localized wavelet based computerized tomography - Semantic Scholar

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In 5], Olson and DeStefano implement a direct recon- struction algorithm in .... order to implement the ltering part using FFT we sam- pled the continuously deĀ ...
LOCALIZED WAVELET BASED COMPUTERIZED TOMOGRAPHY F. Rashid-Farrokhi, K. J. R. Liu, C. A. Berenstein, D. Walnut Electrical Engineering Department and Institute for Systems Research University of Maryland College Park, MD 20742, USA  Department of Mathematical Sciences George Mason University Fairfax, VA, 22030, USA

ABSTRACT

We develop an algorithm to reconstruct the wavelet coecients of an image from the Radon transform data. The proposed method uses the properties of wavelets to localize the Radon transform and can be used to reconstruct a local region of the cross section of a body, using almost completely local data which signi cantly reduces the amount of exposure and computations in X-ray tomography. For example, for a local region of radius 20 pixels in a 256  256 image the proposed method can reduce the exposure to 12:5% of the conventional ltered backprojection method. Compared to the existing schemes [5], [6], which can only reduce to 40%.

1. INTRODUCTION

It is well known that in dimension two and in fact in any even dimension the Radon transform is not local, that is, the recovery of image at any xed point requires the knowledge of all projections of the image. This means that a patient would have to be exposed to a relatively large amount of X-rays even if it was desired to view only a small part of his body. Thus, searching for a means to reduce exposure, and at the same time to be able to perfectly reconstruct the region of interest, has been of great interest recently [5], [6]. In [5], Olson and DeStefano implement a direct reconstruction algorithm in which the one dimensional wavelet transform of R f , is computed for each . In [6], Delaney and Bresler compute the two dimensional separable wavelet transform of the image directly from the projection data. Both algorithms take advantage of the observation, made in [1], that the Hilbert transform of a function with many vanishing moments has rapid decay. In fact, the Hilbert transform of a compactly supported wavelet with suciently many vanishing moments has essentially the same support as the wavelet itself. Thus, in both algorithms, the high{ resolution parts of the image are obtained locally, and the low{resolution parts are obtained by global measurements at a few angles. We have made the surprising observation that, in some cases, the Hilbert transform of a compactly supported scaling function also has essentially the same support as the scaling function itself. We take advantage of this observation to reconstruct the low{resolution parts of the image as well as the high{resolution parts using almost local data plus a small margin for the support of the lters. This gives This work was supported in part by the ONR grant N00014-9310566 and the NSF NYI Award MIP9457397.

substantial savings in exposure and computation over the methods in [5] and [6]. The proposed method calculates the wavelet coecients of the reconstructed image with the same complexity as the conventional ltered backprojection method. The wavelet coecients are obtained directly from the projection data, which saves the computations required to obtain the wavelet coecients from the reconstructed image. This is useful in applications where the wavelet coecients of the reconstructed image are used. Compared to the method proposed in [6] it is more general and can be applied to any wavelet basis even in the cases where the basis is not separable and there exists no multiresolution approach to obtain the wavelet coecients. The main features of our algorithm are: Reduced exposure compared to previous algorithms ([6], [5]). Computationally more ecient than other algorithms because we use smaller exposure lengths. Uniform exposure at all angles which allows for easier implementation in hardware. Ability to reconstruct o {center or even multiple regions of interest, as well as centered reconstruction. It has been noted in a number of places [7], [8], [9], that the interior Radon transform has a non-zero null-space (i.e., interior problem is not uniquely solvable) and that the elements of the null-space do not vary much in the region of interest. In our algorithm this phenomena appears as a constant bias in the reconstructed image. Such a bias is commonly observed in the interior reconstruction problem [7], [9]. In this paper we use the following notations: The d did mensional Euclidean space is denoted by R . We de ne the R Fourier transform in Rd by f^(w~ ) = Rd f (~x)ej2w~ ~x d~x. The space of in nitely di erentiable functions, which all of their derivatives decay faster than any polynomial is denoted as S (Rd ). Both continuous and discrete convolution operators are denoted by .

2. WAVELET RECONSTRUCTION We will brie y introduce the terminology and de nitions required in the subsequent discussion. In computerized tomography, a cross section of the human body is scanned by a non{di racting thin X{ray beam whose intensity loss is recorded by a set of detectors. The Radon transform is a mathematical tool which is used to describe the recorded intensity losses as averages of the tissue density function over hyper{planes which, in dimension two, are lines. Given

f 2 S (R2 ), we de ne the Radon transform of f by Z

R f (s) =



~ x ~=s

f (~x)d~x =

Z

?

f (s~ + y)dy;

(1)

where ~ = (cos; sin);  2 [0; 2), s 2 R and ? is the subspace perpendicular to ~. In order to reconstruct the function f from the projection data we use the ltered backprojection method, which can be implemented in two steps, the ltering part Q (t) = (H@R f )(t); (2) and the backprojection

f (x; y) =

Z

 0

Q (xcos + ysin)d:

(3)

where H is the Hilbert transform and @ represents ordi-2 nary di erentiation. Let be any function de ned on R satisfying 0