On the Global Geometry of Sphere-Constrained Sparse Blind Deconvolution
On the Global Geometry of Sphere-Constrained Sparse Blind Deconvolution
复制标题
DOI：
10.1109/tpami.2019.2939237
复制
发表时间：
2019-01
影响因子：
23.6
通讯作者：
Yuqian Zhang;Yenson Lau;Han-Wen Kuo;S. Cheung;A. Pasupathy;John Wright
中科院分区：
文献类型：
--
作者：
Yuqian Zhang;Yenson Lau;Han-Wen Kuo;S. Cheung;A. Pasupathy;John Wright
盲反卷积是从卷积 $\boldsymbol{y} = \boldsymbol{a}_0 \circledast \boldsymbol{x}_0$y=a0⊛x0 中恢复卷积核 $\boldsymbol{a}_0$a0 和激活信号 $\boldsymbol{x}_0$x0 的问题。如果没有进一步的约束或先验，这个问题是不适定的。本文研究了激活信号中非零条目稀疏且随机填充的情况。我们将卷积核归一化为单位 Frobenius 范数，并将稀疏盲反卷积问题转化为球面上的非凸优化问题。通过这种球形约束，在某些假设下，每个虚假局部最小值都接近于地面实况的某种有符号移位截断。这种良性特性激发了一种有效的两阶段算法，该算法从次优局部最小值提供的部分信息中恢复基本事实。这种受几何启发的算法可以恢复某些显微镜问题的基本事实，并且在更具挑战性的图像去模糊问题中也表现出有希望的性能。我们对全局几何和两阶段算法的见解扩展到卷积字典学习问题，其中观察到多个卷积信号的叠加。
Blind deconvolution is the problem of recovering a convolutional kernel $\boldsymbol{a}_0$a0 and an activation signal $\boldsymbol{x}_0$x0 from their convolution $\boldsymbol{y} = \boldsymbol{a}_0 \circledast \boldsymbol{x}_0$y=a0⊛x0. 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.