里斯变换和可控小波的衰减性质
Decay properties of Riesz transforms and steerable wavelets
摘要：
里斯变换是希尔伯特变换的自然多维扩展，由于其良好的数学性质，多年来一直是人们研究的对象。最近，里斯变换和它的变体已被用来构建复杂的小波和可控小波框架在更高的维度。然而，这种方法的另一面是小波的里斯变换通常具有缓慢的衰减。然而，人们可以克服这个问题，要求原始小波有足够的光滑度，衰减，消失的时刻。在本文中，我们推导出必要条件在这三个属性，保证了里斯变换及其变种的衰减，并作为一个应用程序，我们展示了如何流行的Simoncelli小波的衰减可以通过适当修改其傅里叶变换。通过将里斯变换应用于这些新小波，我们得到了具有快速衰减的可操纵框架。
英文摘要：
The Riesz transform is a natural multi-dimensional extension of the Hilbert transform, and it has been the object of study for many years due to its nice mathematical properties. More recently, the Riesz transform and its variants have been used to construct complex wavelets and steerable wavelet frames in higher dimensions. The flip side of this approach, however, is that the Riesz transform of a wavelet often has slow decay. One can nevertheless overcome this problem by requiring the original wavelet to have sufficient smoothness, decay, and vanishing moments. In this paper, we derive necessary conditions in terms of these three properties that guarantee the decay of the Riesz transform and its variants, and as an application, we show how the decay of the popular Simoncelli wavelets can be improved by appropriately modifying their Fourier transforms. By applying the Riesz transform to these new wavelets, we obtain steerable frames with rapid decay.
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