Generalized methods and solvers for noise removal from piecewise constant signals. I. Background theory

Removing noise from piecewise constant (PWC) signals is a challenging signal processing problem arising in many practical contexts. For example, in exploration geosciences, noisy drill hole records need to be separated into stratigraphic zones, and in biophysics, jumps between molecular dwell states have to be extracted from noisy fluorescence microscopy signals. Many PWC denoising methods exist, including total variation regularization, mean shift clustering, stepwise jump placement, running medians, convex clustering shrinkage and bilateral filtering; conventional linear signal processing methods are fundamentally unsuited. This paper (part I, the first of two) shows that most of these methods are associated with a special case of a generalized functional, minimized to achieve PWC denoising. The minimizer can be obtained by diverse solver algorithms, including stepwise jump placement, convex programming, finite differences, iterated running medians, least angle regression, regularization path following and coordinate descent. In the second paper, part II, we introduce novel PWC denoising methods, and comparisons between these methods performed on synthetic and real signals, showing that the new understanding of the problem gained in part I leads to new methods that have a useful role to play.

Generalized methods and solvers for noise removal from piecewise constant signals. II. New methods

Removing noise from signals which are piecewise constant (PWC) is a challenging signal processing problem that arises in many practical scientific and engineering contexts. In the first paper (part I) of this series of two, we presented background theory building on results from the image processing community to show that the majority of these algorithms, and more proposed in the wider literature, are each associated with a special case of a generalized functional, that, when minimized, solves the PWC denoising problem. It shows how the minimizer can be obtained by a range of computational solver algorithms. In this second paper (part II), using this understanding developed in part I, we introduce several novel PWC denoising methods, which, for example, combine the global behaviour of mean shift clustering with the local smoothing of total variation diffusion, and show example solver algorithms for these new methods. Comparisons between these methods are performed on synthetic and real signals, revealing that our new methods have a useful role to play. Finally, overlaps between the generalized methods of these two papers and others such as wavelet shrinkage, hidden Markov models, and piecewise smooth filtering are touched on.

Lobe growth variation and the maintenance of symmetry in foliose lichen thalli

The aim of this study was to determine how thallus symmetry could be maintained in foliose lichens when variation in the growth of individual lobes may be high. Hence, the radial growth of a sample of lobes was studied monthly, over 22 months, in 7 thalli of Parmelia conspersa (Ehrh. Ex Ach.) Ach. And 5 thalli of P. glabratula ssp fuliginosa (fr. ex Duby) Laund. The degree of variation in the total radial growth of different lobes within a thallus over 22 months varied between thalli. Individual lobes showed a fluctuating pattern of radial growth from month to month with alternating periods of fast and slow growth. Monthly variations in radial growth of different lobes were synchronized in some but not in all thalli. Few significant correlations were found between the radial growth of individual lobes and total monthly rainfall or shortwave radiation. The levels of ribitol, arabitol and mannitol were measured in individual lobes. All three polyols varied significantly between lobes within a thallus suggesting that variations in algal phostosynthesis and in the partitioning of fungal polyols may contribute to lobe growth variation. The effect on thallus symmetry of lobes which grew radially either consistently faster or slower than average was studied. Slow growing lobes were overgrown, and gaps in the perimeter were eliminated by the growth of neighbouring lobes, in approximately 7 to 9 months. However, a rapidly growing lobe, with its neighbours removed on either side, continued to grow radially at the same rate as rapidly growing control lobes. The results suggested that lobe growth variation results from a combination of factors which may include the origin of the lobes, lobe morphology and the patterns of algal cell division and hyphal elongation in different lobes. No convincing evidence was found to suggest that exchange of carbohydrate occurred between lobes which would tend to equalize their radial growth. Hence, the fluctuating pattern of lobe growth observed may be sufficient to maintain a degree of symmetry in most thalli. In addition, slow growing lobes would tend to be overgrown by faster growing neighbours thus preventing the formation of indentations in the thallus perimeter.

Variation in hypothallus width and the growth of the lichen Rhizocarpon geographicum (L.) DC.

Variations in hypothallus width were studied in relation to radial growth in the lichen Rhizocarpon geographicum (L.) DC. in South Gwynedd, Wales, UK. Variations were present both within and between thalli and in successive three-month growth periods, but there was no significant variation associated with thallus size. In individual thalli, there were increases and reductions in hypothallus width in successive three-month growth periods attributable to hypothallus growth and changes at the margin of the areolae. Total radial growth over 18 months was positively correlated with initial hypothallus width. These results suggest: 1) individual thalli of similar size vary considerably in hypothallus width, 2) fluctuations in the location of the margin of the areolae in successive three month periods is an important factor determining this variability, 3) hypothallus width predicts subsequent radial growth over 18 months, and 4) variation in hypothallus; width is a factor determining between thallus variability in radial growth rates in yellow-green species of Rhizocarpon.