  Photo-Acoustic Tomography (PAT) can reconstruct a distribution of optical
absorbers acting as instantaneous sound sources in subcutaneous
microvasculature of a human breast. Adjoint methods for PAT, typically
Time-Reversal (TR) and Back-Projection (BP), are ways to refocus time-reversed
acoustic signals on sources by wave propagation from the position of sensors.
TR and BP have different treatments for received signals, but they are
equivalent under continuously sampling on a closed circular sensor array in two
dimensions. Here, we analyze image quality with discrete under-sampled sensors
in the sense of the Shannon sampling theorem. We investigate resolution and
contrast of TR and BP, respectively in one source-sensor pair configuration and
the frequency domain. With Hankel's asymptotic expansion to the integrands of
imaging functions, our main contribution is to demonstrate that TR and BP have
better performance on contrast and resolution, respectively. We also show that
the integrand of TR includes additional side lobes which degrade axial
resolution whereas that of BP conversely has relatively small amplitudes.
Moreover, omnidirectional resolution is improved if more sensors are employed
to collect the received signals. Nevertheless, for the under-sampled sensors,
we propose the Truncated Back-Projection (TBP) method to enhance the contrast
of BP using removing higher frequency components in the received signals. We
conduct numerical experiments on the two-dimensional projected phantom model
extracted from OA-Breast Database. The experiments verify our theories and show
that the proposed TBP possesses better omnidirectional resolution as well as
contrast compared with TR and BP with under-sampled sensors.
