Motion synthesis through 1D affine matching

We present the study of a data-driven motion synthesis approach based on a 1D affine image-matching equation. We start by deriving the relevant properties of the exact matching operator, such as the existence of a singular point. Next, we approximate such operator by the Green’s function of a second-order differential equation, finding that it leads to a more compelling motion impression, due to the incorporation of blur. We then proceed to show that, by judicious choice of the matching parameters, the 1D affine Green’s filter allows the simulation of a broad class of effects, such as zoom-in and zoom-out, and of complex nonrigid motions such as that of a pulsating heart.

Analytical and experimental validations of a numerical band-limited Green’s function approach for modelling acoustic emission waves

Dynamic Green’s functions are essential for modelling acoustic emission (AE) wave propagation and for the quantitative characterisation of AE sources. In this work, a method for evaluating the Green’s function of a body using the finite element method is presented. The advantage of the proposed method is that it can be used to model realistic geometries, material properties and sources that cannot be treated analytically. The numerical results presented in this paper are compared with known analytical solutions of the Green’s function for an infinite isotropic plate and also with experimental measurements of AE waves generated by known artificial AE sources (ball impact and pencil lead break).     

Shape from shading through photometric motion

We present a new shape from shading algorithm, extending to the single-input case, a recently introduced approach to the photometric motion process. As proposed by Pentland, photometric motion is based on the intensity variation, due to the motion, at a given point on a rotating surface. Recently, an alternative formulation has also appeared, based on the intensity change at a fixed image location. Expressing this as a function of reflectance-map and motion-field parameters, a constraint on the shape of the imaged surface can be obtained. Coupled with an affine matching constraint, this has been shown to yield a closed-form expression for the surface function. Here, we extend such formulation to the single-input case, by using the Green’s function of an affine matching equation to generate an artificial pair to the input image, corresponding to an approximate rendition of the imaged surface under a rotated view. Using this, we are able to obtain high quality shape-from-shading estimates, even under conditions of unknown reflectance map and light source direction, as demonstrated here by an extensive experimental study.     

Solutions and Green’s functions for some linear second-order three-point boundary value problems

In this paper we consider the Green’s functions for a second-order linear ordinary differential equation with some three-point boundary conditions. We give exact expressions of the unique solutions for the linear three-point boundary problems by the Green’s functions. As applications, we study the iterative solutions for some nonlinear singular second-order three-point boundary value problems.     

Hopfield neural networks for affine invariant matching

The affine transformation, which consists of rotation, translation, scaling, and shearing transformations, can be considered as an approximation to the perspective transformation. Therefore, it is very important to find an effective means for establishing point correspondences under affine transformation in many applications. In this paper, we consider the point correspondence problem as a subgraph matching problem and develop an energy formulation for affine invariant matching by a Hopfield type neural network. The fourth-order network is investigated first, then order reduction is done by incorporating the neighborhood information in the data. Thus we can use the second-order Hopfield network to perform subgraph isomorphism invariant to affine transformation, which can be applied to an affine invariant shape recognition problem. Experimental results show the effectiveness and efficiency of the proposed method.     

Guards,Bounds, and Generalized Semantics

Some initial motivations for the Guarded Fragment still seem of interest in carrying its program further. First, we stress the equivalence between two perspectives: (a) satisfiability on standard models for guarded first-order formulas, and (b) satisfiability on general assignment models for arbitrary first-order formulas. In particular, we give a new straightforward reduction from the former notion to the latter. We also show how a perspective shift to general assignment models provides a new look at the fixed-point extension LFP(FO) of first-order logic, making it decidable. Next, we relate guarded syntax to earlier quantifier restriction strategies for achieving effective axiomatizability in second-order logic – pointing at analogies with ‘persistent’ formulas, which are essentially in the Bounded Fragment of many-sorted first-order logic. Finally, we look at some further unexplored directions, including the systematic use of ‘quasi-models’ as a semantics by itself.     

Flooding and Drying in Discontinuous Galerkin Finite-Element Discretizations of Shallow-Water Equations. Part 1: One Dimension

Free boundaries in shallow-water equations demarcate the time-dependent water line between ‘‘flooded’’ and ‘‘dry’’ regions. We present a novel numerical algorithm to treat flooding and drying in a formally second-order explicit space discontinuous Galerkin finite-element discretization of the one-dimensional or symmetric shallow-water equations. The algorithm uses fixed Eulerian flooded elements and a mixed Eulerian–Lagrangian element at each free boundary. When the time step is suitably restricted, we show that the mean water depth is positive. This time-step restriction is based on an analysis of the discretized continuity equation while using the HLLC flux. The algorithm and its implementation are tested in comparison with a large and relevant suite of known exact solutions. The essence of the flooding and drying algorithm pivots around the analysis of a continuity equation with a fluid velocity and a pseudodensity (in the shallow water case the depth). It therefore also applies, for example, to space discontinuous Galerkin finite-element discretizations of the compressible Euler equations in which vacuum regions emerge, in analogy of the above dry regions. We believe that the approach presented can be extended to finite-volume discretizations with similar mean level and slope reconstruction.This revised version was published online in July 2005 with corrected volume and issue numbers.     

Synthesis of rewrite programs by higher-order and semantic unification

This paper presents a framework for synthesizing rewrite programs using higher-order and semantic unification. Many problems in computer science and artificial intelligence can be formalized as problems of higher-order unification. Among such problems is first-order anti-unification. In this paper, we show that first-order anti-unification can be regarded as a second-order matching problem and solved by the algorithm for higher-order unification.     

The Girard–Reynolds isomorphism

The second-order polymorphic lambda calculus, F2, was independently discovered by Girard and Reynolds. Girard additionally proved a Representation Theorem: every function on natural numbers that can be proved total in second-order intuitionistic predicate logic, P2, can be represented in F2. Reynolds additionally proved an Abstraction Theorem: for a suitable notion of logical relation, every term in F2 takes related arguments into related results. We observe that the essence of Girard’s result is a projection from P2 into F2, and that the essence of Reynolds’s result is an embedding of F2 into P2, and that the Reynolds embedding followed by the Girard projection is the identity. The Girard projection discards all first-order quantifiers, so it seems unreasonable to expect that the Girard projection followed by the Reynolds embedding should also be the identity. However, we show that in the presence of Reynolds’s parametricity property that this is indeed the case, for propositions corresponding to inductive definitions of naturals or other algebraic types.