We propose a useful formulation for ill-posed inverse problems in Hilbert spaces with nonlinear clipping effects. Ill-posed inverse problems are often formulated as optimization problems, and nonlinear clipping effects may cause nonconvexity or nondifferentiability of the objective functions in the case of commonly used regularized least squares. To overcome these difficulties, we present a tractable formulation in which the objective function is convex and differentiable with respect to optimization variables, on the basis of the Bregman divergence associated with the primitive function of the clipping function. By using this formulation in combination with the representer theorem, we need only to deal with a finite-dimensional, convex, and differentiable optimization problem, which can be solved by well-established algorithms. We also show two practical examples of inverse problems where our theory can be applied, estimation of band-limited signals and time-harmonic acoustic fields, and evaluate the validity of our theory by numerical simulations.

A new real-time sound field transmission system is presented. To construct this system, a large listening area needs to be reproduced at not less than a constant height. Additionally, the driving signals of the loudspeakers should be obtained only from received signals of microphones. Wave field reconstruction (WFR) filtering for linear arrays of microphones and loudspeakers is considered to be suitable for this kind of system. An experimental system was developed to show the feasibility of real-time sound field transmission using the WFR filter. Experiments to measure the reproduced sound field and a subjective listening test of sound localization were conducted to evaluate the proposed system. Although the reproduced sound field included several artifacts such as spatial aliasing and faster amplitude decay, the experimental results indicated that the proposed system was able to provide sound localization accuracy for virtual sound sources comparable to that for real sound sources in a large listening area.