Wavelet localization of the Radon transform in even dimensions

DeStefano, J.; Olson, T.

doi:10.1109/tftsa.1992.274217

Cited by 21 publications

(17 citation statements)

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“…The other traditional approach to regularizing the noisy data problem is statistically based. This method starts with a statistical model for the noisy observations based on (2): yk = Tkf + nk (18) where nk is taken as an additive noise vector at angle k. This observation model is then coupled with a 2-D Markov random field (MRF) prior model [25,26] for f to yield a direct MAP estimate of the object f. While statistically based, thus allowing the systematic inclusion of prior information, the 2-D spatially-local MRF prior models used for the object generally lead to optimization problems that are extremely computationally complex. As a result, these methods have traditionally not found favor in practical applications.…”

Section: Multiscale Regularized Reconstructionsmentioning

confidence: 99%

“…As in Section 3, we then allow the resulting projection domain coefficients to induce a 2-D object representation through the back-projection and summation operations. To this end we start with an observation equation relating the noisy data Yk to the FBP object coefficients zk, rather than the corresponding 2-D object f as is done in (18). Such a relationship may be found in the FBP relationship (4), which in the presence of noise in the data becomes:…”

Section: Multiscale Regularized Reconstructionsmentioning

confidence: 99%

“…Again as in [15], in [16,17] the object is represented in the original spatial domain by a 2-D wavelet basis which is much different than our multiscale object representation. DeStefano and Olson [18], and Berenstein and Walnut [19] have also used wavelets for tomographic reconstruction problems, in particular to localize the Radon transform in even dimensions. Through this localization the radiation exposure can be reduced when a local region of the object is to be imaged.…”

Section: Introductionmentioning

confidence: 99%

“…Through this localization the radiation exposure can be reduced when a local region of the object is to be imaged. The work in [18] and [19] does not provide a framework for multiscale reconstruction, however, which is the central theme here. In addition, our reconstruction procedure also localizes the Radon transform, though we do not stress this particular application in this work.…”

Section: Introductionmentioning

confidence: 99%

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A Wavelet-Based Method for Multiscale Tomographic Reconstruction

Bhatia¹,

Karl²,

Willsky³

1993

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Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. AbstractWe represent the standard ramp filter operator of the filtered back-projection (FBP) reconstruction in different bases composed of Haar and Daubechies compactly supported wavelets. The resulting multiscale representation of the ramp filter matrix operator is approximately diagonal. The accuracy of this diagonal approximation becomes better as wavelets with larger number of vanishing moments are used. This wavelet-based representation enables us to formulate a multiscale tomographic reconstruction technique wherein the object is reconstructed at multiple scales or resolutions. A complete reconstruction is obtained by combining the reconstructions at different scales. Our multiscale reconstruction technique has the same computational complexity as the FBP reconstruction method. It differs from other multiscale reconstruction techniques in that 1) the object is defined through a multiscale transformation of the projection domain, and 2) we explicitly account for noise in the projection data by calculating maximum aposteriori probability (MAP) multiscale reconstruction estimates based on a chosen fractal prior on the multiscale object coefficients. The computational complexity of this MAP solution is also the same as that of the FBP reconstruction. This is in contrast to commonly used methods of statistical regularization which result in computationally intensive optimization algorithms. The framework for multiscale reconstruction presented here can find application in object feature recognition directly from projection data, and regularization of imaging problems where the projection data are noisy.

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Section: Multiscale Regularized Reconstructionsmentioning

confidence: 99%

Section: Multiscale Regularized Reconstructionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Wavelet-Based Method for Multiscale Tomographic Reconstruction

Bhatia¹,

Karl²,

Willsky³

1993

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“…This is due to the expectation that by employing a function having a sufficient amount of zero moments, the support will remain essentially unchanged. Wavelet filters are functions with compact support and can be constructed with a certain amount of zero moments [3], [28], [29]. …”

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confidence: 99%

Tiled-Block Image Reconstruction by Wavelet- Based, Parallel-Filtered Back-Projection

Escobedo

Ozanyan

2016

IEEE Sensors J.

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Abstract-We demonstrate an algorithm, relevant to tomography sensor systems, to obtain images from the parallel reconstruction of essentially localized elements at different scales. This is achieved by combining methodology to reconstruct images from limited and/or truncated data, with the time-frequency capabilities of the Wavelet Transform. Multi-scale, as well as time-frequency, localization properties of the separable twodimensional wavelet transform are exploited as an approach for faster reconstruction. The speed up is realized not only by reducing the computation load on a single processor, but also by achieving the parallel reconstruction of several tiled blocks. With tiled-block image reconstruction by wavelet-based, parallel filtered back-projection we measure more than 36 times gain in speed, compared to standard filtered back-projection.Index Terms-image reconstruction, computed tomography, data processing algorithms, parallel processing, wavelet transform. I. INTRODUCTIONCROSS many applications, it is common to identify the need of information about an object, without altering its physical structure. Fortunately, there are numerous methods allowing radiation, either emitted or transmitted, to be employed to obtain cross-sectional images characterizing the inner structure of an object. The mathematical foundation behind such an approach was developed by Johann Radon in 1917 [1] and several decades later, in 1972, experimentally implemented by Hounsfield [2] resulting in the demonstration of the first Computed Tomography (CT) scanner.The main motivation for this work was the existing body of knowledge and achievements on the reconstruction of reduced-area full-resolution images, originally encouraged by the radiation dose exposure reduction in medical imaging and where the Wavelet Transform (WT), along its different representations, proved to be an effective tool given its timefrequency localization capabilities [4], [5]. Such research has been commonly named as Wavelet-based local reconstruction, and has been reported to be useful in other application areas such as Nano and Micro Tomography [6], [7]; Terahertz Tomography [8], and Dental Radiology [9].Of special interest is the 2D fast wavelet transform (2D FWT), employed in [5] and [10]. In addition to achieving local Manuscript received October 15, 2015. Jorge Guevara Escobedo (jorge.guevaraescobedo@manchester.ac.uk) and Krikor B. Ozanyan (k.ozanyan@manchester.ac.uk) are with the School of Electrical and Electronic Engineering, The University of Manchester, M13 9PL, Manchester, UK. JGE wishes to acknowledge the financial support of Consejo Nacional de Ciencia y Tecnología (CONACyT), Mexico, for a doctoral studentship.reconstruction, it allows projection data to be processed in a multi-resolution scheme. Such a feature proves to be of great benefit, when realizing its similarities with the parallel algorithm proposed in [11]. In that algorithm, a frequential decomposition of projection data is performed with the aim to back-project every componen...

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A Novel Computed Tomography Scanning Mode and Local Image Reconstruction of Impurities in Pipeline

Zhang

Zeng

2018

Lecture Notes in Electrical Engineering

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Wavelet localization of the Radon transform in even dimensions

Cited by 21 publications

References 1 publication

A Wavelet-Based Method for Multiscale Tomographic Reconstruction

A Wavelet-Based Method for Multiscale Tomographic Reconstruction

Tiled-Block Image Reconstruction by Wavelet- Based, Parallel-Filtered Back-Projection

A Novel Computed Tomography Scanning Mode and Local Image Reconstruction of Impurities in Pipeline

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