This paper reviews how image formation or reconstruction in synthetic aperture radar (SAR) in two and three dimensions can be viewed as the inversion of the circular and spherical Radon transforms, respectively. The advantage of viewing image formation in this way is that it could be used in situations where more standard methods could fail, such as high squint and ultra-wideband SAR. The inversion formulae for the circular and spherical Radon transforms employ Hankel and Fourier transforms along with coordinate mappings. We show how the theory extends to include finite bandwidth signals and finite synthetic apertures.

The paper investigates the impact of SAR image enhancement on the performance of small target detection in SAR images. Three SAR image enhancement algorithms are evaluated on large SAR image data-sets. The evaluation results show that image enhancement can greatly improve the performance of false alarm mitigation, and that the level of performance improvement is correlated with the resolution and background suppression of the enhanced images. The higher the resolution and the level of background suppression, the higher the level of performance improvement.

This paper investigates the impact of SAR image enhancement on the performance of small target detection in SAR images. Three SAR image enhancement algorithms are evaluated on large SAR image data-sets. The evaluation results show that image enhancement can greatly improve the performance of false alarm mitigation, and that the level of performance improvement is correlated with the resolution and background suppression of the enhanced images. The higher the resolution and the level of background suppression, the higher the level of performance improvement.

Implicit polynomials (i.e., multinomials) have a number of properties that make them attractive for modeling curves and surfaces in computer vision. The paper considers the problem of finding the best fitting implicit polynomial (or algebraic curve) to a collection of points in the plane using an orthogonal distance metric. Approximate methods for orthogonal distance regression have been shown by others to be prone to the problem of cusps in the solution and this is confirmed here. Consequently, this work focuses on exact methods for orthogonal distance regression. The most difficult and costly part of exact methods is computing the closest point on the algebraic curve to an arbitrary point in the plane. The paper considers three methods for achieving this in detail. The first is the standard Newton's method, the second is based on resultants which are making a resurgence in computer graphics, and the third is a novel technique based on successive circular approximations to the curve. It is shown that Newton's method is the quickest, but that it can fail sometimes even with a good initial guess. The successive circular approximation algorithm is not as fast, but is robust. The resultant method is the slowest of the three, but does not require an initial guess. The driving application of this work was the fitting of implicit quartics in two variables to thinned oblique ionogram traces.

Orthogonal distance regression (ODR) is widely used to fit a curve through scattered points in the plane. It generally produces good fits, but intuition and practice suggest it can give biased results when fitting closed convex curves or corners. ODR gives the maximum likelihood estimate of the best curve under a straightforward stochastic model for data generation. The paper proposes an extension of this model to include a reasonable partial prior for the distribution of samples along the curve. Although the associated optimisation problem for the maximum likelihood estimator under the extended model no longer has a simple form, it lends itself naturally to asymptotic approximation. The first order approximation is essentially ODR: the second order approximation has an extra contribution involving the local curvature. Finally the results are applied to the illustrative problem of fitting a circle through scattered data: ODR is shown to indeed give a biased estimate of the true radius, and this bias is reduced under the second order estimator.

This paper presents a method for extracting features from images using implicit quartics in two variables. These features have been particularly designed to be appropriate for the image understanding problem involving oblique ionograms, but they are more generally applicable. Attention is given to the problem of choosing a suitable fitting metric, and the paper presents a robust procedure (exact orthogonal distance regression) for achieving a good fit.

We present a new approach to analysing the continuous Gabor transform and two critically sampled discretizations of it: the periodic finite-discrete and non-periodic finite-discrete versions. In particular, we distinguish between the analysis and synthesis forms of the transform, and introduce an intermediate operation that decomposes both transforms to collections of independent Toeplitz operators. In the continuous and the periodic finite-discrete case this decomposition allows us to show that for appropriate window functions the analysis and synthesis transforms are inverses of each other. In the nonperiodic finite-discrete case this relation no longer holds, but we are still able to use the decomposition and results on Toeplitz matrices to show that both transforms and their inverses are computable in O(PlogP) operations (after a setup cost of O(Plog/sup 2/P)) for P discrete samples. Moreover the decomposition also facilitates analysis of the conditioning of finite versions of the transform.>