Abstract-We consider a common frequency-domain procedure hilbert for generating discrete-time analytic signals and show how it fails for a specific class of signals. A new frequency-domain technique ehilbert is formulated that solves the defect. Moreover, the new technique is applicable to all discrete-time real signals of even length. It is implemented by the introduction of one additional zero of the continuous spectrum of the analytic signal hilbert at a negative frequency. Both frequency-domain methods generate equal length discrete-time analytic signals. The new analytic signal preserves the original signal (real part) and also the zeros of the discrete spectrum hilbert in the negative frequencies. The greater attenuation at the negative frequencies affects the degree of aliasing of the analytic signal. It is measured by applying the analytic signal to an orthogonal wavelet transform and determining the improved transform shiftability.

In this paper we consider a common procedure [2] for generating analytic signals and show how it fails for specific discrete-time real signals. A new frequency domain technique is formulated that solves the defect. Both methods have the same redundancy. The new analytic signal preserves the original signal (real part) as also the zeros of its discrete spectrum in the negative frequencies. The superiority of the new method is in the introduction of one additional zero of the continuous spectrum of the original signal at a negative frequency and a corresponding reduction in shiftability.

We propose a simple, novel and efficient method for generating complex half-band FIR filters, which we use in the generation of discrete-time analytic (DTA) signals. These filters have properties of linear phase and real-time implementation while the DTA signals generated are orthogonal and invertible (the original real signal is recoverable). The filter design, in contrast to some other methods, is easily scalable and stable. The new method is evaluated for performance (i.e. aliasing) by comparing its shiftability [10] property with that of other transforms. Using a total variation measure for determining function variation, we see that its shiftability either matches or exceeds that of other methods. Furthermore, this design method lends itself to an enhancement [1], thereby allowing additional improvement in shiftability. We prove an important theoretical aspect of the new method: the amplitude spectrum of the length N filter converges almost everywhere to the ideal complex half-band amplitude spectrum as N → +∞, thereby assuring shiftability.

Methods for detecting edges, be they multidimensional or multiresolution, ultimately reduce to finding extremal points, first derivatives or zeros of second derivatives. However, problems such as missing edges, weak edges due to thresholding, derivatives not existing and false edge generation, are some of the consequences. We adopt a new formalism: Edges are singularities of the mathematically smoothest function possible -the complex analytic function. We embed a real image into the real part of an analytic function. After solving the conjugate harmonic problem, edges in discrete images are identified from the imaginary part. The analytic function model is inherently two-dimensional and an invariant measure. Comparisons are made with other standard edge detection methods. We outline issues that need to be considered for establishing analytic functions for edge detection.