Iterative algorithms for deblurring and deconvolution with constraints
Iterative algorithms for deblurring and deconvolution with constraints
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DOI：
10.1088/0266-5611/14/6/006
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发表时间：
1998-12-01
期刊：
影响因子：
2.1
通讯作者：
Byrne, C
中科院分区：
文献类型：
--
作者：
Byrne, C
去模糊问题是从（噪声）值积分h（s，t）c（t）dr = d（s）恢复函数c = c（t）的问题。该问题的离散有限版本是求解线性方程组He = d for c，其中H是矩阵，d和c是向量。当核Iz（s，t）是差（s-t）的函数时，去模糊问题变成去卷积问题。对于非负核函数h的情况，Snyder等人已经提出了使用迭代算法来实现受c上的非负约束的去模糊。在本文中，我们扩展这些算法，包括所需的解决方案的条目上的上限和下限。我们表明，任何涉及一个真实的内核h的线性去模糊问题可以转化为一个涉及非负内核的线性去模糊问题。因此，我们的算法适用于一般的去模糊和反卷积问题。这些算法收敛于方程组y = Pr的解，其中P = [P-ij] P-ij大于或等于0，其中i = 1，.，I，j = 1，...，J，满足向量不等式a小于或等于x小于或等于B，只要存在这样的解。当没有满足约束条件的解时，同时版本收敛到一个近似解，该近似解最小化与Kullback-Leibler交叉熵和Fermi-Dirac广义熵相关的成本函数。
The deblurring problem is that of recovering the function c = c(t) from (noisy) values integral h(s, t)c(t) dr = d(s). The discrete finite version of the problem is to solve the system of linear equations He = d for c, where H is a matrix and d and c are vectors. When the kernel Iz(s, t) is a function of the difference (s - t), the deblurring problem becomes a deconvolution problem. The use of iterative algorithms to effect deblurring subject to non-negativity constraints on c has been presented by Snyder er al for the case of non-negative kernel function h. In this paper we extend these algorithms to include upper and lower bounds on the entries of the desired solution. We show that any linear deblurring problem involving a real kernel h can be transformed into a linear deblurring problem involving a non-negative kernel. Therefore our algorithms apply to general deblurring and deconvolution problems. These algorithms converge to a solution of the system of equations y = Pr, with P = [P-ij] P-ij greater than or equal to 0 for i = 1,..., I, j = 1,..., J, satisfying the vector inequalities a less than or equal to x less than or equal to b, whenever such a solution exists. When there is no solution satisfying the constraints the simultaneous versions converge to an approximate solution that minimizes a cost function related to the Kullback-Leibler cross-entropy and the Fermi-Dirac generalized entropy.