Browsing by Author "Fernandes, Felix"
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Item Complex Wavelet Transforms with Allpass Filters(2003-08-20) Fernandes, Felix; Selesnick, Ivan W.; van Spaendonck, Rutger; Burrus, C. Sidney; Digital Signal Processing (http://dsp.rice.edu/)Complex discrete wavelet transforms have significant advantages over real wavelet transforms for certain signal processing problems. Two approaches to the implementation of complex wavelet transforms have been proposed earlier. Both approaches require discrete-time allpass systems having approximately linear-phase and (fractional) delay. This paper compares the results when different allpass systems are used. In the earlier work, maximally flat delay allpass systems were used. In this paper, it is shown that an allpass system designed according to the minimax criterion yields improvements for the complex discrete wavelet transforms.Item Directional Complex-Wavelet Processing(2000-08-20) Fernandes, Felix; van Spaendonck, Rutger; Burrus, C. Sidney; Digital Signal Processing (http://dsp.rice.edu/)Poor directional selectivity, a major disadvantage of the separable 2D discrete wavelet transform (DWT), has previously been circumvented either by using highly redundant, nonseparable wavelet transforms or by using restrictive designs to obtain a pair of wavelet trees. In this paper, we demonstrate that superior directional selectivity may be obtained with no redundancy in any separable wavelet transform. We achieve this by projecting the wavelet coefficients to separate approximately the positive and negative frequencies. Subsequent decimation maintains non-redundancy. A novel reconstruction step guarantees perfect reconstruction within this critically-sampled framework. Although our transform generates complex-valued coefficients, it may be implemented with a fast algorithm that uses only real arithmetic. We also explain how redundancy may be judiciously introduced into our transform to benefit certain applications. To demonstrate the efficacy of our projection technique, we show that it achieves state-of-the-art performance in a seismic image-processing application.Item Directional, Shift-Insensitive, Complex Wavelet Transforms with Controllable Redundancy(2001-08-20) Fernandes, Felix; Digital Signal Processing (http://dsp.rice.edu/)Although the Discrete Wavelet Transform (DWT) is a powerful tool for signal and image processing, it has three serious disadvantages. First, the DWT is shift sensitive because input-signal shifts generate unpredictable changes in DWT coefficients. Second, the DWT suffers from poor directionality because DWT coefficients reveal only three spatial orientations. Third, DWT analysis lacks the phase information that accurately describes non-stationary signal behavior. To overcome these disadvantages, we introduce the notion of projection-based complex wavelet transforms. These two-stage, projection-based complex wavelet transforms consist of a projection onto a complex function space followed by a DWT of the complex projection. 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 projection stage offers a balance between degree of shift sensitivity and transform redundancy. This allows us to create a directional, non-redundant, 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 non-redundant. The second advantage of our approach is the flexibility to use any DWT in the transform implementation. We exploit this flexibility to create the Complex Double-density DWT (CDDWT): a shift-insensitive, directional, complex wavelet transform with a low redundancy of (3^m - 1)/(2^m - 1) in m dimensions. To the best of our knowledge, no other transform achieves all these properties at a lower redundancy. Besides the mitigation of DWT shortcomings, our transforms have unique properties that will potentially benefit a variety of signal processing applications. As an example, we demonstrate that our projection-based complex wavelet transforms achieve state-of-the-art results in a seismic signal-processing application.Item M-Band Multiwavelet Systems(1999-03-20) Burrus, C. Sidney; Fernandes, Felix; Digital Signal Processing (http://dsp.rice.edu/)In this paper we investigate multiwavelet systems whose scaling functions have disjoint support. We demonstrate that, with the exception of a trivial case, this property may not be attained by two band multiwavelet systems. We show that to enjoy this property, it is indeed necessary to invoke M-band multiwavelet systems. This indicates the existence of tilings of the time-frequency plane that may be obtained with M--band multiwavelet systems but not with two band multiwavelet systems. Hence M--band multiwavelet systems are inherently more powerful than two band multiwavelet systems and deserve a thorough investigation. Finally, we derive KthM--band multiwavelet systems. These conditions will enable M--band multiwavelet systems to be used for practical digital signal processing applications.Item A New Framework for Complex Wavelet Transforms(2003-06-20) Fernandes, Felix; van Spaendonck, Rutger; Burrus, C. Sidney; Digital Signal Processing (http://dsp.rice.edu/)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 two-stage 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, non-redundant, 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 non-redundant. The second advantage of our approach is the flexibility to use any DWT in the transform implementation. As an example, we can exploit this flexibility to create the Complex Double-density DWT (CDDWT): a shift-insensitive, directional, complex wavelet transform with a low redundancy of (3m - 1)/(2m - 1) in m dimensions. To the best of our knowledge, no other transform achieves all these properties at a lower redundancy.Item Non-Redundant, Linear-Phase, Semi-Orthogonal, Directional Complex Wavelets(2004-05-01) Fernandes, Felix; Wakin, Michael; Baraniuk, Richard G.; Digital Signal Processing (http://dsp.rice.edu/)The directionality and phase information provided by non-redundant complex wavelet transforms (NCWTs) provide significant potential benefits for image/video processing and compression applications. However, because existing NCWTs are created by downsampling filtered wavelet coefficients, the finest scale of these transforms has resolution 4x lower than the real input signal. In this paper, we propose a linear-phase, semi-orthogonal, directional NCWT design using a novel triband filter bank. At the finest scale, the resulting transform has resolution 3x lower than the real input signal. We provide a design example to demonstrate three important properties for image/video processing applications: directionality, magnitude coherency, and phase coherency.