1,429 research outputs found
Fast Mojette Transform for Discrete Tomography
A new algorithm for reconstructing a two dimensional object from a set of one
dimensional projected views is presented that is both computationally exact and
experimentally practical. The algorithm has a computational complexity of O(n
log2 n) with n = N^2 for an NxN image, is robust in the presence of noise and
produces no artefacts in the reconstruction process, as is the case with
conventional tomographic methods. The reconstruction process is approximation
free because the object is assumed to be discrete and utilizes fully discrete
Radon transforms. Noise in the projection data can be suppressed further by
introducing redundancy in the reconstruction. The number of projections
required for exact reconstruction and the response to noise can be controlled
without comprising the digital nature of the algorithm. The digital projections
are those of the Mojette Transform, a form of discrete linogram. A simple
analytical mapping is developed that compacts these projections exactly into
symmetric periodic slices within the Discrete Fourier Transform. A new digital
angle set is constructed that allows the periodic slices to completely fill all
of the objects Discrete Fourier space. Techniques are proposed to acquire these
digital projections experimentally to enable fast and robust two dimensional
reconstructions.Comment: 22 pages, 13 figures, Submitted to Elsevier Signal Processin
The curvelet transform for image denoising
We describe approximate digital implementations of two new mathematical transforms, namely, the ridgelet transform and the curvelet transform. Our implementations offer exact reconstruction, stability against perturbations, ease of implementation, and low computational complexity. A central tool is Fourier-domain computation of an approximate digital Radon transform. We introduce a very simple interpolation in the Fourier space which takes Cartesian samples and yields samples on a rectopolar grid, which is a pseudo-polar sampling set based on a concentric squares geometry. Despite the crudeness of our interpolation, the visual performance is surprisingly good. Our ridgelet transform applies to the Radon transform a special overcomplete wavelet pyramid whose wavelets have compact support in the frequency domain. Our curvelet transform uses our ridgelet transform as a component step, and implements curvelet subbands using a filter bank of a` trous wavelet filters. Our philosophy throughout is that transforms should be overcomplete, rather than critically sampled. We apply these digital transforms to the denoising of some standard images embedded in white noise. In the tests reported here, simple thresholding of the curvelet coefficients is very competitive with "state of the art" techniques based on wavelets, including thresholding of decimated or undecimated wavelet transforms and also including tree-based Bayesian posterior mean methods. Moreover, the curvelet reconstructions exhibit higher perceptual quality than wavelet-based reconstructions, offering visually sharper images and, in particular, higher quality recovery of edges and of faint linear and curvilinear features. Existing theory for curvelet and ridgelet transforms suggests that these new approaches can outperform wavelet methods in certain image reconstruction problems. The empirical results reported here are in encouraging agreement
Recovering missing slices of the discrete fourier transform using ghosts
The discrete Fourier transform (DFT) underpins the solution to many inverse problems commonly possessing missing or unmeasured frequency information. This incomplete coverage of the Fourier space always produces systematic artifacts called Ghosts. In this paper, a fast and exact method for deconvolving cyclic artifacts caused by missing slices of the DFT using redundant image regions is presented. The slices discussed here originate from the exact partitioning of the Discrete Fourier Transform (DFT) space, under the projective Discrete Radon Transform, called the discrete Fourier slice theorem. The method has a computational complexity of O(n\log-{2}n) (for an n=N\times N image) and is constructed from a new cyclic theory of Ghosts. This theory is also shown to unify several aspects of work done on Ghosts over the past three decades. This paper concludes with an application to fast, exact, non-iterative image reconstruction from a highly asymmetric set of rational angle projections that give rise to sets of sparse slices within the DFT
Fast Digital Convolutions using Bit-Shifts
An exact, one-to-one transform is presented that not only allows digital
circular convolutions, but is free from multiplications and quantisation errors
for transform lengths of arbitrary powers of two. The transform is analogous to
the Discrete Fourier Transform, with the canonical harmonics replaced by a set
of cyclic integers computed using only bit-shifts and additions modulo a prime
number. The prime number may be selected to occupy contemporary word sizes or
to be very large for cryptographic or data hiding applications. The transform
is an extension of the Rader Transforms via Carmichael's Theorem. These
properties allow for exact convolutions that are impervious to numerical
overflow and to utilise Fast Fourier Transform algorithms.Comment: 4 pages, 2 figures, submitted to IEEE Signal Processing Letter
On random tomography with unobservable projection angles
We formulate and investigate a statistical inverse problem of a random
tomographic nature, where a probability density function on is
to be recovered from observation of finitely many of its two-dimensional
projections in random and unobservable directions. Such a problem is distinct
from the classic problem of tomography where both the projections and the unit
vectors normal to the projection plane are observable. The problem arises in
single particle electron microscopy, a powerful method that biophysicists
employ to learn the structure of biological macromolecules. Strictly speaking,
the problem is unidentifiable and an appropriate reformulation is suggested
hinging on ideas from Kendall's theory of shape. Within this setup, we
demonstrate that a consistent solution to the problem may be derived, without
attempting to estimate the unknown angles, if the density is assumed to admit a
mixture representation.Comment: Published in at http://dx.doi.org/10.1214/08-AOS673 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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