Figure 1 Riemann-Liouville Fractional Differential Operator
Certainly:
Given the practical implementations of the Riemann-Liouville (RL)
fractional differentiation, its operational order tends to be confined
predominantly between 0.1 to 0.2, as highlighted by Ni et al. [19].
Beyond this defined range, images tend to compromise their inherent
details. This manifests as an uptick in noise levels, rendering the
algorithm overly susceptible to noise perturbations. This noise
susceptibility, coupled with the stringent constraints on the fractional
differentiation order, often results in the sharpening effects falling
short of desired outcomes in a multitude of scenarios.
Recognizing this intrinsic limitation, and with the overarching goal of
amplifying visual fidelity whilst concurrently tempering the operator’s
noise impact, it becomes imperative to recalibrate the operator to be
less noise-sensitive. Delving into the nuances of differentiation
operators, it is discernible that the coefficient associated with the
template’s central pixel is represented as 8. This
coefficient invariably wields the most profound influence on the image
enhancement process. As one moves further from this pivotal central
pixel, its corresponding influence on the image undergoes a gradual
attenuation.
In pursuit of diminishing the blurring effect and further curtailing the
noise footprint, this study introduces a nuanced modification. We
recalibrate the template coefficients of the RL fractional differential
operator, anchoring our approach on a weighted scheme. The precise
methodology for this modification unfolds as:
The modified differential operator with adjusted coefficients is
illustrated in Figure 2 below: