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Additional file 1: Table S1. Bioinformatics-based identification of the AsCEP112 protein. Figure S1. Detection of AsCEP112 Protein Signal Peptide. Figure S3. The signal peptide (SP) of AsCEP112 is functional. The validation of the function of AsCEP112SP with yeast signal trap assay. The YTK12 yeast strain containing pSUC2 is able to grow on a CMD−W medium without tryptophan, but not on YPRAA medium. AsCEP112SP can grow on both CMD−W and YPRAA media. The SP of Avr1b was used as positive control. Figure S4. Transient expression of AsCEP112 in N. benthamiana leaves. The upper left and upper right corners of the leaf were injected with control (EV) and AsCEP112 (FL/NSP), respectively. The lower left and right corners were respectively injected with INF1 and AsCEP112 (FL/NSP) coupled with INF1. Figure S5. Figure S6. The colony areas and growth radii of the WT, AsCEP112 mutant and revertant strains. Figure S7. Determination of the phenotypes of the wild-type, AsCEP112 mutant and revertant strains. The colony phenotypes of Alternaria solani wild-type (left 1), AsCEP112 mutant (middle 2, 3) and revertant (right 4, 5) strains cultured on PDA medium for 7 d at 25°C in the dark. Figure S8. Pathogenicity detection of AsCEP112 gene. The isolated potato leaves were inoculated with the spore suspensions of AsCEP112 mutant strains (left, leaf tip to petiole direction), wild-type strains (upper right, leaf tip to petiole direction) and revertant strains (lower right, leaf tip to petiole direction). Figure S9. Full-length gel of Figure S5. The red line is the intercepted part. Figure S10. Subcellular localization of AsCEP112 proteins in N. benthamiana. The Agrobacterium strain EHA105 containing the pCAMBIA1301 vector, as the control, and the AsCEP112 gene were independently transiently expressed in N. benthamiana leaves. Bar = 20 μm. Figure S11. Full-length gel of Figure 4B. The red box is the intercepted part. Figure S12. Subcellular localization of AsCEP112 proteins in N. benthamiana. All images are AsCEP112.

Additional file 2. Figure S1. Schematic diagram of long-range synteny between genomes of strains NL03003 and HWC168. Figure S2. Multiple sequence alignment of AsCEP20 and its homologs. Figure S3. Transient expression of AsCEP19 and AsCEP20 in Nicotiana benthamiana leaves. Figure S4. PCR validation of the gene pair (AsCEP19 and AsCEP20) knockout mutant of Alternaria solani HWC168.

The l1-norm regularization has attracted attention for image reconstruction in computed tomography. The l0-norm of the gradients of an image provides a measure of the sparsity of gradients of the image. In this paper, we present a new combined l1-norm and l0-norm regularization model for image reconstruction from limited projection data in computed tomography. We also propose an algorithm in the algebraic framework to solve the optimization effectively using the nonmonotone alternating direction algorithm with hard thresholding method. Numerical experiments indicate that this new algorithm makes much improvement by involving l0-norm regularization.

The nonmonotone alternating direction algorithm (NADA) was recently proposed for effectively solving a class of equality-constrained nonsmooth optimization problems and applied to the total variation minimization in image reconstruction, but the reconstructed images suffer from the artifacts. Though by the ${l}_{\mathrm{0}}$ -norm regularization the edge can be effectively retained, the problem is NP hard. The smoothed ${l}_{\mathrm{0}}$ -norm approximates the ${l}_{\mathrm{0}}$ -norm as a limit of smooth convex functions and provides a smooth measure of sparsity in applications. The smoothed ${l}_{\mathrm{0}}$ -norm regularization has been an attractive research topic in sparse image and signal recovery. In this paper, we present a combined smoothed ${l}_{\mathrm{0}}$ -norm and ${l}_{\mathrm{1}}$ -norm regularization algorithm using the NADA for image reconstruction in computed tomography. We resolve the computation challenge resulting from the smoothed ${l}_{\mathrm{0}}$ -norm minimization. The numerical experiments demonstrate that the proposed algorithm improves the quality of the reconstructed images with the same cost of CPU time and reduces the computation time significantly while maintaining the same image quality compared with the ${l}_{\mathrm{1}}$ -norm regularization in absence of the smoothed ${l}_{\mathrm{0}}$ -norm.