Project Overview

1
Project Overview

• Reconstruction in Diffracted Ultrasound
  Tomography
• Tali Meiri Tali Saul
• Supervised by
• Dr. Michael Zibulevsky
• Dr. Haim Azhari
• Alexander Michael Bronstein

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Project Goals

• Different methods of image reconstruction for
  diffraction ultrasound tomography.
• Computing an image of a slice of an object from
  projections.
• More specifically
• In acoustic imaging, diffraction must be taken
  into consideration when modeling the interaction
  of radiation with matter. Diffraction is caused
  by the wave nature of the radiation.

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Terminology Definition

• Tomographic Reconstruction A method of computing
  a sliced image of an object from data collected
  from projections.
• Projection Data acquired from a single
  illumination at a specific angle. The
  illumination is by electromagnetic radiation or
  acoustic waves, and the intensity of the
  radiation traversing the object is measured.
• Sinogram A picture of a set of projections at
  different angles placed one on top of the other.
• Straight-ray tomography Radiation illuminating
  the object has the nature of straight rays.
• Diffraction tomography Radiation illuminating
  the object has the nature of waves. In such case,
  wave phenomena such as diffraction should not be
  overlooked.

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System Overview
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Straight-Ray tomography

• In straight-ray tomography the radiation
  illuminating the object has the nature of
  straight rays. As such, the wavelength of the
  radiation is infinitely small compared to the
  dimensions of the illuminated object.
• Radon transform
• The Radon function is the projection of the
  image intensity along a radial line oriented at a
  specific angle. The Radon transform of an object
  represents the image intensity along many radial
  lines oriented at different angles.
• Tomographic scan
• It would be fair to define a tomographic scan as
  a discretization of the Radon transform of an
  object since the projections are taken at
  discrete angles around the object.

r is the radial axis oriented at angle
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Straight-Ray Tomography

• Fourier Slice Theorem
• This theorem connects the Radon transform with
  the Fourier transform
• 1D Fourier transform of the Radon transform
  equals the 2D Fourier transform of the object.

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Straight-Ray Tomography
w2
Fourier transform
w1
Frequency domain
DFT of the discrete tomographic projections gives
the values of the 2D Fourier transform of the
objects on radial straight lines oriented at the
same angles as the projections. Reconstruction is
usually using Filtered Back-Projection.
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Diffraction Tomography

• Fourier Diffraction Theorem
• DFT of the discrete tomographic projections
  gives the values of the 2D Fourier transform of
  the object along a semi-circular arc in the
  frequency domain.

The radius of the arc is proportional to the
frequency of the incident wave
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Diffraction Tomography

• Monochromatic Illumination
• The object is rotated and the scattered field
  for different orientations is measured. For each
  orientation the object is illuminated with a
  monochromatic wave. This produces an estimate of
  the objects Fourier transform along a circular
  arc rotated at the same angle as the object.

Each arc contains 32 sampling points.
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Diffraction Tomography

• Broadband Illumination
• Transmitting a superposition of monochromatic
  waves at different frequencies. This allows
  obtaining more information from a single
  projection since the frequency domain is sampled
  on several arcs simultaneously. Consequently, a
  smaller amount of projections should suffice for
  covering the entire frequency domain.

Each arc contains 32 sampling points.
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Methods of reconstruction

• Methods used in straight-ray tomography are not
  applicable to tomography with diffractive
  sources. In this work we introduce and compare
  the performances between three different methods
  of reconstruction which will include the
  following
• Reconstruction using inverse NUFT
• Straightforward computation of the forward and
  the inverse NUFT by creating the transform matrix
  and applying it to the picture in column stack.
• Reconstruction using frequency domain
  interpolation
• Frequency interpolation of the non-uniform data
  to a uniform Cartesian grid using bilinear
  interpolation.
• Reconstruction using Non-uniform Fast Fourier
  Transform
• A method equivalent to a convolution regridding
  method on an over sampled grid using an optimal
  selection of a Gaussian kernel.

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Inverse NUFT (2D)

• 2D Non Uniform Fourier Transform
• Our goal is to move from non uniform frequency
  samples in frequency domain to uniform samples in
  space domain.
• The 2D NUFT matrix will be built in the
  following way
• for every frequency sample which will be
  represented in frequency domain
• as , the basis pictures will be built such
  that
• and will represent the columns of . (n,m)
  are the reconstructed samples in space domain.

Pseudo inverse
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Inverse NUFT (2D)

Since this is the straight forward computation
without using any approximations, it is the
most accurate reconstruction technique but is
also computationally extensive and requires
operations as it is equivalent to matrix
multiplication. When the signal is large
(typically 64 X 64 4096 and above),
straightforward inversion is practically
impossible.
Example of applying the transformation on a 2D
sinc function in Frequency domain to get a 2D
step function in space domain. In this example
the frequency samples where uniformly spaced.
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Inverse NUFT (2D)

Example of applying the transformation on a 2D
sinc function in Frequency domain to get a 2D
step function in space domain. In this example
the frequency samples where randomly spaced.
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Inverse NUFT (2D)

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Freq. domain interpolation

• The algorithm by Kak and Slaney
• Start from a Cartezian grid.
• For each point (w1,w2) find its representation in
  (w, ) coordinates.
• Use bilinear interpolation to find the most
  accurate value of the signal at the new location.
• where
• After computing at each point on the rectangular
  grid, the object is obtained by a simple 2-D
  inverse FFT.
• Complexity

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Freq. domain interpolation

• Bilinear interpolation

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Freq. domain interpolation

• Shepp-Logan phantom
• In order to avoid forward-projection errors, and
  analytic Shepp-Logan phantom was used. This
  phantom is a superposition of ellipses
  representing features of the human brain. The
  advantage of such a phantom is that its Fourier
  Transform has an analytical expression. The
  Fourier Transform of an ellipse is given by
• where is the first order Bessel function
  of the first kind, is the center of
  the ellipse, its intensity, its
  orientation and lastly A and B are the lengths of
  horizontal and vertical semi-axes respectively.

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Freq. domain interpolation
Monochromatic illumination using a 64X64 phantom
picture with 64 projections.
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Freq. domain interpolation
Monochromatic illumination using a 64X64 phantom
picture with 64 projections and with different
addition of Gaussian noise.
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Freq. domain interpolation
Calculated mean squared error and the max error
between reconstructed picture and original
picture as function of the number of projections.
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Freq. domain interpolation
Less coverage of samples in the high frequency
region between 17 projections and 18 projections
due to multiple equal samples.
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Freq. domain interpolation
Broadband illumination using a 64X64 phantom
picture with 15 different angles and 5 different
frequencies.
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Freq. domain interpolation
Calculated mean squared error and the max error
between reconstructed picture and original
picture as function of the number of different
frequencies in each projection
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The NUFFT method (1-D)

• Definition of the problem
• The input parameters is a vector
  of samples of a signal
  sampled in non-uniformly distributed frequencies
• .
  Our objective is to reconstruct the signal from
  its non-uniform frequency samples using a method
  which takes a non-uniformed data in frequency
  domain and transforms it to a uniform data in
  space domain.
• Method
• Using the method of Fast Fourier Transform
  approximation for non-equispaced data suggested
  by A. Dutt and V. Rokhlin. This method uses
  interpolation of the data on some over-sampled
  Cartesian grid using a Gaussian kernel. Once the
  data is uniformly spaced on the rectangular grid,
  the signal
• can be
  obtained by a simple inverse FFT.
• Complexity of algorithm
• O(NlogNNq) where q is a constant.

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The NUFFT method (1-D)

• The algorithm
• For a given signal
  in frequency domain, the inverse
  transformation is defined by the formula
• where is the non-uniformed frequency. The
  algorithm approximates this formula by finding a
  suitable approximation for any expression of the
  form
• using a q number of expressions of
  the form where .
• It is proven that the error between the
  reconstructed signal and the original signal
  obeys the following inequality
• where

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The NUFFT method (1-D)
DFT of a non-uniformly sampled set of N data
points may be computed with an ordinary FFT of
length mN with a precision that depends on the
selection of m. Usually a choice of m2 is
sufficient for most practical applications.
w
w
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The NUFFT (1-D) results
NUFFT using an analytic sinc function. Red dots
non-uniformed samples of the sinc function.
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The NUFFT (1-D) results
UFFT using an analytic sinc function. Red dots
uniformed samples of the sinc function.
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The Sarty NUFFT (2-D)

• Definition of the problem
• The input parameters is a CS vector of samples
  of a 2_D signal sampled in non-uniformly
  distributed frequencies on the 2_D range.
• Method
• Voronoi areas.
• Extension of the 1-D NUFFT algorithm to 2-D.

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The NUFFT method (2-D)

• Direct computation using Voronoi areas
• In this case the equation is
• where S(p) is the vector of the samples in CS of
  the signal, W(p) is the CS vector of the
  correlated weights derived from the Voronoi areas
  associated with each sample point and
  are the non-uniformed frequencies in CS.
• This straight forward computation requires
  multiplications and additions. It
  may take hours of computational time for a
  typical 256X256 signal picture. The fast
  algorithm requires only
  while using the FFT algorithm.

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The NUFFT method (2-D)

• Data representation using Voronoi area

The Voronoi areas associated with a k-space point
is the area of the set whose points are closer to
the given point than all the other k-spaced
sample points.
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The NUFFT method (2-D)

• Efficient implementation using the D-R algorithm
• Let and
• Furthermore we use
• The image reconstruction is computed as
• where

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The NUFFT (2-D) results
Monochromatic illumination using a 64X64 phantom
picture with 64 projections.
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The NUFFT (2-D) results
Monochromatic illumination using a 64X64 phantom
picture with 64 projections and with different
addition of Gaussian noise.
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Conclusion


For 32X32 pictures
Complexity Time of calculation (seconds) Mean squared error Infinite Error
Gridding 17.925 6.4572 7.8607
Direct transformation 81.07 0.2812 0.096
NUFFT 15.8020 3.8445 8.8118