Department of Electrical and Computer Engineering, Rice University, Houston, TX 77005, USA.
IEEE Trans Image Process. 2012 Feb;21(2):588-600. doi: 10.1109/TIP.2011.2165551. Epub 2011 Aug 22.
The inefficiency of separable wavelets in representing smooth edges has led to a great interest in the study of new 2-D transformations. The most popular criterion for analyzing these transformations is the approximation power. Transformations with near-optimal approximation power are useful in many applications such as denoising and enhancement. However, they are not necessarily good for compression. Therefore, most of the nearly optimal transformations such as curvelets and contourlets have not found any application in image compression yet. One of the most promising schemes for image compression is the elegant idea of directional wavelets (DIWs). While these algorithms outperform the state-of-the-art image coders in practice, our theoretical understanding of them is very limited. In this paper, we adopt the notion of rate-distortion and calculate the performance of the DIW on a class of edge-like images. Our theoretical analysis shows that if the edges are not "sharp," the DIW will compress them more efficiently than the separable wavelets. It also demonstrates the inefficiency of the quadtree partitioning that is often used with the DIW. To solve this issue, we propose a new partitioning scheme called megaquad partitioning. Our simulation results on real-world images confirm the benefits of the proposed partitioning algorithm, promised by our theoretical analysis.
可分离小波在表示平滑边缘方面的效率低下，导致人们对新的二维变换的研究产生了极大的兴趣。分析这些变换的最流行的标准是逼近能力。具有接近最优逼近能力的变换在去噪、增强等许多应用中非常有用。然而，它们对于压缩并不一定是好的。因此，大多数近乎最优的变换，如曲线波和轮廓波，尚未在图像压缩中找到任何应用。图像压缩最有前途的方案之一是方向小波（DIW）的优雅思想。虽然这些算法在实践中优于最先进的图像编码器，但我们对它们的理论理解非常有限。在本文中，我们采用率失真的概念，并计算 DIW 在一类边缘图像上的性能。我们的理论分析表明，如果边缘不“尖锐”，那么 DIW 将比可分离小波更有效地压缩它们。它还证明了 DIW 中常用的四叉树分区的效率低下。为了解决这个问题，我们提出了一种称为 megaquad 分区的新分区方案。我们在真实图像上的仿真结果证实了我们的理论分析所承诺的所提出的分区算法的好处。
