  We investigate the problem of reconstructing signals from a subsampled
convolution of their modulated versions and a known filter. The problem is
studied as applies to specific imaging systems relying on spatial phase
modulation by randomly coded "masks." The diversity induced by the random masks
is deemed to improve the conditioning of the deconvolution problem while
maintaining sampling efficiency.
  We analyze a linear model of the system, where the joint effect of the
spatial modulation, blurring, and spatial subsampling is represented by a
measurement matrix. We provide a bound on the conditioning of this measurement
matrix in terms of the number of masks, the dimension of the image, and certain
characteristics of the blurring kernel and subsampling operator. The derived
bound shows that stable deconvolution is possible with high probability even if
the total number of (scalar) measurements is within a logarithmic factor of the
image size. Furthermore, beyond a critical number of masks determined by the
extent of blurring and subsampling, every additional mask improves the
conditioning of the measurement matrix.
  We also consider a more interesting scenario where the target image is
sparse. We show that under mild conditions on the blurring kernel, with high
probability the measurement matrix is a restricted isometry when the number of
masks is within a logarithmic factor of the sparsity of the image. Therefore,
the image can be reconstructed using many sparse recovery algorithms such as
the basis pursuit. The bound on the required number of masks is linear in
sparsity of the image but it is logarithmic in its dimension. The bound
provides a quantitative view of the effect of the blurring and subsampling on
the required number of masks, which is critical for designing efficient imaging
systems.
