Larkin K G, Bone D J, Oldfield M A
Canon Information Systems Research Australia, North Ryde, Sydney, NSW.
J Opt Soc Am A Opt Image Sci Vis. 2001 Aug;18(8):1862-70. doi: 10.1364/josaa.18.001862.
It is widely believed, in the areas of optics, image analysis, and visual perception, that the Hilbert transform does not extend naturally and isotropically beyond one dimension. In some areas of image analysis, this belief has restricted the application of the analytic signal concept to multiple dimensions. We show that, contrary to this view, there is a natural, isotropic, and elegant extension. We develop a novel two-dimensional transform in terms of two multiplicative operators: a spiral phase spectral (Fourier) operator and an orientational phase spatial operator. Combining the two operators results in a meaningful two-dimensional quadrature (or Hilbert) transform. The new transform is applied to the problem of closed fringe pattern demodulation in two dimensions, resulting in a direct solution. The new transform has connections with the Riesz transform of classical harmonic analysis. We consider these connections, as well as others such as the propagation of optical phase singularities and the reconstruction of geomagnetic fields.
在光学、图像分析和视觉感知领域，人们普遍认为希尔伯特变换在一维以上不能自然且各向同性地扩展。在图像分析的某些领域，这种观点限制了解析信号概念在多维度上的应用。我们表明，与这种观点相反，存在一种自然、各向同性且优雅的扩展。我们根据两个乘法算子开发了一种新颖的二维变换：一个螺旋相位谱（傅里叶）算子和一个取向相位空间算子。将这两个算子结合起来就得到了一个有意义的二维正交（或希尔伯特）变换。这种新变换被应用于二维封闭条纹图案解调问题，从而得到一个直接的解决方案。这种新变换与经典调和分析中的里斯变换有关。我们考虑这些联系，以及其他一些联系，比如光学相位奇点的传播和地磁场的重建。
