Generalised morphological image diffusion

International audienceRelationships between linear and morphological scale-spaces have been considered by various previous works. The aim of this paper is to study how to generalise the diffusion-based approaches in order to introduce nonlinear filters whose effects mimic the asymmetric behaviour of morphological dilation and erosion, as well as other evolved morphological filters. A methodology based on the counter-harmonic mean is adopted here. Details of numerical implementation are discussed and results are provided to illustrate the various studied cases: isotropic, nonlinear and coherence-enhancing diffusion. We also found a new way to derive the classical link between Gaussian scale-space and dilation/erosion scale-spaces based on quadratic structuring functions. We have included some preliminary applications of the generalised morphological diffusion to solve image processing problems such as denoising and image enhancement in the case of asymmetric bright/dark image properties

Mathematical Imaging and Surface Processing

Within the last decade image and geometry processing have become increasingly rigorous with solid foundations in mathematics. Both areas are research fields at the intersection of different mathematical disciplines, ranging from geometry and calculus of variations to PDE analysis and numerical analysis. The workshop brought together scientists from all these areas and a fruitful interplay took place. There was a lively exchange of ideas between geometry and image processing applications areas, characterized in a number of ways in this workshop. For example, optimal transport, first applied in computer vision is now used to define a distance measure between 3d shapes, spectral analysis as a tool in image processing can be applied in surface classification and matching, and so on. We have also seen the use of Riemannian geometry as a powerful tool to improve the analysis of multivalued images. This volume collects the abstracts for all the presentations covering this wide spectrum of tools and application domains

Morphological Network: How Far Can We Go with Morphological Neurons?

In recent years, the idea of using morphological operations as networks has received much attention. Mathematical morphology provides very efficient and useful image processing and image analysis tools based on basic operators like dilation and erosion, defined in terms of kernels. Many other morphological operations are built up using the dilation and erosion operations. Although the learning of structuring elements such as dilation or erosion using the backpropagation algorithm is not new, the order and the way these morphological operations are used is not standard. In this paper, we have theoretically analyzed the use of morphological operations for processing 1D feature vectors and shown that this gets extended to the 2D case in a simple manner. Our theoretical results show that a morphological block represents a sum of hinge functions. Hinge functions are used in many places for classification and regression tasks (Breiman (1993)). We have also proved a universal approximation theorem -- a stack of two morphological blocks can approximate any continuous function over arbitrary compact sets. To experimentally validate the efficacy of this network in real-life applications, we have evaluated its performance on satellite image classification datasets since morphological operations are very sensitive to geometrical shapes and structures. We have also shown results on a few tasks like segmentation of blood vessels from fundus images, segmentation of lungs from chest x-ray and image dehazing. The results are encouraging and further establishes the potential of morphological networks.Comment: 35 pages, 19 figures, 7 table

Locally Adaptive Frames in the Roto-Translation Group and their Applications in Medical Imaging

Locally adaptive differential frames (gauge frames) are a well-known effective tool in image analysis, used in differential invariants and PDE-flows. However, at complex structures such as crossings or junctions, these frames are not well-defined. Therefore, we generalize the notion of gauge frames on images to gauge frames on data representations $U:\mathbb{R}^{d} \rtimes S^{d-1} \to \mathbb{R}$ defined on the extended space of positions and orientations, which we relate to data on the roto-translation group $SE(d)$, $d=2,3$. This allows to define multiple frames per position, one per orientation. We compute these frames via exponential curve fits in the extended data representations in $SE(d)$. These curve fits minimize first or second order variational problems which are solved by spectral decomposition of, respectively, a structure tensor or Hessian of data on $SE(d)$. We include these gauge frames in differential invariants and crossing preserving PDE-flows acting on extended data representation $U$ and we show their advantage compared to the standard left-invariant frame on $SE(d)$. Applications include crossing-preserving filtering and improved segmentations of the vascular tree in retinal images, and new 3D extensions of coherence-enhancing diffusion via invertible orientation scores

Jump-sparse and sparse recovery using Potts functionals

We recover jump-sparse and sparse signals from blurred incomplete data corrupted by (possibly non-Gaussian) noise using inverse Potts energy functionals. We obtain analytical results (existence of minimizers, complexity) on inverse Potts functionals and provide relations to sparsity problems. We then propose a new optimization method for these functionals which is based on dynamic programming and the alternating direction method of multipliers (ADMM). A series of experiments shows that the proposed method yields very satisfactory jump-sparse and sparse reconstructions, respectively. We highlight the capability of the method by comparing it with classical and recent approaches such as TV minimization (jump-sparse signals), orthogonal matching pursuit, iterative hard thresholding, and iteratively reweighted $\ell^1$ minimization (sparse signals)