Transport of Intensity phase imaging by intensity spectrum fitting of exponentially spaced defocus planes

We propose an alternative method for solving the Transport of Intensity equation (TIE) from a stack of through-focus intensity images taken by a microscope or lensless imager. Our method enables quantitative phase and amplitude imaging with improved accuracy and reduced data capture, while also being computationally efficient and robust to noise. We use prior knowledge of how intensity varies with propagation in the spatial frequency domain in order to constrain a fitting algorithm [Gaussian process (GP) regression] for estimating the axial intensity derivative. Solving the problem in the frequency domain inspires an efficient measurement scheme which captures images at exponentially spaced focal steps, significantly reducing the number of images required. Low-frequency artifacts that plague traditional TIE methods can be suppressed without an excessive number of captured images. We validate our technique experimentally by recovering the phase of human cheek cells in a brightfield microscope.