Multiresolution Monogenic Signal Analysis Using the Riesz–Laplace Wavelet Transform
Multiresolution Monogenic Signal Analysis Using the Riesz–Laplace Wavelet Transform
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DOI：
10.1109/tip.2009.2027628
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发表时间：
2009-11
影响因子：
10.6
通讯作者：
M. Unser;D. Sage;D. Ville
中科院分区：
文献类型：
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作者：
M. Unser;D. Sage;D. Ville
单基因信号是一维解析信号的自然二维对应物。我们建议通过考虑Riesz变换的复化版本将这个概念转置到小波域，该变换具有将L2（R2）的实值（初级）小波基映射到复小波基的显著性质。Riesz算子在某种意义上也是可操纵的，因为它可以得到信号沿任何方向的希尔伯特变换。在建立了这些基础之后，我们指定了L2（R2）的一个主要的多谐样条小波基，它涉及一个类似墨西哥帽的母小波（b样条的拉普拉斯）。重要的一点是，我们的初级小波是准各向同性的：它们的行为就像分数阶拉普拉斯算子的多尺度版本，这保证了它们的可操作性。我们建议将这些实值基函数与它们的复Riesz对应物配对，以指定多分辨率单基因信号分析。这产生了一个表示，其中每个小波指数与一个局部方向，幅度和相位相关联。给出了一种相应的小波域方法来估计潜在的瞬时频率。我们还提供了一种改善小波分解的移位和旋转不变性的机制，并展示了如何使用完美重构滤波器组有效地实现变换。我们说明了表征的具体特征提取能力，并提出了小波域处理的新例子；特别是，一个鲁棒的，基于张量的定向图像模式分析，干涉图的解调，和数字全息图的重建。
The monogenic signal is the natural 2D counterpart of the 1D analytic signal. We propose to transpose the concept to the wavelet domain by considering a complexified version of the Riesz transform which has the remarkable property of mapping a real-valued (primary) wavelet basis of L2(R2) into a complex one. The Riesz operator is also steerable in the sense that it give access to the Hilbert transform of the signal along any orientation. Having set those foundations, we specify a primary polyharmonic spline wavelet basis of L2(R2) that involves a single Mexican-hat-like mother wavelet (Laplacian of a B-spline). The important point is that our primary wavelets are quasi-isotropic: they behave like multiscale versions of the fractional Laplace operator from which they are derived, which ensures steerability. We propose to pair these real-valued basis functions with their complex Riesz counterparts to specify a multiresolution monogenic signal analysis. This yields a representation where each wavelet index is associated with a local orientation, an amplitude and a phase. We give a corresponding wavelet-domain method for estimating the underlying instantaneous frequency. We also provide a mechanism for improving the shift and rotation-invariance of the wavelet decomposition and show how to implement the transform efficiently using perfect-reconstruction filterbanks. We illustrate the specific feature-extraction capabilities of the representation and present novel examples of wavelet-domain processing; in particular, a robust, tensor-based analysis of directional image patterns, the demodulation of interferograms, and the reconstruction of digital holograms.