The contourlet transform: An efficient directional multiresolution image representation
The contourlet transform: An efficient directional multiresolution image representation
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DOI：
10.1109/tip.2005.859376
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发表时间：
2005-12-01
影响因子：
10.6
通讯作者：
Vetterli, M
中科院分区：
文献类型：
--
作者：
Do, MN;Vetterli, M
通常使用的一维变换的可分离扩展，例如傅立叶变换和小波变换，在捕捉图像边缘的几何形状方面的局限性是众所周知的。在这篇文章中，我们追求一种“真正的”二维变换，它可以捕捉到视觉信息中关键的内在几何结构。探索图像中的几何图形的主要挑战来自数据的离散性质。因此，与曲线波等其他方法不同的是，我们的方法首先在连续域中进行变换，然后对采样数据进行离散化，而我们的方法从离散域结构开始，然后研究其收敛到连续域中的展开。具体地说，我们使用不可分离的滤波器组来构造离散域的多分辨率和多方向展开式，这与小波从滤波器组得到的方式非常相似。这种构造利用轮廓段实现了灵活的多分辨率、局部和方向性的图像扩展，因此，它被称为轮廓波变换。离散轮廓波变换具有快速迭代滤波器组算法，该算法需要对N像素图像进行N阶运算。此外，我们通过一个定向多分辨率分析框架在所开发的滤波器组和相关的连续域轮廓波展开之间建立了精确的联系。我们证明了在抛物线标度和足够的方向消失矩的情况下，Contourlet对沿两次连续可微曲线具有间断的分段光滑函数获得了最优的逼近速度。最后，我们给出了一些数值实验，展示了轮廓波在几种图像处理应用中的潜力。
The limitations of commonly used separable extensions of one-dimensional transforms, such as the Fourier and wavelet transforms, in capturing the geometry of image edges are well known. In this paper, we pursue a "true" two-dimensional transform that can capture the intrinsic geometrical structure that is key in visual information. The main challenge in exploring geometry in images comes from the discrete nature of the data. Thus, unlike other approaches, such as curvelets, that first develop a transform in the continuous domain and then discretize for sampled data, our approach starts with a discrete-domain construction and then studies its convergence to an expansion in the continuous domain. Specifically, we construct a discrete-domain multiresolution and multidirection expansion using nonseparable filter banks, in much the same way that wavelets were derived from filter banks. This construction results in a flexible multiresolution, local, and directional image expansion using contour segments, and, thus, it is named the contourlet transform. The discrete contourlet transform has a fast iterated filter bank algorithm that requires an order N operations for N-pixel images. Furthermore, we establish a precise link between the developed filter bank and the associated continuous-domain contourlet expansion via a directional multiresolution analysis framework. We show that with parabolic scaling and sufficient directional vanishing moments, contourlets achieve the optimal approximation rate for piecewise smooth functions with discontinuities along twice continuously differentiable curves. Finally, we show some numerical experiments demonstrating the potential of contourlets in several image processing applications.