Abstract: We consider two aspects of the action of the extended metaplectic representation of the group G of affine, measure and orientation preserving maps of the time-frequency plane on L2 functions on the line. On the one hand, we list, up to equivalence, all possible reproducing formulas that arise by restricting the representation to connected Lie subgroups of G. On the other hand, we describe, in terms of Weyl calculus, the commutative von Neumann algebras generated by restriction to one-parameter subgroups.

Abstract: This paper presents a general solution for the problem of affine point pattern matching (APPM). Formally, given two sets of two-dimensional points (x,y) which are related by a general affine transformation (up to small deviations of the point coordinates and maybe some additional outliers). Then we can determine the six parameters a ik of the transformation using new Hu point-invariants which are invariant with respect to affine transformations. With these invariants we compute a weighted point reference list. The affine parameters can be calculated using the method of the least absolute differences (LAD method) and using linear programming. In comparison to the least squares method, our approach is very robust against noise and outliers. The algorithm works in O(n) average time and can be used for translation and/or rotations, isotropic and non-isotropic scalings, shear transformations and reflections.

Abstract: This paper presents a general solution for the problem of affine point pattern matching (APPM). Formally, given two sets of two-dimensional points (x,y) which are related by a general affine transformation (up to small deviations of the point coordinates and maybe some additional outliers). Then we can determine the six parameters aik of the transformation using new Hu point-invariants which are invariant with respect to affine transformations. With these invariants we compute a weighted point reference list. The affine parameters can be calculated using the method of the least absolute differences (LAD method) and using linear programming. In comparison to the least squares method, our approach is very robust against noise and outliers. The algorithm works in O(n) average time and can be used for translation and/or rotations, isotropic and non-isotropic scalings, shear transformations and reflections.

Abstract: A new definition of affine invariant erosion of 3D surfaces is introduced. Instead of being based in terms of Euclidean distances, the volumes enclosed between the surface and its chords are used. The resulting erosion is insensitive to noise, and by construction, it is affine invariant. We prove some key properties about this erosion operation, and we propose a simple method to compute the erosion of implicit surfaces. We also discuss how the affine erosion can be used to define 3D affine invariant robust skeletons.

Abstract: We characterize conic sections in the Euclidean plane in terms of appropriate eigenfunctions of second order on their spherical image; we clarify the affine background of such results.

Abstract: We propose a Hopfield network for affine invariant object recognition. Affine transformation can be considered as an approximation to the weak perspective transformation. Therefore, it is desirable to derive an effective means to establish point correspondence under affine transformation in many applications. By considering the point correspondence as a sub-graph matching problem, a new energy function for affine invariant matching by Hopfield network has been derived. Due to the high order connections, it can only be solved by a fourth order network. However, the order of the network can be reduced by incorporating the neighborhood information available in the data when the proposed method is applied to solve affine invariant shape matching problems. The experimental results show that the proposed method is effective in object recognition under affine transformation.

Abstract: The number of points on affine diagonal curves aX m + bY n = c over finite fields can be computed in terms of cyclotomic numbers. The approach of Berndt, Evans and Williams [1] is to express the number of points in terms of generalized Jacobi sums, then to relate the Jacobi sums Jr(χ u , χ v ) to cyclotomic numbers. In this article we present the direct elementary method for the number of points on the affine curves aX m + bY n = c over finite fields in terms of cyclotomic numbers. This approach is applicable when explicit formulas are already known for cyclotomic numbers, and circumvents the use of Jacobi sums. It generalizes to the determination of the number of points on affine diagonal hypersurfaces of higher dimension. The curves for which this method applies includes examples of elliptic and hyperelliptic curves which are of interest for public-key cryptosystems, coding theory and the design and analysis of sequences.

Abstract: The affine group is a set of transformations in space that leave straight lines as straight lines. (These can be thought of as transformations of the coordinate system, leaving the objects unchanged, or as transformations of the objects, leaving the coordinate system unchanged. We shall take the latter viewpoint, since it is the one most closely associated with computer graphics.) Since one such transformation followed by another results in straight lines going to straight lines going to straight lines, the result of two affine transformations in succession is a third affine transformation. Further, one can reverse any such transformation, to get back to the original lines, so each affine transformation has an inverse transformation. This is all that is required to form a group in mathematics:

Abstract: In this paper we describe a four dimensional family of special rational elliptic surfaces admitting an involution with isolated fixed points. For each surface in this family we calculate explicitly the action of a spectral version of the involution (namely of its Fourier-Mukai conjugate) on global line bundles and on spectral data. The calculation is carried out both on the level of cohomology and in the derived category. We find that the spectral involution behaves like a fairly simple affine transformation away from the union of those fiber components which do not intersect the zero section. These results are the key ingredient in the construction of Standard-Model bundles in [DOPWa]. MSC 2000: 14D20, 14D21, 14J60