Orthogonality is superiority in piecewise-polynomial signal segmentation and denoising

Abstract

Segmentation and denoising of signals often rely on the polynomial model which assumes that every segment is a polynomial of a certain degree and that the segments are modeled independently of each other. Segment borders (breakpoints) correspond to positions in the signal where the model changes its polynomial representation. Several signal denoising methods successfully combine the polynomial assumption with sparsity. In this work, we follow on this and show that using orthogonal polynomials instead of other systems in the model is beneficial when segmenting signals corrupted by noise. The switch to orthogonal bases brings better resolving of the breakpoints, removes the need for including additional parameters and their tuning, and brings numerical stability. Last but not the least, it comes for free!

Image Edges Resolved Well When Using an Overcomplete Piecewise-Polynomial Model

“…3) Coins, higher noise, orthonormal basis: The same image was processed using the orthonormal basis instead of the standard basis [23]. This time, the model error was set to δ = 8 000 and the respective thresholds are 0.15, 0.29, 0.14 and 0.14.…”

“…where x k are the expansion coefficients in such a basis. The system {p k (t)} K k=0 is only required to form a basis, but additional properties such as orthogonality can sometimes be beneficial, see [23]. In a discrete setting, the individual elements of a polynomial signal are represented as…”

Joint Stochastic Spline and Autoregressive Identification Aiming Order Reduction Based on Noisy Sensor Data

“…Over the last few decades, the spline functions theory has been applied in many fields of science and engineering, such as: signal processing [ 1 ], computer graphics [ 2 ], system modeling [ 3 ] and identification [ 4 ], statistics [ 5 ], industrial design [ 6 ], geodetics [ 7 ], etc. In parallel, substantial effort has been made towards developing the mathematical bases of spline functions [ 8 ].…”

“…Moreover, the condition of unitary sum for the connected piecewise polynomials in each point seems unnatural. In [ 1 ], orthogonal polynomial based approximation is introduced, but the case study refers to signals with abrupt changes; without this feature, the data segments corresponding to each polynomial are difficult to set. The article [ 18 ] is concerned with nonlinearity identification of a complex dynamic system, by using uniformly sampled knots and a nonlinear basis of functions, which have to be appropriately selected.…”

Joint Demosaicing and Denoising Based on Interchannel Nonlocal Mean Weighted Moving Least Squares Method

“…The work in [ 24 ] applied a nonlinear moving least-squares projection method for the denoising of high-dimensional noisy scattered data, while the work in [ 25 ] applied a biquadratic polynomial approximation for denoising of medical images. In [ 26 ], a modified singular value thresholding with minimizing error constraints is used for the segmentation of noisy signals by an orthogonal polynomial approximation.…”