Noise reduction, smoothing and time interval segmentation of noisy signals using an energy optimisation method

Abstract

Noise reduction and time interval segmentation of a noise-contaminated piecewise continuous signal is considered by the authors as a non-linear optimisation problem. The mathematical framework of this method is presented both in continuous-time and discrete-time domains. The smoothed signal and segmented time intervals of the original noisy signal are calculated as an optimised solution for an energy functional. An algorithm similar to the level set method is developed to find the optimised solution. In this algorithm, the discontinuity points separating consecutive continuous signals are preserved while the noise is reduced. Therefore this method fundamentally exhibits a better performance compared with a traditional low-pass filter suppressing high frequency components, including discontinuity points. The results also demonstrate a better quality in noise reduction in comparison to the median and Gaussian filters. IntroductionThe purpose of this paper is to introduce a new method based on energy optimisation in signal processing for noise reduction and signal smoothing. Similar methods have long been developed and used in image processing and computer vision [1 -7]. Image restoration, also known as the 'inverse problem', was developed by Rudin et al. [7] as an optimisation method based on the concept of total variation. In another development within computer vision, a method known as the 'snake' algorithm was introduced for object segmentation in images by Kass et al. [14,15]. The objective of this spectral method is then to recover the segmented signals from artefacts introduced by the Gibbs phenomenon, by using the Gegenbauer reconstruction algorithm [16 -18], whereas in our case, noise reduction of the segmented signals is the main aim of the proposed method. Furthermore in our approach, segmentation and noise reduction are achieved simultaneously in contrast to the spectral method, where segmentation and reconstruction are performed separately. The difficulty of investigating the non-linear functional considered in this paper is the lack of differentiability in discontinuities. Hence, EulerLagrange equations cannot be employed in this optimisation problem. A signal g(t) is considered as a timed sequence of separate continuous signals that are subject to channel noise and degradation. g(t) can be approximated as a piecewise continuous function f (t) consisting of time series of at least class C 2 functions f i (t) over a time interval (t i21 , t i ). These continuous functions f i (t) are optimal solutions of the following energy functional Eð f ; SÞ ¼ 1 2where E( f, S) is the energy functional to be optimised, f i (t) the smoothed function approximating g(t) over the time interval (t i21 , t i ), m the non-negative parameter, S i (t) the segmented time interval, which is a rectangular window function in the time domain and can be defined as followsIn (1), the first term, ( f i (t) 2 g(t)) 2 , is a data fidelity term whose minimisation indicates that f i (t) approximates g(t).

Contour evolution scheme for variational image segmentation and smoothing

“…Functional (13) was initially developed for one-dimensional signals [24,25], and it was demonstrated in [25] that its second variation with respect to points representing discontinuities is positive. This indicates that points representing discontinuities are minimisers of such a functional.…”

Geometrical-based algorithm for variational segmentation and smoothing of vector-valued images

“…Mahmoodi et al proposed a generic approach for signal and image segmentation and smoothing applicable to signals with any dimension, based on the second variation with respect to points and contours that represent discontinuities [26][27][28][29]. It was demonstrated that points or contours that represent discontinuities were minimisers of the functional proposed in [26][27][28][29].…”

“…It was demonstrated that points or contours that represent discontinuities were minimisers of the functional proposed in [26][27][28][29]. A variational-based solution to the functional for piecewise continuous low-pass signals was proposed in [27].…”

“…It was demonstrated that points or contours that represent discontinuities were minimisers of the functional proposed in [26][27][28][29]. A variational-based solution to the functional for piecewise continuous low-pass signals was proposed in [27]. An operator-based geometrical approach being faster and more robust in the presence of excessive noise was proposed in [28] to implement the functional introduced in [27].…”

“…A variational-based solution to the functional for piecewise continuous low-pass signals was proposed in [27]. An operator-based geometrical approach being faster and more robust in the presence of excessive noise was proposed in [28] to implement the functional introduced in [27]. This operator-based approach was generalised for grey-scale images based on the geometrical properties of surfaces (smoothed images) [26].…”