31 research outputs found
New method of probability density estimation with application to mutual information based image registration
We present a new, robust and computationally efficient method for estimating the probability density of the intensity values in an image. Our approach makes use of a continuous representation of the image and develops a relation between probability density at a particular intensity value and image gradients along the level sets at that value. Unlike traditional sample-based methods such as histograms, minimum spanning trees (MSTs), Parzen windows or mixture models, our technique expressly accounts for the relative ordering of the intensity values at different image locations and exploits the geometry of the image surface. Moreover, our method avoids the histogram binning problem and requires no critical parameter tuning. We extend the method to compute the joint density between two or more images. We apply our density estimation technique to the task of affine registration of 2D images using mutual information and show good results under high noise. 1
Signal Recovery in Perturbed Fourier Compressed Sensing
In many applications in compressed sensing, the measurement matrix is a
Fourier matrix, i.e., it measures the Fourier transform of the underlying
signal at some specified `base' frequencies , where is the
number of measurements. However due to system calibration errors, the system
may measure the Fourier transform at frequencies
that are different from the base frequencies and where
are unknown. Ignoring perturbations of this nature can lead to major errors in
signal recovery. In this paper, we present a simple but effective alternating
minimization algorithm to recover the perturbations in the frequencies \emph{in
situ} with the signal, which we assume is sparse or compressible in some known
basis. In many cases, the perturbations can be expressed
in terms of a small number of unique parameters . We demonstrate that
in such cases, the method leads to excellent quality results that are several
times better than baseline algorithms (which are based on existing off-grid
methods in the recent literature on direction of arrival (DOA) estimation,
modified to suit the computational problem in this paper). Our results are also
robust to noise in the measurement values. We also provide theoretical results
for (1) the convergence of our algorithm, and (2) the uniqueness of its
solution under some restrictions.Comment: New theortical results about uniqueness and convergence now included.
More challenging experiments now include
Unlabelled Sensing with Priors: Algorithm and Bounds
In this study, we consider a variant of unlabelled sensing where the
measurements are sparsely permuted, and additionally, a few correspondences are
known. We present an estimator to solve for the unknown vector. We derive a
theoretical upper bound on the reconstruction error of the unknown
vector. Through numerical experiments, we demonstrate that the additional known
correspondences result in a significant improvement in the reconstruction
error. Additionally, we compare our estimator with the classical robust
regression estimator and we find that our method outperforms it on the
normalized reconstruction error metric by up to in the high permutation
regimes . Lastly, we showcase the practical utility of our framework
on a non-rigid motion estimation problem. We show that using a few manually
annotated points along point pairs with the key-point (SIFT-based) descriptor
pairs with unknown or incorrectly known correspondences can improve motion
estimation.Comment: 14 pages, 6 figure
Analysis of Tomographic Reconstruction of 2D Images using the Distribution of Unknown Projection Angles
It is well known that a band-limited signal can be reconstructed from its
uniformly spaced samples if the sampling rate is sufficiently high. More
recently, it has been proved that one can reconstruct a 1D band-limited signal
even if the exact sample locations are unknown, but given just the distribution
of the sample locations and their ordering in 1D. In this work, we extend the
analytical bounds on the reconstruction error in such scenarios for
quasi-bandlimited signals. We also prove that the method for such a
reconstruction is resilient to a certain proportion of errors in the
specification of the sample location ordering. We then express the problem of
tomographic reconstruction of 2D images from 1D Radon projections under unknown
angles with known angle distribution, as a special case for reconstruction of
quasi-bandlimited signals from samples at unknown locations with known
distribution. Building upon our theoretical background, we present asymptotic
bounds for 2D quasi-bandlimited image reconstruction from 1D Radon projections
in the unknown angles setting, which commonly occurs in cryo-electron
microscopy (cryo-EM). To the best of our knowledge, this is the first piece of
work to perform such an analysis for 2D cryo-EM, even though the associated
reconstruction algorithms have been known for a long time