Local scales on curves and surfaces

In this paper, we extend a previous work on the study of local scales of a function to studying local scales of a d-dimensional surface. In the case of a function, the scale functions are computed by convolving the function with a symmetric kernel of zero mean and zero first moments of various scales. From the goodness of fit point of view, this convolution can be viewed as measuring locally the deviations from a linear function at various scales. The local scales are defined as points where deviation from a linear function reaches a local maximum. In the case of a d-dimensional surface, the analogy of the scale functions is to compute local deviations from a d-plane at various scales (this is related to Jones beta number). This analogy is realized through convolving the (surface) measure with a symmetric kernel of zero mean and zero first moments. We then apply the theory of singular integral operators on d-dimensional surfaces to show useful properties of local scales. We also show that the defined local scales are consistent in the sense that the number of local scales are invariant under dilation, and that one can relate the local scales of the original object with its dilated version via the dilating factor. In addition, with the assumption that the d-dimensional surface enjoys a certain degree of smoothness, we prove that our local scales are related to curvatures. Furthermore, this connection makes apparent that our local scales are intimately related to the change in deviation from flatness. Computational examples are also presented. In shape analysis, the local scales and the scale functions on the boundary can be used as local signatures or descriptors.