  We introduce an efficient method for the reconstruction of the correlation
between a compressively measured image and a phase-only filter. The proposed
method is based on two properties of phase-only filtering: such filtering is a
unitary circulant transform, and the correlation plane it produces is usually
sparse. Thanks to these properties, phase-only filters are perfectly compatible
with the framework of compressive sensing. Moreover, the lasso-based recovery
algorithm is very fast when phase-only filtering is used as the compression
matrix. The proposed method can be seen as a generalisation of the
correlation-based pattern recognition technique, which is hereby applied
directly to non-adaptively acquired compressed data. At the time of
measurement, any prior knowledge of the target object for which the data will
be scanned is not required. We show that images measured at extremely high
compression rates may still contain sufficient information for target
classification and localization, even if the compression rate is high enough,
that visual recognition of the target in the reconstructed image is no longer
possible. The method has been applied by us to highly undersampled measurements
obtained from a single-pixel camera, with sampling based on randomly chosen
Walsh-Hadamard patterns.
