Narrow Results

We introduce a general framework for consistent linear reconstruction in infinite-dimensional Hilbert spaces. We study stable reconstructions in terms of Riesz bases and frames, and generalize the notion of oblique dual frames to infinte-dimensional frames. As we show, the linear reconstruction scheme coincides with the so-called oblique projection, which turns into an ordinary orthogonal projection when adapting the inner product. The inner product of interest is, in general, not unique. We characterize the inner products and the corresponding positive operators for which this geometrical interpretation applies.

We introduce a general framework for consistent linear reconstruction in infinite-dimensional Hilbert spaces. We study stable reconstructions in terms of Riesz bases and frames, and generalize the notion of oblique dual frames to infinite-dimensional frames. As we show, the linear reconstruction scheme coincides with the so-called oblique projection, which turns into an ordinary orthogonal projection when adapting the inner product. The inner product of interest is, in general, not unique. We characterize the inner products and corresponding positive operators for which the new geometrical interpretation applies.

A short review of mathematical methods of image analysis is provided. Mathematical models of shapes of signals and images are formulated, and mathematical methods and numerical algorithms for calculating characteristics of closeness (connectedness and dependence) of two signal or image shapes using oblique projection technique are considered. Examples of using morphological image analysis to solve applied problems are given. In addition, elements of mathematical formalism of subjective modeling of categories of fuzziness and uncertainty that reflect incompleteness and unreliability of information used in construction of a subjective morphological model to take into account researcher’s subjective notions. As their applications, the mathematical foundations of morphological image analysis, oblique projection approach and analysis of dependency of relative image shapes and connectedness of their absolute shapes are shown to have an interpretation in terms of subjective modeling, with examples of subjective morphological models and optimization of decisions based on them using oblique projection and subjective optimization. It is shown that optimal solutions of morphological problems of image analysis and image interpretation can be formulated in terms of oblique image projection and subjective optimization.

Traditional beamforming algorithms have relatively poor performance in conditions of small snapshot numbers, high signal-to-noise-ratio and coherent sources. To address this issue, an oblique projection-based beamforming algorithm is modified in this paper. In this algorithm, the oblique projector is utilized to eliminate the noise of the array input data and enhance the robustness of algorithm. Furthermore, the transformation-based linear constraint matrix may eliminate the interferences, and Chebyshev window function is utilized to suppress the side-lobe level. Simulation results demonstrate that the modified algorithm has good robustness and performance.