Abstract. We build a conceptual bridge between mathematical morphology and max-min sampling operators, offering new insights into the convergence behavior of discrete morphological transformations on multivariate grids as the spatial resolution of the data increases. To this end, we introduce and analyze nonlinear sampling operators rooted in the arithmetic operations of the max-min algebra. These new sampling operators can be interpreted as approximate morphological operations relying on the function values on the sampling grid and a specific kernel. We investigate the convergence behavior of these operators, alongside with the respective properties of their morphological operations as dilation, erosion, opening, and closing. In the two-dimensional setting, we show how these morphological sampling operators can be effectively applied to various image processing tasks, as upscaling, denoising, or segmentation.
Abstract. Low-contrast medical images pose significant challenges for segmentation due to weak intensity gradients and ambiguous boundaries (e.g., in prostate or brain tumor regions). Traditional methods, such as thresholding or edge detection, often fail to simultaneously address joint registration, segmentation, and bias correction, leading to misalignment and inaccurate tissue delineation of low-contrast medical images. To address this challenge, we propose a novel variational model with synergistic mechanism that handles segmentation, registration, and bias correction simultaneously, with its innovation lying in utilizing image segmentation methods from simple scenarios to assist segmentation of low-contrast and other complex scenario images through registration and bias field correction. The theoretical analysis of the proposed model includes the existence of the solution and the convergence of the alternating minimization method (AMM). Numerical experiments validate that the model’s innovative unified framework and logarithmic transformation design significantly improve the accuracy of segmentation and registration for low-contrast images compared with state-of-the-art methods. The code of the algorithm in this paper is publicly available at https://github.com/zzyhp/S-R-B-SIIMS.git .
Abstract. In a variational denoising model, weight in the data fidelity term plays the role of enhancing the noise-removal capability. It is profoundly correlated with noise information while also balancing the data fidelity and regularization terms. However, assigning weight is expected to be challenging when the noise pattern is beyond independent identical Gaussian distribution, e.g., impulse noise, stripe noise, or a mixture of several patterns, etc. Furthermore, how to leverage weight to balance the data fidelity and regularization terms is even less evident. In this work, we propose a data-driven loss weighting (DLW) scheme to address these issues. Specifically, DLW trains a parameterized weight function (i.e., a neural network) that maps the noisy image to the weight. The training is achieved by a bilevel optimization framework, where the lower level problem is solving several denoising models with the same weight predicted by the weight function, and the upper level problem minimizes the distance between the restored image and the clean image. In this way, information from both the noise and the regularization can be efficiently extracted to determine the weight function. DLW also facilitates the easy implementation of a trained weight function on denoising models. Numerical results verify the remarkable performance of DLW on improving the ability of various variational denoising models to handle different complex noise. This implies that DLW has the ability to transfer the noise knowledge at the model level to heterogeneous tasks beyond the training models. We studythe generalization theory underlying DLW i, validating its intrinsic transferability.
Texture analysis consists of a classical and everlasting task in image processing, often involved in a broad range of applications, possibly very different in nature. For large classes of textures, scale-free (or fractal) spatial dynamics as well as anisotropy constitute key properties. However, the local (pixelwise) joint estimation of anisotropy and scale-free attributes constitutes a difficult challenge, as both consist of nonlocal properties. Yet accurately detecting variations of these attributes across the image is often crucial. The overarching goal of the present work is thus to propose an inverse problem formulation for the analysis of piecewise homogeneous textures, grounded jointly on scale-free dynamics and anisotropy, and to study its performance for local assessment of textures. The formulation combines several major contributions. First, piecewise homogeneous Gaussian fields are defined as texture mixture models, with prescribed scale-free and anisotropy properties. Second, multiband complex wavelet coefficients, implemented via (nondecimated) dual-tree fast algorithms, are theoretically shown to be sensitive to both scale-free dynamics and anisotropy. Notably, it is shown that the squared-modulus of the wavelet coefficients behaves locally (or pixelwise) approximately as power-laws, with both scaling exponent and intercept jointly sensitive to anisotropy and scale-free dynamics. Thus, a key originality of the present work is to propose to perform local analysis of intrinsically nonlocal properties without having recourse to local averages, which would significantly impair accurate segmentation or accurate local characterization. Third, these local power-law-like behaviors, combined with regularization terms enforcing piecewise homogeneity, are embedded into a minimization procedure, solved by an optimization algorithm, whose convergence conditions are theoretically well-studied, and practically implemented through an efficient proximal algorithm due to strong convexity of the resulting minimization problem. Segmentation performance achieved by the proposed procedure is quantified and compared with respect to the difficulty of the segmentation task, for synthetic piecewise homogeneous Gaussian fields. The potential of the proposed texture analysis tool is also illustrated at work on real-world textures.
Abstract. Minkowski tensors, also known as tensor valuations, provide robust [Formula: see text]-point information for a wide range of random spatial structures. Local estimators for point clouds, e.g., representing voxelized data, however, are unavoidably biased even in the limit of infinitely high resolution. Here, we substantially improve a recently proposed, asymptotically unbiased algorithm to estimate Minkowski tensors from point clouds. Our improved algorithm is more robust and efficient. Moreover we generalize the theoretical foundations for an asymptotically bias-free estimation of the interfacial tensors, among others, to the case of finite unions of compact sets with positive reach, which is relevant for many applications like rough surfaces or composite materials. As a realistic test case of random spatial structures, we consider random (beta) polytopes. We first derive explicit expressions of the expected Minkowski tensors, which we then compare to our simulation results. We obtain precise estimates with relative errors of a few percent for practically relevant resolutions. Finally, we apply our methods to real data of metallic grains and nanorough surfaces, and we provide an open-source Python package, which works in any dimension.
