## 5 Conclusions  
We proposed effective smoothing methods based on minimizing the \(\ell_{0}\) gradient using novel gradient constraints. To suppress the pseudo-edge artifacts such as the staircasing effect, box- and ball-type gradient constraints were introduced. These constraints limited the gradient intensity of an output image to the range of a reference image. By solving the \(\ell_{0}\) gradient minimization problem based on the proposed gradient constraints via ADMM for nonconvex optimization, we effectively preserved the strong edges and removed weak edges without yielding pseudo-edges and halo artifacts. It was confirmed that the smoothed image obtained by the proposed method can generate better contrast-enhanced images and tone map images without pseudo-edges or halos. These results suggest the potential for delivering more dependable image diagnosis and recognition outcomes in medical imaging and autonomous driving applications, where image enhancement applications are essential. Throughout all our experiments, we consistently observed the effectiveness of the proposed methods in comparison to existing \(\ell_{0}\) gradient-based methods. In future work, we will apply our proposed gradient constraints for other optimization problems, e.g., intrinsic image decomposition, reflection removal, and flash/no-flash image blending. Methodology, R.M. and M.O. ; software, R.M. ; validation, R.M. and M.O. ; formal analysis, R.M. and M.O. ; investigation, R.M. and M.O. ; data curation, R.M. ; writing--original draft preparation, R.M. and M.O. ; writing--review and editing, R.M. and M.O. ; project administration,  
Figure 10: Results of clip-art JPEG artifact removal and their PSNR [dB]. (**a**) Ground truth, (**b**) Input image, (**c-1**) [31], 31.51, (**c-2**) [31], 32.18, (**d-1**) [37], 31.19, (**d-2**) [37], 31.89, (**e-1**) Ours, 31.81, (**e-2**) Ours, 32.23. Figure 9: Comparison PSNR and SSIM for clip-art JPEG artifact removal. (**a**) Peak signal-to-noise ratio (PSNR) [dB]. (**b**) Structural similarity (SSIM) [65]. R.M. and M.O. ; funding acquisition, R.M. and M.O. All authors have read and agreed to the published version of the manuscript. * [1][PERSON]; [PERSON]; [PERSON] Nonlinear Total Variation Based Noise Removal Algorithms. _Physics D_**1992**, _60_, 259-268. [CrossRef]
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