This paper provides a definition of back-propagation through geometric correspondences for morphological neural networks. In addition, dilation layers are shown to learn probe geometry by erosion of layer inputs and outputs. A proof-of-principle is provided, in which predictions and convergence of morphological networks significantly outperform convolutional networks.

Abstract We design a computational method to align pairs of counter-fitting fracture surfaces of digitized archaeological artefacts. The challenge is to achieve an accurate fit, even though the data is inherently lacking material through abrasion, missing geometry of the counterparts, and may have been acquired by different scanning practices. We propose to use the non-linear complementarity-preserving properties of Mathematical Morphology to guide the pairwise fitting in a manner inherently insensitive to these aspects. In our approach, the fracture surface is tightly bounded by a concise set of characteristic multi-local morphological features. Such features and their descriptors are computed by analysing the discrete distance transform and its causal scale-space information. This compact morphological representation provides the information required for accurately aligning the fracture surfaces through applying a RANSAC-based algorithm incorporating weighted Procrustes to the morphological features, followed by ICP on morphologically selected ‘flank’ regions. We propose new criteria for evaluating the resulting pairwise alignment quality, taking into consideration both penetration and gap regions. Careful quantitative evaluation on real terracotta fragments confirms the accuracy of our method under the expected archaeological noise. We show that our morphological method outperforms a recent linear pairwise alignment method and briefly discuss our limitations and the effects of variations in digitization and abrasion on our proposed alignment technique.

The well-known Procrustes method determines the optimal rigid body motion that registers two point clouds by minimizing the square distances of the residuals. In this paper, we perform the first order error analysis of this method for the 3D case, fully specifying how directional noise in the point clouds affects the estimated parameters of the rigid body motion. These results are much more specific than the error bounds which have been established in numerical analysis. We provide an intuitive understanding of the outcome to facilitate direct use in applications.

The problem of finding collision-free placements for an object amid obstacles has two well-known solutions: the task space approach and the configuration space approach. In this correspondence, we study the mathematical structure of the placement problem, and show that Minkowski decomposition of the object produces a hierarchy of intermediate reformulations. This provides the mathematical foundation for common approximate solution methods already used in applications. In particular, it provides a recipe for discretizing rotations consistently. The methods discussed are particularly effective for simple shapes.

A* graph search effectively computes the optimal solution path from start nodes to goal nodes in a graph, using a heuristic function. In some applications, the graph may change slightly in the course of its use and the solution path then needs to be updated. Very often, the new solution will differ only slightly from the old. Rather than perform the full A* on the new graph, we compute the necessary OPEN nodes from which the revised solution can be obtained by A*. In this "Differential A*" algorithm, the graph topology, transition costs, or start/goals may change simultaneously. We develop the algorithm and discuss when it gives an improvement over simply reapplying A*. We briefly discuss an application to robot path planning in configuration space, where such graph changes naturally arise.

We present a directional boundary representation which deals locally and consistently with the boundary's "inside". We show that collision and wave propagation are reduced to addition on the spectrum of directions, and we derive transformation laws for differential geometrical properties such as directed curvature.

Driving a car involves simultaneous consideration of events at different spatio-temporal scales. Proper interpretation and planning then leads to behaviour such as the parallel parking manoeuvre, the three-point turn, free Euclidean driving in a desert, following a road, and translationally passing other vehicles at high speed. In the study of autonomous systems, it is desirable to find a representation in which such different behaviours of a single system can be related to each other, and to find precisely how and under what conditions a change of representation and corresponding choice of motions occurs. In this paper, we formulate an abstraction mechanism based on approximations of flows of commutators of vector fields (the paper also contains an explanation of these concepts). We apply it to the goal-directed motion of a car and show how the environmental constraints induce, through this abstraction mechanism, a recognizable hierarchy of descriptions of the car's motion. This suggests that this method of analyzing the behaviour to derive a suitable level of description is conceptually correct, and that its implementation would give the right level of computational representation on which to apply one of the standard path planning algorithms.

This paper considers the problem of measuring properties of digitized straight lines from the viewpoint of measurement methodology. The measurement and estimation process is described in detail, revealing the importance of a step called ``characterization'' which was not recognized explicitly before. Using this new concept, BLUE (Best Linear Unbiased) estimators are found. These are calculated for various properties of digitized straight lines, and are briefly compared to previous work.

Using diagrams called ``spirographs'' a general theory is developed with which one can easily perform calculations on various aspects of digitized straight lines. The mathematics of the theory establishes a link between digitized straight lines and the theory of numbers (Farey series, continued fractions). To show that spirograph theory is a useful unification, we derive two previously known advanced results within the framework of the theory, and new results concerning the accuracy in position of a digitized straight line as a function of its slope and length.

If a continuous straight line segment is digitized on a regular grid, obviously a loss of information occurs. As a result, the discrete representation obtained (e.g., a chaincode string) can be coded more conveniently than the continuous line segment, but measurements of properties (such as line length) performed on the representation have an intrinsic inaccuracy due to the digitization process. In this paper, two fundamental properties of the quantization of straight line segments are treated. 1) It is proved that every ``straight'' chaincode string can be represented by a set of four unique integer parameters. Definitions of these parameters are given. 2) A mathematical expression is derived for the set of all continuous line segments which could have generated a given chaincode string. The relation with the chord property is briefly discussed.