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