Fast Wavelet Transform Using Piecewise Polynomial Segmentation
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Solution Overview
Problem
Continuous wavelet transforms are computationally intensive and resource-consuming, limiting their use in real-time and low-energy applications, especially for neuronal signal processing, as existing fast algorithms require high-order polynomial approximations and recursive calculations that are not optimized for independent frequency analysis.
Innovation Solution
A method that determines wavelet coefficients using a piecewise polynomial mother wavelet with a small maximum order, calculating primitives and convolving them with coefficients dependent on the wavelet's extremity points, allowing independent determination of coefficients for different frequencies and spectral bands, implemented by a processor for efficient real-time processing.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If conventional continuous wavelet transform is implemented by replacing the integral with a discrete sum using piecewise constant signal approximation, then the wavelet coefficients can be calculated, but the computational cost becomes very high requiring (2α-1) multiplications and (2α-2) sums per coefficient where α can be between 100 and 3000
Solution Approach 1:
The method segments the computation by dividing the signal processing into discrete intervals and using piecewise polynomial approximation. The support of the mother wavelet is subdivided into J intervals, and the signal is processed in segments corresponding to these intervals, allowing parallel and efficient computation of wavelet coefficients without requiring the full (2α-1) multiplications per coefficient.
Solution Approach 2:
The invention changes the approximation parameter from piecewise constant (order 0) to piecewise polynomial of order N≥1. This parameter change in the approximation order allows for more accurate representation of the signal while maintaining computational efficiency through the structured polynomial basis, reducing the computational burden compared to the conventional approach.
2Productivity
If fast CWT method using polynomial approximation of mother wavelet is used, then computational cost is reduced to O(ds), but high-order polynomial approximation is required which increases computational complexity
Solution Approach 1:
Instead of using a single high-order polynomial to approximate the mother wavelet, the method segments the wavelet support into J intervals and uses piecewise polynomial approximation. This segmentation allows using lower-order polynomials (N≥1) on each interval while maintaining overall accuracy, thereby reducing the polynomial order requirement and associated computational complexity.
Solution Approach 2:
The method employs a composite approximation approach by combining multiple low-order polynomial segments to form the overall wavelet representation. Each interval uses a polynomial of maximum level N, and these segments are combined to create the complete piecewise polynomial mother wavelet, achieving accuracy equivalent to high-order polynomials without the computational burden.
3Productivity
If recursive calculation method is used where Ws(τ) is calculated from Ws-1(τ), then computational cost is reduced, but coefficients for different frequencies cannot be determined independently
Solution Approach 1:
The method segments the frequency analysis by allowing independent calculation of wavelet coefficients for different scales and translations. The piecewise polynomial formulation with pre-defined coefficients Cik(s) enables each Ws(τ) to be computed independently without requiring recursive dependency on previous coefficients, providing versatility for targeted spectral band analysis.
Solution Approach 2:
The method performs preliminary action by pre-calculating and storing the coefficients Cik(s) that are dependent only on the mother wavelet and scale factor. These pre-computed coefficients are then used in the convolution step to determine wavelet coefficients independently for different frequencies, eliminating the need for recursive calculation while maintaining computational efficiency.
4Ease of manufacture
If piecewise constant signal approximation is used, then the calculation can be performed with discrete sums, but the approximation accuracy is limited
Solution Approach 1:
The invention changes the approximation parameter from piecewise constant (order 0) to piecewise polynomial of order N≥1. This parameter change in the polynomial order improves the signal approximation accuracy by capturing more nuanced signal variations within each interval, while still maintaining the simplicity of discrete sum calculations through the structured polynomial basis.
Solution Approach 2:
The method uses a composite approximation structure combining multiple polynomial segments of order N≥1 across different intervals. This composite piecewise polynomial approximation provides higher accuracy than piecewise constant approximation by better representing the signal's local behavior, while maintaining computational simplicity through the systematic formulation with pre-defined coefficients.
Data Source
AI summary
Method for determining at least one wavelet coefficient Ws(τ) of a wavelet transform of a signal in which the mother wavelet of the transform has a support subdivided into J≧1 intervals bound by (J+1) extremity points, and is defined by a polynomial of a maximum level N≧1 on each interval. The method includes calculating all or some of the primitives of the signal of order k between 2 and N+1, at least at (J+1) points corresponding to extremity points of the intervals of the wavelet support dilated by a factor of s and translated by a time τ; calculating the convolution of said or each primitive sampled in this way with a respective succession of (J+1) coefficients Cik(s), dependent upon said wavelet; and determining the wavelet coefficient by calculating a linear combination of convolutions. Steps a) to c) are implemented by a processor configured or programmed in an appropriate manner.