On the Global Geometry of Sphere-Constrained Sparse Blind Deconvolution
| Autor: | Sky C. Cheung, Yenson Lau, John Wright, Abhay Pasupathy, Han-Wen Kuo, Yuqian Zhang |
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| Rok vydání: | 2017 |
| Předmět: |
FOS: Computer and information sciences
Blind deconvolution Computer Science - Machine Learning Deblurring Optimization problem Computer Vision and Pattern Recognition (cs.CV) Computer Science - Computer Vision and Pattern Recognition Matrix norm Geometry 010103 numerical & computational mathematics 02 engineering and technology 01 natural sciences Machine Learning (cs.LG) Artificial Intelligence 0202 electrical engineering electronic engineering information engineering 0101 mathematics Image restoration Mathematics Ground truth Applied Mathematics 020206 networking & telecommunications Computational Theory and Mathematics Kernel (image processing) 020201 artificial intelligence & image processing Computer Vision and Pattern Recognition Deconvolution Software |
| Zdroj: | CVPR |
| DOI: | 10.1109/cvpr.2017.466 |
| Popis: | Blind deconvolution is the problem of recovering a convolutional kernel $\boldsymbol{a}_0$ a 0 and an activation signal $\boldsymbol{x}_0$ x 0 from their convolution $\boldsymbol{y} = \boldsymbol{a}_0 \circledast \boldsymbol{x}_0$ y = a 0 ⊛ x 0 . This problem is ill-posed without further constraints or priors. This paper studies the situation where the nonzero entries in the activation signal are sparsely and randomly populated. We normalize the convolution kernel to have unit Frobenius norm and cast the sparse blind deconvolution problem as a nonconvex optimization problem over the sphere. With this spherical constraint, every spurious local minimum turns out to be close to some signed shift truncation of the ground truth, under certain hypotheses. This benign property motivates an effective two stage algorithm that recovers the ground truth from the partial information offered by a suboptimal local minimum. This geometry-inspired algorithm recovers the ground truth for certain microscopy problems, also exhibits promising performance in the more challenging image deblurring problem. Our insights into the global geometry and the two stage algorithm extend to the convolutional dictionary learning problem, where a superposition of multiple convolution signals is observed. |
| Databáze: | OpenAIRE |
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