Two Dimensional Image Reconstruction Algorithms

Two Dimensional Image Reconstruction Algorithms-By,Srihari K. Malagi,Reg No. 090907471Roll No. 53Section ADept. of Electronics & CommunicationManipal Institute of Technology.

Image Courtesy: Advanced Electron Microscopy Techniques on Semiconductor Nanowires: from Atomic Density of States Analysis to 3D Reconstruction Models, by Sonia Conesa-Boj, Sonia Estrade, Josep M. Rebled, Joan D. Prades, A. Cirera, Joan R. Morante, Francesca Peiro and Jordi Arbiol

Data FlowIntroductionParallel Beam ProjectionsFan Beam ProjectionsTruncated ProjectionsConvolution Back-Projection AlgorithmDigital ImplementationResultsApplicationsPresent ResearchConclusionReferences

IntroductionWhat are Projections?How to obtain Projections?What is Image Reconstruction?What are Truncated Projections?

Image- Courtesy: Fundamentals of Digital Image Processing, by Anil K. Jain

Parallel Beam ProjectionsImage- Courtesy: Computed Tomography, Principles of Medical Imaging, by Prof. Dr.PhilippeCattin, MIAC,University of Basel

Fan Beam Projections

Image- Courtesy: Matlab, Image Processing Toolbox

Radon Transform

Image- Courtesy: Matlab, Image Processing Toolbox

Inverse Radon Transformf(x,y) = For reconstruction of the image, we define Inverse Radon Transform (IRT) which helps us achieve in defining the image from its projection data. Inverse Radon Transform is defined as:

Reconstruction of an Image: Algorithm

RebinningFan Beam Projections can be related to parallel beam projection data as:s = Dsin ; = + ;Therefore,g(s,) = b(sin-1 s/D, - sin-1 s/D);Hence to obtain g(sm,m) we interpolate b(,).This process is called Rebinning.

Block Diagram of the System

RebinningConvolution Back ProjectionFan Beam ProjectionsReconstructed Image(RAM-LAK, SHEPP LOGAN, LOWPASS COSINE, GENRALIZED HAMMING Filter can be used).

Filters

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Results

Results

MAE = 0.177CBP using RAM-LAK Filter

Results

MAE = 0.167CBP using SHEPP-LOGAN Filter

ResultsMAE = 99.2961

CBP using No Filter

Results

CBP for Truncated Projections (wrt s)

Results

CBP for Truncated Projections using extrapolation Technique

Results

CBP algorithm using less number of projections

ApplicationsDigital image reconstruction is a robust means by which the underlying images hidden in blurry and noisy data can be revealed.Reconstruction algorithms derive an image of a thin axial slice of the object, giving an inside view otherwise unobtainable without performing surgery. Such techniques are important in medical imaging (CT scanners), astronomy, radar imaging, geological exploration, and non-destructive testing of assemblies.

Image- Courtesy: Fundamentals of Digital Image Processing, by Anil K. Jain

Present ResearchPresently, the key concern is on Reconstruction of objects using limited data such as truncated projections, limited projections etc Filtered Back-projection (FBP) Algorithms have been implemented since the system is faster when compared to CBP Algorithm. Also new techniques such as Discrete Radon Transform (DRT) Techniques have been implemented to achieve the goal.Also Fan Beam projections are considered for 2D image reconstructions, since less number of projections will be required when compared to parallel beam projections. Also from the conventional fixed focal length Fan-Beam projections, we have observed that the research is moved onto defining variable focal length Fan-Beam Projections.

ConclusionImage reconstruction is unfortunately an ill-posed problem. Mathematicians consider a problem to be well posed if its solution (a) exists, (b) is unique, and (c) is continuous under innitesimal changes of the input. The problem is ill posed if it violates any of the three conditions.In image reconstruction, the main challenge is to prevent measurement errors in the input data from being amplied to unacceptable artifacts in the reconstructed image.New techniques are being implemented, and tested to overcome these problems.

ReferencesSoumekh, M., IEEE Transactions on Acoustics, Speech and Signal Processing, Image reconstruction techniques in tomographic imaging systems, Aug 1986, ISSN :0096-3518.Matej, S.,Bajla, I.,Alliney, S., IEEE Transactions on Medical Imaging, On the possibility of direct Fourier reconstruction from divergent-beam projections, Jun 1993, ISSN :0278-0062.You, J., Liang, Z.,Zeng, G.L., IEEE Transactions on Medical Imaging, A unified reconstruction framework for both parallel-beam and variable focal-length fan-beam collimators by a Cormack-type inversion of exponential Radon transform, Jan. 1999, ISBN: 0278-0062.

ReferencesClackdoyle, R., Noo, F.,Junyu Guo.,Roberts, J.A.,IEEE Transactions on Nuclear Science, Quantitative reconstruction from truncated projections in classical tomography, Oct. 2004, ISSN :0018-9499.O'Connor, Y.Z., Fessler, J.A., IEEE Transactions on Medical Imaging, Fourier-based forward and back-projectors in iterative fan-beam tomographic image reconstruction, May 2006, ISSN :0278-0062.Wang, L., IEEE Transactions on Computers, Cross-Section Reconstruction with a Fan-Beam Scanning Geometry, March 1977, ISSN :0018-9340.

ReferencesAnil K. Jain, Fundamentals of Digital Image Processing, Prentice Hall, Englewood Cliffs, NJ 07632, ISBN 0-13-336165-9.Avinash C. Kak and Malcolm Slaney, Principles of Computerized Tomographic Imaging, Society for Industrial and Applied Mathematics, Philadelphia, ISBN 0-89871-494-X.G. Van Gompel, Department of Physics, University of Antwerp, Antwerp, Towards accurate image reconstruction from truncated X-ray CT projections, Publication Type: Thesis, 2009.

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