Fundamentals Of Statistical Signal Processing

Brain-computer interfaces for communication and control.

The paper discusses the theory behind the dual-tree transform, shows how complex wavelets with good properties can be designed, and illustrates a range of applications in signal and image processing The authors use the complex number symbol C in CWT to avoid confusion with the often-used acronym CWT for the (different) continuous wavelet transform The four fundamentals, intertwined shortcomings of wavelet transform and some solutions are also discussed Several methods for filter design are described for dual-tree CWT that demonstrates with relatively short filters, an effective invertible approximately analytic wavelet transform can indeed be implemented using the dual-tree approach

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The dual-tree complex wavelet transform

The brain's electrical signals enable people without muscle control to physically interact with the world.

Brain-computer interfaces for communication and control

Stable non-Gaussian random processes , by G. Samorodnitsky and M. S. Taqqu. Pp. 632. £49.50. 1994. ISBN 0-412-05171-0 (Chapman and Hall).

Preface. Acknowledgments. 1. Continuous Wavelet Transform. Introduction. Continuous-Time Wavelets. Definition of the CWT. The CWT as a Correlation. Constant Q-Factor Filtering Interpretation and Time-Frequency Resolution. The CWT as an Operator. Inverse CWT. Problems. 2. Introduction to the Discrete Wavelet Transform and Orthogonal Wavelet Decomposition. Introduction. Approximations of Vectors in Nested Linear Vector Subspaces. Example of Approximating Vectors in Nested Subspaces of a Finite-Dimensional Linear Vector Space. Example of Approximating Vectors in Nested Subspaces of an Infinite-Dimensional Linear Vector Space. Example of an MRA. Bases for the Approximation Subspaces and Haar Scaling Function. Bases for the Detail Subspaces and Haar Wavelet. Digital Filter Implementation of the Haar Wavelet Decomposition. Problems. 3. MRA, Orthonormal Wavelets, and Their Relationship to Filter Banks. Introduction. Formal Definition of an MRA. Construction of a General Orthonormal MRA. Scaling Function and Subspaces. Implications of the Dilation Equation and Orthogonality. A Wavelet Basis for the MRA. Two-scale Relation for psi (t). Basis for the Detail Subspaces. Direct Sum Decomposition. Digital Filtering Interpretation. Decomposition Filters. Reconstructing the Signal. Examples of Orthogonal Basis-Generating Wavelets. Daubechies D4 Scaling Function and Wavelet. Bandlimited Wavelets. Interpreting Orthonormal MRAs for Discrete-Time Signals. Continuous-time MRA Interpretation for the DTWT. Discrete-Time MRA. Basis Functions for the DTWT. Miscellaneous Issues Related to PRQMF Filter Banks. Generating Scaling Functions and Wavelets from Filter Coefficients. Problems. 4. Alternative Wavelet Representations. Introduction. Biorthogonal Wavelet Bases. Filtering Relationship for Biorthogonal Filters. Examples of Biorthogonal Scaling Functions and Wavelets. Two-Dimensional Wavelets. Nonseparable Multidimensional Wavelets. Wavelet Packets. Problems. 5. Wavelet Transform and Data Compression. Introduction. Transform Coding. DTWT for Image Compression. Image Compression Using DTWT and Run-length Encoding. Embedded Tree Image Coding. Comparison with JPEG. Audio Compression. Audio Masking. Standards Specifying Subband Implementation: ISO/MPEG Coding for Audio. Wavelet-Based Audio Coding. Video Coding Using Multiresolution Techniques: A Brief Introduction. 6. Other Applications of Wavelet Transforms. Introduction. Wavelet Denoising. Speckle Removal. Edge Detection and Object Isolation. Image Fusion. Object Detection by Wavelet Transforms of Projections. Communication Applications. Scaling Functions as Signaling Pulses. Discrete Wavelet Multitone Modulation. 7. Advanced Topics. Introduction. CWTs Revisited. Parseval's Identity for the CWT. Inverse CWT Is a Many-to-One Operation. Wavelet Inner Product as a Projection Operation. Bridging the Gap Between CWTs and DWTs. CWT with an Orthonormal Basis-Generating Wavelet. A Trous Algorithm. Regularity and Convergence. Daubechies Construction of Orthonormal Scaling Functions. Bandlimited Biorthogonal Decomposition. Scaling Function Pair Construction. Wavelet Pair Construction. Design and Selection of Wavelets. The Lifting Scheme. Best Basis Selection. Wavelet Matching. Perfect Reconstruction Circular Convolution Filter Banks. Downsampling. Upsampling. Procedure for Implementation. Conditions for Perfect Reconstruction. Procedure for Constructing PRCC Filter Banks. Interpolators Matched to the Input Process. Interpolation Sampling. Frequency-Sampled Implementation of Bandlimited DWTs. The Scaling Operation and Self-Similar Signals. LTI Systems and Eigenfunctions. Continuous-Time Linear Scale-Invariant System. Scaling in Discrete Time. Discrete-time LSI Systems. Appendix A. Fundamentals of Multirate Systems. The Downsampler. The Upsampler. Noble Identities. Appendix B. Linear Algebra and Vector Spaces. Brief Review of Vector Spaces. Vector Subspace. Linear Independence and Bases. Inner Product Spaces. Hilbert Space and Riesz Bases. Index.

Wavelet transforms : introduction to theory and applications

Wavelet analysis of neuroelectric waveforms: a conceptual tutorial.

Deep neural networks have achieved near-human accuracy levels in various types of classification and prediction tasks including images, text, speech, and video data. However, the networks continue to be treated mostly as black-box function approximators, mapping a given input to a classification output. The next step in this human-machine evolutionary process — incorporating these networks into mission critical processes such as medical diagnosis, planning and control — requires a level of trust association with the machine output. Typically, statistical metrics are used to quantify the uncertainty of an output. However, the notion of trust also depends on the visibility that a human has into the working of the machine. In other words, the neural network should provide human-understandable justifications for its output leading to insights about the inner workings. We call such models as interpretable deep networks. Interpretability is not a monolithic notion. In fact, the subjectivity of an interpretation, due to different levels of human understanding, implies that there must be a multitude of dimensions that together constitute interpretability. In addition, the interpretation itself can be provided either in terms of the low-level network parameters, or in terms of input features used by the model. In this paper, we outline some of the dimensions that are useful for model interpretability, and categorize prior work along those dimensions. In the process, we perform a gap analysis of what needs to be done to improve model interpretability.

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Interpretability of deep learning models: A survey of results

Manual screening procedures for detecting concealed weapons such as handguns, knives, and explosives are common in controlled access settings like airports, entrances to sensitive buildings, and public events. The detection of weapons concealed underneath a person's clothing is an important obstacle to the improvement of the security of the general public as well as the safety of public assets like airports and buildings. It is desirable to be able to detect concealed weapons from a standoff distance, especially when it is impossible to arrange the flow of people through a controlled procedure. The goal is the eventual deployment of automatic detection and recognition of concealed weapons. It is a technological challenge that requires innovative solutions in sensor technologies and image processing. A number of sensors based on different phenomenology as well as image processing support are being developed to observe objects underneath people's clothing. The main aim of this article is to provide a tutorial overview of these developments.

Imaging for concealed weapon detection: a tutorial overview of development in imaging sensors and processing

Algorithms for designing a mother wavelet /spl psi/(x) such that it matches a signal of interest and such that the family of wavelets {2/sup -(j/2)//spl psi/(2/sup -j/x-k)} forms an orthonormal Riesz basis of L/sup 2/(/spl Rscr/) are developed. The algorithms are based on a closed form solution for finding the scaling function spectrum from the wavelet spectrum. Many applications require wavelets that are matched to a signal of interest. Most current design techniques, however, do not design the wavelet directly. They either build a composite wavelet from a library of previously designed wavelets, modify the bases in an existing multiresolution analysis or design a scaling function that generates a multiresolution analysis with some desired properties. In this paper, two sets of equations are developed that allow us to design the wavelet directly from the signal of interest. Both sets impose bandlimitedness, resulting in closed form solutions. The first set derives expressions for continuous matched wavelet spectrum amplitudes. The second set of equations provides a direct discrete algorithm for calculating close approximations to the optimal complex wavelet spectrum. The discrete solution for the matched wavelet spectrum amplitude is identical to that of the continuous solution at the sampled frequencies. An interesting byproduct of this work is the result that Meyer's spectrum amplitude construction for an orthonormal bandlimited wavelet is not only sufficient but necessary. Specific examples are given which demonstrate the performance of the wavelet matching algorithms for both known orthonormal wavelets and arbitrary signals.

Raghuveer M. Rao

Papers

Wavelet transforms : introduction to theory and applications

Wavelet analysis of neuroelectric waveforms: a conceptual tutorial.

Interpretability of deep learning models: A survey of results

Imaging for concealed weapon detection: a tutorial overview of development in imaging sensors and processing

Algorithms for designing wavelets to match a specified signal