Deformable kernels for early vision

Early vision algorithms often have a first stage of linear filtering that 'extracts' from the image information at multiple scales of resolution and multiple orientations. A common difficulty in the design and implementation of such schemes is that one feels compelled to discretize coarsely the space of scales and orientations in order to reduce computation and storage costs. This discretization produces anisotropies due to a loss of traslation-, rotation- scaling- invariance that makes early vision algorithms less precise and more difficult to design. This need not be so: one can compute and store efficiently the response of families of linear filters defined on a continuum of orientations and scales. A technique is presented that allows (1) to compute the best approximation of a given family using linear combinations of a small number of 'basis' functions; (2) to describe all finite-dimensional families, i.e. the families of filters for which a finite dimensional representation is possible with no error. The technique is general and can be applied to generating filters in arbitrary dimensions. Experimental results are presented that demonstrate the applicability of the technique to generating multi-orientation multi-scale 20 edge-detection kernels. The implementation issues are also discussed

Deformable kernels for early vision

Early vision algorithms often have a first stage of linear-filtering that `extracts' from the image information at multiple scales of resolution and multiple orientations. A common difficulty in the design and implementation of such schemes is that one feels compelled to discretize coarsely the space of scales and orientations in order to reduce computation and storage costs. A technique is presented that allows: 1) computing the best approximation of a given family using linear combinations of a small number of `basis' functions; and 2) describing all finite-dimensional families, i.e., the families of filters for which a finite dimensional representation is possible with no error. The technique is based on singular value decomposition and may be applied to generating filters in arbitrary dimensions and subject to arbitrary deformations. The relevant functional analysis results are reviewed and precise conditions for the decomposition to be feasible are stated. Experimental results are presented that demonstrate the applicability of the technique to generating multiorientation multi-scale 2D edge-detection kernels. The implementation issues are also discussed

A high-order Nystrom discretization scheme for boundary integral equations defined on rotationally symmetric surfaces

A scheme for rapidly and accurately computing solutions to boundary integral equations (BIEs) on rotationally symmetric surfaces in R^3 is presented. The scheme uses the Fourier transform to reduce the original BIE defined on a surface to a sequence of BIEs defined on a generating curve for the surface. It can handle loads that are not necessarily rotationally symmetric. Nystrom discretization is used to discretize the BIEs on the generating curve. The quadrature is a high-order Gaussian rule that is modified near the diagonal to retain high-order accuracy for singular kernels. The reduction in dimensionality, along with the use of high-order accurate quadratures, leads to small linear systems that can be inverted directly via, e.g., Gaussian elimination. This makes the scheme particularly fast in environments involving multiple right hand sides. It is demonstrated that for BIEs associated with the Laplace and Helmholtz equations, the kernel in the reduced equations can be evaluated very rapidly by exploiting recursion relations for Legendre functions. Numerical examples illustrate the performance of the scheme; in particular, it is demonstrated that for a BIE associated with Laplace's equation on a surface discretized using 320,800 points, the set-up phase of the algorithm takes 1 minute on a standard laptop, and then solves can be executed in 0.5 seconds.Comment: arXiv admin note: substantial text overlap with arXiv:1012.56301002.200

A fast and exact ww-stacking and ww-projection hybrid algorithm for wide-field interferometric imaging

The standard wide-field imaging technique, the $w$-projection, allows correction for wide-fields of view for non-coplanar radio interferometric arrays. However, calculating exact corrections for each measurement has not been possible due to the amount of computation required at high resolution and with the large number of visibilities from current interferometers. The required accuracy and computational cost of these corrections is one of the largest unsolved challenges facing next generation radio interferometers such as the Square Kilometre Array. We show that the same calculation can be performed with a radially symmetric $w$-projection kernel, where we use one dimensional adaptive quadrature to calculate the resulting Hankel transform, decreasing the computation required for kernel generation by several orders of magnitude, whilst preserving the accuracy. We confirm that the radial $w$-projection kernel is accurate to approximately 1% by imaging the zero-spacing with an added $w$-term. We demonstrate the potential of our radially symmetric $w$-projection kernel via sparse image reconstruction, using the software package PURIFY. We develop a distributed $w$-stacking and $w$-projection hybrid algorithm. We apply this algorithm to individually correct for non-coplanar effects in 17.5 million visibilities over a $25$ by $25$ degree field of view MWA observation for image reconstruction. Such a level of accuracy and scalability is not possible with standard $w$-projection kernel generation methods. This demonstrates that we can scale to a large number of measurements with large image sizes whilst still maintaining both speed and accuracy.Comment: 9 Figures, 19 Pages. Accepted to Ap