Journal of Mathematical Imaging and Vision最新文献

We define morphological operators and filters for directional images whose pixel values are unit vectors. This requires an ordering relation for unit vectors which is obtained by using depth functions. They provide a centre-outward ordering with respect to a specified centre vector. We apply our operators on synthetic directional images and compare them with classical morphological operators for grey-scale images. As application examples, we enhance the fault region in a compressed glass foam and segment misaligned fibre regions of glass fibre-reinforced polymers.

Spectral unmixing is important in analyzing and processing hyperspectral images (HSIs). With the availability of large spectral signature libraries, the main task of spectral unmixing is to estimate corresponding proportions called abundances of pure spectral signatures called endmembers in mixed pixels. In this vein, only a few endmembers participate in the formation of mixed pixels in the scene and so we call them active endmembers. A plethora of sparse unmixing algorithms exploit spectral and spatial information in HSIs to enhance abundance estimation results. Many algorithms, however, treat the abundances corresponding to active and nonactive endmembers in the scene equivalently. In this article, we propose a framework named mixing support detection (MSD) for the spectral unmixing problem. The main idea is first to detect the active and nonactive endmembers at each iteration and then to treat the corresponding abundances differently. It follows that we only focus on the estimation of active abundances with the assumption of zero abundances corresponding to nonactive endmembers. It can be expected to reduce the computational cost, avoid the perturbations in nonactive abundances, and enhance the sparsity of the abundances. We embed the MSD framework in classic alternating direction method of multipliers (ADMM) updates and obtain an ADMM-MSD algorithm. In particular, five ADMM-MSD-based unmixing algorithms are provided. The residual and objective convergence results of the proposed algorithm are given under certain assumptions. Both simulated and real-data experiments demonstrate the efficacy and superiority of the proposed algorithm compared with some state-of-the-art algorithms.

This work describes a Bayesian framework for reconstructing the boundaries that represent targeted features in an image, as well as the regularity (i.e., roughness vs. smoothness) of these boundaries. This regularity often carries crucial information in many inverse problem applications, e.g., for identifying malignant tissues in medical imaging. We represent the boundary as a radial function and characterize the regularity of this function by means of its fractional differentiability. We propose a hierarchical Bayesian formulation which, simultaneously, estimates the function and its regularity, and in addition we quantify the uncertainties in the estimates. Numerical results suggest that the proposed method is a reliable approach for estimating and characterizing object boundaries in imaging applications, as illustrated with examples from high-intensity X-ray CT and image inpainting with Gaussian and Laplace additive noise models. We also show that our method can quantify uncertainties for these noise types, various noise levels, and incomplete data scenarios.

We propose a novel abstraction of the image segmentation task in the form of a combinatorial optimization problem that we call the multi-separator problem. Feasible solutions indicate for every pixel whether it belongs to a segment or a segment separator, and indicate for pairs of pixels whether or not the pixels belong to the same segment. This is in contrast to the closely related lifted multicut problem, where every pixel is associated with a segment and no pixel explicitly represents a separating structure. While the multi-separator problem is np-hard, we identify two special cases for which it can be solved efficiently. Moreover, we define two local search algorithms for the general case and demonstrate their effectiveness in segmenting simulated volume images of foam cells and filaments.

Sliced optimal transport, which is basically a Radon transform followed by one-dimensional optimal transport, became popular in various applications due to its efficient computation. In this paper, we deal with sliced optimal transport on the sphere (mathbb {S}^{d-1}) and on the rotation group (textrm{SO}(3)). We propose a parallel slicing procedure of the sphere which requires again only optimal transforms on the line. We analyze the properties of the corresponding parallelly sliced optimal transport, which provides in particular a rotationally invariant metric on the spherical probability measures. For (textrm{SO}(3)), we introduce a new two-dimensional Radon transform and develop its singular value decomposition. Based on this, we propose a sliced optimal transport on (textrm{SO}(3)). As Wasserstein distances were extensively used in barycenter computations, we derive algorithms to compute the barycenters with respect to our new sliced Wasserstein distances and provide synthetic numerical examples on the 2-sphere that demonstrate their behavior for both the free- and fixed-support setting of discrete spherical measures. In terms of computational speed, they outperform the existing methods for semicircular slicing as well as the regularized Wasserstein barycenters.

This paper proposes an edge-preserving regularization technique to solve the color image demosaicing problem in the realistic case of noisy data. We enforce intra-channel local smoothness of the intensity (low-frequency components) and inter-channel local similarity of the depth of object borders and textures (high-frequency components). Discontinuities of both the low-frequency and high-frequency components are accounted for implicitly, i.e., through suitable functions of the proper derivatives. For the treatment of even the finest image details, derivatives of first, second, and third orders are considered. The solution to the demosaicing problem is defined as the minimizer of an energy function, accounting for all these constraints plus a data fidelity term. This non-convex energy is minimized via an iterative deterministic algorithm, applied to a family of approximating functions, each implicitly referring to geometrically consistent image edges. Our method is general because it does not refer to any specific color filter array. However, to allow quantitative comparisons with other published results, we tested it in the case of the Bayer CFA and on the Kodak 24-image dataset, the McMaster (IMAX) 18-image dataset, the Microsoft Demosaicing Canon 57-image dataset, and the Microsoft Demosaicing Panasonic 500-image dataset. The comparisons with some of the most recent demosaicing algorithms show the good performance of our method in both the noiseless and noisy cases.

Variational level set method has become a powerful tool in image segmentation due to its ability to handle complex topological changes and maintain continuity and smoothness in the process of evolution. However its evolution process can be unstable, which results in over flatted or over sharpened contours and segmentation failure. To improve the accuracy and stability of evolution, we propose a high-order level set variational segmentation method integrated with molecular beam epitaxy (MBE) equation regularization. This method uses the crystal growth in the MBE process to limit the evolution of the level set function. Thus can avoid the re-initialization in the evolution process and regulate the smoothness of the segmented curve and keep the segmentation results independent of the initial curve selection. It also works for noisy images with intensity inhomogeneity, which is a challenge in image segmentation. To solve the variational model, we derive the gradient flow and design a scalar auxiliary variable scheme, which can significantly improve the computational efficiency compared with the traditional semi-implicit and semi-explicit scheme. Numerical experiments show that the proposed method can generate smooth segmentation curves, preserve segmentation details and obtain robust segmentation results of small objects. Compared to existing level set methods, this model is state-of-the-art in both accuracy and efficiency.

To study convexity properties of digital planar objects, the minimum perimeter polygon (MPP) was defined in the 1970 s in articles by Sklansky, Chazin, Hansen, Kibler, and Kim, where pixels were identified with polygonal tiles in mosaics, and two algorithms (1972, 1976) were proposed to determine the MPP vertices. These algorithms are based on constructing and iteratively restricting visibility cones, the MPP vertices result as special vertices of the tiles. The present paper proposes a novel MPP algorithm for objects given as regular complexes in rectangular mosaics, which are edge-adjacency-connected sets of tiles that have neither end tiles nor holes and whose boundaries not necessarily are simple. The new algorithm takes as input the canonical boundary path, we also propose a boundary tracing algorithm to obtain this path. We review the two classic MPP algorithms for rectangular tiles and a simplified adaptation for square tiles that is recommended in widely used modern textbooks on digital image analysis (2018, 2020) to produce approximations of simple digital 4-contours. We show that all these algorithms fail and that their mathematical basis is flawed, we correct the errors to develop the new MPP algorithm. Our MPP algorithm is illustrated using examples and its correctness is proved. Under our assumptions, the MPP coincides with the relative convex hull of a set A with respect to a polygon (Bsupset A), where A is not necessarily a polygon, not even connected.

This paper develops an in-depth treatment concerning the problem of approximating the Gaussian smoothing and the Gaussian derivative computations in scale-space theory for application on discrete data. With close connections to previous axiomatic treatments of continuous and discrete scale-space theory, we consider three main ways of discretizing these scale-space operations in terms of explicit discrete convolutions, based on either (i) sampling the Gaussian kernels and the Gaussian derivative kernels, (ii) locally integrating the Gaussian kernels and the Gaussian derivative kernels over each pixel support region, to aim at suppressing some of the severe artefacts of sampled Gaussian kernels and sampled Gaussian derivatives at very fine scales, or (iii) basing the scale-space analysis on the discrete analogue of the Gaussian kernel, and then computing derivative approximations by applying small-support central difference operators to the spatially smoothed image data.

We study the properties of these three main discretization methods both theoretically and experimentally and characterize their performance by quantitative measures, including the results they give rise to with respect to the task of scale selection, investigated for four different use cases, and with emphasis on the behaviour at fine scales. The results show that the sampled Gaussian kernels and the sampled Gaussian derivatives as well as the integrated Gaussian kernels and the integrated Gaussian derivatives perform very poorly at very fine scales. At very fine scales, the discrete analogue of the Gaussian kernel with its corresponding discrete derivative approximations performs substantially better. The sampled Gaussian kernel and the sampled Gaussian derivatives do, on the other hand, lead to numerically very good approximations of the corresponding continuous results, when the scale parameter is sufficiently large, in most of the experiments presented in the paper, when the scale parameter is greater than a value of about 1, in units of the grid spacing. Below a standard deviation of about 0.75, the derivative estimates obtained from convolutions with the sampled Gaussian derivative kernels are, however, not numerically accurate or consistent, while the results obtained from the discrete analogue of the Gaussian kernel, with its associated central difference operators applied to the spatially smoothed image data, are then a much better choice.

Diffusion probabilistic models excel at sampling new images from learned distributions. Originally motivated by drift-diffusion concepts from physics, they apply image perturbations such as noise and blur in a forward process that results in a tractable probability distribution. A corresponding learned reverse process generates images and can be conditioned on side information, which leads to a wide variety of practical applications. Most of the research focus currently lies on practice-oriented extensions. In contrast, the theoretical background remains largely unexplored, in particular the relations to drift-diffusion. In order to shed light on these connections to classical image filtering, we propose a generalised scale-space theory for diffusion probabilistic models. Moreover, we show conceptual and empirical connections to diffusion and osmosis filters.