Minimal-scan filtered backpropagation algorithms for diffraction tomography

Accurate image reconstruction from few-view and limited-angle data in diffraction tomography

We present a method for obtaining accurate image reconstruction from highly sparse data in diffraction tomography (DT). A practical need exists for reconstruction from few-view and limited-angle data, as this can greatly reduce required scan times in DT. Our method does this by minimizing the total variation (TV) of the estimated image, subject to the constraint that the Fourier transform of the estimated image matches the measured Fourier data samples. Using simulation studies, we show that the TV-minimization algorithm allows accurate reconstruction in a variety of few-view and limited-angle situations in DT. Accurate image reconstruction is obtained from far fewer data samples than are required by common algorithms such as the filtered-backpropagation algorithm. Overall our results indicate that the TV-minimization algorithm can be successfully applied to DT image reconstruction under a variety of scan configurations and data conditions of practical significance.

Investigating the enhancement of three-dimensional diffraction tomography by using multiple illumination planes

The three dimensional (3-D) extension of the two well-known diffraction tomography algorithms, namely, direct Fourier interpolation (DFI) and filtered backpropagation (FBP), are presented and the problem of the data needed for a full 3-D reconstruction is investigated. These algorithms can be used efficiently to solve the inverse scattering problem for weak scatterers in the frequency domain under the first-order Born and Rytov approximations. Previous attempts of 3-D reconstruction with plane-wave illumination have used data obtained with the incident direction restricted at the xy plane. However, we show that this restriction results in the omission of the contribution of certain spatial frequencies near the omegaz axis for the final reconstruction. The effect of this omission is studied by comparing the results of reconstruction with and without data obtained from other incident directions that fill the spatial frequency domain. We conclude that the use of data obtained for incident direction in only the xy plane is sufficient to achieve a satisfactory quality of reconstruction for a class of objects presenting smooth variation along the z axis, while abrupt variations along the z axis cannot be imaged. This result should be taken into account in the process of designing the acquisition geometry of a tomography scanner.

Reconstructions in limited-view thermoacoustic tomography

The limited-view problem is studied for thermoacoustic tomography, which is also referred to as photoacoustic or optoacoustic tomography depending on the type of radiation for the induction of acoustic waves. We define a "detection region," within which all points have sufficient detection views. It is explained analytically and shown numerically that the boundaries of any objects inside this region can be recovered stably. Otherwise some sharp details become blurred. One can identify in advance the parts of the boundaries that will be affected if the detection view is insufficient. If the detector scans along a circle in a two-dimensional case, acquiring a sufficient view might require covering more than a pi-, or less than a pi-arc of the trajectory depending on the position of the object. Similar results hold in a three-dimensional case. In order to support our theoretical conclusions, three types of reconstruction methods are utilized: a filtered backprojection (FBP) approximate inversion, which is shown to work well for limited-view data, a local-tomography-type reconstruction that emphasizes sharp details (e.g., the boundaries of inclusions), and an iterative algebraic truncated conjugate gradient algorithm used in conjunction with FBP. Computations are conducted for both numerically simulated and experimental data. The reconstructions confirm our theoretical predictions.

Short-scan SPECT imaging with non-uniform attenuation and 3D distance-dependent spatial resolution

Image quality and quantitative accuracy in single-photon emission computed tomography (SPECT) can be degraded by, e.g., the effects of photon attenuation and finite spatial resolution. It is generally considered that adequate compensation for such effects on SPECT images requires data acquired over 2 pi. Recently, using the existing consistency condition on the data function, Noo and Wagner (2001 Inverse Problems 17 1357-72) have shown analytically that data acquired over only pi can be used to correct completely for the effect of uniform attenuation in SPECT. It remains unknown, however, whether data acquired only over pi in SPECT with non-uniform attenuation and/or 3D distance-dependent spatial resolution (DDSR) contain complete information for accurate image reconstruction. In this work, we develop a heuristic perspective, which is referred to as the potato peeler perspective to show conceptually that data in SPECT with non-uniform attenuation and/or 3D DDSR acquired over 2 pi contain redundant information and that such information can be used to reduce the scanning angle in SPECT. Specifically, we show heuristically that, in SPECT with only non-uniform attenuation, the scanning angle can be reduced from 2 pi to pi and that, in SPECT with both non-uniform attenuation and DDSR with a physically realistic form, the scanning angle can be reduced from 2 pi to pi in a practical sense. We conduct computer simulation studies, and the results from these studies corroborate the observations obtained based upon the heuristic potato peeler perspective.

On a limited-view reconstruction problem in diffraction tomography

Diffraction tomography (DT) is an inversion technique that reconstructs the refractive index distribution of a scattering object. We previously demonstrated that by exploiting the redundant information in the DT data, the scattering object could be exactly reconstructed using measurements taken over the angular range [0, phimin], where pi < phimin < or = 3pi/2. In this paper, we reveal a relationship between the maximum scanning angle and image resolution when a filtered backpropagation (FBPP) reconstruction algorithm is employed for image reconstruction. Based on this observation, we develop short-scan FBPP algorithms that reconstruct a low-pass filtered scattering object from measurements acquired over the angular range [0, phi(c)], where phi(c) < phimin.

Full- and minimal-scan reconstruction algorithms for fan-beam diffraction tomography

Diffraction tomography (DT) is a tomographic inversion technique that reconstructs the spatially variant refractive-index distribution of a scattering object. In fan-beam DT, the interrogating radiation is not a plane wave but rather a cylindrical wave front emanating from a line source located a finite distance from the scattering object. We reveal and examine the redundant information that is inherent in the fan-beam DT data function. Such redundant information can be exploited to reduce the reconstructed image variance or, alternatively, to reduce the angular scanning requirements of the fan-beam DT experiment. We develop novel filtered backpropagation and estimate-combination reconstruction algorithms for full-scan and minimal-scan fan-beam DT. The full-scan algorithms utilize measurements taken over the angular range 0 </= phi </= 2pi, whereas the minimal-scan reconstruction algorithms utilize only measurements taken over the angular range 0 </= phi </= phi(min), where pi </= phi(min) </= 3pi/2 is a specified function that describes the fan-beam geometry. We demonstrate that the full- and minimal-scan fan-beam algorithms are mathematically equivalent. An implementation of the algorithms and numerical results obtained with noiseless and noisy simulated data are presented.