Computationally efficient filtered-backprojection algorithm for tomographic image reconstruction using Walsh transform

Abstract

In this paper, we discuss the implementation of the filtered-backprojection (FBP) algorithm for tomographic image reconstruction using Walsh transform to exploit its fast computational ability. Walsh transform is the fastest unitary transform known so far. The major advantage of Walsh transform is that it involves only real additions and subtractions whereas Fourier transform involves complex multiplications and additions. Implementation of the proposed algorithm necessitates the design of an appropriate filter in Walsh domain. In this research, the known Fourier filter coefficients have been transformed into Walsh domain, thereby the 1 × N Fourier filter coefficients were converted into an N × N sparse matrix with nonzero elements in a special pattern. The proposed algorithm has been implemented by taken into account of the special nature of the Walsh domain filter coefficients and tested for its performance using the well-known ‘Shepp-Logan head phantom’ test image. The results demonstrate that the reconstruction strategy has comparable performance with a significant reduction of computing time. For example, with a 128 × 128-pixel image and 180 views, the speedup achieved is fourfold, with reconstructions qualitatively and visually the same as that of FBP algorithm in the Fourier domain.

Introduction

The problem of reconstructing a two/three-dimensional function from measurements of line integrals has applications in such diverse areas as computerized tomography (CT), electron microscopy, radioastronomy, optical interferometry, earth resources exploration, nondestructive testing, radar, sonar, and ultrasonic imaging. The inverse Radon transform is an exact solution to this problem given by Radon in 1917 [1], for the case of continuous exact parallel line integrals taken over a continuous rotation about an origin. In CT-scanning the number of X-ray photons transmitted through an object of interest is measured by an array of detectors. This array is rotated by a fixed angular increment between each set of measurements. From these measurements the total attenuation along each measurement paths is derived and an image of the linear attenuation coefficients may be reconstructed. A detailed description of this application and several approach to solutions are discussed in [2], [3].

In many of the potential applications of tomographic medical imaging, it is of the interest to achieve ‘real-time’ imaging. As always, the meaning of ‘real-time’ is application dependent. In medical image reconstruction, where the data collection devices tend to be very expensive, it is important that the utilization of the devices is not impaired by the time required to compute the images. It is also often desirable to have these images available while the patient is still in the device, to avoid having to recall the patient due to the initial imaging procedure not meeting the diagnostic need. Even in applications where real-time imaging is not of the essence, fast computation of images is desired so that the imaging procedure can be evaluated and optimized without outrageous demands on computational resources [4].

The filtered-backprojection (FBP) algorithm [5] is the most common method to reconstruct a two-dimensional image from its parallel line projections as the detector rotates 180° around the object. In this algorithm, the projection data are first convolved with a ramp-filter kernel at each projection angle, then backprojected into the image space. This algorithm is efficient and provides a stable image with stationary noise propagation. In well-controlled environments, it is generally believed that the FBP algorithm provides a reconstructed image of high quality with normally available computer capacity and computation times [6], [7]. Traditionally, for medical applications, backprojection has been accomplished by special hardware that exploits parallelism of the backprojection process to achieve near real-time reconstruction of a single slice [8]. However, there still exists a lag in reconstruction time that is becoming increasingly important with the emergence of new technologies able to acquire data at ever-faster rates such as in fifth generation CT-scanners. With the increasing data rates, the FBP algorithm becomes the bottleneck [9] in the reconstruction and display process, and presents a barrier to real-time imaging. Hence computational speed of the reconstruction algorithms is of great importance in the tomographic imaging field [10]. In this aspect, we have attempted to implement the FBP algorithm using Walsh transform in place of the widely accepted Fourier transform to exploit its fast computational ability.

Walsh transform is the fastest unitary transform known so far. The transform involves only real additions and subtractions and can be computed using fast algorithm of complexity O (N log₂N) [11], [12]. A filter has been generated in the Walsh domain using the known Fourier filter coefficients [13]. Computer simulation of the FBP algorithm has been carried out both in the Fourier and Walsh domains and the results are compared. We limit ourselves to the case with parallel beams. Modern machines use the so called fan-beam projections [14], where many nonparallel beams emitted from one point in the form of a fan are considered. The fan-beam geometry has the advantage over parallel-beam geometry that it can collect multiple angles simultaneously in a short time interval. The fan-beam algorithm in the Walsh domain is left as a future work.

The organization of the paper is as follows: In Section 2, we discuss the theoretical background of the tomographic image reconstruction problem along with the widely accepted FBP algorithm. Section 3 deals with the implementation of FBP algorithm in the Walsh domain. The comparative study of the algorithms both in the Fourier and Walsh domain is presented in Section 4 followed by a conclusion in Section 5.