Atomic force microscopy (AFM) is typically used for analysis of relatively flat surfaces with topographic features smaller than the height of the AFM tip. On flat surfaces, it is relatively easy to find the object of interest and deconvolute imaging artifacts resulting from the finite size of the AFM tip. In contrast, AFM imaging of three-dimensional objects much larger than the AFM tip height is rarely attempted although it could provide topographic information that is not readily available from two-dimensional imaging, such as scanning electron microscopy. In this paper, we report AFM measurements of a vertically-mounted razor blade, which is taller and sharper than the AFM tip. In this case, the AFM height data, except for the data collected around the cutting edge of the blade, reflect the shape of the AFM tip. The height data around the apex area are effectively the convolution of the AFM tip and the blade cutting edge. Based on computer simulations mimicking an AFM tip scanning across a round sample, a simple algorithm is proposed to deconvolute the AFM height data of an object taller and sharper than the AFM tip and estimate its effective curvature.

In this paper, we introduce the anisotropic Sobolev capacity with fractional order and develop some basic properties for this new object. Applications to the theory of anisotropic fractional Sobolev spaces are provided. In particular, we give geometric characterizations for a nonnegative Radon measure $\mu $ that naturally induces an embedding of the anisotropic fractional Sobolev class $\dot{\Lambda }_{\alpha ,K}^{1,1}$ into the $\mu $-based-Lebesgue-space $L_{\mu }^{n/\beta }\,\text{with}\,0<\beta \le n$. Also, we investigate the anisotropic fractional $\alpha $-perimeter. Such a geometric quantity can be used to approximate the anisotropic Sobolev capacity with fractional order. Estimation on the constant in the related Minkowski inequality, which is asymptotically optimal as $\alpha \to {{0}^{+}}$, will be provided.

To realize the automation of biomaterial cutting, the interaction between the blade and biomaterial deserves insight understanding. In this paper, a blade-compression cutting model is developed. The influences from the material properties, deformation, and blade properties on the cutting force are thoroughly explained. An approach to describe the sharpness of a blade is provided and its applicability is experimentally demonstrated. Based on the material property and knife sharpness property, the required force to realize compression cut can be predicted. This provides the reference force trajectory for the automation of biomaterial compression cutting.