The blind image deconvolution is a challenging, highly ill-posed nonlinear inverse problem. We introduce a Multiscale Hierarchical Decomposition Method (MHDM) that is iteratively solving variational problems with adaptive data and regularization parameters, towards obtaining finer and finer details of the unknown kernel and image. We establish convergence of the residual in the noise-free data case, and then in the noisy data case when the algorithm is stopped early by means of a discrepancy principle. Fractional Sobolev norms are employed as regularizers for both kernel and image, with the advantage of computing the minimizers explicitly in a pointwise manner. In order to break the notorious symmetry occurring during each minimization step, we enforce a positivity constraint on the Fourier transform of the kernels. Numerical comparisons with a single-step variational method and a non-blind MHDM show that our approach produces comparable results, while less laborious parameter tuning is necessary at the price of more computations. Additionally, the scale decomposition of both reconstructed kernel and image provides a meaningful interpretation of the involved iteration steps.

翻译：盲图像去卷积是一个具有挑战性的、高度不适定的非线性逆问题。本文提出一种多尺度层次分解方法（MHDM），该方法通过迭代求解具有自适应数据和正则化参数的变分问题，以逐步获取未知核与图像的更精细细节。我们在无噪声数据情况下建立了残差的收敛性，进而在有噪声数据情况下，通过差异原则提前终止算法时证明了收敛性。采用分数阶Sobolev范数作为核与图像的正则项，其优势在于能以逐点方式显式计算极小化子。为打破每次最小化步骤中出现的对称性难题，我们对核的傅里叶变换施加了非负约束。与单步变分方法及非盲MHDM的数值比较表明，本方法在减少参数调优工作量的同时获得了可比的结果，但需要更多的计算量。此外，重建核与图像的尺度分解为迭代步骤提供了有意义的解释。