Supremal multiscale signal analysis
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We introduce a novel approach to nonlinear signal analysis, which is referred to as supremal multiscale analysis. The proposed approach provides a rigorous mathematical foundation for a class of nonlinear multiscale signal analysis schemes and leads to a decomposition that can effectively be used in signal processing and analysis. Moreover, it is related to the supremal scale-spaces proposed by Heijmans and van den Boomgaard and is similar in flavor to the well-known linear multiresolution theory of Mallat and Meyer. In this framework, linear concepts such as vector spaces, projections, and linear operators are replaced by conceptually analogous nonlinear notions. We use supremal multiscale analysis to construct a multiscale image decomposition scheme based on two mathematical concepts that play a key role in the analysis and interpretation of images by vision systems, namely, regional maxima and connectivity. The resulting scheme is referred to as skyline supremal multiscale analysis and satisfies several useful properties desired by any multiscale image analysis tool. It is grayscale invariant, as well as translation and scale invariant. Moreover, it progressively removes connected components from the level sets of an image without introducing new ones. But, most importantly, it decomposes the regional maxima of an image in a natural causal hierarchy by gradually removing these maxima without introducing new ones. Image decomposition by skyline supremal multiscale analysis can be used to construct nonlinear tools for image processing and analysis that provide solutions to problems where traditional linear techniques are ineffective. We discuss one such tool and illustrate its use in object-based extraction and denoising. 2004 Society for Industrial and Applied Mathematics.
