Innovation modelling and wavelet analysis of fractal processes in bio-imaging

Abstract

Growth and form in biology are often associated with some level of fractality. Fractal characteristics have also been noted in a number of imaging modalities. These observations make fractal modelling relevant in the context of bio-imaging.In this paper, we introduce a simple and yet rigorous innovation model for multi-dimensional fractional Brownian motion (fBm) and provide the computational tools for the analysis of such processes in a multi-resolution framework. The key point is that these processes can be whitened by application of the appropriate fractional Laplacian operator which has a corresponding polyharmonic wavelet. We examine the case of MRI and mammography images through comparison with theoretical results, which underline the suitability of fractal models in the study of bio-textures.

Fractal modelling and analysis of flow-field images

“…operand, achieved by means of an operator E that projects its operand onto its curl-free component 2 and has Fourier symbol ωω T / ω 2 . In symbols:…”

“…2 The complement, Id − E, is a projection onto the divergence-free component 3 The action of an operator on a random field finds a rigorous interpretation in the framework of the theory of generalized random processes of Gelfand and Vilenkin [15]. …”

“…It is a well-known fact that wavelet transform coefficients can be used to estimate the Hurst exponent of scalar fBm processes and fields [2,[10][11][12][13][14]16]. This property of wavelets holds true also in the vector setting, in the following manner: A wavelet transform applied independently to each of the vectorial components of a fractional Brownian vector field can be used, in almost exactly the same fashion as in the scalar setting, to estimate H. We refer the reader to Tafti & al.…”

“…Stochastic fractal models are commonly used in a range of applications where some form of self-similarity or scale-invariance is observed (examples include image processing, seismology, and the study of growth processes [1,2]). The quintessential stochastic fractal is the fractional Brownian motion (fBm)-so named by Mandelbrot and Van Ness [3] but already considered by Kolmogorov [4] and others before them-which can be defined by means of the structure of its variogram (variance of increments) [5]:…”

“…We shall employ wavelets to address the second aspect (statistical study of the models). Wavelet-based techniques have been used with efficacy in the past in the statistical analysis of fBm and, in particular, in the estimation of the Hurst exponent [2,[10][11][12][13][14]. The effectiveness of wavelet analysis for this purpose relies on two facts: First, wavelets essentially behave as low-frequency differentiators, and this, by virtue of Eqn (1), means that a wavelets analysis of fBm (which is non-stationary) yields coefficients that correspond to (stationary) filtered white noise.…”