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We propose an unsupervised method for eigen tree hierarchies and quantisation group association for segmentation of corrosion in marine vessel hull inspection via camera images. Our unsupervised approach produces image segments that are examined to decide on defect recognition. The method generates a binary decision tree, which, by means of bottom-up pruning, is revised, and dominant leaf nodes predict the areas of interest. Our method is compared with other techniques, and the results indicate that it achieves better performance for true- vs. false-positive area against ideal (ground truth) coverage.

Hertzian mechanics is the most frequently used theory for data processing in Atomic Force Microscopy (AFM) indentation experiments on soft biological samples, due to its simplicity and significant scientific results previously published. For instance, using the Hertz model, it has been proven that there are significant differences in the mechanical properties of normal and cancerous tissues and that cancer cells’ invasive properties are correlated with their nanomechanical properties. However, many scientists are skeptical regarding the applicability of the Hertz theory to biological materials, as they are highly heterogeneous. The main critical question to be addressed is “what do we calculate” when fitting the force-indentation data to Hertz equations. Previous studies have shown that when using cylindrical, parabolic, or conical indenters, the fitting parameter is the average Young’s modulus. In this paper, it is demonstrated that it is also valid to fit equations derived from Hertzian mechanics to force-indentation data when testing soft, heterogeneous samples for any indenter geometry. The fitting factor calculated through this approach always represents the average Young’s modulus for a specific indentation depth. Therefore, Hertzian mechanics can be extended to soft heterogeneous materials, regardless of the indenter’s shape.

Atomic Force Microscopy (AFM) is a powerful tool for determining the mechanical properties of soft materials at the nanoscale. When using spherical indenters, the data is usually fitted to the classic Hertzian equation which is valid for parabolic indenters, or spherical indenters for small indentation depths compared to the tip radius (i.e. h/R«1). However, for bigger indentation depths, more complex equations should be used for data fitting. In this paper, a straightforward energy-based approach is proposed to determine the Young's modulus of soft samples without the need for a fitting procedure. By calculating the integral of the function that relates the force to the indentation depth for deep spherical indentations, an equation was derived that relates the work done by the indenter (W) to the h/R ratio. The Young's modulus (E) can be easily determined by calculating the work done by the indenter (which is equal to the area under the force-indentation data) and using a function that depends on the h/R ratio (i.e., X = f (h/R)). In this dataset article, the function X = f (h/R) and a table including the X-values corresponding to different h/R values in the domain 0 ≤ h/R ≤ 5 are presented. An example demonstrating indentation experiments on an agarose gel is also included. This proposed approach represents the simplest way to generate Young's modulus maps, as it only requires calculating the area under the force-indentation data and using the X-values presented in this dataset article.

Solving nonlinear oscillations is a challenging task due to the mathematical complexity of the related differential equations. In many cases, determining the oscillation’s period requires the solution of complicated integrals using numerical methods. To avoid the complexity, there are many empirical equations in the literature that can be used instead of rigorous mathematical analysis to provide an acceptable approximation. In this paper, a recently developed method, He’s formulation, is applied to find the period in many different cases of nonlinear oscillators. The cases are those of the Duffing equation, the Helmholtz nonlinear oscillator, the simple pendulum and the case of a vertical oscillation under the influence of a nonlinear elastic force. The results of the method are accurate; thus, He’s formulation is a strong tool for solving nonlinear oscillations.

Atomic force microscopy (AFM) is a powerful tool that enables imaging and nanomechanical properties characterization of biological materials. Nanofibers are the structural units of many biological systems and their role in the development of advanced biomaterials is crucial. AFM methods have proven to be effective towards the characterization of fibers with respect to biological and bioengineering applications at the nanoscale. However, both the topographical and mechanical properties’ nanocharacterizations of single fibers using AFM are challenging procedures. In particular, regarding imaging procedures, significant artifacts may arise from tip convolution effects. The geometrical characteristics of the AFM tip and the nanofibers, and the fact that they have similar magnitudes, may lead to significant errors regarding the topographical imaging. In addition, the determination of the mechanical properties of nanofibers is also challenging due to their small dimensions and heterogeneity (i.e., the elastic half-space assumption is not valid in most cases). This review elucidates the origins of errors in characterizing individual nanofibers, while also providing strategies to address limitations in experimental procedures and data processing.

Atomic force microscopy (AFM) is a powerful tool for characterizing biological materials at the nanoscale utilizing the AFM nanoindentation method. When testing biological materials, spherical indenters are typically employed to reduce the possibility of damaging the sample. The accuracy of determining Young’s modulus depends, among other factors, on the calibration of the indenter, i.e., the determination of the tip radius. This paper demonstrates that the tip radius can be approximately calculated using a single force–indentation curve on an unknown, soft sample without performing any additional experimental calibration process. The proposed method is based on plotting a tangent line on the force indentation curve at the maximum indentation depth. Subsequently, using equations that relate the applied force, maximum indentation depth, and the tip radius, the calculation of the tip radius becomes trivial. It is significant to note that the method requires only a single force–indentation curve and does not necessitate knowledge of the sample’s Young’s modulus. Consequently, the determination of both the sample’s Young’s modulus and the tip radius can be performed simultaneously. Thus, the experimental effort is significantly reduced. The method was tested on 80 force–indentation curves obtained on an agarose gel, and the results were accurate.

Measuring the mechanical properties (i.e., elasticity in terms of Young’s modulus) of biological samples using Atomic Force Microscopy (AFM) indentation at the nanoscale has opened new horizons in studying and detecting various pathological conditions at early stages, including cancer and osteoarthritis. It is expected that AFM techniques will play a key role in the future in disease diagnosis and modeling using rigorous mathematical criteria (i.e., automated user-independent diagnosis). In this review, AFM techniques and mathematical models for determining the spatial variability of elastic properties of biological materials at the nanoscale are presented and discussed. Significant issues concerning the rationality of the elastic half-space assumption, the possibility of monitoring the depth-dependent mechanical properties, and the construction of 3D Young’s modulus maps are also presented.