IMAGE RECONSTRUCTION

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IMAGE RECONSTRUCTION

• A brief background
• By DINESHBABU V DINAKARABABU
• 15th October 2007

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What is Digital Image Processing?

• Can any one of you come up with a simple
  definition?

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What is Digital Image Processing?

• Processing of a 2-D picture by a digital
  computer.
• In a broader context, it is the digital
  processing of a 2D-data.
• Digital image is an array of real or complex
  numbers represented by a finite number of bits.

Remote Sensing
Robotics
Medical Imaging
Radar
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Classes of problems

• Many applications but lots of problems
  associated with image processing
• 1. Image representation and modeling
• 2. Image Enhancement
• 3. Image restoration
• 4. Image Analysis
• 5. Image Reconstruction
• 6.Image Data Compression

Image Analysis
Image Data Compression
Image Reconstruction
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Image reconstruction from projections

• Special class of image restoration problems
• 2 or higher dimensional object is reconstructed
  from several 1-D projections
• Projection obtained by projecting X-ray beam
  through object
• Important in medical imaging ( CT Scanners ),
  astronomy, radar imaging, geological exploration
  and non-destructive testing of assemblies
• We focus on the medical imaging applications.
• Lots of algorithms available.
• This problem can be set up mathematically in the
  framework of Radon transform theory ( We will
  deal with it in a short while )

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What is projection?

• Shadowgram obtained by illuminating an object by
  penetrating radiation
• Each pixel on the projected image represents the
  total absorption of the X-ray along its path from
  source to detector
• Rotate the source-detector assembly around the
  object projection views for several different
  angles can be obtained.
• Reconstructing a cross-section of an object from
  several images of its transaxial projections an
  important problem to be addressed in image
  processing
• Goal of image reconstruction is to obtain an
  image of a cross-section of the object from these
  projections.

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CT Scanners

• Imaging systems that generate such slice views
  are called computerized tomography scanners
• Resolution lost along path of X-rays while
  obtaining projections
• CT restores this resolution by using information
  from multiple projections.
• Special case of image restoration

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Imaging Techniques

• Transmission Tomography
• Reflection Tomography
• Emission Tomography
• Magnetic Resonance Imaging
• Projection-based image processing

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Radon Transform

• Provides mathematical framework for going back
  and forth between the spatial coordinates (x,y)
  and projection-space coordinates (s,?)
• Radon transform of f(x,y) is g(s,?)
• Line integral along a line inclined at an angle ?
  from the y-axis and at a distance s from the
  origin.
• Mathematical expression

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Radon Transform

• Rotated coordinate system (s,u)
• G(s,?) is called ray-sum
• Represents summation of f(x,y) along a ray at a
  distance s and at an angle ?
• Each point in (s,?) domain corresponds to a line
  in (x,y) domain
• (s,?) Not polar coordinates

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The Back Projection Operator

• Defined by the mathematical expression
• b(x,y) called the back-projection of g(s,?)
• Can be written in polar co-ordinates as
• Represents the accumulation of the ray sums of
  all of the rays that pass through (x,y) or (r,F)

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Parallel-Beam Backprojection An FPGA
ImplementationOptimized for Medical Imaging

• Radon Transform and Back-Projection Operator
• Mathematical Frameworks needed for Image
  Reconstruction
• Lets proceed to the Reconstruction Algorithm and
  the FPGA implementation

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What the authors present?

• FPGA implementation of the parallel-beam
  backprojection algorithm
• Explore a number of quantization issues
• Minimizing error while maximizing efficiency
• Reference
• Parallel-Beam Backprojection An FPGA
  Implementation Optimized for Medical Imaging
• by Srdjan Coric, Miriam Leeser, Eric Miller
• Department of Electrical and Computer
  Engineering
• Northeastern University
• Boston, MA 02115

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Filtered Backprojection

• Called Filtered Backprojection (FBP) as we are
  filtering the projections using a high-pass
  filter before the actual backprojection.
• Most commonly used approach for image
  reconstruction
• Dense projection data
• Parallel-beam and fan-beam variations depends
  on x-ray beam
• we focus on parallel-beam Backprojection
• Computationally intensive process
• Highly parallelizable process

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Challenges facing FBP

• Computationally intensive process
• - an image of size n n complexity is O(n3)
• - upto order n2log2 n
• Difficulty in producing high resolution images
• - density of each projection
• - total number of projections be large
• High precision reconstructions difficult
• - analyze the effects of quantization
• - fixed-point implementation with properly
  chosen bit-widths
• - bit reduction of intermediate results for
  different rounding schemes

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Parallel-Beam Backprojection

• Finds widespread application in medical imaging
• Computationally intensive algorithms requiring
  the rapid
• processing of large amounts of data
• Can benefit greatly from hardware acceleration
• Reconfigurable hardware has not been applied to
  this important area
• Efficient implementation and maximum speedup -
  fixed-point implementations are required
• Associated quantization errors must be carefully
  balanced against the requirements
• Visual quality of image not compromised.

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PARALLEL-BEAM FILTEREDBACKPROJECTION

• Parallel-beam CT scanning system
• an array of equally spaced unidirectional sources
  of focused X-ray beams
• radiation reaches a collinear array of detectors
• Spatial variation of the absorbed energy in the
  two-dimensional plane through the object is
  expressed by the attenuation coefficient µ(x, y).
• set of values given by all detectors comprises a
  one-dimensional projection of the attenuation
  coefficient, P(t, ?)
• t is the detector distance from the origin
• ? is the angle at which the measurement is taken

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Radon and Inverse Radon Transform

• Io is the source intensity
• Id is the detected intensity
• d() is the Dirac delta function
• Radon transform represents an operator that maps
  an image µ(x, y) to a sinogram P(t, ?)
• Inverse mapping-Inverse Radon transform which
  when applied to a sinogram gives an image
• FBP algorithm uses this mapping

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Discrete formulation of Backprojection and the
algorithm

• ??(t) is a filtered projection at angle ?
• K is the number of projections taken during CT
  scanning at angles ?i over a 180
• Algorithmically, this equation is implemented as
  a triple nested for loop
• Outermost loop is over projection angle, ?
• For each ?, we update every pixel in the image in
  raster-scan order starting in the upper left
  corner and looping first over columns, c, and
  next over rows, r.
• the pixel at location (r,c) is incremented by the
  value of ??(t) where t is a function of r and c

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Linear Interpolation

• The issue here is that the X-ray going through
  the currently reconstructed pixel, in general,
  intersects the detector array between detectors.
• So, use linear interpolation.
• point of intersection is calculated as an address
  corresponding to detectors numbered from 0 to
  1023 (i)
• fractional part of this address is the
  interpolation factor
• interpolation can be performed beforehand in
  software or it can be a part of the
  backprojection hardware itself
• Implementation in hardware done
• - reduces the amount of data that must be
  transmitted to the reconfigurable board
• - interpolation in hardware is much faster
• equation that performs linear interpolation is
  given by

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Illustration of the coordinate system used in
parallel-beam backprojection
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Geometrical explanation of the incremental
spatial address calculation
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Quantization

• Quantization here indicates the conversion of
  floating point variables to fixed point variables
  of proper bit-width.
• Quantize all our data and calculations to
  increase the speed and decrease the resources
  required for implementation.
• Determining allowable quantization is based on a
  software simulation of the tomographic process.
• Quantization error compare a fixed-point image
  reconstruction with a floating a fixed-point
  image reconstruction with a floating-point one.
• Quantization should not result in loss of
  numerical accuracy and hence a change in the
  visual quality of the image.
• Possibilities for bit reduction on the outputs of
  certain functional units

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Representation of fixed point variables

• V is an arbitrary real number
• Va is its fixed-point approximation
• Q is an integer that encodes V
• S is the slope
• B is the bias
• Fixed-point versions of the sinogram and the
  filtered sinogram use slope/bias scaling where
  the slope and bias are calculated to give maximal
  precision

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Quantization of two variables

• ws is the word size in bits of integer Q
• Round represents rounding to nearest
• min(V) min(Q) B 0 for unsigned numbers
• For a given sinogram, S and B are constants and
  they do not show up in the hardware only the
  quantization value Q is a part of the hardware
  implementation
• Interpolation factor is an unsigned fractional
  number - uses radix point-only scaling

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Relative Error

• N is the total number of pixels
• xi and yiFP are the values of the i-th pixel in
  the quantized and floating-point reconstructions

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Simulation steps for quantization

• Reproject Generates a sinogram of 1024
  projections and 1024 samples per projection.
• Filter Sinogram data convolved with the impulse
  response of the ramp filter generating a filtered
  sinogram.
• Backproject Backprojecting the filtered
  sinogram to give a reconstructed image.

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Detailed flowchart of the simulation process

• Different quantization steps possible are shown
  in the figure above
• Each path represents a separate simulation cycle.

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Hardware Organization

• Hardware acceleration in reconfigurable hardware
  - parallel processing
• Two basic sources of parallelism
• Pixel parallelism - image can be divided into
  subsections which can be reconstructed
  simultaneously
• Projection parallelism simultaneously
  processing individual projections
• Pixel parallelism limited by available memory
  bandwidth
• We concentrate on projection parallelism.

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Data flow of backprojection hardware

• Three flows of data going on simultaneously
• - Pipeline flow
• - Sinogram data flow
• - Accumulation data flow

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Datapath implementation of the non-parallel
backprojection hardware
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Datapath of the implemented 4-way parallel
backprojection hardware
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Image comparison grayscale range mapped to a
part of the pixel value range
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RESULTS AND PERFORMANCE
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CONCLUSION

• FPGA implementation of parallel-beam
  backprojection algorithm
• Analysis of quantization effects caused by finite
  bit-widths
• Paid special attention not to compromise the high
  precision requirements
• 20 times speed-up over a similar software
  implementation
• Worst case relative error of 0.015 compared to a
  floating-point implementation
• Real time image reconstruction is easily
  attainable by exploiting parallelism.
• Hardware architecture presented can easily be
  modified to different bit-widths for different
  sensors and applications