Multidimensional, mapping-based complex wavelet transforms

Although the discrete wavelet transform (DWT) is a powerful tool for signal and image processing, it has three serious disadvantages: shift sensitivity, poor directionality, and lack of phase information. To overcome these disadvantages, we introduce multidimensional, mapping-based, complex wavelet transforms that consist of a mapping onto a complex function space followed by a DWT of the complex mapping. Unlike other popular transforms that also mitigate DWT shortcomings, the decoupled implementation of our transforms has two important advantages. First, the controllable redundancy of the mapping stage offers a balance between degree of shift sensitivity and transform redundancy. This allows us to create a directional, nonredundant, complex wavelet transform with potential benefits for image coding systems. To the best of our knowledge, no other complex wavelet transform is simultaneously directional and nonredundant. The second advantage of our approach is the flexibility to use any DWT in the transform implementation. As an example, we exploit this flexibility to create the complex double-density DWT: a shift-insensitive, directional, complex wavelet transform with a low redundancy of (3/sup M/-1)/(2/sup M/-1) in M dimensions. No other transform achieves all these properties at a lower redundancy, to the best of our knowledge. By exploiting the advantages of our multidimensional, mapping-based complex wavelet transforms in seismic signal-processing applications, we have demonstrated state-of-the-art results.

I. Compelling Reasons for Multidimensional, Mapping-Based Complex Wavelet Transforms

ALTHOUGH the discrete real wavelet transform (RWT)

The acronym RWT refers specifically to the discrete wavelet transform of a real-valued function. Since we only consider wavelet transforms implemented with real-coefficient filters, RWT coefficients are always real. On the other hand, the acronym DWT refers to the discrete wavelet transform implemented with real-coefficient filters on a complex-valued function.

is a powerful signal-processing tool, it has three disadvantages that undermine its usage in many applications. First, it is shift sensitive because input-signal shifts generate unpredictable changes in RWT coefficients. Second, RWT analysis lacks the phase information that accurately describes nonstationary signal behavior. Third, the two-dimensional (2-D), separable RWT suffers from poor directionality because RWT coefficients reveal only three spatial edge orientations, as illustrated in Fig. 1. In Fig. 1(a), we show the image of a disc to which we apply a level-one RWT and obtain the , , and subbands shown in Fig. 1(b)–(d). Observe that the RWT subbands differentiate between only three edge orientations: horizontal, vertical, and diagonal features. To overcome the first two disadvantages of the RWT, in an earlier report [1], we demonstrated the shift insensitivity, explicit phase content and controllable redundancy of one-dimensional (1-D), mapping-based complex wavelet transforms. In this paper, we shall extend the mapping-based framework to multidimensional complex wavelet transforms and then show that improved directionality and flexibility are powerful new advantages. In Fig. 11(a), we demonstrate the improved directionality of our 2-D complex wavelet transform applied to the disc image [Fig. 1(a)]. The directional subbands in this figure clearly distinguish between six edge orientations: −75°, −45°, −15°, +15°, +45°, and +75°. In addition, we shall prove that the flexibility to apply the mapping-based framework to any real-valued wavelet transform allows us to create the directional, shift-insensitive, complex double-density wavelet transform (CDDWT) with a low redundancy factor of 2.67 in two dimensions. To the best of our knowledge, no other transform achieves all these properties at a lower redundancy. We shall also demonstrate that the controllable redundancy of the mapping-based framework allows us to create a directional, nonredundant, complex wavelet transform. No other complex wavelet transform is simultaneously directional and nonredundant, to the best of our knowledge. Fig. 1. (a) Disc image. (b) RWT HL subband: vertical edges. (c) RWT LH subband: horizontal edges. (d) RWT HH subband: diagonal edges.