This nonlinear inverse problem arises in a variety of applications such as microscopy [3, 4], crystallography [5], optical imaging [6, 7], and astronomy [8]. In its most commonly encountered form known as Fourier phase retrieval, the available measurements are Fourier intensities. Due to its nonlinear and ill-posed nature, solving the phase retrieval problem, particularly Fourier phase retrieval, is challenging even though a unique solution can be almost always guaranteed in various practical scenarios of interest [9]. Although several solution approaches exist for the phase retrieval problems, each of them has its own drawbacks. Classical approaches are alternating projection-based methods such as the popular Gerchberg-Saxton (GS), error-reduction and hybrid input-output (HIO) algorithms, and their variants [10, 11]. Such projection-based methods are widely used due to their computational e!ciency, simple implementation and flexibility to be used for di"erent phase retrieval problems. These methods jointly utilize available intensity measurements and information known a priori, and alternate between space and measurement domains to impose these constraints through projections [10-13].

Gram staining has been one of the most frequently used staining protocols in microbiology for over a century, utilized across various fields, including diagnostics, food safety, and environmental monitoring. Its manual procedures make it vulnerable to staining errors and artifacts due to, e.g., operator inexperience and chemical variations. Here, we introduce virtual Gram staining of label-free bacteria using a trained deep neural network that digitally transforms darkfield images of unstained bacteria into their Gram-stained equivalents matching brightfield image contrast. After a one-time training effort, the virtual Gram staining model processes an axial stack of darkfield microscopy images of label-free bacteria (never seen before) to rapidly generate Gram staining, bypassing several chemical steps involved in the conventional staining process. We demonstrated the success of the virtual Gram staining workflow on label-free bacteria samples containing Escherichia coli and Listeria innocua by quantifying the staining accuracy of the virtual Gram staining model and comparing the chromatic and morphological features of the virtually stained bacteria against their chemically stained counterparts. This virtual bacteria staining framework effectively bypasses the traditional Gram staining protocol and its challenges, including stain standardization, operator errors, and sensitivity to chemical variations.

The integration of deep learning techniques with biophotonic setups has opened new horizons in bioimaging. A compelling trend in this field involves deliberately compromising certain measurement metrics to engineer better bioimaging tools in terms of cost, speed, and form-factor, followed by compensating for the resulting defects through the utilization of deep learning models trained on a large amount of ideal, superior or alternative data. This strategic approach has found increasing popularity due to its potential to enhance various aspects of biophotonic imaging. One of the primary motivations for employing this strategy is the pursuit of higher temporal resolution or increased imaging speed, critical for capturing fine dynamic biological processes. This approach also offers the prospect of simplifying hardware requirements/complexities, thereby making advanced imaging standards more accessible in terms of cost and/or size. This article provides an in-depth review of the diverse measurement aspects that researchers intentionally impair in their biophotonic setups, including the point spread function, signal-to-noise ratio, sampling density, and pixel resolution. By deliberately compromising these metrics, researchers aim to not only recuperate them through the application of deep learning networks, but also bolster in return other crucial parameters, such as the field-of-view, depth-of-field, and space-bandwidth product. Here, we discuss various biophotonic methods that have successfully employed this strategic approach. These techniques span broad applications and showcase the versatility and effectiveness of deep learning in the context of compromised biophotonic data. Finally, by offering our perspectives on the future possibilities of this rapidly evolving concept, we hope to motivate our readers to explore novel ways of balancing hardware compromises with compensation via AI.