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: