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.
The contemporary artist David Hockney has hypothesized that some early Renaissance painters secretly projected optical
images onto their supports (canvas, paper, oak panel, ...), directly traced these projections, and then filled in the tracings with
paint. Hockney has presented somewhat impressionistic image evidence for this claim, but he and thin-film physicist Charles
Falco also point to perspective anomalies, to the fidelity of passages in certain paintings, and to historical documents in
search of support for this direct tracing claim.
Key visual evidence adduced in support of this tracing claim is a pair of portraits by Jan van Eyck of Cardinal Niccolo
Albergati - a small informal silverpoint study of 1431 and a slightly larger formal work in oil on panel of 1432. The
contours in these two works bear striking resemblance in shape (after being appropriately scaled) and there are at least two
"relative shifts" - passages that co-align well after a spatial shift of one of the images . This evidence has led the
theory's proponents to claim that van Eyck copied the silverpoint by means of an optical projector, or epidiascope, the
relative shifts due to him accidentally <i>bumping </i>the setup during the copying.
Previous tests of the tracing theory for these works considered four candidate methods van Eyck might have used to
copied and enlarged the image in the silverpoint study: unaided ("by eye"), mechanical, grid, and the optical projection
method itself . Based on the full evidence, including the recent discovery of tiny pinprick holes in the silverpoint, reenactments,
material culture and optical knowledge in the early 15th century, the mechanical method was judged most
plausible and optical method the least plausible.
However, this earlier work did not adequately test whether a trained artist could "re-enact" the copying by mechanical
methods: "Although we have not explicitly verified that high fidelities can be achieved through the use of a <i>Reductionszirkel</i>(or compass, protractor and ruler), there are no significant challenges in this regard". Our work here seeks to complete the
test of the direct tracing claim. As we shall see, a talented realist artist can indeed achieve fidelity comparable to that found in
these works, a result that re-affirms the earlier conclusion that when copying and enlarging the silverpoint image, it is more
likely that van Eyck used a well-known, simple, mechanical method than a then unknown, secret and complicated optical
<i>Compressive Sensing</i> 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 <i>Compressive Imaging</i> 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.