Compressive sensing (CS) is a technique used to recover a signal from under-sampled data. Traditionally, the Nyquist rate has been used to determine the minimum sample rate based on the highest frequency component of a signal. CS works with signals which have only a few non-zero frequency components (called sparse), and can reconstruct signals accurately with far fewer measurements than the Nyquist rate predicts is necessary. CS requires measurements which are uncorrelated with eachother, termed incoherent. Of particular interest is how to generate sequences which are incoherent. A simple and effective method is to choose random points, however this method suffers from several operational drawbacks, notably the unpredictable and potentially large travel distance between points. Our work focuses on improving the methods for choosing measurement locations for CS reconstruction. This work is aimed at mobile sensing platforms, for which the energy used to travel between sampling locations is a large factor in overall mission performance. Publications
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Example of compressive sensing (CS) in one dimension. The signal, shown in the top plot, is the sum of seven sinusoidal signals, with the maximum frequency equal to 40 Hz. Twenty measurements are taken within ten seconds, either randomly chosen or equi-spaced. CS reconstruction assumes the signal is sparse, that is, many of the frequency components are zero. The CS reconstruction of the frequency coeffcients, shown in the bottom plot, is exact, while that from the equi-spaced measurements is incorrect. Without exploiting sparsity, the signal would need to be measured above the Nyquist rate of 80 Hz. |
1D Compressive Sensing Example |
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Experimental results from sea trials in Buzzards Bay, MA. Water depth was measured with an autonomous kayak performing a TSP tour of 100 random points. Sea floor field reconstruction was performed using the Fourier basis. In performing the reconstruction, sensor measurement error must be estimated, shown increasing from left to right. Additionally, the number of basis coeffcients is shown increasing from top to bottom. Measurements are shown as circles. The L1 norm and sparsity, S, are given below each result. The trends can be inferred as follows. With low estimated measurement noise, the reconstruction is tightly constrained by the data, leading to a high sparsity and L1 norm and a reconstructed field that contains extreme values in places where no measurements were taken. Increasing the noise estimate relaxes this constraint, allowing for a lower sparsity and L1 norm and a more realistic reconstructed field. The number of basis coefficients determines the highest frequency component that is available for the reconstruction. Its effect on reconstruction is unclear, however it does act to decrease the L1 norm of the result. | |
Experimental Results with Autonomous Kayak |