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Quantitative Image Analysis Does Deconvolution Help - Electrical

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					  BMED-4800/ECSE-4800
Introduction to Subsurface
     Imaging Systems
            Lecture 6: Introduction to CT Scanners
             Kai E. Thomenius1 & Badri Roysam2

        1
          Chief Technologist, Imaging Technologies,
           General Electric Global Research Center
        2
          Professor, Rensselaer Polytechnic Institute



Center for Sub-Surface Imaging & Sensing
                                                        Slide no. 1
         Outline of Course Topics

• THE BIG PICTURE                        • PULSE ECHO METHODS
  – What is subsurface sensing &           – Examples
    imaging?
                                         • MRI
  – Why a course on this topic?
                                           – A different sensing modality from 
• EXAMPLE:  THROUGH 
                                             the others
  TRANSMISSION SENSING
  – X-Ray Imaging                          – Basics of MRI
  – Computer Tomography                  • MOLECULAR IMAGING
• COMMON FUNDAMENTALS                      – What is it?
  – propagation of waves                   – PET & Radionuclide Imaging
  – interaction of waves with targets    • IMAGE PROCESSING & CAD
    of interest 




                                                                       Slide no. 2
                                Recap 
                                             Modulation Transfer Function:
• Three aspects to x-ray 
  imaging
  – x-ray sources
  – x-ray interactions with matter            Contrast-to-noise ratio:
  – detectors (analog / digital)
• Performance metrics (MTF, 
  SNR, CNR, DQE) for x-ray                Detective quantum efficiency:
  performance were given.                         signal
  –Justification for digital detectors 
   was based on these.
• ROC curves for hypothesis 
  testing.                                Noise power        Exposure
  – TP, FP, TN, FN                         spectrum            (mR)
                                                                     Incident energy
                                                                    density(mR/mm2)
                                                                          Slide no. 3
         Lambert-Beer Equation: 
                Review
where
   – I(z) is the x-ray intensity at the 
     measurement plane
   – Io is the x-ray intensity at the 
     source plane                          Io                I(z)
   – z is the distance between the 
     source & measurement planes
   –  m is the attenuation coefficient.             z
• Also known as Beer-Lambert 
  or Beer’s Law
• The goal of the CT scanner is              In general, m is a
  to apply the Lambert-Beer                 function of z, this
  Equation to find the m-values            complicates matters.
  across the cross section.

                                                              Slide no. 4
    X-Ray Imaging to CT Imaging
• Standard X-ray’s limitations
  – 3D structures are collapsed into 
    2D images
  – Low soft-tissue contrast, great for 
    bones
  – Not very quantitative
• X-ray CT
  – Take a large number of x-rays at 
    multiple angles
  – Calculate the 3D image
     • Similar hardware to ordinary x-ray
     • Image of a slice - extendable to 3D
     • But, heavy computational load



                                             Slide no. 5
     Brief History of CT Scanners
• 1972 - CT scanning invented by Godfrey 
  Hounsfield, a UK scientist:
   – Announced computed axial transverse 
     scanning
   – Initially known as CAT scanners, this 
     caused predictable cartoons to show up
   – Presented initial cross-sectional images of 
     the brain, etc.
   – The invention was shown to have excellent 
     diagnostic potential immediately.
• Hounsfield shared the 1979 Nobel Prize 
  with Alan Cormack, who derived 
  alternative CT reconstruction algorithms.
• Unfortunately, Dr. Hounsfield’s company, 
  EMI, failed to capitalize on the invention 
  and has since then left medical imaging.
• Dr. Hounsfield passed away in 2004.
                                                                               From CafePress.com
        http://nobelprize.org/medicine/laureates/1979/hounsfield-lecture.pdf
                                                                                             Slide no. 6
Hounsfield’s Apparatus




                         Slide no. 7
                    CT Scanners
• This is what they look like 
  today …
• Basically rotating x-ray 
  tubes & detectors.
  – With a lot of computing power.
• CT scanners are being used 
  for:
  – All types of medical 
    diagnostics
  – Airport luggage inspection
  – Nondestructive evaluation of 
    materials.


                                     Slide no. 8
CT System Components




                       Slide no. 9
Types of Images, Viewing Modes




         Courtesy of Tom Toth, GEHC
                                      Slide no. 10
    Hi-Res Volumetric CT (VCT)




Note: this kind of display comes after a lot of graphics processing
                                                             Slide no. 11
    Image Reconstruction Problem
• CT images generated by a 
  reconstruction from projections.
   – Projections can be understood on 
     the basis of Lambert-Beer.
   – How can we generate an image 
     given a set of such projections?
• The image shows a single                      Io
  measurement of attenuation                                      I
  through a brain section.
   – In CT scanning, many parallel 
     measurements are made to form a 
     projection of the attenuation.
• Let us define our measurement, g, 
  as                                     Now, let us suppose we
                                         have an unknown object and
                                         projection data. How do we
                                         figure out the contents of such
                                         an object?
                                                                  Slide no. 12
   CT Reconstruction Problem
• Let us make a model of the unknown object:
  – n boxes of same size but different attenuation coefficients.
• Now applying Lambert-Beer, we get:




      Io
                                             …               I

                        Dx
                                                                   Slide no. 13
  CT Reconstruction Problem
• Using our definition, g, we have:



• And in the limit as Dx goes to zero,




     Io
                                         …   I

                       Dx
                                                 Slide no. 14
  CT Reconstruction Problem
• Our intuition tells us that a solution is possible.
• Here’s a quick demo to show that this indeed is the case:
   – Let us do four experiments in which we transmit four x-ray beams of 
     intensity Io and receive four times with intensities I1, I2, I3, and I4.
   – We can express the results mathematically with the Lambert-Beer 
     relation:




                                        Thus we have 4 equations,
                                       4 unknowns, problem solved.
                                                                           Slide no. 15
    CT Reconstruction Problem
 • Unfortunately the real world is not as kind as the example implies.
 • For clinical utility, we need at least a 512 by 512 pixel grid.
 • Thus we would need at least 264,144 equations to solve for that 
   many unknowns.
 • This is challenging to say the least.
    – when CT scanners were first introduced in the late 1970s, this 
      approach was used.
    – Images had  to be calculated with fewer pixels; 
      people gave up on resolution so as to achieve 
      reasonable reconstruction times.




Today’s systems use a reconstruction
approach called “Filtered Backprojection”,
to be discussed shortly.                                                Slide no. 16
  Projections & Sinograms




                                              Sinogram

Projection: all rays in                     Sinogram: 2D plot of
   direction q are                            all projections as a
summed along the rays.                         function of q and
                                               projection width.


  After: http://dolphin.radiology.uiowa.edu/ge/Slides/CTPhys1/index.htm   Slide no. 17
CT Image & Its Sinogram




                          Slide no. 18
        Basics:  2D Fourier Transform
    • 2D FFT of an image f(x,y):
         – A good way to understand CT 
           reconstruction
         – Actually, whole bunch of 1D FFTs
         – In the image shown, most energy in 
           low frequencies - why?
         – Assuming image size of S, what is 
           the frequency increment in FT?
         – With N pixels, what is the largest 
           frequency in FT?
         – Transformed image is in “k-space”
             • Spatial frequencies




                                                                                        Slide no. 19
After: http://thayer.dartmouth.edu/~bpogue/ENGG167/12%2520ROC%2520Analysis.pdf&e=7620
                       Fourier Slice Theorem
                                             X-ray attenuation profile

                                                                                  Fourier transform of projection,
                                                                                    possible w. some filtering



                                                                                  The result forms one line of the 2D
                                                                                     FFT of the original image.

         Data acquisition along one angle, say theta,
               repeated for all desired angles




                                                                                                         Slide no. 20
After: http://thayer.dartmouth.edu/~bpogue/ENGG167/12%2520ROC%2520Analysis.pdf&e=7620
           Fourier Slice Theorem




• Fourier Slice Theorem can be used to derive a superior
  reconstruction approach. Here’s what it is:
   • The 1D FT of a projection is equal to a radial slice of the 2D FT of the
     image.
   • With enough 1D FT’s of the projections, one can estimate the 2D FT of
     the image, & by taking the inverse, the the final image.
• It also forms the basis for the filtered backprojection algorithm.
            After: http://dolphin.radiology.uiowa.edu/ge/Slides/CTPhys1/index.htm   Slide no. 21
Slide no. 22
Backprojection with a Point Target




                                 Slide no. 23
Backprojection:  Alternative to 
    Fourier Slice Recon




• Based on symmetry relations of 2D FT
• Once projection data are acquired, one can begin the
• These steps are repeated for all projected angles.

                                                     Slide no. 24
 Introduction of Filtering
                      Backprojection reconstruction
                      w. no filtering.




Impact of filter on
Sinogram.




                       Backprojection reconstruction
                       w. filter, compare images.
                                             Slide no. 25
Backprojection – Shepp & 
    Logan Phantom




                        Slide no. 26
Backprojection – Torso




                         Slide no. 27
                     Summary
• Introduction to CT scanners
• Emphasis on reconstruction algorithms
  – Basic problem of reconstruction from 
    projections
  – Simple reconstruction using linear equations.
  – Fourier Slice Theorem
  – Backprojection & filtered backprojection
• References:
  – http://thayer.dartmouth.edu/~bpogue/ENGG167/12%2520ROC%25
    20Analysis.pdf&e=7620
  – http://dolphin.radiology.uiowa.edu/ge/Slides/CTPhys1/index.htm  




                                                                       Slide no. 28
                         Homework Lecture 6:
            Algebraic Reconstruction Technique
• Determine the attenuations in the 
  four pixels as follows:
   – With no prior knowledge, assume 
     uniform attenuation
      • i.e. (I1 + I2) / 4 for each of the four 
        pixels
   – Re-evaluate horizontal line integrals,                               3
     compare new values w. measured 
     results (e.g. [m1+m2] - 3)
   – Split the error evenly w. m1 and m2                                  7
     for top row, m3 and m4 for bottom 
     row. 
   – Repeat the process for the vertical 
     line integrals
   – Repeat until no further 
     improvements.
                                                          4   6
   – Try for different values, preferably 
     double digit.                                 ART was the first recon
• Compare result to one arrived at                 algorithm used, may be
  with the approach from Slide no.                 making a comeback.
  15.
                                                                        Slide no. 29
Instructor Contact Information


       Badri Roysam
       Professor of Electrical, Computer, & Systems Engineering
       Office: JEC 7010
       Rensselaer Polytechnic Institute
       110, 8th Street, Troy, New York 12180
       Phone: (518) 276-8067
       Fax: (518) 276-6261/2433
       Email: roysam@ecse.rpi.edu
       Website: http://www.rpi.edu/~roysab 
       Secretary: Laraine Michaelides, JEC 7012, (518) 276 –8525, 
           michal@rpi.edu 




                                                            Slide no. 30
Instructor Contact Information


       Kai E Thomenius
       Chief Technologist, Ultrasound & Biomedical
       Office: KW-C300A
       GE Global Research
       Imaging Technologies
       Niskayuna, New York 12309
       Phone: (518) 387-7233
       Fax: (518) 387-6170
       Email: thomeniu@crd.ge.com, thomenius@ecse.rpi.edu 
       Secretary: Laraine Michaelides, JEC 7012, (518) 276 –8525, 
            michal@rpi.edu
        




                                                                     Slide no. 31

				
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