Piecewise-polynomial signal segmentation using convex optimization

Abstract

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Orthogonality is superiority in piecewise-polynomial signal segmentation and denoising

“…The comparisons presented in this article will be concerned only with the detection of breakpoints, and thus, in our further analysis, we process no more than the vector d. However, in case we would like to recover the denoised signal, we would proceed as in our former works [11,12], where first a moving median filter is applied to d and subtracted from d, allowing to keep the significant values and at the same time to push small ones toward zero. Put simply, values larger than a selected threshold then indicate the breakpoints positions.…”

“…Within the first of the two classes, i.e., within approaches based on modeling, one can distinguish explicit and implicit types of models. In the "explicit" type, the signal is modeled such that it is a composition of sub-signals which often can be expressed analytically [9][10][11][12][13][14][15][16]. In the "implicit" type of models, the signal is characterized by features that are derived from the signal by using an operator [17][18][19][20][21].…”

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

“…with denoting the elementwise (Hadamard) product. To convert (5) to the form involving the standard matrix-vector product, we define…”

“…The projector proj B2(y,δ) finds the closest point in the 2 -ball {z : y − z 2 ≤ δ} with respect to the input point. Both the operations are computationally cheap, for details see for example [5].…”

“…In our recent works, we dealt with various segmentation algorithms for 1D signals, all of them following the piecewisepolynomial model [5], [22], [23]. In the present work, we generalize the 1D model to 2D, derive a new numerical solver and adapt the way the model output is processed.…”