A new representation and projection model for tomography, based on separable B-splines (original) (raw)

Spline driven: high accuracy projectors for tomographic reconstruction from few projections

IEEE Transactions on Image Processing, 2015

Tomographic iterative reconstruction methods need a very thorough modeling of data. This point becomes critical when the number of available projections is limited. At the core of this issue is the projector design, i.e. the numerical model relating the representation of the object of interest to the projections on the detector. Voxel driven and ray driven projection models are widely used for their short execution time in spite of their coarse approximations. Distance driven model has an improved accuracy but makes strong approximations to project voxel basis functions. Cubic voxel basis functions are anisotropic, accurately modeling their projection is therefore computationally expensive. Smoother and more isotropic basis functions both better represent continuous functions and provide simpler projectors. These considerations have led to the development of spherically symmetric volume elements, called blobs. Set apart their isotropy, blobs are often considered too computationally expensive in practice. In this paper, we consider using separable B-splines as basis functions to represent the object and we propose to approximate the projection of these basis functions by a 2D separable model. When the degree of the B-splines increases, their isotropy improves and projections can be computed regardless of their orientation. The degree and the sampling of the B-splines can be chosen according to a trade-off between approximation quality and computational complexity. We quantitatively measure the good accuracy of our model and compare it with other projectors like distance-driven and the model proposed by Long et al. [1]. From numerical experiments, we demonstrate that our projector with an improved accuracy better preserves the quality of the reconstruction as the number of projections decreases. Our projector with cubic B-splines requires about twice as many operations as a model based on This work was supported by the MiTiV project (Méthodes Inverses pour le Traitement en Imagerie du Vivant), funded by the French ANR (N • ANR-09-EMER-008).

On Visualization and Reconstruction from Non‐Uniform Point Sets using B‐splines

2009

Abstract In this paper we present a novel framework for the visualization and reconstruction from non-uniform point sets. We adopt a variational method for the reconstruction of 3D non-uniform data to a uniform grid of chosen resolution. We will extend this reconstruction to an efficient multi-resolution uniform representation of the underlying data. Our multi-resolution representation includes a traditional bottom-up approach and a novel top-down hierarchy for adaptive hierarchical reconstruction.

Embedded B-spline Reconstruction in the Volume Rendering

2008

This paper deals with reconstruction problem in the visualization. The representation of the object is based on the volumetric data. Volumetric data is sampled on a regular grid. The surface of the object is implicitly defined as a zero value of the volumetric function. In the volume rendering procedure, intersection of the ray with the object is required. Trilinear interpolation is usually used for the reconstruction. In this paper we propose interpolation B-spline embedded directly into the rendering procedure. Different artifacts may appear due to the reconstruction in the rendering procedure. If the quality of the rendered object is of the crucial importance, we have to pay attention to the reconstruction in the rendering procedure. Keywords: visualization, interpolation B-spline, reconstruction, volume rendering 1

B-spline Surface Reconstruction by Inverse Subdivisions

2009 IEEE-RIVF International Conference on Computing and Communication Technologies, 2009

This paper presents a method to reconstruct a B-spline surface from a quadrangular mesh, using an inverse Catmull-Clark subdivision. We want to minimize the surface contraction due to the approximating subdivision scheme. We introduce geometrical operations which minimize the impact of the subdivision approximation and can be used in the parametric surface reconstruction. The quality of the method is evaluated by criteria of distances, curvatures or computing time on experimental results.

Frequency domain analysis of B-spline interpolation

ISIE '99. Proceedings of the IEEE International Symposium on Industrial Electronics (Cat. No.99TH8465), 1999

This paper describes B-spline interpolation and compares it with other reconstruction methods, especially in three-dimensional space. We rst consider the B-spline bases in the terms of convolution in signal processing. Presented analysis requires careful usage of continuous and discrete representation of B-splines. Emphasis is given to important di erence between B-spline interpolation and approximation. The di erence is shown through frequency domain analysis, so we derive frequency responses of the Bspline interpolation and approximation. We conclude by demonstrating the use of several reconstruction lters and appropriate gradient estimators in the volume rendering. Exact reconstruction in the volume visualization is very important in many industrial applications, such as material cavity control.

Parallel image reconstruction using B-spline approximation (PROBER

Magnetic Resonance in Medicine, 2007

A new reconstruction method for parallel MRI called PROBER is proposed. The method PROBER works in an image domain similar to methods based on Sensitivity Encoding (SENSE). However, unlike SENSE, which first estimates the spatial sensitivity maps, PROBER approximates the reconstruction coefficients directly by B-splines. Also, B-spline coefficients are estimated at once in order to minimize the reconstruction error instead of estimating the reconstruction in each pixel independently (as in SENSE). This makes the method robust to noise in reference images. No presmoothing of reference images is necessary. The number of estimated parameters is reduced, which speeds up the estimation process. PROBER was tested on simulated, phantom, and in vivo data. The results are compared with commercial implementations of the algorithms SENSE and GRAPPA (Generalized Autocalibrating Partially Parallel Acquisitions) in terms of elapsed time and reconstruction quality. The experiments showed that PROBER is faster than GRAPPA and SENSE for images wider than 150 × 150 pixels for comparable reconstruction quality. With more basis functions, PROBER outperforms both SENSE and GRAPPA in reconstruction quality at the cost of slightly increased computational time. Magn Reson Med 58:582–591, 2007. © 2007 Wiley-Liss, Inc.

The B-spline Interpolation in Visualization of the Three-dimensional Objects

2000

The volume data is generally in the form of the large array of numbers. In order to render the object hidden in the volume, we need to reconstruct or interpolate data values between the samples. The novelty presented in this paper is B-spline interpolation in the volumetric space. We show that this approach is better then currently used methods. To

Reconstruction of B-spline surfaces from scattered data points

2000

We present a new approach for reconstructing a smooth surface from a set of scattered points in three-dimensional (3D) space. Our algorithm first decomposes a given point set into a quadtree-like data structure known as a strip tree. The strip tree is used to fit a set of least squares quadratic surfaces to the data points. These quadratic surfaces are then degree-elevated to bi-cubic surfaces and blended together to form a set of B-spline surfaces that approximates the given point set.

Fast Parallel MRI Reconstruction Using B-Spline Approximation

Parallel MRI (pMRI) is a way to increase the speed of the MRI acquisition by combining data obtained simultaneously from several receiver coils with distinct spatial sensitivities. The measured data contains additional information about the position of the signal with respect to data obtained by a standard, uniform sensitivity coil. The idea is to speed up the acquisition by sampling more sparsely in the k-space and to compensate the data loss using the additional information obtained by a higher number of receiver coils.