The theory of fractional calculus extends the order of classical integer calculus from integer to non-integer. As a new engineering application tool, it has made many important research achievements in many fields, including image processing. This paper mainly studies the application of fractional calculus theory in image enhancement and denoising, including the basic theory of fractional calculus and its amplitude frequency characteristics, the application of fractional differential operator in image enhancement, and the application of fractional integral operator in image denoising. The experimental results show that the fractional calculus theory has more special advantages in image enhancement and denoising. Compared with the existing integer order image enhancement operators, the fractional differential operator can more effectively enhance the "weak edge" and "strong texture" details of the image. The fractional order integral image denoising operator can not only improve the signal-to-noise ratio of the image compared to traditional denoising methods, but also better preserve detailed information such as edges and textures of the image.

Aluminum (Al) and manganese (Mn) toxicity are the top two constraints to crop production in acid soil. Crops have evolved common and specific mechanisms to tolerate the two stresses. In the present study, the responses (toxicity and tolerance) of wheat near-isogenic lines (ET8 and ES8) and their parents (Carazinho and Egret) to Al and Mn were compared by determining physiolog-ical parameters and transcriptome profiling of roots. The results showed that: (1) Carazinho and ET8 exhibited dual tolerance to Al and Mn as compared with Egret and ES8, indicated by higher relative root elongation and SPAD; (2) After entering into roots, Al was mainly distributed in the root and fixed in the cell wall, while Mn was mainly distributed in the cell sap and then trans-ported to leaves. Both Al and Mn stresses decreased the contents of Ca, Mg, and Zn; Mn stress al-so inhibited the accumulation of Fe, while Al showed an opposite effect; (3) Transcriptomic analy-sis identified 5581 differentially expressed genes (DEGs) under Al stress and 4165 DEGs under Mn stress. Among these, 2774 DEGs were regulated by both Al and Mn stresses, while 2280 and 1957 DEGs were exclusively regulated by Al and Mn stress, respectively. GO and KEGG analyses indi-cated that cell wall metabolism responds exclusively to Al, while nicotianamine synthesis exclu-sively responds to Mn. Pathways such as signaling, phenylpropanoid metabolism, and metal ion transport showed commonality and specificity to Al and Mn. TFs, such as MYB, WRKY, and AP2 families, were also regulated by Al and Mn, and WGCNA identified PODP7, VATB2, and ABCC3 as the hub genes for Al tolerance and NAS for Mn tolerance. The identified genes and pathways can be used as targets for pyramiding genes and breeding multi-tolerant varieties.