data | global | local | total |
orthogonal | 139 | 21 | 160 |
natural | 137 | 24 | 161 |
cartoon | 91 | 15 | 106 |
seismic | 95 | 16 | 111 |
ultrasound | 95 | 18 | 113 |
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 $ \mathit{\boldsymbol{{A}}}\in \mathbb{C}^{M\times N} $. These matrices are of the form $ S_{\mathit{\boldsymbol{{\lambda}}}}(\mathit{\boldsymbol{{u}}}\mathit{\boldsymbol{{v}}}^*) $ where $ \mathit{\boldsymbol{{u}}}\in \mathbb{C}^M $, $ \mathit{\boldsymbol{{v}}}\in \mathbb{C}^N $ and $ \mathit{\boldsymbol{{\lambda}}}\in \mathbb{Z}^N $. The operator $ S_{\mathit{\boldsymbol{{\lambda}}}} $ circularly shifts the $ k $-th column of $ \mathit{\boldsymbol{{u}}}\mathit{\boldsymbol{{v}}}^* $ by $ \lambda_k $.
These kind of shifts naturally appear in applications, where an object $ \mathit{\boldsymbol{{u}}} $ is observed in $ N $ measurements at different positions indicated by the shift $ \mathit{\boldsymbol{{\lambda}}} $. The vector $ \mathit{\boldsymbol{{v}}} $ gives the observation intensity. This model holds for seismic waves that are recorded at $ N $ sensors at different times $ \mathit{\boldsymbol{{\lambda}}} $. Other examples are a car that moves through a video changing its position $ \mathit{\boldsymbol{{\lambda}}} $ in each of the $ N $ frames, or non-destructive testing based on ultrasonic waves that are reflected by defects inside the material.
The main difficulty of the above stated problem lies in finding a suitable shift vector $ \mathit{\boldsymbol{{\lambda}}} $. Once the shift is known, a simple singular value decomposition can be applied to reconstruct $ \mathit{\boldsymbol{{u}}} $ and $ \mathit{\boldsymbol{{v}}} $. We propose a greedy method to reconstruct $ \mathit{\boldsymbol{{\lambda}}} $. By using the formulation of the problem in Fourier domain, a shifted rank-$ 1 $ approximation can be calculated in $ O(NM\log M) $. Convergence to a locally optimal solution is guaranteed. Furthermore, we give a heuristic initial guess strategy that shows good results in the numerical experiments.
We validate our approach in several numerical experiments on different kinds of data. We compare the technique to shift-invariant dictionary learning algorithms. Furthermore, we provide examples from application including object segmentation in non-destructive testing and seismic exploration as well as object tracking in video processing.
Citation: |
Table 1. Mean number of iterations for different kinds of input data
data | global | local | total |
orthogonal | 139 | 21 | 160 |
natural | 137 | 24 | 161 |
cartoon | 91 | 15 | 106 |
seismic | 95 | 16 | 111 |
ultrasound | 95 | 18 | 113 |
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