There is a continuous effort to develop the theory and methods for computing digital distance functions, and to lower the rotational dependency of distance functions. Working on the digital space, e.g., on the square grid, digital distance functions are defined by minimal cost-paths, which can be processed (back-tracked etc.) without any errors or approximations. Recently, digital distance functions defined by weight sequences, which is a concept allowing multiple types of weighted steps combined with neighborhood sequences, were developed. With appropriate weight sequences, the distance between points on the perimeter of a square and the center of the square (i.e., for squares of a given size the weight sequence can be easily computed) are exactly the Euclidean distance for these distances based on weight sequences. However, distances based on weight sequences may not fulfill the triangular inequality. In this paper, continuing the research, we provide a sufficient condition for weight sequences to provide metric distance. Further, we present an algorithm to compute the distance transform based on these distances. Optimization results are also shown for the approximation of the Euclidean distance inside the given square.
Mathematical morphological methods have successfully been applied to filter out (emphasize or remove) different structures of an image. However, it is argued that these methods could be suitable for the task only if the type and order of the filter(s) as well as the shape and size of operator kernel are designed properly. Thus the existing filtering operators are problem (instance) specific and are designed by the domain experts. In this work we propose a morphological network that emulates classical morphological filtering consisting of a series of erosion and dilation operators with trainable structuring elements. We evaluate the proposed network for image de-raining task where the SSIM and mean absolute error (MAE) loss corresponding to predicted and ground-truth clean image is back-propagated through the network to train the structuring elements. We observe that a single morphological network can de-rain an image with any arbitrary shaped rain-droplets and achieves similar performance with the contemporary CNNs for this task with a fraction of trainable parameters (network size). The proposed morphological network(MorphoN) is not designed specifically for de-raining and can readily be applied to similar filtering / noise cleaning tasks. The source code can be found here https://github.com/ranjanZ/2D-Morphological-Network
Rigid motions on 2D digital images were recently investigated with the purpose of preserving geometric and topological properties. From the application point of view, such properties are crucial in image processing tasks, for instance image registration. The known ideas behind preserving geometry and topology rely on connections between the 2D continuous and 2D digital geometries that were established via various notions of regularity on digital and continuous sets. We start by recalling these results; then we discuss the difficulties that arise when extending them from \(\mathbb {Z}^2\) to \(\mathbb {Z}^3\). On the one hand, we aim to provide a discussion on strategies that prove to be successful in \(\mathbb {Z}^2\) and remain valid in \(\mathbb {Z}^3\); on the other hand, we explain why certain strategies cannot be extended to the 3D framework of digitized rigid motions. We also emphasize the relationships that may exist between specific concepts initially proposed in \(\mathbb {Z}^2\). Overall, our objective is to initiate an investigation about the most promising approaches for extending the 2D results to higher dimensions.
Granulometry, by characterizing the distribution of object sizes, is a powerful tool for the analysis of binary images. It may, for example, extract pertinent features in the context of texture classification. The granulometry is similar to a sieving process that filters image details of increasing sizes. Its computation classically relies on morphological openings, i.e. the sequence of an erosion followed by a dilation for each ball radius. It is well known that a distance map can be described as an “erosion transform” that summarizes the erosions with the balls of all radii. Using a Steiner formula, we show how a vector of parameters measured on an eroded contour can be extrapolated to the measures of the dilation. Instead of completing the openings from the distance map by computing a dilation for each ball size, we can estimate their cardinality from the area, perimeter and Euler-Poincaré characteristic of each erosion. We extract these measures for all radii at once from an asymmetric distance map. The result is a fast streaming algorithm that provides an estimate of the granulometry distribution, in a single scan of the image and with very limited memory footprint.
We present a new plane-probing algorithm, i.e., an algorithm that computes the normal vector of a digital plane from a starting point and a predicate “Is a point x in the digital plane?”. This predicate is used to probe the digital plane as locally as possible and decide on-the-fly the next points to consider. We show that this algorithm returns the same vector as another plane-probing algorithm proposed in Lachaud et al. (J. Math. Imaging Vis., 59, 1, 23–39, 2017), but requires fewer probes. The theoretical upper bound is indeed lowered from O(ωlogω ) to O(ω ) calls to the predicate, where ω is the arithmetical thickness of the digital plane, and far fewer calls are experimentally observed on average. This reduction is made possible by a study that shows how to avoid computations that do not contribute to the final solution. In the context of digital surface analysis, this new algorithm is expected to be of great interest for normal estimation and shape reconstruction.
A skeleton is a centered geometric representation of a shape that describes the shape in a simple and intuitive way, typically reducing the dimension by at least one. Skeletons are useful in shape analysis and recognition since they provide a framework for part decomposition, are stable under topology preserving deformation, and supply information about the topology and connectivity of the shape. The main drawback to skeletonization algorithms is their sensitivity to small boundary perturbations: noise on a shape boundary, such as pixelation, will produce many spurious branches within a skeleton. As a result, skeletonizations often require a second pruning step. In this article, we propose a new 2D skeletonization algorithm that directly produces a clean skeleton for a shape, avoiding the creation of noisy branches. The approach propagates a circle inside the shape, maintaining neighborhood-based contact with the boundary and bypassing boundary oscillations below a chosen threshold. By explicitly modeling the scale of noise via two parameters that are shape-independent, the algorithm is robust to noise while preserving important shape details. Neither preprocessing of the shape boundary nor pruning of the skeleton is required. Our method produces skeletons with fewer spurious branches than other state-of-the-art methods, while outperforming them visually and according to error measures such as Hausdorff distance and symmetric difference, as evaluated on the MPEG-7 database (1033 images).
This paper introduces a new definition of entropy for hypergraphs. It takes into account the fine structure of a hypergraph by considering its partial hypergraphs, leading to an entropy vector. This allows for more precision in the description of the underlying complexity of the hypergraph. Properties of the proposed definitions are analyzed.
Hierarchical image segmentation provides a region-oriented scale-space, i.e., a set of image segmentations at different detail levels in which the segmentations at finer levels are nested with respect to those at coarser levels. Guimarães et al. proposed a hierarchical graph based image segmentation (HGB) method based on the Felzenszwalb-Huttenlocher dissimilarity. This HGB method computes, for each edge of a graph, the minimum scale in a hierarchy at which two regions linked by this edge should merge according to the dissimilarity. In order to generalize this method, we first propose an algorithm to compute the intervals which contain all the observation scales at which the associated regions should merge. Then, following the current trend in mathematical morphology to study criteria which are not increasing on a hierarchy, we present various strategies to select a significant observation scale in these intervals. We use the BSDS dataset to assess our observation scale selection methods. The experiments show that some of these strategies lead to better segmentation results than the ones obtained with the original HGB method.
The distance transform (DT) and its many variations are ubiquitous tools for image processing and analysis. In many imaging scenarios, the images of interest are corrupted by noise. This has a strong negative impact on the accuracy of the DT, which is highly sensitive to spurious noise points. In this study, we consider images represented as discrete random sets and observe statistics of DT computed on such representations. We, thus, define a stochastic distance transform (SDT), which has an adjustable robustness to noise. Both a stochastic Monte Carlo method and a deterministic method for computing the SDT are proposed and compared. Through a series of empirical tests, we demonstrate that the SDT is effective not only in improving the accuracy of the computed distances in the presence of noise, but also in improving the performance of template matching and watershed segmentation of partially overlapping objects, which are examples of typical applications where DTs are utilized.
Persistent homology is a popular data analysis technique that is used to capture the changing topology of a filtration associated with some simplicial complex K. These topological changes are summarized in persistence diagrams. We propose two contraction operators which when applied to K and its associated filtration, bound the perturbation in the persistence diagrams. The first assumes that the underlying space of K is a 2-manifold and ensures that simplices are paired with the same simplices in the contracted complex as they are in the original. The second is for arbitrary d-complexes, and bounds the bottleneck distance between the initial and contracted p-dimensional persistence diagrams. This is accomplished by defining interleaving maps between persistence modules which arise from chain maps defined over the filtrations. In addition, we show how the second operator can efficiently compose with itself across multiple contractions. The paper concludes with experiments demonstrating the second operator’s utility on manifolds and a brief discussion of future directions for research.
A compact packing is a set of non-overlapping discs where all the holes between discs are curvilinear triangles. There is only one compact packing by discs of size 1. There are exactly nine values of r which allow a compact packing by discs of sizes 1 and r. We prove here that there are exactly 164 pairs (r, s) allowing a compact packing by discs of sizes 1, r and s.
A set $S \subset \mathbb{Z}^d$ is digital convex if $conv(S) \cap \mathbb{Z}^d = S$, where $conv(S)$ denotes the convex hull of $S$. In this paper, we consider the algorithmic problem of testing whether a given set $S$ of $n$ lattice points is digital convex. Although convex hull computation requires $\Omega(n \log n)$ time even for dimension $d = 2$, we provide an algorithm for testing the digital convexity of $S\subset \mathbb{Z}^2$ in $O(n + h \log r)$ time, where $h$ is the number of edges of the convex hull and $r$ is the diameter of $S$. This main result is obtained by proving that if $S$ is digital convex, then the well-known quickhull algorithm computes the convex hull of $S$ in linear time. In fixed dimension $d$, we present the first polynomial algorithm to test digital convexity, as well as a simpler and more practical algorithm whose running time may not be polynomial in $n$ for certain inputs.
Recent works have indicated the potential of using curvature as a regularizer in image segmentation, in particular for the class of thin and elongated objects. These are ubiquitous in biomedical imaging (e.g. vascular networks), in which length regularization can sometime perform badly, as well as in texture identification. However, curvature is a second-order differential measure, and so its estimators are sensitive to noise. The straightforward extensions to Total Variation are not convex, making them a challenge to optimize. State-of-art techniques make use of a coarse approximation of curvature that limits practical applications. We argue that curvature must instead be computed using a multigrid convergent estimator, and we propose in this paper a new digital curvature flow which mimics continuous curvature flow. We illustrate its potential as a post-processing step to a variational segmentation framework.
Euclidean circle approximation on the square grid is an important problem in digital geometry. Recently several schemes have been proposed for approximation of Euclidean circles based on Neighbourhood Sequences, which correspond to repeated application of the von Neumann and Moore neighbourhoods on a square grid. In this paper we study polygon approximations of the Euclidean circles on the square grid with Broadcasting Sequences which can be seen as a generalization of Neighbourhood Sequences. The polygons generated by Broadcasting Sequences are the Minkowski sums of digital disks defined by a given set of broadcasting radii. We propose a polynomial time algorithm that can generate Broadcasting Sequences which are providing flexible and accurate approximation of Euclidean circles.
Building on [23], we investigate the problem of defining and computing the average of digital curves. Given n digital curves that satisfy compatibility conditions, a set - called the gap - in which the average curve is looked for, is defined. The proposed definition rewrites the classical arithmetic mean for curves by (i) defining the distance between each point of the gap and its projection on each curve and (ii) computing the points that minimize the sum of squared deviations. We show that, algorithmically speaking, computing such projections comes down to computing a distance transform with visibility constraints. We propose a fast algorithm to compute a good approximation of these distance maps and finally show that the average curve can be obtained using classical watershed algorithm.
In this article, we suggest a grid-unfolding of level 1 Menger polycubes of arbitrary size with L holes along the x-axis, M the y-axis and N the z-axis. These polycubes can have a high genus, and most vertices are of degree 6. The unfolding is based mainly on the inner faces (that do not lie on the outer most envelope) except for some outer faces that are needed to connect lines or planes in the object. It is worth noticing that this grid-unfolding algorithm is deterministic and without refinement.
Let P be a set of n points in the plane in general position. The order type of P specifies, for every ordered triple, a positive or negative orientation; and the x-type (a.k.a. crossing type) of P specifies, for every unordered 4-tuple, whether they are in convex position. Geometric algorithms on P typically rely on primitives involving the order type or x-type (i.e., triples or 4-tuples). In this paper, we show that the x-type of P can be reconstructed from the compatible exchange graph \(\mathcal {G}_1(P)\) of noncrossing spanning trees on P. This extends a recent result by Keller and Perles (2016), who proved that the x-type of P can be reconstructed from the exchange graph \(\mathcal {G}_0(P)\) of noncrossing spanning trees, where \(\mathcal {G}_1(P)\) is a subgraph of \(\mathcal {G}_0(P)\). A crucial ingredient of our proof is a structure theorem on the maximal sets of pairwise noncrossing edges (msnes) between two components of a planar straight-line graph on the point set P.
In this paper we are proposing a new way to compute a discrete bisector function, which is an important tool for analyzing and filtering Euclidean skeletons. From a continuous point of view, a point that belongs to the medial axis is the center of a maximal ball that hits the background in more than one point. The maximal angle between those points is expected to be high for most of the object points and corresponds to the bisector angle. This logic is not really applicable in the discrete space since in some configurations we miss some background points leading sometimes to small bisector angles. In this work we use annuli to find the background points in order to compute the bisector angle. The main advantage of this approach is the possibility to change the thickness and therefore to be more flexible while computing the bisector angle.
Component trees are interesting structures of nested connected components, efficiently represented by max-trees, used to implement fast algorithms in Image Processing. In these structures, connected components are constructed using a single neighborhood. In recent years, an extension of component trees, called component-hypertrees, was introduced. It consists of a sequence of component trees, generated from a sequence of increasing neighborhoods, in which their connected components are also hierarchically organized. Although this structure could be useful in applications dealing with clusters of objects, not much attention has been given to component-hypertrees. A naive implementation can be costly both in terms of time and memory. So, in this paper, we present algorithms and data structures to efficiently compute and store these structures without redundancy obtaining a minimal representation of component-hypertrees. Experimental results using our efficient algorithm show that the number of nodes is reduced by approximately 70% in comparison to a naive implementation.
This paper focuses on the classical problem of the control of information loss during the digitization step. The properties proposed in the literature rely on smoothness hypotheses that are not verified by the curves including angular points. The notion of turn introduced by Milnor in the article On the Total Curvature of Knots generalizes the notion of integral curvature to continuous curves. Thanks to the turn, we are able to define the local turn-boundedness. This promising property of curves do not require smoothness hypotheses and shares several properties with the par(r)-regularity, in particular well-composed digitizations. Besides, the local turn-boundedness enables to constraint spatially the continuous curve in function of its digitization.
