Transport of intensity equation from a single intensity image via deep learning

•Deep learning TIE is proposed.•Reducing the inputs required by TIE phase retrieval from multiple to one (simplicity).•Being free from the image boundary problem (tolerance).•Being more insensitive to noise than traditional transport of intensity equation (robustness). The transport of intensity equation (TIE) is an ideal candidate for phase imaging with partially coherent illuminations. TIE has the advantages of simplicity in phase calculation due to its closed-form solution and no requirement for a reference beam and phase unwrapping due to its non-interferometric nature. However, TIE requires multiple through-focus intensity images, and is very sensitive to image boundaries and noise. Thus, in this paper, we combine deep learning with TIE, abbreviated as dTIE. After being trained by TIE phase results, the dTIE retains the advantages of TIE, and overcomes the shortcomings of TIE as follows: (i) only one de-focus intensity image is required for phase imaging while the result is very close to the TIE result with SSIM index reaches 0.95, enabling more efficient phase imaging; (ii) the boundary problem automatically disappears due to the translation invariance of the convolutional networks; (iii) it is insensitive to noise even with very heavy noise. All these enhancements are verified in the application of dTIE for phase imaging of real cells.