Efficient algorithms for linear dynamic inverse problems with known motion

An inverse problem is called dynamic if the object changes during the data acquisition process. This occurs e.g. in medical applications when fast moving organs like the lungs or the heart are imaged. Most regularization methods are based on the assumption that the object is static during the measuring procedure. Hence, their application in the dynamic case often leads to serious motion artefacts in the reconstruction. Therefore, an algorithm has to take into account the temporal changes of the investigated object. In this paper, a reconstruction method that compensates for the motion of the object is derived for dynamic linear inverse problems. The algorithm is validated at numerical examples from computerized tomography.

[1]  Bernadette N. Hahn,et al.  Reconstruction of dynamic objects with affine deformations in computerized tomography , 2013 .

[2]  Cameron J. Ritchie,et al.  Respiratory compensation in projection imaging using a magnification and displacement model , 1996, IEEE Trans. Medical Imaging.

[3]  Alfred K. Louis,et al.  Feature reconstruction in inverse problems , 2011 .

[4]  M Grass,et al.  Motion-compensated iterative cone-beam CT image reconstruction with adapted blobs as basis functions , 2008, Physics in medicine and biology.

[5]  Pierre Grangeat,et al.  Exact reconstruction in 2D dynamic CT: compensation of time-dependent affine deformations. , 2004, Physics in medicine and biology.

[6]  Weiguo Lu,et al.  Tomographic motion detection and correction directly in sinogram space. , 2002, Physics in medicine and biology.

[7]  Peter Maass,et al.  A mollifier method for linear operator equations of the first kind , 1990 .

[8]  A linear regularization scheme for inverse problems with unbounded linear operators on Banach spaces , 2013 .

[9]  A K Louis,et al.  Efficient algorithms for the regularization of dynamic inverse problems: I. Theory , 2002 .

[10]  A J Reader,et al.  Respiratory motion correction for PET oncology applications using affine transformation of list mode data , 2007, Physics in medicine and biology.

[11]  Pierre Grangeat,et al.  Compensation of Some Time Dependent Deformations in Tomography , 2007, IEEE Transactions on Medical Imaging.

[12]  Carsten H. Wolters,et al.  Efficient algorithms for the regularization of dynamic inverse problems: II. Applications , 2002 .

[13]  A. Baskakov,et al.  Analysis of linear differential equations by methods of the spectral theory of difference operators and linear relations , 2013 .

[14]  Yongyi Yang,et al.  Tomographic Reconstruction of Dynamic Cardiac Image Sequences , 2007, IEEE Transactions on Image Processing.

[15]  F. Natterer.  The Mathematics of Computerized Tomography , 1986 .

[16]  Thomas Schuster,et al.  The Method of Approximate Inverse: Theory and Applications , 2007 .

[17]  Alfred K. Louis.  Approximate inverse for linear and some nonlinear problems , 1995 .

[18]  Alexander Katsevich,et al.  An accurate approximate algorithm for motion compensation in two-dimensional tomography , 2010 .

[19]  Simon Rit,et al.  Comparison of Analytic and Algebraic Methods for Motion-Compensated Cone-Beam CT Reconstruction of the Thorax , 2009, IEEE Transactions on Medical Imaging.

[20]  Alexander Katsevich,et al.  Local Tomography and the Motion Estimation Problem , 2011, SIAM J. Imaging Sci..

[21]  Stefan Kindermann,et al.  Regularization by dynamic programming , 2007, ArXiv.

[22]  Alfred K. Louis,et al.  Diffusion reconstruction from very noisy tomographic data , 2010 .

[23]  Stefan Kindermann,et al.  On Regularization Methods for Inverse Problems of Dynamic Type , 2006, ArXiv.

[24]  Bernadette N. Hahn,et al.  Combined reconstruction and edge detection in dimensioning , 2013 .

[25]  Alfred K. Louis,et al.  Combining Image Reconstruction and Image Analysis with an Application to Two-Dimensional Tomography , 2008, SIAM J. Imaging Sci..

[26]  Xiaoyi Jiang,et al.  Motion Correction in Dual Gated Cardiac PET Using Mass-Preserving Image Registration , 2012, IEEE Transactions on Medical Imaging.

[27]  Bernadette N. Hahn,et al.  Reconstruction in the three-dimensional parallel scanning geometry with application in synchrotron-based x-ray tomography , 2012 .

[28]  Pierre Grangeat,et al.  Dynamic X-ray computed tomography , 2003, Proc. IEEE.

[29]  Solving linear operator equations in Banach spaces non-iteratively by the method of approximate inverse , 2010 .

[30]  C. J. Ritchie,et al.  Predictive respiratory gating: a new method to reduce motion artifacts on CT scans. , 1994, Radiology.