{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:35:15Z","timestamp":1760060115108,"version":"build-2065373602"},"reference-count":40,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2025,8,4]],"date-time":"2025-08-04T00:00:00Z","timestamp":1754265600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"NIH","award":["2R15EB024283-03"],"award-info":[{"award-number":["2R15EB024283-03"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In compressed sensing, it is believed that the L0 norm minimization is the best way to enforce a sparse solution. However, the L0 norm is difficult to implement in a gradient-based iterative image reconstruction algorithm. The total variation (TV) norm minimization is considered a proper substitute for the L0 norm minimization. This paper points out that the TV norm is not powerful enough to enforce a piecewise-constant image. This paper uses the limited-angle tomography to illustrate the possibility of using the L0 norm to encourage a piecewise-constant image. However, one of the drawbacks of the L0 norm is that its derivative is zero almost everywhere, making a gradient-based algorithm useless. Our novel idea is to replace the zero value of the L0 norm derivative with a zero-mean random variable. Computer simulations show that the proposed L0 norm minimization outperforms the TV minimization. The novelty of this paper is the introduction of some randomness in the gradient of the objective function when the gradient is zero. The quantitative evaluations indicate the improvements of the proposed method in terms of the structural similarity (SSIM) and the peak signal-to-noise ratio (PSNR).<\/jats:p>","DOI":"10.3390\/axioms14080605","type":"journal-article","created":{"date-parts":[[2025,8,5]],"date-time":"2025-08-05T08:46:55Z","timestamp":1754383615000},"page":"605","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Mitigating the Drawbacks of the L0 Norm and the Total Variation Norm"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0790-6043","authenticated-orcid":false,"given":"Gengsheng L.","family":"Zeng","sequence":"first","affiliation":[{"name":"Department of Computer Science, Utah Valley University, Orem, UT 84058, USA"}]}],"member":"1968","published-online":{"date-parts":[[2025,8,4]]},"reference":[{"key":"ref_1","first-page":"46741","article-title":"Morphing from the TV-norm to the l0-norm","volume":"55","author":"Zeng","year":"2024","journal-title":"Biomed. 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