Loading [MathJax]/jax/output/SVG/jax.js
Research article

Single hyperspectral image super-resolution using a progressive upsampling deep prior network

  • Hyperspectral image super-resolution (SR) aims to enhance the spectral and spatial resolution of remote sensing images, enabling more accurate and detailed analysis of ground objects. However, hyperspectral images have high dimensional characteristics and complex spectral patterns. As a result, it is critical to effectively leverage the spatial non-local self-similarity and spectral correlation within hyperspectral images. To address this, we have proposed a novel single hyperspectral image SR method based on a progressive upsampling deep prior network. Specifically, we introduced the spatial-spectral attention fusion unit (S2AF) based on residual connections, in order to extract spatial and spectral information from hyperspectral images. Then we developed the group convolutional upsampling (GCU) to efficiently utilize the spatial and spectral prior information inherent in hyperspectral images. To address the challenges posed by the high dimensionality of hyperspectral images and limited training dataset, we implemented a parameter-sharing grouped convolutional upsampling framework within the GCU to ensure model stability and enhance performance. The experimental results on three benchmark datasets demonstrated that the proposed single hyperspectral image SR using a progressive upsampling deep prior network (PUDPN) method effectively improves the reconstruction quality of hyperspectral images and achieves promising performance.

    Citation: Haijun Wang, Wenli Zheng, Yaowei Wang, Tengfei Yang, Kaibing Zhang, Youlin Shang. Single hyperspectral image super-resolution using a progressive upsampling deep prior network[J]. Electronic Research Archive, 2024, 32(7): 4517-4542. doi: 10.3934/era.2024205

    Related Papers:

    [1] Chunyan Luo, Yuping Yu, Tingsong Du . Estimates of bounds on the weighted Simpson type inequality and their applications. AIMS Mathematics, 2020, 5(5): 4644-4661. doi: 10.3934/math.2020298
    [2] Sabir Hussain, Javairiya Khalid, Yu Ming Chu . Some generalized fractional integral Simpson’s type inequalities with applications. AIMS Mathematics, 2020, 5(6): 5859-5883. doi: 10.3934/math.2020375
    [3] Muhammad Tariq, Hijaz Ahmad, Soubhagya Kumar Sahoo, Artion Kashuri, Taher A. Nofal, Ching-Hsien Hsu . Inequalities of Simpson-Mercer-type including Atangana-Baleanu fractional operators and their applications. AIMS Mathematics, 2022, 7(8): 15159-15181. doi: 10.3934/math.2022831
    [4] Xuexiao You, Fatih Hezenci, Hüseyin Budak, Hasan Kara . New Simpson type inequalities for twice differentiable functions via generalized fractional integrals. AIMS Mathematics, 2022, 7(3): 3959-3971. doi: 10.3934/math.2022218
    [5] Maimoona Karim, Aliya Fahmi, Shahid Qaisar, Zafar Ullah, Ather Qayyum . New developments in fractional integral inequalities via convexity with applications. AIMS Mathematics, 2023, 8(7): 15950-15968. doi: 10.3934/math.2023814
    [6] Saad Ihsan Butt, Artion Kashuri, Muhammad Umar, Adnan Aslam, Wei Gao . Hermite-Jensen-Mercer type inequalities via Ψ-Riemann-Liouville k-fractional integrals. AIMS Mathematics, 2020, 5(5): 5193-5220. doi: 10.3934/math.2020334
    [7] Areej A. Almoneef, Abd-Allah Hyder, Fatih Hezenci, Hüseyin Budak . Simpson-type inequalities by means of tempered fractional integrals. AIMS Mathematics, 2023, 8(12): 29411-29423. doi: 10.3934/math.20231505
    [8] Naila Mehreen, Matloob Anwar . Some inequalities via Ψ-Riemann-Liouville fractional integrals. AIMS Mathematics, 2019, 4(5): 1403-1415. doi: 10.3934/math.2019.5.1403
    [9] Ghulam Farid, Saira Bano Akbar, Shafiq Ur Rehman, Josip Pečarić . Boundedness of fractional integral operators containing Mittag-Leffler functions via (s,m)-convexity. AIMS Mathematics, 2020, 5(2): 966-978. doi: 10.3934/math.2020067
    [10] Yousaf Khurshid, Muhammad Adil Khan, Yu-Ming Chu . Conformable fractional integral inequalities for GG- and GA-convex functions. AIMS Mathematics, 2020, 5(5): 5012-5030. doi: 10.3934/math.2020322
  • Hyperspectral image super-resolution (SR) aims to enhance the spectral and spatial resolution of remote sensing images, enabling more accurate and detailed analysis of ground objects. However, hyperspectral images have high dimensional characteristics and complex spectral patterns. As a result, it is critical to effectively leverage the spatial non-local self-similarity and spectral correlation within hyperspectral images. To address this, we have proposed a novel single hyperspectral image SR method based on a progressive upsampling deep prior network. Specifically, we introduced the spatial-spectral attention fusion unit (S2AF) based on residual connections, in order to extract spatial and spectral information from hyperspectral images. Then we developed the group convolutional upsampling (GCU) to efficiently utilize the spatial and spectral prior information inherent in hyperspectral images. To address the challenges posed by the high dimensionality of hyperspectral images and limited training dataset, we implemented a parameter-sharing grouped convolutional upsampling framework within the GCU to ensure model stability and enhance performance. The experimental results on three benchmark datasets demonstrated that the proposed single hyperspectral image SR using a progressive upsampling deep prior network (PUDPN) method effectively improves the reconstruction quality of hyperspectral images and achieves promising performance.



    Critical materials are garnering many research interests from academia, industry, and defense sectors due to their increasing demand for clean-energy solutions and the potential for significant supply risks. These materials include cobalt, lithium, manganese, and rare earth elements (REEs), which have limited production capacities but are increasingly used in lithium-ion batteries (LIBs) and neodymium–iron–boron (Nd–Fe–B) magnets for electric vehicles (EVs), and renewable energy generation and/or storage. To achieve net zero emissions by 2050, the demand for these critical materials is projected to surpass the supply. Increased mining of critical materials from ores creates extra burdens or disturbances on the environment and affected communities, and recycling of these valuable materials from end-of-life (EOL) products can be a promising alternative from both economic and environmental perspectives. This special issue aims to gather up-to-date knowledge related to cutting-edge research in the broad scientific area of critical materials for clean energy applications.

    This special issue consists of five articles. In the first paper of this special issue, Alipanah et al. [1] presented a review of emerging technologies and pathways such as refurbishing, direct recycling (i.e., cathode-to-cathode), and hydrometallurgical and pyrometallurgical processes for critical materials recovery from spent LIBs. The study revealed the economic and environmental advantages of LIB reuse over materials recycling, though significant research and infrastructure developments are required. Among the materials recycling methods, direct recycling is superior in closing the loop with less chemicals and energy consumed. However, high operational costs and changes in battery chemistry over time could limit the widespread application of direct recycling. To this end, this paper also reviewed the government policies adopted by Europe and the US for promoting LIB recycling.

    In the second review paper, Ji et al. [2] provided a comprehensive review on the recent advancements in each step of the direct recycling process, namely, harvesting cathode materials, separation of cathode active materials from other components through thermal and floatation processes, and regeneration of degraded electrochemical performance of homogenous cathode materials through relithiation (e.g., solid-state relithiation, hydrothermal relithiation). The authors emphasized complete separation of cathode materials from binders and carbon, as the presence of residue affected the electrochemical performance of regenerated cathode materials. Moreover, they suggested future endeavors to minimize fluoride emissions during the separation process.

    The special issue includes another mini-review paper on recent advances in acid-free dissolution and separation of REEs from Nd–Fe–B and samarium–cobalt (Sm–Co) magnet wastes by Inman et al. [3]. The research was motivated by the fact that acid-based hydrometallurgical processes generate substantial amounts of hazardous waste, which needs to be controlled to avoid environmental hazards for recovering REEs. A promising solution is to dissolve magnet materials using an aqueous solution of a copper (Ⅱ) salt, which transfers pertinent REEs to the dissolved solution. With further filtration, precipitation, and calcination procedures, mixed rare earth oxides were produced with a yield of > 98%. Separation of heavy REEs (e.g., Dy) from light REEs (e.g., Nd, Pr) was also investigated, highlighting the research need to develop economically and environmentally sound alternatives to traditional solvent extraction (SX) route.

    Alongside these review papers, the special issue includes two research articles. In the first article, Maria et al. [4] emphasized the inclusion of temporal information while calculating the environmental impacts of buildings. The authors performed life cycle analysis to evaluate both static and dynamic global warming impact for two newly developed construction materials: (ⅰ) goethite-based inorganic polymers (GIP), and (ⅱ) stainless steel slag-based alkali-activated aerated blocks (SSSaer), compared to traditional autoclaved aerated ordinary Portland cement (OPC) concrete. Although both static and dynamic approaches provided similar results, the latter allowed a more informed analysis of emission flows over time. According to their analysis, GIP presents the highest global warming impact at any time horizon, both for the static and the dynamic approach, while SSSaer has the lowest impact.

    As the final article of the special issue, Nguyen et al. [5] presented a market-oriented critical-materials database, which aimed to help material researchers gain a better understanding of the market for 29 critical materials. The database provided insightful information regarding the most impactful applications of each element, as well as their industry specifications, prices, product composition, and global consumption.

    We would like to thank all the authors and reviewers who have contributed their exceptional work to this special issue of Critical Materials for Low Carbon Society. We also appreciate the technical and administrative support from the editors and editorial board members of the Journal of Clean Technologies and Recycling. We hope this special issue provides an archive of stimulating articles that contribute to industrial decarbonization.

    The authors have no conflicts of interest to declare.



    [1] B. Lu, P. D. Dao, J. Liu, Y. He, J. Shang, Recent advances of hyperspectral imaging technology and applications in agriculture, Remote Sens., 12 (2020), 2659. https://doi.org/10.3390/rs12162659 doi: 10.3390/rs12162659
    [2] B. P. Banerjee, S. Raval, P. J. Cullen, UAV-hyperspectral imaging of spectrally complex environments, Int. J. Remote Sens., 41 (2020), 4136–4159. https://doi.org/10.1080/01431161.2020.1714771 doi: 10.1080/01431161.2020.1714771
    [3] M. Shimoni, R. Haelterman, C. Perneel, Hyperspectral imaging for military and security applications: combining myriad processing and sensing techniques, IEEE Geosci. Remote Sens. Mag., 7 (2019), 101–117. https://doi.org/10.1109/MGRS.2019.2902525 doi: 10.1109/MGRS.2019.2902525
    [4] J. M. Bioucas-Dias, A. Plaza, G. Camps-Valls, P. Scheunders, N. Nasrabadi, J. Chanussot, Hyperspectral remote sensing data analysis and future challenges, IEEE Geosci. Remote Sens. Mag., 1 (2013), 6–36. https://doi.org/10.1109/MGRS.2013.2244672 doi: 10.1109/MGRS.2013.2244672
    [5] W. Xie, X. Jia, Y. Li, J. Lei, Hyperspectral image super-resolution using deep feature matrix factorization, IEEE Trans. Image Process., 57 (2019), 6055–6067. https://doi.org/10.1109/TGRS.2019.2904108 doi: 10.1109/TGRS.2019.2904108
    [6] W. Dong, F. Fu, G. Shi, X. Gao, J. Wu, G. Li, et al., Hyperspectral image super-resolution via non-negative structured sparse representation, IEEE Trans. Image Process., 25 (2016), 2337–2352. https://doi.org/10.1109/TIP.2016.2542360 doi: 10.1109/TIP.2016.2542360
    [7] W. Wan, W. Guo, H. Huang, J. Liu, Nonnegative and nonlocal sparse tensor factorization-based hyperspectral image super-resolution, IEEE Trans. Geosci. Remote Sensing, 58 (2020), 8384–8394. https://doi.org/10.1109/TGRS.2020.2987530 doi: 10.1109/TGRS.2020.2987530
    [8] S. C. Park, M. K. Park, M. G. Kang, Super-resolution image reconstruction: a technical overview, IEEE Signal Process. Mag., 20 (2003), 21–36. https://doi.org/10.1109/MSP.2003.1203207 doi: 10.1109/MSP.2003.1203207
    [9] Q. Wei, N. Dobigeon, J. Y. Tourneret, Bayesian fusion of multiband images, IEEE J. Sel. Top. Signal Process., 9 (2015), 1117–1127. https://doi.org/10.1109/JSTSP.2015.2407855 doi: 10.1109/JSTSP.2015.2407855
    [10] N. Yokoya, T. Yairi, A. Iwasaki, Coupled nonnegative matrix factorization unmixing for hyperspectral and multispectral data fusion, IEEE Trans. Geosci. Remote Sensing, 50 (2011), 528–537. https://doi.org/10.1109/TGRS.2011.2161320 doi: 10.1109/TGRS.2011.2161320
    [11] N. Akhtar, F. Shafait, A. Mian, Sparse spatio-spectral representation for hyperspectral image super-resolution, in Proceedings of the European Conference on Computer Vision (ECCV), Springer, (2014), 63–78. https://doi.org/10.1007/978-3-319-10584-0_5
    [12] Y. Zhou, A. Rangarajan, P. D. Gader, An integrated approach to registration and fusion of hyperspectral and multispectral images, IEEE Trans. Geosci. Remote Sensing, 58 (2020), 3020–3033. https://doi.org/10.1109/TGRS.2019.2946803 doi: 10.1109/TGRS.2019.2946803
    [13] S. He, H. Zhou, Y. Wang, W. Cao, Z. Han, Super-resolution reconstruction of hyperspectral images via low rank tensor modeling and total variation regularization, in 2016 IEEE International Geoscience and Remote Sensing Symposium (IGARSS), IEEE, (2016), 6962–6965. https://doi.org/10.1109/IGARSS.2016.7730816
    [14] R. Dian, L. Fang, S. Li, Hyperspectral image super-resolution via non-local sparse tensor factorization, in 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, (2017), 3862–3871. https://doi.org/10.1109/CVPR.2017.411
    [15] H. Huang, J. Yu, W. Sun, Super-resolution mapping via multi-dictionary based sparse representation, in 2014 IEEE International Conference on Acoustics, Speech Signal Processing (ICASSP), IEEE, (2014), 3523–3527. https://doi.org/10.1109/ICASSP.2014.6854256
    [16] Q. Li, Q. Wang, X. Li, Mixed 2D/3D convolutional network for hyperspectral image super-resolution, Remote Sens., 12 (2020), 1660. https://doi.org/10.3390/rs12101660 doi: 10.3390/rs12101660
    [17] J. Hou, Z. Zhu, J. Hou, H. Zeng, J. Wu, J. Zhou, Deep posterior distribution-based embedding for hyperspectral image super-resolution, IEEE Trans. Image Process., 31 (2022), 5720–5732. https://doi.org/10.1109/TIP.2022.3201478 doi: 10.1109/TIP.2022.3201478
    [18] S. Mei, X. Yuan, J. Ji, Y. Zhang, S. Wan, Q. Du, Hyperspectral image spatial super-resolution via 3d full convolutional neural network, Remote Sens., 9 (2017), 1139. https://doi.org/10.3390/rs9111139 doi: 10.3390/rs9111139
    [19] J. Jiang, H. Sun, X. Liu, J. Ma, Learning spatial-spectral prior for super-resolution of hyperspectral imagery, IEEE Trans. Comput. Imaging, 6 (2020), 1082–1096. https://doi.org/10.1109/TCI.2020.2996075 doi: 10.1109/TCI.2020.2996075
    [20] Y. Long, X. Wang, M. Xu, S. Zhang, S. Jiang, S. Jia, Dual self-attention swin transformer for hyperspectral image super-resolution, IEEE Trans. Geosci. Remote Sensing, 61 (2023), 5512012. https://doi.org/10.1109/TGRS.2023.3275146 doi: 10.1109/TGRS.2023.3275146
    [21] M. Zhao, J. Ning, J. Hu, T. Li, Attention-driven dual feature guidance for hyperspectral super-resolution, IEEE Trans. Geosci. Remote Sensing, 61 (2023), 5525116. https://doi.org/10.1109/TGRS.2023.3318013 doi: 10.1109/TGRS.2023.3318013
    [22] Y. Li, L. Zhang, C. Dingl, W. Wei, Y. Zhang, Single hyperspectral Image super-resolution with grouped deep recursive residual network, in 2018 IEEE Fourth International Conference on Multimedia Big Data (BigMM), IEEE, (2018), 1–4. https://doi.org/10.1109/BigMM.2018.8499097
    [23] Q. Wei, J. Bioucas-Dias, N. Dobigeon, J. Y. Tourneret, Hyperspectral and multispectral image fusion based on a sparse representation, IEEE Trans. Geosci. Remote Sensing, 53 (2015), 3658–3668. https://doi.org/10.1109/TGRS.2014.2381272 doi: 10.1109/TGRS.2014.2381272
    [24] Y. Xu, Z. Wu, J. Chanussot, Z. Wei, Nonlocal patch tensor sparse representation for hyperspectral image super-resolution, IEEE Trans. Image Process., 28 (2019), 3034–3047. https://doi.org/10.1109/TIP.2019.2893530 doi: 10.1109/TIP.2019.2893530
    [25] X. H. Han, B. Shi, Y. Zheng, Self-similarity constrained sparse representation for hyperspectral image super-resolution, IEEE Trans. Image Process., 27 (2018), 5625–5637. https://doi.org/10.1109/TIP.2018.2855418 doi: 10.1109/TIP.2018.2855418
    [26] L. Zhang, W. Wei, C. Bai, Y. Gao, Y. Zhang, Exploiting clustering manifold structure for hyperspectral imagery super-resolution, IEEE Trans. Image Process., 27 (2018), 5969–5982. https://doi.org/10.1109/TIP.2018.2862629 doi: 10.1109/TIP.2018.2862629
    [27] M. A. Veganzones, M. Simoes, G. Licciardi, N. Yokoya, J. M. BioucasDias, J. Chanussot, Hyperspectral super-resolution of locally low rank images from complementary multisource data, IEEE Trans. Image Process., 25 (2015), 274–288. https://doi.org/10.1109/TIP.2015.2496263 doi: 10.1109/TIP.2015.2496263
    [28] R. Dian, S. Li, Hyperspectral image super-resolution via subspacebased low tensor multi-rank regularization, IEEE Trans. Image Process., 28 (2019), 5135–5146. https://doi.org/10.1109/TIP.2019.2916734 doi: 10.1109/TIP.2019.2916734
    [29] Q. Xie, M. Zhou, Q. Zhao, D. Meng, W. Zuo, Z. Xu, Multispectral and hyperspectral image fusion by ms/hs fusion net, in 2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), IEEE, (2019), 1585–1594. https://doi.org/10.1109/CVPR.2019.00168
    [30] Z. W. Pan, H. L. Shen, Multispectral image super-resolution via RGB image fusion and radiometric calibration, IEEE Trans. Image Process., 28 (2019), 1783–1797. https://doi.org/10.1109/TIP.2018.2881911 doi: 10.1109/TIP.2018.2881911
    [31] C. Dong, C. C. Loy, K. He, X. Tang, Learning a deep convolutional network for image super-resolution, in Computer Vision-ECCV 2014, Springer, (2014), 184–199. https://doi.org/10.1007/978-3-319-10593-2_13
    [32] L. Liebel, M. Körner, Single-image super resolution for multispectral remote sensing data using convolutional neural networks, Int. Arch. Photogramm. Remote Sens. Spatial Inf. Sci., 41 (2016), 883–890. https://doi.org/10.5194/isprs-archives-XLI-B3-883-2016 doi: 10.5194/isprs-archives-XLI-B3-883-2016
    [33] Y. Yuan, X. Zheng, X. Lu, Hyperspectral image superresolution by transfer learning, IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens., 10 (2017), 1963–1974. https://doi.org/10.1109/JSTARS.2017.2655112 doi: 10.1109/JSTARS.2017.2655112
    [34] S. Woo, J. Park, J. Y. Lee, I. S. Kweon, CBAM: convolutional block attention module, in Proceedings of the European conference on computer vision (ECCV), IEEE, (2018), 3–19.
    [35] V. Lempitsky, A. Vedaldi, D. Ulyanov, Deep image prior, in 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition, IEEE, (2018), 9446–9454. https://doi.org/10.1109/CVPR.2018.00984
    [36] O. Sidorov, J. Y. Hardeberg, Deep hyperspectral prior: single-image denoising, inpainting, super-resolution, in 2019 IEEE/CVF International Conference on Computer Vision Workshop (ICCVW), IEEE, (2019), 3844–3851. https://doi.org/10.1109/ICCVW.2019.00477
    [37] C. Dong, C. C. Loy, K. He, X. Tang, Image super-resolution using deep convolutional networks, IEEE Trans. Pattern Anal. Mach. Intell., 38 (2016), 295–307. https://doi.org/10.1109/TPAMI.2015.2439281 doi: 10.1109/TPAMI.2015.2439281
    [38] D. Liu, J. Li, Q. Yuan, A spectral grouping and attention-driven residual dense network for hyperspectral image super-resolution, IEEE Trans. Geosci. Remote Sensing, 59 (2021), 7711–7725. https://doi.org/10.1109/TGRS.2021.3049875 doi: 10.1109/TGRS.2021.3049875
    [39] X. Wang, Q. Hu, J. Jiang, J. Ma, A group-based embedding learning and integration network for hyperspectral image super-resolution, IEEE Trans. Geosci. Remote Sensing, 60 (2022), 5541416. https://doi.org/10.1109/TGRS.2022.3217406 doi: 10.1109/TGRS.2022.3217406
    [40] T. Liu, Y. Liu, C. Zhang, L. Yuan, X. Sui, Q. Chen, Hyperspectral image super-resolution via dual-domain network based on hybrid convolution, IEEE Trans. Geosci. Remote Sensing, 62 (2024), 5512518. https://doi.org/10.1109/TGRS.2024.3370107 doi: 10.1109/TGRS.2024.3370107
    [41] S. Chen, L. Zhang, L. Zhang, Cross-scope spatial-spectral information aggregation for hyperspectral image super-resolution, preprint, arXiv: 2311.17340.
    [42] M. Zhang, C. Zhang, Q. Zhang, J. Guo, X. Gao, J. Zhang, Essaformer: efficient transformer for hyperspectral image super-resolution, in 2023 IEEE/CVF International Conference on Computer Vision (ICCV), IEEE, (2023), 23016–23027. https://doi.org/10.1109/ICCV51070.2023.02109
    [43] X. Huang, L. Zhang, A comparative study of spatial approaches for urban mapping using hyperspectral rosis images over pavia city, Int. J. Remote Sens., 30 (2009), 3205–3221. https://doi.org/10.1080/01431160802559046 doi: 10.1080/01431160802559046
    [44] F. Yasuma, T. Mitsunaga, D. Iso, S. K. Nayar, Generalized assorted pixel camera: postcapture control of resolution, dynamic range, and spectrum, IEEE Trans. Image Process., 19 (2010), 2241–2253. https://doi.org/10.1109/TIP.2010.2046811 doi: 10.1109/TIP.2010.2046811
    [45] N. Yokoya, A. Iwasaki, Airborne hyperspectral data over chikusei, Space Appl. Lab., Univ. Tokyo, Tokyo, 5 (2016), 1–6. https://doi.org/10.1109/TIP.2010.2046811 doi: 10.1109/TIP.2010.2046811
    [46] H. Hou, H. Andrews, Cubic splines for image interpolation and digital filtering, IEEE Trans. Acoust. Speech Signal Process., 26 (1978), 508–517. https://doi.org/10.1109/TASSP.1978.1163154 doi: 10.1109/TASSP.1978.1163154
    [47] Z. Liu, Y. Lin, Y. Cao, H. Hu, Y. Wei, Z. Zhang, et al., Swin transformer: hierarchical vision transformer using shifted windows, in 2021 IEEE/CVF International Conference on Computer Vision (ICCV), IEEE, (2021), 9992–10002. https://doi.org/10.1109/ICCV48922.2021.00986
  • This article has been cited by:

    1. Saima Rashid, Muhammad Aslam Noor, Khalida Inayat Noor, Farhat Safdar, Yu-Ming Chu, Hermite-Hadamard Type Inequalities for the Class of Convex Functions on Time Scale, 2019, 7, 2227-7390, 956, 10.3390/math7100956
    2. Saima Rashid, Fahd Jarad, Muhammad Aslam Noor, Humaira Kalsoom, Yu-Ming Chu, Inequalities by Means of Generalized Proportional Fractional Integral Operators with Respect to Another Function, 2019, 7, 2227-7390, 1225, 10.3390/math7121225
    3. Saima Rashid, Fahd Jarad, Muhammad Aslam Noor, Grüss-type integrals inequalities via generalized proportional fractional operators, 2020, 114, 1578-7303, 10.1007/s13398-020-00823-5
    4. Muhammad Samraiz, Fakhra Nawaz, Sajid Iqbal, Thabet Abdeljawad, Gauhar Rahman, Kottakkaran Sooppy Nisar, Certain mean-type fractional integral inequalities via different convexities with applications, 2020, 2020, 1029-242X, 10.1186/s13660-020-02474-x
    5. Thabet Abdeljawad, Saima Rashid, Zakia Hammouch, İmdat İşcan, Yu-Ming Chu, Some new Simpson-type inequalities for generalized p-convex function on fractal sets with applications, 2020, 2020, 1687-1847, 10.1186/s13662-020-02955-9
    6. Zeynep ŞANLI, SIMPSON TYPE INTEGRAL INEQUALITIES FOR HARMONIC CONVEX FUNCTIONS VIA CONFORMABLE FRACTIONAL INTEGRALS, 2020, 2147-1630, 10.37094/adyujsci.780433
    7. Thabet Abdeljawad, 2020, 9781119654223, 133, 10.1002/9781119654223.ch5
    8. Shuang-Shuang Zhou, Saima Rashid, Saima Parveen, Ahmet Ocak Akdemir, Zakia Hammouch, New computations for extended weighted functionals within the Hilfer generalized proportional fractional integral operators, 2021, 6, 2473-6988, 4507, 10.3934/math.2021267
    9. Thabet Abdeljawad, Saima Rashid, Zakia Hammouch, Yu-Ming Chu, Some new local fractional inequalities associated with generalized (s,m)-convex functions and applications, 2020, 2020, 1687-1847, 10.1186/s13662-020-02865-w
    10. Humaira Kalsoom, Saima Rashid, Muhammad Idrees, Yu-Ming Chu, Dumitru Baleanu, Two-Variable Quantum Integral Inequalities of Simpson-Type Based on Higher-Order Generalized Strongly Preinvex and Quasi-Preinvex Functions, 2019, 12, 2073-8994, 51, 10.3390/sym12010051
    11. Saima Rashid, Fahd Jarad, Zakia Hammouch, Some new bounds analogous to generalized proportional fractional integral operator with respect to another function, 2021, 0, 1937-1179, 0, 10.3934/dcdss.2021020
    12. Saima Rashid, Fahd Jarad, Muhammad Aslam Noor, Khalida Inayat Noor, Dumitru Baleanu, Jia-Bao Liu, On Grüss inequalities within generalized K-fractional integrals, 2020, 2020, 1687-1847, 10.1186/s13662-020-02644-7
    13. Saad Ihsan Butt, Mehroz Nadeem, Shahid Qaisar, Ahmet Ocak Akdemir, Thabet Abdeljawad, Hermite–Jensen–Mercer type inequalities for conformable integrals and related results, 2020, 2020, 1687-1847, 10.1186/s13662-020-02968-4
    14. Saima Rashid, Muhammad Amer Latif, Zakia Hammouch, Yu-Ming Chu, Fractional Integral Inequalities for Strongly h -Preinvex Functions for a kth Order Differentiable Functions, 2019, 11, 2073-8994, 1448, 10.3390/sym11121448
    15. SAIMA RASHID, ZAKIA HAMMOUCH, FAHD JARAD, YU-MING CHU, NEW ESTIMATES OF INTEGRAL INEQUALITIES VIA GENERALIZED PROPORTIONAL FRACTIONAL INTEGRAL OPERATOR WITH RESPECT TO ANOTHER FUNCTION, 2020, 28, 0218-348X, 2040027, 10.1142/S0218348X20400277
    16. Fatih Hezenci, Hüseyin Budak, Hasan Kara, New version of fractional Simpson type inequalities for twice differentiable functions, 2021, 2021, 1687-1847, 10.1186/s13662-021-03615-2
    17. Hüseyin Budak, Seda Kilinç Yildirim, Hasan Kara, Hüseyin Yildirim, On new generalized inequalities with some parameters for coordinated convex functions via generalized fractional integrals, 2021, 44, 0170-4214, 13069, 10.1002/mma.7610
    18. Muhammad Aamir Ali, Hasan Kara, Jessada Tariboon, Suphawat Asawasamrit, Hüseyin Budak, Fatih Hezenci, Some New Simpson’s-Formula-Type Inequalities for Twice-Differentiable Convex Functions via Generalized Fractional Operators, 2021, 13, 2073-8994, 2249, 10.3390/sym13122249
    19. Humaira Kalsoom, Hüseyin Budak, Hasan Kara, Muhammad Aamir Ali, Some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals, 2021, 19, 2391-5455, 1153, 10.1515/math-2021-0072
    20. İmdat İşcan, Erhan Set, Ahmet Ocak Akdemir, Alper Ekinci, Sinan Aslan, 2023, 9780323909532, 157, 10.1016/B978-0-32-390953-2.00017-7
    21. MIAO-KUN WANG, SAIMA RASHID, YELIZ KARACA, DUMITRU BALEANU, YU-MING CHU, NEW MULTI-FUNCTIONAL APPROACH FOR κTH-ORDER DIFFERENTIABILITY GOVERNED BY FRACTIONAL CALCULUS VIA APPROXIMATELY GENERALIZED (ψ, ℏ)-CONVEX FUNCTIONS IN HILBERT SPACE, 2021, 29, 0218-348X, 2140019, 10.1142/S0218348X21400193
    22. Saima Rashid, Dumitru Baleanu, Yu-Ming Chu, Some new extensions for fractional integral operator having exponential in the kernel and their applications in physical systems, 2020, 18, 2391-5471, 478, 10.1515/phys-2020-0114
    23. Xuexiao You, Fatih Hezenci, Hüseyin Budak, Hasan Kara, New Simpson type inequalities for twice differentiable functions via generalized fractional integrals, 2022, 7, 2473-6988, 3959, 10.3934/math.2022218
    24. YONG-MIN LI, SAIMA RASHID, ZAKIA HAMMOUCH, DUMITRU BALEANU, YU-MING CHU, NEW NEWTON’S TYPE ESTIMATES PERTAINING TO LOCAL FRACTIONAL INTEGRAL VIA GENERALIZED p-CONVEXITY WITH APPLICATIONS, 2021, 29, 0218-348X, 2140018, 10.1142/S0218348X21400181
    25. Miguel Vivas-Cortez, Pshtiwan O. Mohammed, Y. S. Hamed, Artion Kashuri, Jorge E. Hernández, Jorge E. Macías-Díaz, On some generalized Raina-type fractional-order integral operators and related Chebyshev inequalities, 2022, 7, 2473-6988, 10256, 10.3934/math.2022571
    26. Yu-Ming Chu, Saima Rashid, Fahd Jarad, Muhammad Aslam Noor, Humaira Kalsoom, More new results on integral inequalities for generalized K-fractional conformable Integral operators, 2021, 14, 1937-1179, 2119, 10.3934/dcdss.2021063
    27. Nassima Nasri, Badreddine Meftah, Abdelkader Moumen, Hicham Saber, Fractional 3/8-Simpson type inequalities for differentiable convex functions, 2024, 9, 2473-6988, 5349, 10.3934/math.2024258
    28. Fatih Hezenci, Hüseyin Budak, A note on fractional Simpson-like type inequalities for functions whose third derivatives are convex, 2023, 37, 0354-5180, 3715, 10.2298/FIL2312715H
    29. Zeynep Sanlı, Simpson type Katugampola fractional integral inequalities via Harmonic convex functions, 2022, 10, 23193786, 364, 10.26637/mjm1004/007
    30. Meriem Merad, Badreddine Meftah, Abdelkader Moumen, Mohamed Bouye, Fractional Maclaurin-Type Inequalities for Multiplicatively Convex Functions, 2023, 7, 2504-3110, 879, 10.3390/fractalfract7120879
    31. Fatih Hezenci, A Note on Fractional Simpson Type Inequalities for Twice Differentiable Functions, 2023, 73, 1337-2211, 675, 10.1515/ms-2023-0049
    32. Hüseyin Budak, Fatih Hezenci, Hasan Kara, Mehmet Zeki Sarikaya, Bounds for the Error in Approximating a Fractional Integral by Simpson’s Rule, 2023, 11, 2227-7390, 2282, 10.3390/math11102282
    33. Nazakat Nazeer, Ali Akgül, Faeem Ali, Study of the results of Hilbert transformation for some fractional derivatives, 2024, 0228-6203, 1, 10.1080/02286203.2024.2371685
    34. Tarek Chiheb, Badreddine Meftah, Abdelkader Moumen, Mouataz Billah Mesmouli, Mohamed Bouye, Some Simpson-like Inequalities Involving the (s,m)-Preinvexity, 2023, 15, 2073-8994, 2178, 10.3390/sym15122178
    35. Gamzenur Köksaldı, Tuba Tunç, Simpson-type inequalities for the functions with bounded second derivatives involving generalized fractional integrals, 2025, 2025, 1687-2770, 10.1186/s13661-025-02040-8
  • Reader Comments
  • © 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1354) PDF downloads(52) Cited by(0)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog