  The transport-of-intensity equation (TIE) is a well-established
non-interferometric phase retrieval approach, which enables quantitative phase
imaging (QPI) of transparent sample simply by measuring the intensities at
multiple axially displaced planes. Nevertheless, it still suffers from two
fundamentally limitations. First, it is quite susceptible to low-frequency
errors (such as \cloudy" artifacts), which results from the poor contrast of
the phase transfer function (PTF) near the zero frequency. Second, the
reconstructed phase tends to blur under spatially low-coherent illumination,
especially when the defocus distance is beyond the near Fresnel region. Recent
studies have shown that the shape of the illumination aperture has a
significant impact on the resolution and phase reconstruction quality, and by
simply replacing the conventional circular illumination aperture with an
annular one, these two limitations can be addressed, or at least significantly
alleviated. However, the annular aperture was previously empirically designed
based on intuitive criteria related to the shape of PTF, which does not
guarantee optimality. In this work, we optimize the illumination pattern to
maximize TIE's performance based on a combined quantitative criterion for
evaluating the \goodness" of an aperture. In order to make the size of the
solution search space tractable, we restrict our attention to binary coded
axis-symmetric illumination patterns only, which are easier to implement and
can generate isotropic TIE PTFs. We test the obtained optimal illumination by
imaging both a phase resolution target and HeLa cells based on a small-pitch
LED array, suggesting superior performance over other suboptimal patterns in
terms of both signal-to-noise ratio (SNR) and spatial resolution.
