Spectral Entropy and Adaptive, Lossy Source Coding

语音、静止图像、高质量音频和视频的高效数字表示，称为有损源压缩，在今天有许多商业应用。这些应用包括数字蜂窝电话、MP3播放器、dvd、高清电视、视频会议、互联网电话以及静态图像的传输/存储。在这些应用程序中，源压缩的最佳方法本质上是自适应的，并且基于一种称为非线性近似的技术。然而，这些压缩方法主要是基于实验设计的，没有任何指导理论。本文研究了一种基于谱熵的数学量的自适应有损源压缩方法。这种新的源压缩方法被称为谱熵（spectral entropy，缩写为SpEnt）方法，它为自适应源压缩提供了一种基本可靠的方法，这是迄今为止所缺少的。这项工作为语音、视频和静止图像开发了基于spentr的有损压缩方法，这些方法应该在许多商业产品中得到应用。目前最成功的有损源压缩方法是样本函数自适应编码器（也称为逐输入自适应或实现自适应）。在样本函数自适应编码器中，不仅在每个块或帧中传输的参数数量可能在块到块（帧到帧）之间变化，而且对于给定数量的传输参数，在每个块中传输的参数可能会变化。对于这样一组基函数固定的编码器，通常说发送最佳n个基函数对应的系数，而不是前n个，这在谐波分析中称为非线性逼近。坎贝尔在1960年推导出他称之为随机过程的系数率的量，他表明系数率取决于谱熵（原始过程的功率谱密度的熵）。没有证明编码定理，也没有说明系数率对源压缩的可能影响。PI和他的学生最近的工作产生了坎贝尔系数率的两个新的推导。一种推导方法允许系数率根据称为过程有效带宽的量来解释。另一种推导揭示了一种新的基于系数率的源压缩方法，该方法适应于源的每种实现。更具体地说，通过研究项乘积级数展开式中的主导项，证明了在一个特定系数的N个样本序列中，需要编码的系数样本数量与系数方差成正比。因此，一个特定的系数是否被编码是块与块之间变化的，因此，基于谱熵的有损压缩显然属于非线性近似方法的范畴。基于这些结果，本研究制定了一种新的有损源压缩方法，称为谱熵（SpEnt）方法，并开发了基于谱熵的编码器，用于宽带语音（50 Hz至7 kHz）、视频、电话带宽语音和静止图像的有损压缩。此外，本研究还考察了谱熵和坎贝尔系数率作为源实现序列自适应编码的基本量的作用。

The efficient digital representation of voice, still images, high quality audio, and video, called lossy source compression, has a host of commercial applications today. These applications include digital cellular telephones, MP3 players, DVDs, HDTV, videoconferencing, Internet telephony, and the transmission/storage of still images. The best approaches to source compression in these applications are adaptive in nature and are based upon a technique called nonlinear approximation. However, these compression methods have been designed primarily based on experiments without any guiding theory. This research investigates a new approach to adaptive lossy source compression based upon a mathematical quantity called the spectral entropy. This new approach to source compression, denoted as the SpEnt (spectral entropy) method, offers a fundamentally sound approach to adaptive source compression that has been missing heretofore. This work develops SpEnt-based lossy compression methods for speech, video, and still images that should find applications in many commercial products.The most successful methods for lossy source compression today are sample-function adaptive coders (also called input-by-input adaptive or realization-adaptive). In sample function adaptive coders, not only might the number of parameters transmitted in each block or frame vary from block-to-block (frame-to-frame), but for a given number of transmitted parameters, which parameters are transmitted in each block may vary. For such coders with a fixed set of basis functions, it is usually said that the coefficients corresponding to the best n basis functions are sent, rather than the first n, and this is called nonlinear approximation in harmonic analysis. Campbell derived the quantity that he called the coefficient rate of a random process in 1960, and he showed that the coefficient rate depends on the spectral entropy (the entropy of the power spectral density of the original process). No coding theorems were proved and no possible implications of coefficient rate for source compression were stated. Recent work by the PI and his students produced two new derivations of Campbell's coefficient rate. One derivation allows coefficient rate to be interpreted with respect to a quantity called the effective bandwidth of the process. The other derivation reveals a new approach to source compression based upon coefficient rate that adapts to each realization of the source. More specifically, by studying the dominant terms in the series expansion of the product of terms, it was shown that in a sequence of N samples of a particular coefficient, the number of coefficient samples that should be coded is proportional to the coefficient variances. Thus, whether a particular coefficient is being coded or not is changing from block-to-block, and thus, lossy compression based upon the spectral entropy clearly falls in the class of nonlinear approximation methods. Motivated by these results, this research formulates a new approach to lossy source compression, called the spectral entropy (SpEnt) method, and develops SpEnt based coders for lossy compression of wideband speech (50 Hz to 7 kHz), video, telephone bandwidth speech, and still images. Further, this work examines the role of spectral entropy and Campbell's coefficient rate as fundamental quantities in adaptive coding of a sequence of source realizations.