The continuous boundary local trigonometric transform

The Local Trigonometric Transform (LTT) provides a nice tool for localizing both a signal and its frequency content. But there are certain properties of this algorithm that make it unattractive for various applications. For example, instability of the folding process on or near an edge can cause increased edge effect which translates recursively to all lower levels of the LTT. This can result in inefficient signal representation, improper segmentation and incorrect analysis of the signal.; In this thesis, some of these disadvantages of the LTT are examined, and a new approach to localized trigonometric analysis is proposed, the Continuous Boundary Local Trigonometric Transform (CBLTT), which attempts to correct these and other shortcomings. The main difference between the LTT and the CBLTT is that the former projects the signal onto smooth overlapping basis functions, whereas the latter decomposes the signal into regions that are assumed to be independent of one another. Each independent subspace then undergoes an invertible, nonlinear transformation to reduce edge effect, and it is immediately projected onto an orthogonal basis; e.g., Fourier basis, Sine/Cosine basis, Wavelet basis. As a tensor product, it can be applied to multi-dimensional data: signals, images, video. Since this new approach can be used to efficiently segment the frequency domain of a signal, it also gives rise to a new version of the Brushlet Transform.; In order to reduce edge effect and increase numerical stability, many variations of this new scheme are derived and presented, and their strengths and weaknesses are examined and contrasted. In addition, their usefulness in various applications, ranging from signal segmentation to compression, are also explored.