Enhanced image approximation using shifted rank-1 reconstruction
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Low rank approximation has been extensively studied in the past. It is most suitable to reproduce rectangular like structures in the data. In this work we introduce a generalization using shifted rank-1 matrices to approximate A∈C^M× N. These matrices are of the form S_λ(uv^*) where u∈C^M, v∈C^N and λ∈Z^N.The operator S_λ circularly shifts the k-th column of uv^* by λ_k. These kind of shifts naturally appear in applications, where an object u is observed in N measurements at different positions indicated by the shift λ. The vector v gives the observation intensity. Exemplary, a seismic wave can be recorded at N sensors with different time of arrival λ; Or a car moves through a video changing its position in every frame. We present theoretical results as well as an efficient algorithm to calculate a shifted rank-1 approximation in O(NM M). The benefit of the proposed method is demonstrated in numerical experiments. A comparison to other sparse approximation methods is given. Finally, we illustrate the utility of the extracted parameters for direct information extraction in several applications including video processing or non-destructive testing.
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