Fundamental Theory and Techniques for Nonlinear Signal Processing

在电信、视频技术和多媒体应用中，一些真正的非线性问题构成了一个真正的挑战，对于这些问题的解决，与线性情况不同，目前还没有最终的统一理论可用。这些非线性问题包括超大规模电子电路和系统(VLSI)中大的非线性失真的消除、噪声图像中对比度的非线性锐化、非线性信号和图像源的建模和识别、以及信号和图像的基于内容的非线性压缩和解压缩。当前项目的目标是开发一个严格而实用的框架，用于设计/综合非线性模拟、数字和混合模式滤波器，以解决上述类型的非线性问题。本项目采用的方法是基于由PI(R.J.P.deFigueiredo)、T.A.Dwyer和L.V.Zyla开发的广义Fock(GF)空间框架。GF空间是具有问题相关加权内积的离散或连续参数Volterra泛函级数的再生核Hilbert空间。在给定的特定应用问题的背景下，通过将期望的滤波器正交投影到由适当的GF空间中的内插、平滑和其他设计约束泛函的表示者所跨越的子空间中来获得最优的非线性滤波器结构。一些分析、综合和实施问题构成了调查的重点。

In telecommunications, video-technology, and multimedia applications, a real challenge is posed by some genuinely nonlinear problems for the solution of which, unlike in the linear case, a final unified theory is not yet available. Such nonlinear problems include cancellation of large nonlinear distortion in very-large-scale electronic circuits and systems (VLSI), nonlinear sharpening of contrast in noisy images, modeling and identification of nonlinear signal and image sources, and content-based nonlinear compression and decompression of signals and images. The goal of the current project is to develop a rigorous and yet practical framework for the design/synthesis of nonlinear analog, digital, and mixed mode filters for the solution of the above types of nonlinear problems. The approach followed in this project is based on an Generalized Fock (GF) space framework developed by the PI (R.J.P. deFigueiredo), T.A. Dwyer, and L.V. Zyla. A GF space is a reproducing kernel Hilbert space of discrete- or continuous- parameter Volterra functional series with a problem-dependent weighted inner product. In the context of a given application-specific problem, the optimal nonlinear filter structure is obtained by an orthogonal projection of the desired filter into the subspace spanned by the representers of interpolating, smoothing, and other design constraint functionals in an appropriate GF space. A number of analysis, synthesis, and implementation issues constitute the focus of the investigation.