The theory of compressive sensing (CS) enables the reconstruction of a sparse or compressible
image or signal from a small set of linear, non-adaptive (even random) projections. However, in
many applications, including object and target recognition, we are ultimately interested in making
a decision about an image rather than computing a reconstruction. We propose here a framework
for compressive classification that operates directly on the compressive measurements without first
reconstructing the image. We dub the resulting dimensionally reduced matched filter the smashed
filter. The first part of the theory maps traditional maximum likelihood hypothesis testing into the
compressive domain; we find that the number of measurements required for a given classification
performance level does not depend on the sparsity or compressibility of the images but only on
the noise level. The second part of the theory applies the generalized maximum likelihood method
to deal with unknown transformations such as the translation, scale, or viewing angle of a target
object. We exploit the fact the set of transformed images forms a low-dimensional, nonlinear
manifold in the high-dimensional image space. We find that the number of measurements required
for a given classification performance level grows linearly in the dimensionality of the manifold but
only logarithmically in the number of pixels/samples and image classes. Using both simulations
and measurements from a new single-pixel compressive camera, we demonstrate the effectiveness
of the smashed filter for target classification using very few measurements.
Compressive Sensing is an emerging field based on the revelation that a small number of linear projections of a compressible signal contain enough information for reconstruction and processing. It has many promising implications and enables the design of new kinds of Compressive Imaging systems and cameras. In this paper, we develop a new camera architecture that employs a digital micromirror array to perform optical calculations of linear projections of an image onto pseudorandom binary patterns. Its hallmarks include the ability to obtain an image with a single detection element while sampling the image fewer times than the number of pixels. Other attractive properties include its universality, robustness, scalability, progressivity, and computational asymmetry. The most intriguing feature of the system is that, since it relies on a single photon detector, it can be adapted to image at wavelengths that are currently impossible with conventional CCD and CMOS imagers.
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