Reza Taheriyoun, A. and Shafie, K. (2022), The estimation of the fractal dimension of a Gaussian field via Euler characteristic. Can J Statistics. https://doi.org/10.1002/cjs.11625

 

A very challenging problem in making some high-tech instruments is the surface smoothness like Copper plating used in a radio frequency quadrupole vane of accelerators. There is a wide range of methods in the engineering of these parts to obtain smooth metal surfaces. The smooth surfaces usually are examined by operating on the accelerators to attest to their smoothness performance. Therefore, before setting up the metal plates, it is worth measuring the smoothness of their surfaces. The purity of employed metals confirms the crystalline structure in precise scales, which is self-similar pattern. The more purity in metal, the more self-similarity and this produces a fractal effect, which is a significant source of roughness. A fractal surface is not a simple 2-dimensional plane and simply speaking, its Euclidean geometric features show contradictions such as the infinite area on a bounded subset of the surface. So, it is not possible to demonstrate the features of this kind of plane as a 2-dimensional object and it is neither a 3-dimensional object on a Cartesian system. The other definitions of dimension, such as Hausdorff or box-counting dimensions, are used on these objects instead. It was shown that the Hausdorff dimension of the mentioned surface is in the form of 2 + 1 α where α is called fractal index and is closely related to the order of Hölder continuity of the surface.  For a differentiable surface that is inevitably smooth, this order is equal to one, and hence this differentiable surface is a 2-dimensional surface with all the features of Euclidean geometry. The fractal index can be numerically computed in many methods. However, the problem becomes worst when the observed surface is only a realization of a random field. We have access to this only realization only on a lattice and not at each arbitrary point of it. In this case, the realized surface is one of the infinitely many possible surfaces generated under the model brought by the random field. Thus, the problem of estimation of the fractal index induced by the model is highly in demand since, for example it is not a wise idea to put the whole of the Copper plates under an atomic force microscopy to determine its smoothness and according to the laboratory restrictions, there is only access to a small piece of this plate’s information on lattice points. The class of index-β Gaussian fields includes almost all the Gaussian fields with this type of realization. This article tries to modify the level crossing approach in one-dimensional fractal objects to N -dimensional and practically employs the results for N = 2, which is the case of observing surfaces as the model’s realizations.  The level crossing method simply counts the number of times a one-dimensional function crosses a threshold. The big number of level crossings is due to the roughness of the function, and there is a hope to find a relationship between the level crossing characteristic and the fractal index. These relationships are kind of scaling laws. Thanks to the Euler characteristic, the concept of level crossing was extended for higher dimensions. Fortunately, a log-log scaling law is obtained for the mean of the Euler characteristics and the bandwidth used to smooth the observed surface using a kernel function. The estimator of the fractal index is based on this scaling law. It is shown that under a sufficient condition, the estimator achieves consistency, which guarantees to tend the estimator to the real unknown value of the fractal index as the number of nodes on the observed lattice increases without any increase in the area of the sampling domain. Although this method is applied to surfaces where usually expected as the data with large sizes, the results show that the estimator’s computational cost is low compared to many other methods. The reader can also see the application of the estimator on a real CNC polished Copper surface.