1,475 research outputs found
A Fast Hadamard Transform for Signals with Sub-linear Sparsity in the Transform Domain
In this paper, we design a new iterative low-complexity algorithm for computing the Walsh-Hadamard transform (WHT) of an N dimensional signal with a K-sparse WHT. We suppose that N is a power of two and K = O(N^α), scales sub-linearly in N for some α ∈ (0,1). Assuming a random support model for the nonzero transform-domain components, our algorithm reconstructs the WHT of the signal with a sample complexity O(K log_2(N/K)) and a computational complexity O(K log_2(K) log_2(N/K)). Moreover, the algorithm succeeds with a high probability approaching 1 for large dimension N. Our approach is mainly based on the subsampling (aliasing) property of the WHT, where by a carefully designed subsampling of the time-domain signal, a suitable aliasing pattern is induced in the transform domain. We treat the resulting aliasing patterns as parity-check constraints and represent them by a bipartite graph. We analyze the properties of the resulting bipartite graphs and borrow ideas from codes defined over sparse bipartite graphs to formulate the recovery of the nonzero spectral values as a peeling decoding algorithm for a specific sparse-graph code transmitted over a binary erasure channel (BEC). This enables us to use tools from coding theory (belief-propagation analysis) to characterize the asymptotic performance of our algorithm in the very sparse (α ∈ (0,1/3]) and the less sparse (α ∈ (1/3,1)) regime. Comprehensive simulation results are provided to assess the empirical performance of the proposed algorithm
Compressive Sensing for Spectroscopy and Polarimetry
We demonstrate through numerical simulations with real data the feasibility
of using compressive sensing techniques for the acquisition of
spectro-polarimetric data. This allows us to combine the measurement and the
compression process into one consistent framework. Signals are recovered thanks
to a sparse reconstruction scheme from projections of the signal of interest
onto appropriately chosen vectors, typically noise-like vectors. The
compressibility properties of spectral lines are analyzed in detail. The
results shown in this paper demonstrate that, thanks to the compressibility
properties of spectral lines, it is feasible to reconstruct the signals using
only a small fraction of the information that is measured nowadays. We
investigate in depth the quality of the reconstruction as a function of the
amount of data measured and the influence of noise. This change of paradigm
also allows us to define new instrumental strategies and to propose
modifications to existing instruments in order to take advantage of compressive
sensing techniques.Comment: 11 pages, 9 figures, accepted for publication in A&
Fast Hadamard transforms for compressive sensing of joint systems: measurement of a 3.2 million-dimensional bi-photon probability distribution
We demonstrate how to efficiently implement extremely high-dimensional
compressive imaging of a bi-photon probability distribution. Our method uses
fast-Hadamard-transform Kronecker-based compressive sensing to acquire the
joint space distribution. We list, in detail, the operations necessary to
enable fast-transform-based matrix-vector operations in the joint space to
reconstruct a 16.8 million-dimensional image in less than 10 minutes. Within a
subspace of that image exists a 3.2 million-dimensional bi-photon probability
distribution. In addition, we demonstrate how the marginal distributions can
aid in the accuracy of joint space distribution reconstructions
Nearly Optimal Deterministic Algorithm for Sparse Walsh-Hadamard Transform
For every fixed constant , we design an algorithm for computing
the -sparse Walsh-Hadamard transform of an -dimensional vector in time . Specifically, the
algorithm is given query access to and computes a -sparse satisfying , for an absolute constant , where is the
transform of and is its best -sparse approximation. Our
algorithm is fully deterministic and only uses non-adaptive queries to
(i.e., all queries are determined and performed in parallel when the algorithm
starts).
An important technical tool that we use is a construction of nearly optimal
and linear lossless condensers which is a careful instantiation of the GUV
condenser (Guruswami, Umans, Vadhan, JACM 2009). Moreover, we design a
deterministic and non-adaptive compressed sensing scheme based
on general lossless condensers that is equipped with a fast reconstruction
algorithm running in time (for the GUV-based
condenser) and is of independent interest. Our scheme significantly simplifies
and improves an earlier expander-based construction due to Berinde, Gilbert,
Indyk, Karloff, Strauss (Allerton 2008).
Our methods use linear lossless condensers in a black box fashion; therefore,
any future improvement on explicit constructions of such condensers would
immediately translate to improved parameters in our framework (potentially
leading to reconstruction time with a reduced exponent in
the poly-logarithmic factor, and eliminating the extra parameter ).
Finally, by allowing the algorithm to use randomness, while still using
non-adaptive queries, the running time of the algorithm can be improved to
Photon counting compressive depth mapping
We demonstrate a compressed sensing, photon counting lidar system based on
the single-pixel camera. Our technique recovers both depth and intensity maps
from a single under-sampled set of incoherent, linear projections of a scene of
interest at ultra-low light levels around 0.5 picowatts. Only two-dimensional
reconstructions are required to image a three-dimensional scene. We demonstrate
intensity imaging and depth mapping at 256 x 256 pixel transverse resolution
with acquisition times as short as 3 seconds. We also show novelty filtering,
reconstructing only the difference between two instances of a scene. Finally,
we acquire 32 x 32 pixel real-time video for three-dimensional object tracking
at 14 frames-per-second.Comment: 16 pages, 8 figure
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