Phase-retrieval is crucial in different applications involving imaging. It not only seeks to reconstruct the missing phases associated with intensity-only measurements, but is also useful in applications characterized by polluted phase information. Coherent diffraction imaging and holography are the main approaches for phase-retrieval problems. Holography reconstructs the missing phases with the aid of a reference beam controlled to create interference fringes within the diffraction pattern that are proportional to the Fourier transform modulus of the object to be imaged. In contrast, coherent diffraction imaging uses no reference beam and forms the images based on intensity patterns which encode the missing phases. The encoded phases can be recovered via iterative phase-retrieval algorithms exploiting data redundancies. For the imaging modalities that produce two-dimensional (2D) diffraction patterns, tomographic methods can be used to assemble the patterns to form depth-resolved images.
Another key problem is the unwanted artifacts in the reconstructed phases that deteriorate the quality of the produced images. The artifacts are due to rotational and translational misalignments and can be avoided by producing holographic data from the intensity-only measurements. Since the data is centered, this approach is alignment-free, and allows for reconstructions even when subjected to random vibrations or constant drifts induced by the mechanical rotations. Additionally, this approach is direct and does not require iterative phase retrieval algorithms or oversampling. It requires, however, high control of the sources’ signals as well as the creation of the narrowband illuminations.
Herein, Professor Miguel Moscoso from Universidad Carlos III de Madrid, Professor Alexei Novikov from Penn State University, Professor George Papanicolaou from Stanford University, and Professor Chrysoula Tsogka from the University of California Merced have developed a new 3D imaging approach from single-element intensity data. The method involved two steps. In the first step, full cross-correlated data was recovered using the polarization identity and frequency diverse illuminations. These field cross-correlations are recovered up to a phase that depends on the source-detector measurement location. The asynchronous cross-correlations were aligned to image coherently in the second step, which aimed to refer all the data to the same global phase. Thus, the problem was addressed by a simple algorithm used to synchronize the cross-correlations at different locations. The work has been published in the research Journal of Optical Society of America A.
In their approach, the sample was scanned using a single moving source-detector pair, which collected intensity-only data at different positions and frequencies. The detector and source could be placed in the reflection mode or transmission mode. Due to the low image dimensional structure, l1-norm minimization was adopted to achieve high-resolution imaging. The authors obtained similar reconstruction results considering different acquisition schemes. Results showed that the source-detector configurations with improved phase diversity in the data provided improved reconstructions. The simulation results also demonstrated that for a source and detector moving along a circle opposite to each other to form angles close to 180 °, the resulting inversion was more unstable and susceptible to the noise in the data. Unlike most of the existing algorithms, the presented method is non-iterative and allows for full phase recovery without any constraint on the unknown except sparsity. Furthermore, it exhibited high image resolution accuracy for highly underdetermined problems where the number of data was significantly smaller than the unknowns.
In summary, a new computational imaging method for obtaining 3D images from intensity-only data was developed. It allowed for the recovery of full-field data up to a single global phase. Moreover, the l1-norm minimization algorithm, given the low dimensional structure of the images, enables for deep, accurate and high-resolution imaging. Based on the results, intensity-only data imaging was consistent with coherent imaging with full data. In a statement to Advances in Engineering, Professor Chrysoula Tsogka explained that the study shows that high resolution 3D imaging reconstructions can be achieved with intensity only data. This is important in applications where phases cannot be measured, or where the data are polluted by phase errors.
Moscoso, M., Novikov, A., Papanicolaou, G., & Tsogka, C. (2020). Three-dimensional imaging from single-element holographic data. Journal of the Optical Society of America A, 38(2), A1-A6.