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Signal decomposition is a classical problem in signal processing, which aims to separate an observed signal into two or more components, each with its own property. Usually, each component is described by its own subspace or dictionary. Extensive research has been done for the case where the components are additive, but in real-world applications, the components are often non-additive. For example, an image may consist of a foreground object overlaid on a background, where each pixel either belongs to the foreground or the background. In such a situation, to separate signal components, we need to find a binary mask which shows the location of each component. Therefore, it requires solving a binary optimization problem. Since most of the binary optimization problems are intractable, we relax this problem to the approximated continuous problem and solve it by alternating optimization technique. We show the application of the proposed algorithm for three applications: separation of text from a background in images, separation of moving objects from a background undergoing global camera motion in videos, and separation of sinusoidal and spike components in 1-D signals. We demonstrate in each case that considering the non-additive nature of the problem can lead to a significant improvement.
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