The Mojette transform is an exact discrete version of the Radon transform that can be exactly implemented from the discrete object with its associated geometry. This exact method requires a very large set of projections that will not be acquired. Then, the goal of this paper is to show how the Mojette projections
set can be interpolated to enlarge the set of projections. The second part of the paper is devoted to recall the sampling geometry both of the reconstructed image and the projections. The third part of the paper presents two Mojette reconstruction algorithms: an exact backprojection filtering Mojette scheme which needs a
(large) finite number of projections and its equivalent FBP-Mojette method. The fourth section presents an angular interpolation method used to generate a suitable set of projections from the known information. The reconstruction results given by this new set of angles used with the two reconstruction methods presented are given and discussed. The quality assessement of the reconstruction algoritms in the case of an insufficient number of projections is done using synthetic phantoms.
Iterative methods are now recognized as powerful tools to solve inverse problems such as tomographic reconstruction. In this paper, the main goal is to present a new reconstruction algorithm made from two components. An iterative algorithm, namely the Conjugate Gradient (CG) method, is used to solve the tomographic problem in the least square (LS) sense for our specific discrete Mojette geometry. The results are compared (with the same geometry) to the corresponding Mojette Filtered Back Projection (FBP) method. In the fist part of the paper, we recall the discrete geometry used to define the projection M and backprojection M* operators. In the second part, the CG algorithm is presented within the context of the Mojette geometry. Noise is then added onto these Mojette projections with respect to the sampling and reconstructions are performed. Finally the Toeplitz block Toeplitz (TBT) character of M*M is demonstrated.
The goal of this paper is to characterize the noise properties of a spline Filtered BackProjection (denoted as FBP) reconstruction scheme. More specifically, the paper focuses on angular and radial sampling of projection data and on assumed local properties of the function to be reconstructed. This new method is visually and quantitatively compared to standard sampling used for FBP scheme. In the second section, we recall the sampling geometry adapted to the discrete geometry of the reconstructed image. Properties of the discrete zero order Spline Ramp filter for classic angles and discrete angles generated from Farey’s series reconstruction are used to generate their equivalent representations for first order Spline filters. Digital phantoms are used to assess the results and the correctness of the linearity and shift-invariantness assumption for the discrete reconstructions. The filter gain has been studied in the Mojette case since the number of projections can be very
different from one angle to another. In the third section, we describe the Spline filter implementation and the continuous/discrete correspondence. In section 4, Poisson noise is added to noise-free onto the projections. The reconstructions between classic angle distribution and Mojette acquisition geometry are compared. Even if the number of bins per projections is fixed for classic FBP while it varies for the Mojette geometry (leading to very different noise behavior per bin) the results of both algorithms are very close. The discussion allows for a general comparison between classic FBP reconstruction and Mojette FBP. The very encouraging results obtained for the Mojette case conclude for the developments of future acquisition devices modeled with the Mojette geometry.
The Filtered BackProjection is still questionable since
many discrete versions have been derived from the continuous Radon formalism.
From a continuous point of view, a previous work has made a link between continuous and discrete FBP versions
denoted as Spline 0-FBP model leading to a regularization of the infinite Ramp filter by the Fourier transform of a
trapezoidal shape. However, projections have to be oversampled (compared to the pixel
size) to retrieve the theoretical properties of Sobolev and Spline spaces. Here we obtain a novel version of the Spline 0 FBP
algorithm with a complete continuous/discrete correspondence using a specific discrete Radon transform, the Mojette transform.
From a discrete point of view, the links toward the FBP algorithm are shaped with the morphological
description and the extended use of discrete projection angles. The resulting equivalent FBP scheme uses a selected set of angles
which covers all the possible discrete Katz's directions issued from the pixels of the (square) shape under reconstruction: this is
implemented using the corresponding Farey's series. We present a new version of a discrete FBP method using a finite number
of projections derived from discrete geometry considerations.
This paper makes links between these two approaches.
In this paper, we propose to perform a novel discrete implementation of the filtered back projection algorithm. For this, we use a version of the discrete exact Radon transform called the Mojette transform that has been developed in our team for few years. The initial questioning was centered about the angular distribution needed for the continuous Radon reconstruction. Because of the discrete set of angles used in the FBP algorithm, discrete angles issued from Farey's series were used. Our version of the FBP algorithm is compared with the classical FBP algorithm. The choice of the set of projection angles is discussed in order to produce a good and efficient angular sampling. Finally, the very different behaviors between the classical FBP and our algorithm justify our study.