623 research outputs found
Local Measurement and Reconstruction for Noisy Graph Signals
The emerging field of signal processing on graph plays a more and more
important role in processing signals and information related to networks.
Existing works have shown that under certain conditions a smooth graph signal
can be uniquely reconstructed from its decimation, i.e., data associated with a
subset of vertices. However, in some potential applications (e.g., sensor
networks with clustering structure), the obtained data may be a combination of
signals associated with several vertices, rather than the decimation. In this
paper, we propose a new concept of local measurement, which is a generalization
of decimation. Using the local measurements, a local-set-based method named
iterative local measurement reconstruction (ILMR) is proposed to reconstruct
bandlimited graph signals. It is proved that ILMR can reconstruct the original
signal perfectly under certain conditions. The performance of ILMR against
noise is theoretically analyzed. The optimal choice of local weights and a
greedy algorithm of local set partition are given in the sense of minimizing
the expected reconstruction error. Compared with decimation, the proposed local
measurement sampling and reconstruction scheme is more robust in noise existing
scenarios.Comment: 24 pages, 6 figures, 2 tables, journal manuscrip
Spherical Slepian functions and the polar gap in geodesy
The estimation of potential fields such as the gravitational or magnetic
potential at the surface of a spherical planet from noisy observations taken at
an altitude over an incomplete portion of the globe is a classic example of an
ill-posed inverse problem. Here we show that the geodetic estimation problem
has deep-seated connections to Slepian's spatiospectral localization problem on
the sphere, which amounts to finding bandlimited spherical functions whose
energy is optimally concentrated in some closed portion of the unit sphere.
This allows us to formulate an alternative solution to the traditional damped
least-squares spherical harmonic approach in geodesy, whereby the source field
is now expanded in a truncated Slepian function basis set. We discuss the
relative performance of both methods with regard to standard statistical
measures as bias, variance and mean-square error, and pay special attention to
the algorithmic efficiency of computing the Slepian functions on the region
complementary to the axisymmetric polar gap characteristic of satellite
surveys. The ease, speed, and accuracy of this new method makes the use of
spherical Slepian functions in earth and planetary geodesy practical.Comment: 14 figures, submitted to the Geophysical Journal Internationa
Average sampling of band-limited stochastic processes
We consider the problem of reconstructing a wide sense stationary
band-limited process from its local averages taken either at the Nyquist rate
or above. As a result, we obtain a sufficient condition under which average
sampling expansions hold in mean square and for almost all sample functions.
Truncation and aliasing errors of the expansion are also discussed
Spectral estimation on a sphere in geophysics and cosmology
We address the problem of estimating the spherical-harmonic power spectrum of
a statistically isotropic scalar signal from noise-contaminated data on a
region of the unit sphere. Three different methods of spectral estimation are
considered: (i) the spherical analogue of the one-dimensional (1-D)
periodogram, (ii) the maximum likelihood method, and (iii) a spherical analogue
of the 1-D multitaper method. The periodogram exhibits strong spectral leakage,
especially for small regions of area , and is generally unsuitable
for spherical spectral analysis applications, just as it is in 1-D. The maximum
likelihood method is particularly useful in the case of nearly-whole-sphere
coverage, , and has been widely used in cosmology to estimate
the spectrum of the cosmic microwave background radiation from spacecraft
observations. The spherical multitaper method affords easy control over the
fundamental trade-off between spectral resolution and variance, and is easily
implemented regardless of the region size, requiring neither non-linear
iteration nor large-scale matrix inversion. As a result, the method is ideally
suited for most applications in geophysics, geodesy or planetary science, where
the objective is to obtain a spatially localized estimate of the spectrum of a
signal from noisy data within a pre-selected and typically small region.Comment: Submitted to the Geophysical Journal Internationa
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