A Theory for Multiresolution Signal Decomposition: The Wavelet Representation
IEEE Transactions on Pattern Analysis and Machine Intelligence
Fractional Splines and Wavelets
SIAM Review
Signal Processing for Computer Vision
Signal Processing for Computer Vision
Shift invariant properties of the dual-tree complex wavelet transform
ICASSP '99 Proceedings of the Acoustics, Speech, and Signal Processing, 1999. on 1999 IEEE International Conference - Volume 03
The fractional spline wavelet transform: definition end implementation
ICASSP '00 Proceedings of the Acoustics, Speech, and Signal Processing, 2000. on IEEE International Conference - Volume 01
Analyzing Image Structure by Multidimensional Frequency Modulation
IEEE Transactions on Pattern Analysis and Machine Intelligence
A Wavelet Tour of Signal Processing, Third Edition: The Sparse Way
A Wavelet Tour of Signal Processing, Third Edition: The Sparse Way
A new framework for complex wavelet transforms
IEEE Transactions on Signal Processing
IEEE Transactions on Signal Processing
The design of approximate Hilbert transform pairs of wavelet bases
IEEE Transactions on Signal Processing
IEEE Transactions on Signal Processing
Hypercomplex signals-a novel extension of the analytic signal tothe multidimensional case
IEEE Transactions on Signal Processing
Hilbert transform pairs of biorthogonal wavelet bases
IEEE Transactions on Signal Processing - Part I
Image analysis using a dual-tree M-band wavelet transform
IEEE Transactions on Image Processing
Coherent Multiscale Image Processing Using Dual-Tree Quaternion Wavelets
IEEE Transactions on Image Processing
Multiresolution monogenic signal analysis using the Riesz-Laplace wavelet transform
IEEE Transactions on Image Processing
On the shiftability of dual-tree complex wavelet transforms
IEEE Transactions on Signal Processing
Analytical footprints: compact representation of elementary singularities in wavelet bases
IEEE Transactions on Signal Processing
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We propose a novel method for constructing Hilbert transform (HT) pairs of wavelet bases based on a fundamental approximation-theoretic characterization of scaling functions--the B-spline factorization theorem. In particular, starting from well-localized scaling functions, we construct HT pairs of biorthogonal wavelet bases of L2 (R) by relating the corresponding wavelet filters via a discrete form of the continuous HT filter. As a concrete application of this methodology, we identify HT pairs of spline wavelets of a specific flavor, which are then combined to realize a family of complex wavelets that resemble the optimally-localized Gabor function for sufficiently large orders. Analytic wavelets, derived from the complexification of HT wavelet pairs, exhibit a one-sided spectrum. Based on the tensor-product of such analytic wavelets, and, in effect, by appropriately combining four separable biorthogonal wavelet bases of L2(R2), we then discuss a methodology for constructing 2-D directional-selective complex wavelets. In particular, analogous to the HT correspondence between the components of the 1-D counterpart, we relate the real and imaginary components of these complex wavelets using a multidimensional extension of the HT--the directional HT. Next, we construct a family of complex spline wavelets that resemble the directional Gabor functions proposed by Daugman. Finally, we present an efficient fast Fourier transform (FFT)-based filterbank algorithm for implementing the associated complex wavelet transform.