The Noise2Noise method allows for training machine learning-based denoisers with pairs of input and target images where both the input and target can be noisy. This removes the need for training with clean target images, which can be difficult to obtain. However, Noise2Noise training has a major limitation: nonlinear functions applied to the noisy targets will skew the results. This bias occurs because the nonlinearity makes the expected value of the noisy targets different from the clean target image. Since nonlinear functions are common in image processing, avoiding them limits the types of preprocessing that can be performed on the noisy targets. Our main insight is that certain nonlinear functions can be applied to the noisy targets without adding significant bias to the results. We develop a theoretical framework for analyzing the effects of these nonlinearities, and describe a class of nonlinear functions with minimal bias.

This paper describes an implementation and tests of a blending scheme for regularly sampled data interpolation, and in particular studies the order of approximation for the method. This particular implementation is a special case of an earlier scheme by Fang for fitting a parametric surface to interpolate the vertices of a closed polyhedron with n-sided faces, where a surface patch is constructed for each face of the polyhedron, and neighbouring faces can meet with a user specified order of continuity. The specialization described in this paper considers functions of the form z=f(x,y) with the patches meeting with C2 continuity. This restriction allows for investigation of order of approximation, and it is shown that the functional version of Fang's scheme has polynomial precision.

A-patches are a form of representation of an algebraic curve or surface over a simplex. The A-patch conditions can be used as the basis for an adaptive subdivision style marching tetrahedra algorithm whose advantage is that it guarantees that we do not miss features of the algebraic: singularities are localized, and in regions around nearby multiple sheets, the subdivision process continues until the sheets are separated.Unfortunately, the A-patch single sheet conditions are too strict: for some algebraics, the subdivision process converges slowly or fails to converge. In this paper, I give an additional single sheet condition for curves that allows for convergence of this process. I also give additional conditions for surfaces that trades off some of the single sheet guarantees for improved convergence.

A piecewise quintic interpolation scheme with approximate G1 continuity is presented. For a given triangular mesh of arbitrary topology, one quintic triangular Bézier patch is constructed for each data triangle. Although the resulting surface has G1 continuity at the data vertices, we only require approximate G1 continuity along the patch boundaries so as to lower the patch degree. To reduce the normal discontinuity along boundaries, neighbouring patches are adjusted to have identical normals at the middle point of their common boundary. In most cases, the surfaces generated by this scheme have the same level of visual smoothness compared to an existing sextic G1 continuous interpolation scheme. Further, using the new boundary construction method presented in this paper, better shape quality is observed for sparse data sets than the surfaces of the original G1 continuous scheme, upon which the new scheme is based.

In this paper, we present a piecewise cubic, parametric surface scheme to interpolate positions and normals on a triangulated data set. For each data triangle, we fit three triangular cubic patches in a Clough-Tocher like arrangement. However, while we construct the micro-patches to meet each other C1, we only require approximate G1 continuity across macro-patches boundaries. To control the normal discontinuity on the macro-patch boundaries, neighbouring patches are constructed to interpolate the position and normals at the ends of their common boundary, as well as to have equal normals at additional points on the boundary. The resulting scheme constructs patches with similar shape to the quartic Shirman-Séquin construction, and has better shape than Peters' G1 cubic scheme on near singular data.

We introduce symSpline: a symmetric, dual-mouse technique for the manipulation of spline curves. In symSpline, two cursors control the positions of the ends of the tangent to an edit point. By moving the tangent with both mice, the tangent and the edit point can be translated while the curvature of the spline is adjusted simultaneously, according to the length and angle of the tangent. We compare the symSpline technique to two asymmetric dual-mouse spline manipulation techniques and to a standard single-mouse technique. In a spline matching experiment, symSpline outperformed the two asymmetric dual-mouse techniques and all three dual-mouse techniques proved to be faster than the single-mouse technique. Additionally, symSpline was the technique most preferred by test participants.

A high efficiency and linearity PA solution is presented based on hybrid linearisation schemes that employ complementary techniques to overcome the weakness (or constraints) of either method on its own. In particular, the high linearity of Cartesian loop is combined with the high efficiency of envelope elimination and restoration (EER). The results of a prototype hybrid linearised PA built using digital generation of the phase and amplitude signals for the EER amplifier, and utilizing a digital Cartesian feedback loop are presented. Limitations of the amplitude modulator on system performance are demonstrated using PWM and delta-modulated switching amplifiers.

Critical points of a vector field are key to their characterization. Their positions as well as their indexes are crucial for understanding vector fields. Considerable work exists in 2D, but less is available for 3D or higher dimensions. Geometric algebra is a derivative of Clifford algebra that not only enables a succinct definition of the index of a critical point in higher dimension; it also provides insight and computational pathways for calculating the index. We describe the problems in terms of geometric algebra and present an octree based solution using the algebra for finding critical points and their index in a 3D vector field.

A runtime analysis is made of a previously published algorithm for polynomial composition. The relationship between this composition algorithm and Sablonnie`re's algorithm is explored. This composition algorithm is then made optimal aby first performing a change of basis.

In view of the fundamental role that functional composition plays in mathematics, it is not surprising that a variety of problems in geometric modeling can be viewed as instances of the following composition problem: given representations for two functions F and G , compute a representation of the function H = F o G . We examine this problem in detail for the case when F and G are given in either Be´zier or B-spline form. Blossoming techniques are used to gain theoretical insight into the structure of the solution which is then used to develop efficient, tightly codable algorithms. From a practical point of view, if the composition algorithms are implemented as library routines, a number of geometric-modeling problems can be solved with a small amount of additional software.

Generating accurate radiosity solutions of real world environments is user-intensive and requires significant knowledge of the method. As a result, few end-users such as architects and designers use it. The output of most commercial modeling packages must be substantially "cleaned up" to satisfy the geometrical and topological criteria imposed by radiosity solution algorithms. Furthermore, the mesh used as the basis of the radiosity computation must meet several additional requirements for the solution to be accurate.A set of geometrical and topological requirements is formalized that when satisfied yields an accurate radiosity solution. A series of algorithms is introduced that automatically processes raw model databases to meet these requirements. Thus, the end-user can concentrate on the design rather than on the details of the radiosity solution process. These algorithms are generally independent of the radiosity solution technique used, and thus apply to all mesh based radiosity methods.