoothers fail, whereas our approach produces good results. In particular, unlike many other well established approaches, the proposed operator is robust to degradations of the input image such as blurring and noise contamination.

In natural images, luminance changes occur both on object contours and on textures. Often, the latter are stronger than the former, thus standard edge detectors fail in isolating object contours from texture. To overcome this problem, we propose a multiresolution contour detector motivated by biological principles. At each scale, texture is suppressed by using a bipolar surround inhibition process. The binary contour map is obtained by a contour selection criterion that is more effective than the classical hysteresis thresholding. Robustness to noise is achieved by Bayesian gradient estimation.