The Ricker and Gabor wavelets and their Hilbert transforms are widely used in many physical/engineering fields. In seismology, they are commonly utilized to describe source functions involved in the computation of synthetic seismograms in acoustic or elastic media, or inversely, the extraction of source wavelets from real seismic traces. We provide analytical formulae of the Ricker and Gabor wavelets’ Hilbert transforms, yielding to analytic expressions of the envelope, instantaneous frequency, and phase attributes of these wavelets. We analyze the plausibility of these wavelets by revealing the regions of positive instantaneous frequency and nonzero envelope.

In many instances, numerical integration of space-scale PDEs is the most time consuming operation of image processing. This is because the scale step is limited by conditional stability of explicit schemes. In this work, we introduce the unconditionally stable semi-implicit linearized difference scheme that is fashioned after additive operator split (AOS) [1], [2] for Beltrami and the subjective surface computation. The Beltrami flow [3], [4], [5] is one of the most effective denoising algorithms in image processing. For gray-level images, we show that the flow equation can be arranged in an advection-diffusion form, revealing the edge-enhancing properties of this flow. This also suggests the application of AOS method for faster convergence. The subjective surface [6] deals with constructing a perceptually meaningful interpretation from partial image data by mimicking the human visual system. However, initialization of the surface is critical for the final result and its main drawbacks are very slow convergence and the huge number of iterations required. In this paper, we first show that the governing equation for the subjective surface flow can be rearranged in an AOS implementation, providing a near real-time solution to the shape completion problem in 2D and 3D. Then, we devise a new initialization paradigm where we first "condition" the viewpoint surface using the Fast-Marching algorithm. We compare the original method with our new algorithm on several examples of real 3D medical images, thus revealing the improvement achieved.