14,810 research outputs found
Image reconstruction/synthesis from nonuniform data and zero/threshold crossings
We address the problem of reconstructing functions from their nonuniform data and zero/threshold crossings. We introduce a deterministic process via the Gram-Schmidt orthonormalization procedure to reconstruct functions from their nonuniform data and zero/threshold crossings. This is achieved by first introducing the nonorthogonal basis functions in a chosen 2-D domain (e.g., for a band-limited signal, a possible choice is the 2-D Fourier domain of the image) that span the signal subspace of the nonuniform data. We then use the Gram-Schmidt procedure to construct a set of orthogonal basis functions that span the linear signal subspace defined by the nonorthogonal basis functions. Next, we project the N-dimensional measurement vector (N is the number of nonuniform data or threshold crossings) onto the newly constructed orthogonal basis functions. Finally, the function at any point can be reconstructed by projecting the representation with respect to the newly constructed orthonormal basis functions onto the reconstruction basis functions that span the signal subspace of the evenly spaced sampled data. The reconstructed signal gives the minimum mean square error estimate of the original signal. This procedure gives error-free reconstruction provided that the nonorthogonal basis functions that span the signal subspace of the nonuniform data form a complete set in the signal subspace of the original band-limited signal. We apply this algorithm
to reconstruct functions from their unevenly spaced sampled data and zero crossings and also apply it to solve the problem of synthesis of a 2-D band-limited function with the prescribed level crossings
Reconstruction of Binary Functions and Shapes from Incomplete Frequency Information
The characterization of a binary function by partial frequency information is
considered. We show that it is possible to reconstruct binary signals from
incomplete frequency measurements via the solution of a simple linear
optimization problem. We further prove that if a binary function is spatially
structured (e.g. a general black-white image or an indicator function of a
shape), then it can be recovered from very few low frequency measurements in
general. These results would lead to efficient methods of sensing,
characterizing and recovering a binary signal or a shape as well as other
applications like deconvolution of binary functions blurred by a low-pass
filter. Numerical results are provided to demonstrate the theoretical
arguments.Comment: IEEE Transactions on Information Theory, 201
Spherical deconvolution of multichannel diffusion MRI data with non-Gaussian noise models and spatial regularization
Spherical deconvolution (SD) methods are widely used to estimate the
intra-voxel white-matter fiber orientations from diffusion MRI data. However,
while some of these methods assume a zero-mean Gaussian distribution for the
underlying noise, its real distribution is known to be non-Gaussian and to
depend on the methodology used to combine multichannel signals. Indeed, the two
prevailing methods for multichannel signal combination lead to Rician and
noncentral Chi noise distributions. Here we develop a Robust and Unbiased
Model-BAsed Spherical Deconvolution (RUMBA-SD) technique, intended to deal with
realistic MRI noise, based on a Richardson-Lucy (RL) algorithm adapted to
Rician and noncentral Chi likelihood models. To quantify the benefits of using
proper noise models, RUMBA-SD was compared with dRL-SD, a well-established
method based on the RL algorithm for Gaussian noise. Another aim of the study
was to quantify the impact of including a total variation (TV) spatial
regularization term in the estimation framework. To do this, we developed TV
spatially-regularized versions of both RUMBA-SD and dRL-SD algorithms. The
evaluation was performed by comparing various quality metrics on 132
three-dimensional synthetic phantoms involving different inter-fiber angles and
volume fractions, which were contaminated with noise mimicking patterns
generated by data processing in multichannel scanners. The results demonstrate
that the inclusion of proper likelihood models leads to an increased ability to
resolve fiber crossings with smaller inter-fiber angles and to better detect
non-dominant fibers. The inclusion of TV regularization dramatically improved
the resolution power of both techniques. The above findings were also verified
in brain data
Significant edges in the case of a non-stationary Gaussian noise
In this paper, we propose an edge detection technique based on some local
smoothing of the image followed by a statistical hypothesis testing on the
gradient. An edge point being defined as a zero-crossing of the Laplacian, it
is said to be a significant edge point if the gradient at this point is larger
than a threshold s(\eps) defined by: if the image is pure noise, then
\P(\norm{\nabla I}\geq s(\eps) \bigm| \Delta I = 0) \leq\eps. In other words,
a significant edge is an edge which has a very low probability to be there
because of noise. We will show that the threshold s(\eps) can be explicitly
computed in the case of a stationary Gaussian noise. In images we are
interested in, which are obtained by tomographic reconstruction from a
radiograph, this method fails since the Gaussian noise is not stationary
anymore. But in this case again, we will be able to give the law of the
gradient conditionally on the zero-crossing of the Laplacian, and thus compute
the threshold s(\eps). We will end this paper with some experiments and
compare the results with the ones obtained with some other methods of edge
detection
Cosmic-Ray Rejection by Laplacian Edge Detection
Conventional algorithms for rejecting cosmic-rays in single CCD exposures
rely on the contrast between cosmic-rays and their surroundings, and may
produce erroneous results if the Point Spread Function (PSF) is smaller than
the largest cosmic-rays. This paper describes a robust algorithm for cosmic-ray
rejection, based on a variation of Laplacian edge detection. The algorithm
identifies cosmic-rays of arbitrary shapes and sizes by the sharpness of their
edges, and reliably discriminates between poorly sampled point sources and
cosmic-rays. Examples of its performance are given for spectroscopic and
imaging data, including HST WFPC2 images.Comment: Accepted for publication in the PASP (November 2001 issue). The
algorithm is implemented in the program L.A.Cosmic, which can be obtained
from http://www.astro.caltech.edu/~pgd/lacosmic
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