二代曲线波图像恢复模型及其算法

Image restoration model and algorithms with second-generation curvelets

假定图像属于Besov空间,可以利用小波系数和Besov范数的等价性来度量图像小波分解的稀疏性.但是,小波并不能十分有效地表示光滑函数的边界,而曲线波对于具有C2奇异性的光滑函数能够达到渐进最优的稀疏表示.基于上述分析,提出了新的二代曲线波的图像恢复模型,运用曲线型分解空间来刻画二代曲线波系数的稀疏性.此外,利用广义条件梯度法得到了迭代硬阈值算法,并给出了解的收敛性定理和停止准则.实验表明新算法比已有方法具有更高的信噪比和更好的主观视觉效果.

Suppose that an image belongs to Besov spaces, one can measure sparse decompositions on wavelet basis by the fact that the Besov norms have equivalent descriptions by means of wavelet coefficients. But as is well known, wavelets fail to very efficiently represent smooth objects with edges while curvelets provide an optimally sparse representation of objects with singularities along C2 edges. Based on the analysis above, a novel model using second-generation curvelets is proposed for restoration of the image. Especially, curvelets-type decomposition spaces are employed for characterizing the sparsity of second-generation curvelet coefficients. On the other hand, an iterative hard shrinkage algorithm is obtained by using the generalized conditional gradient method, as well as convergent theorems for solutions and stopping criterion. Finally, experiments show that the proposed algorithm produces better results in terms of both signal-to-noise ratio and subjective visual quality than methods available.