Point source reconstruction principle of linear inverse problems
Description
Exact point source reconstruction for underdetermined linear inverse problems with a block-wise structure was studied. In a block-wise problem, elements of a source vector are partitioned into blocks. Accordingly, a leadfield matrix, which represents the forward observation process, is also partitioned into blocks. A point source is a source having only one nonzero block. An example of such a problem is current distribution estimation in electroencephalography and magnetoencephalography, where a source vector represents a vector field and a point source represents a single current dipole. In this study, the block-wise norm, a block-wise extension of the lp-norm, was defined as the family of cost functions of the inverse method. The main result is that a set of three conditions was found to be necessary and sufficient for block-wise norm minimization to ensure exact point source reconstruction for any leadfield matrix that admit such reconstruction. The block-wise norm that satisfies the conditions is the sum of the cost of all the observations of source blocks, or in other words, the block-wisely extended leadfield-weighted l1-norm. Additional results are that minimization of such a norm always provides block-wisely sparse solutions and that its solutions form cones in source space.
Journal
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- Inverse Problems
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Inverse Problems 26 (11), 115016-, 2010-10-15
IOP Publishing
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Details 詳細情報について
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- CRID
- 1361137046127986944
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- ISSN
- 13616420
- 02665611
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- Data Source
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- Crossref
- OpenAIRE