Monday, January 18, 2010

CS: This week's big entry.

This week we have several papers looking at what can be detected in the measurement world before performing any type of reconstruction, enjoy:

Compressed sensing (CS) has the potential to reduce MR data acquisition time. There are two fundamental tenets to CS theory: (1) the signal of interest is sparse or compressible in a known representation, and (2) the measurement scheme has good mathematical properties (e.g., restricted isometry or incoherence properties) with respect to this representation. While MR images are often compressible, the second requirement is often only weakly satisfied with respect to commonly used Fourier encoding schemes. This paper investigates the possibility of improving CS-MRI performance using random encoding, in an effort to emulate the "universal'' encoding schemes suggested by the theoretical CS literature. This random encoding is achieved experimentally with tailored spatially-selective RF pulses. Simulation and experimental results show that the proposed encoding scheme has different characteristics than more conventional Fourier-based schemes, and could prove useful in certain contexts.
Hyperspectral target detection from incoherent projections by Kalyani Krishnamurthy, Maxim Raginsky and Rebecca Willett. The abstract reads:
This paper studies the detection of spectral targets corrupted by a colored Gaussian background from noisy, incoherent projection measurements. Unlike many detection methods designed for incoherent projections, the proposed approach a) is computationally efficient, b) allows for spectral backgrounds behind potential targets, and c) yields theoretical guarantees on detector performance. In particular, the theoretical performance bounds highlight fundamental tradeoffs among the number of measurements collected, the spectral resolution of targets, the amount of background signal present, signal-tonoise ratio, and the similarity between potential targets in a dictionary.
Compressive-sensing cameras are an important new class of sensors that have different design constraints than standard cameras. Surprisingly, little work has explored the relationship between compressive-sensing measurements and differential image motion. We show that, given modest constraints on the measurements and image motions, we can omit the computationally expensive compressive-sensing reconstruction step and obtain more accurate motion estimates with significantly less computation time. We also formulate a compressive-sensing reconstruction problem that incorporates known image motion and show that this method outperforms the state-of-the-art in compressive-sensing video reconstruction.
On unusual pixel shapes and image motion by Nathan Jacobs, Stephen Schuh, and Robert Pless.The abstract reads:
We introduce the integral-pixel camera model, where measurements integrate over large and potentially overlapping parts of the visual field. This models a wide variety of novel camera designs, including omnidirectional cameras, compressive sensing cameras, and novel programmable-pixel imaging chips. We explore the relationship of integral-pixel measurements with image motion and find (a) that direct motion estimation using integral-pixels is possible and in some cases quite good, (b) standard compressive-sensing reconstructions are not good for estimating motion, and (c) when we design image reconstruction algorithms that explicitly reason about image motion, they outperform standard compressive-sensing video reconstruction. We show experimental results for a variety of simulated cases, and have preliminary results showing a prototype camera with integral-pixels whose design makes direct motion estimation possible.
We consider the problem of sampling piecewise sinusoidal signals. Classical sampling theory does not enable perfect reconstruction of such signals since they are not bandlimited. However, they can be characterized by a finite number of parameters namely the frequency, amplitude and phase of the sinusoids and the location of the discontinuities. In this paper, we show that under certain hypotheses on the sampling kernel, it is possible to perfectly recover the parameters that define the piecewise sinusoidal signal from its sampled version. In particular, we show that, at least theoretically, it is possible to recover piecewise sine waves with arbitrarily high frequencies and arbitrarily close switching points. Extensions of the method are also presented such as the recovery of combinations of piecewise sine waves and polynomials. Finally, we study the effect of noise and present a robust reconstruction algorithm that is stable down to SNR levels of 7 [dB].
This paper studies the low-rank matrix completion problem from an information theoretic perspective. The completion problem is rephrased as a communication problem of an (uncoded) low-rank matrix source over an erasure channel. The paper then uses achievability and converse arguments to present order-wise optimal bounds for the completion problem.

1 comment:

Dick Gordon said...

Vision as Compressive Sensing

The visual systems of humans, other vertebrates, and some invertebrates (octopus) acquire images in strips called receptive fields. These are generally elongate, and at various orientations and widths. They each integrate over large subsets of the photoreceptors, and overlap one another. The orientations available can be influenced by experience shortly after birth:

Hirsch, H.V. & D.N. Spinelli (1970). Visual experience modifies distribution of horizontally and vertically oriented receptive fields in cats. Science 168(933), 869-871.

These receptive fields were postulated to be rays in a computed tomography scheme, what we now call compressive sensors, in:

Gordon, R. & H.V.B. Hirsch (1977). Vision begins with direct reconstruction of the retinal image, how the brain sees and stores pictures. In: Gegenstrom, Für Helmut Hirsch zum Siebzigsten/Against the Stream, for Helmut Hirsch on His 70th Birthday. Eds.: H. Schallenberger & H. Schrey. Wuppertal, Peter Hammer Verlag GmbH: 201-214.

Gordon, R. & D. Tweed (1983). Quantitative reconstruction of visual cortex receptive fields. Univ. Manitoba Med. J. 53(2), 75.

If correct, this means that our vision works by compressive sensing. Copies available on request.
Yours, -Dick Gordon