The Research on Nonlinear Data Compression and Reconstruction Based on MRA

Wavelet transform is a power-balance linear transform, which can decompose a signal into space and scale level, i.e. time and frequency level .At the same time, it makes the best of different resolution, solving the conflict between time field and frequency field. When met the high-frequency signal, WT changes into wide window using high resolution in low frequency field and low resolution in high frequency field, and when met the low-frequency signal, the situation is converse. Thus it provides the foundation of MRA and is widely used in the digital signal processing field. As an important aspect in signal processing, signal compression needs to express the source signal using data as little as possible to shrink the storage space, transferring time and occupied width. This thesis aims at combining the idea of MR, which comes from MRA in wavelet theory and digital signal compression application. As to the digital curve signals, this thesis designs a nonlinear algorithm to compress them. In the algorithm, interpolation forecasting method is used under the frame of MR. Then an example is presented using the algorithm to compress the perfect smooth and closed curve signal, numerical experiment operated in MATLAB shows the compress-ratio and result is satisfactory. At the same time, because of the character of MR, it is easy to choose the appropriate level of compression in consideration of the request of compression-ratio and losing constriction. So, if set a control in the choosing procedure, this algorithm becomes self-adjust and optimal. Furthermore, in the numerical experiment, the compression using wavelet is also compared with the nonlinear method presented in this thesis. And the results show that this method is efficient and operable.

Alternate abstract:

小波变换（wavelet transform WT）是一个满足能量守恒关系的线性变换，它能够将一个信号分解成具有频带特性的细节信号，巧妙地利用了非均匀分布的分辨率，较好地解决了时间和频率分辨率的矛盾：在低频段采用扁平的时频窗；在高频段采用窄长的时频窗，多分辨分析（MRA）建立了统一的时频分析方法。小波变换克服了FT的不足，因此在数字信号领域得到了广泛的应用。数据压缩作为数字信号处理的一个重要方面，要求以最少的数码表示信源所发出的信号，减少数据的存储容量，传输时间和占有带宽，即要设法压缩给定信息集合所占用的空间域，时间域和频率域资源。本文就是基于将小波变换中多分辨分析框架和数字信号压缩应用相结合的思想，针对空间中的曲线信号，应用多分辨分析框架中的非线性算法对之进行压缩处理，其中非线性算法选择的是数值分析中常用的插值预测方法，离散数据的多分辨分析框架是小波变换中的多分辨分析理论更一般的表达形式。然后针对理想状态下一维光滑信号和闭合曲线信号设计数值实验，数值实验的结果表明，该算法对于曲线信号有较好的压缩效果。同时由于多分辨分析框架的性质，数据分层压缩可以根据对于压缩比和失真度的要求，选择最合适的压缩层数和压缩方法，其中分解层数选择的过程中同时可以进行最优化处理，因此该方法也可以构成一种自适应算法。 在数值实验中，针对一维曲线信号应用本文提出的压缩方法和小波压缩结果比较，可以看到该算法具有更好的压缩效果。在针对有闭合曲线信号研究中我们应用了极坐标变换对之进行预处理，然后沿用针对一维的可用单值函数拟合的信号的算法进行压缩。实验结果证明该算法具有一定的实用性，可操作性。