In this paper, a general methodology is introduced for the determination of potential prototype curves used for the drawing of prehistoric wall paintings. The approach includes 1) preprocessing of the wall-paintings contours to properly partition them, according to their curvature, 2) choice of prototype curves families, 3) analysis and optimization in 4-manifold for a first estimation of the form of these prototypes, 4) clustering of the contour parts and the prototypes to determine a minimal number of potential guides, and 5) further optimization in 4-manifold, applied to each cluster separately, in order to determine the exact functional form of the potential guides, together with the corresponding drawn contour parts. The methodology introduced simultaneously deals with two problems: 1) the arbitrariness in data-points orientation and 2) the determination of one proper form for a prototype curve that optimally fits the corresponding contour data. Arbitrariness in orientation has been dealt with a novel curvature based error, while the proper forms of curve prototypes have been exhaustively determined by embedding curvature deformations of the prototypes into 4-manifolds. Application of this methodology to celebrated wall paintings excavated at Tyrins, Greece, and the Greek island of Thera manifests that it is highly probable that these wall paintings were drawn by means of geometric guides that correspond to linear spirals and hyperbolae. These geometric forms fit the drawings’ lines with an exceptionally low average error, less than 0.39 mm. Hence, the approach suggests the existence of accurate realizations of complicated geometric entities more than 1,000 years before their axiomatic formulation in the Classical Ages.

A novel methodology is introduced here that exploits 2D images of arbitrary elastic body deformation instances so as to quantify mechanoelastic characteristics that are deformation invariant. Determination of such characteristics allows for developing methods offering an image of the undeformed body. General assumptions about the mechanoelastic properties of the bodies are stated which lead to two different approaches for obtaining bodies' deformation invariants. One was developed to spot a deformed body's neutral line and its cross sections, while the other solves deformation PDEs by performing a set of equivalent image operations on the deformed body images. Both of these processes may furnish a body-undeformed version from its deformed image. This was confirmed by obtaining the undeformed shape of deformed parasites, cells (protozoa), fibers, and human lips. In addition, the method has been applied to the important problem of parasite automatic classification from their microscopic images. To achieve this, we first apply the previous method to straighten the highly deformed parasites, and then, apply a dedicated curve classification method to the straightened parasite contours. It is demonstrated that essentially different deformations of the same parasite give rise to practically the same undeformed shape, thus confirming the consistency of the introduced methodology. Finally, the developed pattern recognition method classifies the unwrapped parasites into six families, with an accuracy rate of 97.6 percent.

In this paper a new general methodology is introduced for the determination of the method of construction of prehistoric wall-paintings. The approach consists of algorithms that perform a) preprocessing of the contours of the figures appearing in the wall-paintings, b) determination of pattern repetitions in the contours of the depicted entities, c) classification of these repeated patterns into proper geometric prototypes, d) curve fitting. By means of this methodology, the authors demonstrated that a number of wall-paintings excavated at Mycenae and in Aegean islands, made before the 13th century B.C., were drawn via the use of guides. These stencils - guides were very precise geometric figures and in particular linear spirals and hyperbolae, thus indicating knowledge of constructing these complicated schemes more than 1000 years before the appearance of classical age mathematics.

This paper introduces a novel methodology for the classification of ancient Greek inscriptions according to the writer who carved them. Inscription writer identification is crucial for dating the written content, which in turn is of fundamental importance in the sciences of history and archaeology. To achieve this, we first compute an ideal or "platonic" prototype for the letters of each inscription separately. Next, statistical criteria are introduced to reject the hypothesis that two inscriptions are carved by the same writer. In this way, we can determine the number of distinct writers who carved a given ensemble of inscriptions. Next, maximum likelihood considerations are employed to attribute all inscriptions in the collection to the respective writers. The method has been applied to 24 Ancient Athenian inscriptions and attributed these inscriptions to six different identified hands in full accordance with expert epigraphists' opinions.

A novel methodology is introduced here that exploits microscopic images of domestic animalspsila parasites in arbitrary deformation instances, so as to verify assumptions about their mechanoelastic properties. The obtained knowledge of these properties manifests parasite body characteristics that are deformation invariant, thus allowing for unwrapping them. Next, we have stated differential equations that govern the parasite body deformation and we have solved them by performing a set of equivalent image operations on the deformed body images. This process furnishes the parasite undeformed version from its deformed image. The method has been applied to a dataset of 193 microscopic images of highly deformed parasites. It is demonstrated that different orientations and deformations of the same parasite give rise to practically the same undeformed shape, thus confirming the consistency of the approach.

In this paper, a color image segmentation method and a pattern analysis are presented, in connection with the extraordinary 1650 B.C. wall paintings found in the Greek island of Thera. These wall paintings are usually reconstructed from thousands of fragments widely scattered in the excavated site. The fragments' depiction manifests inhomogeneous color decay, cracks, added extraneous material, etc. The proposed color image segmentation method takes into account the decay problems and offers a very good approximation of the initial fragment depiction as the artist drew it in the late Bronze Age. The algorithm performs essentially better than other standard segmentation schemes as extensive qualitative tests indicate. Moreover, it offers clear-cut color regions and region borders for each fragment depiction. The whole approach is based on classifying the pixels into a number of regions where each region is described by a normal distribution, followed by fragment-decay reduction and edge refining. Extensive pattern analysis to the obtained region borders leads to the conclusion that 3650 years ago, the artist most probably used advanced geometrical methods in order to construct handcrafted ldquoFrench curvesrdquo (stencils or templates) and use them to draw certain figures. On the basis of the aforementioned results, specific pattern matching techniques are employed for the reconstruction of wall paintings, depicting spirals, from their constituent fragments.

In this paper, a methodology of general applicability is presented for answering the question if an artist used a number of archetypes to draw a painting or if he drew it freehand. In fact, the contour line parts of the drawn objects that potentially correspond to archetypes are initially spotted. Subsequently, the exact form of these archetypes and their appearance throughout the painting is determined. The method has been applied to celebrated Thera Late Bronze Age wall paintings with full success. It has been demonstrated that the artist or group of artists has used seven geometrical archetypes and seven corresponding well-constructed stencils (four hyperbolae, two ellipses, and one Archimedes' spiral) to draw the wall painting "Gathering of Crocus" in 1650 B.C. This method of drawing seems to be unique in the history of arts and of great importance for archaeology, and the history of mathematics and sciences, as well.

In this paper, a new decisively important factor in both the perceptual and the automated piano-guitar identification process is introduced. This factor is determined by the nontonal spectral content of a note, while it is, in practice, totally independent of the note spectrum tonal part. This conclusion and all related results are based on a number of extended acoustical experiments, performed over the full pitch range of each instrument. The notes have been recorded from six different performers each of whom played a different instrument. Next, a number of powerful criteria for the classification between guitar and piano is proposed. Using these criteria, automated classification between 754 piano and guitar test notes has been achieved with a 100% success rate.

In this paper, an original general methodology is introduced to establish whether a handmade shape corresponds to a given geometrical prototype. Using this methodology, one can decide if an artist had the intention of drawing a specific mathematical prototype or not. This analysis is applied to the 1650 B.C. wall paintings from the prehistoric settlement on Thera, and inferences of great archaeological and historical importance are made. In particular, strong evidence is obtained suggesting that the spirals depicted on the wall paintings correspond to linear (Archimedes) spirals, certain shapes correspond to canonical 48-gon and 32-gon, while other shapes correspond to parts of ellipses. It seems that the presented wall paintings constitute the earliest archaeological findings on which these geometrical patterns appear with such remarkable accuracy.

In this paper a new computational scheme is introduced for performing recursive least squares adaptive filtering. The proposed algorithm is far more robust than all the already existing RLS schemes, in the sense that it is drastically less sensitive in the numerical error due to the finite precision with which all operations are executed. Hence, it has a lifetime tens of times greater than all the previous RLS schemes. Moreover, the algorithm introduced here has excellent tracking abilities and, due to its particular structure, it is parallelizable. When it is executed in parallel by four processors, it is faster than all the existing RLS algorithms, and in particular, it is by m steps faster, where m is the system order, than the FAEST and the FTF computational schemes.>

An accurate analysis of the finite precision error in the FAEST-5p and the fast transversal algorithm (FTF) is undertaken here, on the basis of an entirely new methodology and practice. The cause of the numerical instability and divergence of these two schemes is clearly pointed out and demonstrated. In particular, it is proved that, out of all the formulas for these two algorithms, only four (4) specific formulas generate an amount of finite precision error that consistently makes the algorithms fail after a certain number of iterations. Moreover, it is shown that there is a limited number of specific formula that transmit this generated finite precision error. Finally, a general methodology is introduced that allows for the development of new a posteriori algorithms that are intrinsically free of the corresponding finite precision numerical problems and that therefore are, in practice, fully stable.>

The error propagation in various useful order- and time-recursive DSP algorithms is studied. It is demonstrated that in all these algorithms there are two kinds of error due to finite precision: an erratic and a systematic one. Examples of both kinds of error are provided, and special emphasis is given to the study of the systematic truncation or round-off error. It is shown that in the Levinson-type and Schur-type algorithms for the solution of the FLP (forward linear prediction) and the FIR (finite impulse response) problem, there is one dominant source of systematic error, while in the l-step ahead case there are two sources of such error. Moreover, it is pointed out that in the fast Kalman algorithms there are two kinds of systematic error. Precise indicators of the exact magnitude of the finite precision error are given, and possible recovery techniques are proposed.>

The numerical behavior of the order-recursive algorithms for the solution of a number of typical Toeplitz systems is studied. Use is made of a novel methodology that gives a direct interpretation of the positive definiteness that leads to the explicit computation of the minimum absolute bounds of the 1-step and l-step-ahead linear predictors. Ill-conditioning is discussed, and an algorithmic scheme with improved numerical behavior is given for the l-step-ahead LP case. It is proved that the Levinson algorithm, when it is used for the solution of the least-squares finite-impulse-response filtering problem, manifests a fundamentally different numerical behavior than when it is used for the l-step-ahead problem.>