Algorithms for Piecewise Constant Signal Approximations

Abstract

We consider the problem of finding optimal piecewise constant approximations of one-dimensional signals. These approximations should consist of a specified number of segments (samples) and minimise the mean squared error to the original signal. We formalise this goal as a discrete nonconvex optimisation problem, for which we study two algorithms. First we reformulate a recent adaptive sampling method by Dar and Bruckstein in a compact and transparent way. This allows us to analyse its limitations when it comes to violations of its three key assumptions: signal smoothness, local linearity, and error balancing. As a remedy, we propose a direct optimisation approach which does not rely on any of these assumptions and employs a particle swarm optimisation algorithm. Our experiments show that for nonsmooth signals or low sample numbers, the direct optimisation approach offers substantial qualitative advantages over the Dar-Bruckstein method. As a more general contribution, we disprove the optimality of the principle of error balancing for optimising data in the ℓ 2 norm.

Optimised Sparse Image Reconstruction Via Densification and Relocation

“…-Polygonal finite elements [20,53] -Stochastic hill climbing [52] -Optimal control approaches [19,37] -Particle swarm optimisation [12,13] Table 1: Taxonomy of Spatial Data Optimisation Algorithms…”

“…Finally, the method by Dar and Bruckstein (2019) [24] finds an optimised mask for piecewise constant approximations of 1D signals. This approach has three key assumptions: smoothness of the input signal f , local linearity of f , and a balanced error distribution over the mask pixels P (Bergerhoff et al, 2019; Dar and Bruckstein, 2019) [13,24].…”

“…Consequently, various authors (Chizhov and Weickert, 2021;Demaret and Iske, 2004;Peter, 2019) [21,26,55] see the tonal optimisation as a postprocessing step in minimising the MSE (1). In contrast, the spatial optimisation problem in P is known to be nonconvex (Bergerhoff et al, 2019) [13] and NP-Hard (Agarwal and Suri, 1998) [3]. Moreover, the computational cost of evaluating a reconstruction operator u is not necessarily cheap (Chizhov and Weickert, 2021;Hoffmann et al, 2015;Kämper and Weickert, 2022) [21,39,43].…”

A Systematic Evaluation of Coding Strategies for Sparse Binary Images

“…4) Unconstrained mask approaches: While there are numerous publications that deal with optimal selection of sparse known data for inpainting [4,[11][12][13][14][42][43][44][45][46][47] , only few use them for actual compression. Hoffman et al [29] primarily use a regular grid as already mentioned, but it also uses some unconstrained mask points to further enhance quality which are encoded using JBIG .…”

Signal Modeling with IG Noise and Parameter Estimation Based on RJMCMC

“…This can be seen from the pattern that has not decreased from year to year in the last 18 years. Studies related to signal modeling with PC models can be found in various kinds of literature, for example [5][6][7]. Based on the type of noise, some PC models also use different types of noise, for example, Gaussian [8], Poisson [9], Gamma [10], and Rayleigh [11].…”