Discrete Geometry and Mathematical Morphology

We introduce and investigate the concept of a discrete circle in the triangular grid, taking into account the presence of two distinct types of triangles, depending on their orientation. We propose an analytical definition of a discrete circle in this grid, incorporating a non-constant thickness that depends on both the type of the considered triangle and the position of the circle’s center. This varying thickness ensures the edge connectivity of the resulting circle. Building on this analytical definition, we also propose an incremental generation algorithm that is linear in the number of triangles on the circle.

This paper studies how well we can infer the geometry of a (smooth or not) convex shape X from the convex hull $$Y_h$$ Y h of its Gauss digitization with a given gridstep h. Without smoothness constraint, we first present results concerning the proximity of facet normal vectors to the shape normal vectors, as well as a relation between the number of lattice points just above a facet and its area. Then, further results can be obtained when X is smooth, that are valid in arbitrary dimension d. More precisely, we show that the boundary of $$Y_h$$ Y h is Hausdorff-close to the boundary of X with distance less than $$\sqrt{d}h$$ d h , and that the vertices of $$Y_h$$ Y h are even much closer (some $$O(h^{\frac{2d}{d+1}})$$ O ( h 2 d d + 1 ) ). Finally we show that the geometric normal vectors to the facets of $$Y_h$$ Y h tend to the smooth shape normals with a speed $$O(h^{\frac{1}{2}})$$ O ( h 1 2 ) , and the bound is tight.

Computing visibility on a geometric object requires heavy computations since it requires to identify pairs of points that are visible to each other, i.e. there is a straight segment joining them that stays in the close vicinity of the object boundary. We propose to exploit a specific representation of digital sets based on lists of integral intervals in order to compute efficiently the complete visibility graph between lattice points of the digital shape. As a quite direct application, we show then how we can use visibility to estimate the normal vector field of a digital shape in an accurate and convergent manner while staying aware of the salient and sharp features of the shape.

In this paper, we consider the geometrical analysis of oriented digital surfaces, which form the boundary of connected voxel sets. We present a method for the detection and reconstruction of its convex and concave parts. The proposed method composes two tools: the first one is a probing approach for the extraction of locally extremal points, the second one, based on the notion of visibility, joins some pairs of points whenever the straight segment between them stays sufficiently close to the input surface without crossing it. Finally, outer and inner candidate segments are assembled separately into facets, thus building polyhedral approximations of the local convex and concave zones.

Information loss induced by digitization is inevitable, and constitutes a classical problem in digital geometry. Indeed, preserving the geometric and topological features of an object is crucial for image processing tasks. To ensure this preservation when converting continuous objects to discrete representations, different classes of shapes have been introduced. However, the current state of the art still leaves room for the definition of non smooth classes that have digitization preservation properties. To that end, we focus on Locally Turn-Bounded (LTB) Curves, introduced by É. Le Quentrec et al. in 2019. While LTB curves has shown promising results in two dimensions, their extension to higher dimensions is not trivial. This paper introduces the Locally Turn-Bounded Surfaces, a generalization of LTB curves to the 3D Euclidean space. As first results, we not only show that our definition preserves basic properties from LTB curves to LTB surfaces, but also demonstrate that LTB surfaces impose conditions on topology, notably preventing the surface from containing small holes.

Full convexity is an interesting alternative to classical digital convexity since it guarantees connectedness and even simple connectedness in digital spaces $$\mathbb {Z}^d$$ Z d , for any dimension d. This paper aims at giving a better understanding of the monotonicity properties of fully convex digital sets, since earlier works showed that the question was difficult for thin fully convex sets. To decipher the hierarchy of fully convex sets ordered by inclusion, we study how we can peel a fully convex set progressively while keeping its full convexity. We provide a characterization of peelable points and fast algorithms to identify them. Furthermore, we show that fully convex set can be peeled one point at a time till reduced to the empty set, similarly to digitally convex sets in the classical sense. The peeling of a fully convex set can be seen as an analog to homotopic thinning processes, but with an additional geometric property.

A configuration of the plane is a subset (both finite or infinite) A of points in $$\mathbb {Z}^2$$ Z 2 . Provided a finite window probe W, i.e., a finite set of positions of $$\mathbb {Z}^2$$ Z 2 , we shift W in all possible positions on $$\mathbb {Z}^2$$ Z 2 and count the points of A that fit in its positions. In case all the values are equal, say h, then we say that A is h-homogeneous w.r.t. W. When W has a tile shape, i.e., a shape that can be used to tile the plane by translation, h-homogeneous configurations related to W can be decomposed into disjoint, 1-homogeneous ones, in some cases. Very few is known about the properties that a tile has to satisfy in order to allow such a decomposition. In this paper, we shed some light on the topic, by investigating the decomposability of a class of windows, single squares, i.e., those tiles that can be surrounded in one only possible way by four copies of themselves in order to produce a tiling of the plane, that constitute one of the simplest classes of tiles. We also introduce other classes of tiles that allow homogeneous, non-decomposable planar configurations. Finally, we define the notion of reducible configuration, that concerns the possibility of decomposing a configuration into smaller, non 1-homogeneous ones, moving toward the classification of tiles w.r.t. the decomposability property.

We consider two classes of puzzles Aquarium and Tilepaint connected to the field of Discrete Tomography. The input mixes a tiling of a rectangular grid with projections prescribing the number of squared cells that have to be filled in each row and column. In Tilepaint, the tiles are either entirely filled or remain empty while in Aquarium, the tiles are filled like aquariums with an horizontal level of water. Tilepaint is known to be NP-complete with a proof showing that Aquarium is also NP-complete, even with tiles of size at most 3. In this paper, we investigate the complexity of the two puzzles in the special case of regular tilings with small tiles. We show that Tilepaint can be solved in polynomial time while Aquarium is NP-complete for regular triomino tilings and is polynomial time for some regular domino tilings. As Aquarium is NP-complete, we propose, in a second part, a physical zero-knowledge proof (ZKP) protocol for Aquarium, using decks of playing cards (using a total of $$2mn+2$$ 2 m n + 2 cards for an $$m\times n$$ m × n grid).

This paper addresses the problem of converting a 2d digital object S (a finite set of points in $$\mathbb {Z}^2$$ Z 2 ) into a finite set $$\mathcal {B}$$ B of balls centered on $$\mathbb R^2$$ R 2 , such that the balls of $$\mathcal {B}$$ B cover all points of S and no point of $$\mathbb {Z}^2\backslash S$$ Z 2 \ S , and the cardinality of $$\mathcal {B}$$ B is minimum. In a previous work, we showed that the problem was polynomial for the specific class of 2d hole-free digital objects. In this article, we show that the problem is NP-complete in the general case using a reduction from the 3-Planar Vertex Cover problem.

The medial axis of a smoothly embedded surface in $${{\mathbb R}}^3$$ R 3 consists of all points for which the Euclidean distance function on the surface has at least two global minima. We generalize this notion to the mid-sphere axis, which consists of all points for which the Euclidean distance function has two interchanging saddles that swap their partners in the pairing by persistent homology. It offers a discrete-algebraic multi-scale approach to computing ridge-like structures on the surface. As a proof of concept, an algorithm that computes stair-case approximations of the mid-sphere axis is provided.

Thinning is an iterative object reduction to extract skeleton-like shape features from binary objects. Parallel 3D curve-thinning algorithms are composed of parallel reductions to produce centerlines. In subfield-based parallel thinning the digital space is partitioned into $$s\ge 2$$ s ≥ 2 subfields which are alternatively activated, and only certain points in the active subfield can be deleted. This paper presents the first two subfield-based parallel 3D curve-thinning algorithms acting on the unconventional face-centered cubic (FCC) grid. The proposed algorithms use partitionings of the the FCC grid into four and eight subfields.

Computing homology and cohomology is at the heart of many recent works and a key issue for topological data analysis. Among homological objects, homology generators are useful to locate or understand holes (especially for geometric objects). The present paper provides a characterization of the class of homology bases that are computed by standard algorithmic methods. The proof of this characterization relies on the Homological Discrete Vector Field, a combinatorial structure for computing homology, which encompasses several standard methods (persistent homology, tri-partitions, Smith Normal Form, discrete Morse theory). These results refine the combinatorial homology theory and provide novel ideas to gain more control over the computation of homology generators.

We rely on the framework of Morse sequences to enable the direct computation of gradient vector fields on simplicial complexes. A Morse sequence is a filtration from a subcomplex L to a complex K via elementary expansions and fillings, naturally encoding critical and regular simplexes. Maximal increasing and minimal decreasing schemes allow constructing these sequences, and are linked to algorithms like Random Discrete Morse and Coreduction. Extending the approach to cosimplicial complexes ( $$S=K\setminus L$$ S = K \ L ) allows for efficient computation using reductions, perforations, coreductions, and coperforations. We further generalize to F-sequences, which are Morse sequences weighted by an arbitrary stack function F, and provide algorithms to compute maximal and minimal sequences. A particular case is when the stack function is given through a vertex map, common in topological data analysis. For injective maps, the complex decomposes into lower stars, recovering established methods and enabling parallel computation; for non-injective maps, our approach applies directly without requiring perturbations. Thus, the paper adopts Morse sequences as a framework that simplifies and connects some important existing propagation-based methods, while also introducing new schemes that extend their scope and practical applicability.

We recently introduced the topological tree of shapes as a new structure in the family of morphological trees. The topological tree of shapes is both a topological invariant of grey-level images and a hierarchical model of these images. Building upon these two properties, we explain how the topological tree of shapes can be used to relevantly design connected operators. In this theoretical article, we show that such operators present algebraic properties related to openings/closings, while also guaranteeing the preservation of the topological structure of the processed images. Software implementation available at https://github.com/jmendesf/TopologicalTreeOfShapes .

Alternating Sequential Filters (ASFs) rely on the iterated application of openings and closings of increasing strength. Since connected openings and closings can be designed from the component tree, it is possible to define Connected ASFs (CASFs) by alternatively pruning the branches of the dual max- and min-trees of an image. The main drawback of this approach is that pruning one of the trees modifies not only the image but also the dual tree. Recomputing a component tree at each iteration of the process is computationally expensive. In this article, we show how a component tree can be efficiently updated (avoiding reconstruction) when its dual tree is pruned. We build upon this algorithmic scheme to propose a computationally efficient CASF algorithm.

Spectrograms offer a time-frequency representation well-suited for audio analysis, where structured signals manifest as horizontal or vertical patterns and noise appears as irregular textures. In this paper, we propose a spectrogram denoising method based on mathematical morphology, and more precisely on max-trees. We treat the spectrogram as a grayscale image and apply a sequence of filters on its max-tree representation. We propose a method to isolate signal regions by filtering components according to contrast and shape. A binary mask is constructed from the remaining components, which is used to reconstruct the denoised signal by inverting the masked short-time Fourier transform (STFT). The method is highly interpretable and preserves the signal structure while effectively removing noise. We demonstrate its effectiveness on synthetic and real audio signals.

Maximizing influence in networks is a widely studied problem with applications across various domains. In this paper, we approach this problem by considering influence maximization in directed tree structures. Given a tree T, each node is associated with a binary label, obtaining $$1-$$ 1 - nodes and $$0-$$ 0 - nodes. The influence of $$1-$$ 1 - nodes over T is then computed as the total number of $$0-$$ 0 - nodes in their neighbourhood. Given an integer k, our objective is to identify a subset of k $$1-$$ 1 - nodes that maximizes the influence over T. After analysing heuristic approaches and their limitations, we present a dynamic programming algorithm for solving this problem exactly when the maximum degree and k are fixed. Our findings provide new insights into influence maximization in directed tree structures and establish a foundation for further exploration of constrained optimization problems.

Human vision is dynamic: body, head, and eye movements change the view of the actual environment. Regions and features close to the boundary are likely to disappear while the probability in the center is high that they re-appear after the movements. We therefore exploit a re-arrangement of the pixels that orders the pixels such that the center of the image is on one end of the sequence and the boundary pixels are on the other end. A discrete spiral that connects a corner of the image with the center and contains all the pixels satisfies these constraints. In addition it is a strict total order. This order enables a max-link strategy generating efficiently a spanning forest of the image. It allows the efficient contraction for the levels of an irregular pyramid. The max-link strategy on the spiral pushes the surviving vertices geometrically towards the center of the image. We explore the properties of the proposed combination and their integration into the contrast pyramid.

In this work, we study several implementations of the variational formulation of classical morphological operators such as erosion, dilation, opening, and closing, as well as arbitrary rank filters using convex optimization over graphs. Since the structuring element can naturally vary along nodes in the graph, this formulation allows for an effective implementation of spatially-variant mathematical morphology. It is particularly efficient on massively parallel architectures. We focus on solving the associated optimization problems using various algorithms from convex analysis. The problem being non-smooth, we consider algorithms based on the proximal operator, such as ADMM and PPXA+. We tested these approaches on both images and graph data. Our results show that proximal algorithms can handle the problems efficiently. We believe that this variational framework is a promising direction to combine mathematical morphology with modern computational tools, and is a direct avenue for solving inverse problems with mathematical morphology operators in the formulation.

Max-trees are hierarchical structures for anti-extensive attribute filtering. A node attribute is a function of the connected component of a threshold set represented by the node. Filtering is carried out by removing nodes based on their attributes. In this paper we present a new algorithm for parallel attribute computation and filtering using the subtractive rule and image reconstruction on a GPU. Results show up to $$30.4\times $$ 30.4 × speed-up compared to sequential algorithms, and increased numerical stability by several orders of magnitude.

We study a certain class of greedy algorithms for combinatorial optimization, and present a generalized version of this algorithmic pattern encompassing several previously published methods. We analyze the properties of the solutions produced by such algorithms, and provide proofs of their optimality through the concept of lexicographic max-ordering. By presenting a unified formulation of this class of greedy algorithms, we hope to facilitate the development of new such algorithms, and to provide a deeper understanding on the properties of existing algorithms.

The tree of shapes (ToS) is a self-dual and contrast invariant hierarchical representation of images, making it highly suitable for various image processing tasks such as filtering, segmentation, and object detection. The existing linear algorithm for computing the ToS is effective for low-quantized images, but exhibits increased complexity for high-quantized images due to the reliance on data structures like red-black trees, which have logarithmic complexity for insertion and deletion operations. This paper introduces a novel algorithm based on the fast marching method to compute a distance map from which the ToS is extracted. This method takes advantage of the distribution of the gradient values of the image pixels to make use of an efficient priority queue structure that lowers the computational complexity of the ToS on high-dynamic range images.

We present a linear-time algorithm for computing sliding window maximum and minimum filters over one-dimensional signals. The algorithm is based on a single forward pass using a double-ended queue (deque) and requires only a small amount of memory. It supports in-place and streaming implementations. We provide a self-contained description of the algorithm, including illustrative examples, pseudocode, a practical implementation in C, and an analysis of its runtime behavior. We compare the performance of the algorithm against the well-known Van Herk–Gil–Werman (HGW) algorithm, which performs three scans over the data and offers data-independent performance.

This article presents an algorithm to construct the binary partition tree by altitude ordering based on Borvka’s minimum spanning tree strategy. The binary partition tree by altitude ordering is a data structure to represent an image in the form of a hierarchy of segmentations that is widely used in mathematical morphology processing. Recently, the authors of PANDORA [21] have proposed a novel parallel algorithm for computing single linkage clustering from the minimum spanning tree of a point cloud on a GPU. In this paper, we show that their method can indeed be nicely cast in the framework of watershed cuts and waterfalls for computing the binary partition tree by altitude ordering. More precisely, the method consists in performing a sequence of watershed cuts-basins contractions, seen as a variant of Borvka’s algorithm with a complexity of $$O(n\log (n))$$ O ( n log ( n ) ) , followed by a dedicated post-processing. Furthermore, we show that this algorithm can be extended to process any edge-weighted graphs and not only minimum-spanning trees. These results open a new path towards massively parallel algorithms for hierarchical watershed algorithms.

Replacing convolution with morphological operations in trainable layers has received significant attention lately. Among the various strategies that have emerged, smooth morphological layers have shown strong potential and flexibility, as a single layer can behave either like a (pseudo-)erosion or a (pseudo-)dilation depending on the sign and value of its trainable control parameter. In this work, we build upon the so-called $$\mathcal {S}$$ S Morph layer by introducing a harmonized formulation that addresses previously identified asymptotic limitations when learning grayscale erosion and dilation. We also investigate and compare two strategies (a novel penalty term in the training loss and shared-weight layers) to improve the learning of grayscale opening and closing operations in two-layer networks. Finally, we evaluate the performance of this improved $$\mathcal {S}$$ S Morph layer on a salt-and-pepper denoising task in a four-layer network architecture, and compare it with other morphological and convolutional networks.

Scattering networks involve cascading wavelet transforms, modulus operators, and low-pass filtering. They provide a sound mathematical proxy to model and understand aspects of deep learning such as invariance, stability, hierarchical representation, role of nonlinear activation functions, etc. Linear translation-equivariant filtering based on convolution can be seen as a morphological erosion on a complete inf-semilattice defined by a specific partial ordering in the Fourier domain. In this case, ideal linear filters correspond to morphological openings. Using this framework, we provide a morphological interpretation of the scattering networks from the Fourier-based inf-semilattice. Then, we use the morphological universal representation theory to justify the universal representation of scattering networks of increasing operators on the Fourier-based inf-semilattice. Finally, using our theory we discuss the relevance of learning operators more general than wavelets in the frequency domain.

As a combination of two successful paradigms in image processing, namely Mathematical Morphology and Deep Learning, Morphological Networks seem very promising for image analysis. However, their practical applications remain constrained due to possible limitations of expressivity and difficulties related to gradient descent based optimization algorithms. In this paper, we focus on neural architectures inspired by the morphological representation theory, which proves their expressivity. We investigate their optimization difficulties and find out that, rather than the non-differentiability of morphological layers, the sparsity of their gradient (or subgradient) is what limits them the most, and requires an appropriate initialization. Furthermore, we propose a method to reduce the number of model parameters after training, which produces a minimal equivalent operator.

A morphological perceptron is a multilayer feedforward neural network in which neurons perform elementary operations from mathematical morphology. For multiclass classification tasks, a morphological perceptron with a competitive layer (MPCL) is obtained by integrating a winner-take-all output layer into the standard morphological architecture. The non-differentiability of morphological operators renders gradient-based optimization methods unsuitable for training such networks. Consequently, alternative strategies that do not depend on gradient information are commonly adopted. This paper proposes the use of the convex-concave procedure (CCP) for training MPCL networks. The training problem is formulated as a difference of convex (DC) functions and solved iteratively using CCP, resulting in a sequence of linear programming subproblems. Computational experiments demonstrate the effectiveness of the proposed training method in addressing classification tasks with MPCL networks.

The spatial derivatives in Hamilton-Jacobi partial differential equations for the definition of morphological operations such as dilation and erosion for grey-value images are replaced by fractional derivatives of arbitrary positive order. Focus is laid on geometric invariance with respect to reflections and rotations so that directional bias towards the coordinate directions is avoided. Discretisation of directional fractional derivatives via truncated general power series ultimately leads to an optimisation problem for the advection direction. We numerically compare the proposed fractional morphological operations with conventional counterparts and a simpler fractional-order alternative on grey-value images to show interesting phenomena and gain insights into the effects of the non-local nature of the fractional derivatives which merit further investigation.

This paper introduces the class of grey-scale image stack operators as those that (a) map binary-images into binary-images and (b) commute on average with cross-sectioning. Equivalently, stack operators are 1-Lipchitz extensions of set operators which can be represented by applying a characteristic set operator to the cross-sections of the image and adding. In particular, they are a generalisation of stack filters, for which the characteristic set operators are increasing. Our main result is that stack operators inherit lattice properties of the characteristic set operators. We focus on the case of translation-invariant and locally defined stack operators and show the main result by deducing the characteristic function, kernel, and basis representation of stack operators. The results of this paper have implications on the design of image operators, since imply that to solve some grey-scale image processing problems it is enough to design an operator for performing the desired transformation on binary images, and then considering its extension given by a stack operator. We leave many topics for future research regarding the machine learning of stack operators and the characterisation of the image processing problems that can be solved by them.

Levelings are morphological filters that are used to simplify and segment images. Depending on the scale, different amounts of detail disappear while the contours of the remaining objects remain sharp.In this work, we will consider two colour morphological methods to perform the dilation and erosion operations necessary for leveling. The first one computes a direct supremum based on the spectral decomposition of symmetric matrices, which offers some favourable properties, such as transitivity. Based on this, the second method generates a colour ordering of the input colours. We compare the levelings resulting from these two methods experimentally using natural colour images.

Mathematical morphology provides a nonlinear framework for image and spatial data processing and analysis. Although there have been many successful applications of mathematical morphology to vector-valued images, such as color and hyperspectral images, there is still no consensus on the most suitable vector ordering for constructing morphological operators. This paper addresses this issue by examining a reduced ordering approximating the Condorcet ranking derived from a set of vector orderings. Inspired by voting problems, the Condorcet ordering ranks elements from most to least voted, with voters representing different orderings. In this paper, we develop a machine learning approach that learns a reduced ordering that approximates the Condorcet ordering. Preliminary computational experiments confirm the effectiveness of learning the reduced mapping to define vector-valued morphological operators for color images.

Discrete Tomography fits in the broad framework of inverse problems. One of its main issues concerns the faithful reconstruction of discrete homogeneous objects from internal quantitative information known as projections.In this paper, we consider specific types of projections that count, for each position, the number of points of the discrete objects that lie in its 8 or 4 neighbourhood, including the point itself.Our final goal is to reconstruct the object from the projections, if possible uniquely, or to determine the minimal number of its points that have to be revealed to do that. This problem is commonly addressed as the Minimum Surgical Probing problem (MSP).Our contribution is to study MSP with the addition of convexity constraint. In particular, we focus on a specific class of objects, commonly involved in Discrete Tomography, namely hv-convex polyominoes, which are connected sets of points with the further constraint of horizontal and vertical convexity. These constraints allow us to achieve unique reconstruction (except in some corner cases). Additionally, they allow for faster reconstruction algorithms.

Pattern spectra are a powerful tool for characterizing image content at both local and global levels. They have been successfully applied to image or patch retrieval, enabling efficient processing of large satellite imagery, for instance. In Earth Observation, multitemporal imaging, or satellite image time series, provides not only spatial but also temporal information. In this context, retrieval aims to identify similar spatio-temporal patterns. In this paper, we address the problem of efficient content-based time series retrieval using pattern spectra. To this end, we revisit the pattern spectra retrieval methodology originally defined in the spatial domain and adapt it to the spatio-temporal case. Preliminary results on the SECOND dataset demonstrate its effectiveness compared to both a deep learning baseline and a uniform guessing baseline, especially when using small patch sizes.

The aim of our work is to analyze size distributions of particles and their agglomerates in imagery from astrophysical microgravity experiments. The data acquired in these experiments are given by sequences consisting of several hundred images. It is desirable to establish an automated routine that helps to assess size distributions of important image structures and their dynamics in a statistical way.The main technique we adopt to this end is the morphological granulometry. After preprocessing steps that facilitate granulometric analysis, we show how to extract useful information on size of particle agglomerates as well as underlying dynamics. At hand of the discussion of two different microgravity key experiments we demonstrate that the granulometric analysis enables to assess important experimental aspects. We conjecture that our developments are a useful basis for the quantitative assessment of microgravity particle experiments.

The discovery of patterns in a point-set representation of music consists in identifying repeating subsets of points. In this task, musical symbolic data is modeled as a discrete set of points, usually in $$\mathbb {R}^n$$ R n , where each point represents a musical note and its coordinates represent the characteristics of the note, such as its onset or its pitch value. While numerous algorithms have been developed to discover all exact repetitions and extract musically relevant patterns, recent research has turned toward the discovery of patterns that repeat with some variations. Because the morphological erosion of a pattern provides its occurrences, we propose an adaptation of this operation to also obtain its variations with respect to a given approximation. This approach not only reveals certain variations of the pattern, but also enables to associate specific points to the pattern despite the fact that they were not initially present due to the constraints of strict repetition. We demonstrate that the proposed formalism satisfies certain fundamental properties for the musical pattern discovery task, such as the fact that iterating erosion produces cycles of patterns and its translation values. Finally, we apply these operations to the corpus of fugues from Bach’s Well-Tempered Clavier, highlighting the usefulness of the proposed approach.

A digital blurred plane is a finite set of points with controlled thickness. This notion can be used to characterize a real world object surface, considered as planar at a large scale, but actually quite irregular at finer scale. In this paper, we introduce a new framework to detect a blurred plane in a point set, based on the analysis of two orthogonal height profiles using digital geometry tools. It is applied to the interactive extraction of relict charcoal platforms from an airborne LiDAR point cloud. This process was tested on two sets of geolocated platforms obtained by on-site prospecting or visual survey of the LiDAR digital terrain model. Experimental validations show that a large amount of them is successfully extracted and that their estimated extent closely matches the ground truth.