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The two-dimensional sample entropy marks a significant advance in evaluating the regularity and predictability of images in the information domain. Unlike the direct computation of sample entropy, which incurs a time complexity of O (N 2) for the series with N length, the Monte Carlo-based algorithm for computing one-dimensional sample entropy (MCSampEn) markedly reduces computational costs by minimizing the dependence on N. This paper extends MCSampEn to two dimensions, referred to as MCSampEn2D. This new approach substantially accelerates the estimation of two-dimensional sample entropy, outperforming the direct method by more than a thousand fold. Despite these advancements, MCSampEn2D encounters challenges with significant errors and slow convergence rates. To counter these issues, we have incorporated an upper confidence bound (UCB) strategy in MCSampEn2D. This strategy involves assigning varied upper confidence bounds in each Monte Carlo experiment iteration to enhance the algorithm's speed and accuracy. Our evaluation of this enhanced approach, dubbed UCBMCSampEn2D, involved the use of medical and natural image data sets. The experiments demonstrate that UCBMCSampEn2D achieves a 40 % reduction in computational time compared to MCSampEn2D. Furthermore, the errors with UCBMCSampEn2D are only 30 % of those observed in MCSampEn2D, highlighting its improved accuracy and efficiency. [ABSTRACT FROM AUTHOR]