A curve evolution approach to object-based tomographic reconstruction

In this paper, we develop a new approach to tomographic reconstruction problems based on geometric curve evolution techniques. We use a small set of texture coefficients to represent the object and background inhomogeneities and a contour to represent the boundary of multiple connected or unconnected objects. Instead of reconstructing pixel values on a fixed rectangular grid, we then find a reconstruction by jointly estimating these unknown contours and texture coefficients of the object and background. By designing a new "tomographic flow", the resulting problem is recast into a curve evolution problem and an efficient algorithm based on level set techniques is developed. The performance of the curve evolution method is demonstrated using examples with noisy limited-view Radon transformed data and noisy ground-penetrating radar data. The reconstruction results and computational cost are compared with those of conventional, pixel-based regularization methods. The results indicate that the curve evolution methods achieve improved shape reconstruction and have potential computation and memory advantages over conventional regularized inversion methods.

[1]  A. U.S.,et al.  Polynomial Representation of Pictures , 2003 .

[2]  F. Santosa,et al.  Reconstruction of a two-dimensional binary obstacle by controlled evolution of a level-set , 1998 .

[3]  David A. Castanon,et al.  Curve evolution object-based techniques for image reconstruction and segmentation , 2002 .

[4]  J. Fessler,et al.  Object-based 3-D reconstruction of arterial trees from magnetic resonance angiograms. , 1991, IEEE transactions on medical imaging.

[5]  W. Clem Karl,et al.  Tomographic Reconstruction of Polygons from Knot Location and Chord Length Measurements , 1996, CVGIP Graph. Model. Image Process..

[6]  Kenneth M. Hanson,et al.  Tomographic reconstruction based on flexible geometric models , 1994, Proceedings of 1st International Conference on Image Processing.

[7]  Peyman Milanfar,et al.  A moment-based variational approach to tomographic reconstruction , 1996, IEEE Trans. Image Process..

[8]  James A. Sethian,et al.  Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid , 2012 .

[9]  Uday B. Desai,et al.  Direct parametric object detection in tomographic images , 1998, Image Vis. Comput..

[10]  Jerry L Prince,et al.  A geometric projection-space reconstruction algorithm , 1990 .

[11]  Anthony J. Yezzi,et al.  A geometric snake model for segmentation of medical imagery , 1997, IEEE Transactions on Medical Imaging.

[12]  A. Willsky,et al.  Reconstruction from projections based on detection and estimation of objects--Parts I and II: Performance analysis and robustness analysis , 1984 .

[13]  W. Karl,et al.  Underground imaging based on edge-preserving regularization , 1999, Proceedings 1999 International Conference on Information Intelligence and Systems (Cat. No.PR00446).

[14]  Jeffrey A. Fessler,et al.  A Bayesian Approach to Reconstruction from Incomplete Projections of a Multiple Object 3D Domain , 1989, IEEE Trans. Pattern Anal. Mach. Intell..

[15]  F. Santosa A Level-set Approach Inverse Problems Involving Obstacles , 1995 .

[16]  Sanjeev R. Kulkarni,et al.  Convex set estimation from support line measurements and applications to target reconstruction from laser radar data , 1990, Photonics West - Lasers and Applications in Science and Engineering.

[17]  Jerry L Prince Geometric model-based estimation from projections , 1988 .

[18]  Eric L. Miller,et al.  A new shape-based method for object localization and characterization from scattered field data , 2000, IEEE Trans. Geosci. Remote. Sens..

[19]  Sanjeev R. Kulkarni,et al.  Convex-polygon estimation from support-line measurements and applications to target reconstruction from laser-radar data , 1992 .

[20]  S. Osher,et al.  Algorithms Based on Hamilton-Jacobi Formulations , 1988 .

[21]  Michael Unser,et al.  Polynomial representation of pictures , 1986 .

[22]  Jayant Shah Riemannian Drums, Anisotropic Curve Evolution and Segmentation , 1999, Scale-Space.

[23]  Peyman Milanfar,et al.  Reconstructing polygons from moments with connections to array processing , 1995, IEEE Trans. Signal Process..

[24]  D. Mumford,et al.  Optimal approximations by piecewise smooth functions and associated variational problems , 1989 .

[25]  Sanjeev R. Kulkarni,et al.  Local tests for consistency of support hyperplane data , 1996, Journal of Mathematical Imaging and Vision.

[26]  A Tikhonov,et al.  Solution of Incorrectly Formulated Problems and the Regularization Method , 1963 .

[27]  Yoram Bresler,et al.  Three-dimensional reconstruction from projections with incomplete and noisy data by object estimation , 1987, IEEE Trans. Acoust. Speech Signal Process..

[28]  Yoram Bresler,et al.  Estimation Of 3-D Shape Of Blood Vessels From X-Ray Images. , 1984, Other Conferences.

[29]  T. Chan,et al.  A Variational Level Set Approach to Multiphase Motion , 1996 .

[30]  Jerry L. Prince,et al.  Reconstructing Convex Sets from Support Line Measurements , 1990, IEEE Trans. Pattern Anal. Mach. Intell..

[31]  Tony F. Chan,et al.  Active contours without edges , 2001, IEEE Trans. Image Process..

[32]  Peyman Milanfar,et al.  Reconstructing Binary Polygonal Objects from Projections: A Statistical View , 1994, CVGIP Graph. Model. Image Process..

[33]  Ken D. Sauer,et al.  Image reconstruction from a limited number of projections using multiple object detection/estimation , 1990, International Conference on Acoustics, Speech, and Signal Processing.

[34]  E. Miller,et al.  A shape reconstruction method for electromagnetic tomography using adjoint fields and level sets , 2000 .

[35]  Kaleem Siddiqi,et al.  Area and length minimizing flows for shape segmentation , 1998, IEEE Trans. Image Process..

[36]  Eric L. Miller,et al.  Object-based reconstruction using coupled tomographic flows , 2000, Proceedings 2000 International Conference on Image Processing (Cat. No.00CH37101).

[37]  Yves Bizais,et al.  3D attenuation map reconstruction using geometrical models and free form deformations , 2000, IEEE Trans. Medical Imaging.

[38]  Jayant Shah,et al.  A common framework for curve evolution, segmentation and anisotropic diffusion , 1996, Proceedings CVPR IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[39]  Jayant Shah Riemannian Drums, Anisotropic Curve Evolution, and Segmentation , 2000, J. Vis. Commun. Image Represent..

[40]  Kenneth M. Hanson,et al.  Tomographic Reconstruction Of Axially Symmetric Objects From A Single Radiograph , 1985, Other Conferences.

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

[42]  Carey M. Rappaport,et al.  A general method for FDTD modeling of wave propagation in arbitrary frequency-dispersive media , 1997 .

[43]  W. Clem Karl,et al.  Tomographic reconstruction using curve evolution , 2000, Proceedings IEEE Conference on Computer Vision and Pattern Recognition. CVPR 2000 (Cat. No.PR00662).

[44]  J. Sethian,et al.  Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations , 1988 .

[45]  Til Aach,et al.  Coding of segmented images using shape-independent basis functions , 1998, IEEE Trans. Image Process..

[46]  Song Wang,et al.  Model-based reconstruction of multiple circular and elliptical objects from a limited number of projections , 1996, IEEE Trans. Image Process..

[47]  Anthony J. Yezzi,et al.  A statistical approach to snakes for bimodal and trimodal imagery , 1999, Proceedings of the Seventh IEEE International Conference on Computer Vision.

[48]  Y. Bizais,et al.  Three-dimensional attenuation map reconstruction using geometrical models and free-form deformations [SPECT application] , 2000, IEEE Transactions on Medical Imaging.

[49]  Alfred O. Hero,et al.  Model-based estimation for dynamic cardiac studies using ECT , 1994, IEEE Trans. Medical Imaging.

[50]  Avinash C. Kak,et al.  Principles of computerized tomographic imaging , 2001, Classics in applied mathematics.

[51]  Kaleem Siddiqi,et al.  Area and length minimizing flows for shape segmentation , 1997, Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[52]  K M Hanson,et al.  Tomographic reconstruction using 3D deformable models. , 1998, Physics in medicine and biology.