Reconstruction from Irregular Fourier Samples

1
Reconstruction from Irregular Fourier Samples

• Long Wang
• Southern Polytechnic State Univ.
• Yang Wang
• Georgia Institute of Technology
• May 22, 2007

2
Reconstruction from Irregular Fourier Samples

• Introduction
• Fourier Frames and Reconstruction
• Discrete Singular Convolution Technique
• Examples

3
Introduction

• Let be a compactly supported function in
• Given or some sample points of
  how can we reconstruct
• Of course, if is known for every
  then is readily obtained by the inverse
  Fourier transform

4
Introduction

• Suppose that If
  is known for every , where
• then is given by the Fourier series
  inversion formula

5
Introduction

• However, we may not know on a lattice
  which is a regular set. For example, in MRI,
  Fourier transform is sampled along several paths,
  with each path being a curve as a spiral, a
  circle or a line. These sampled points do not
  contain a regular set ( a lattice). So the image
  reconstruction in MRI must begin with irregular
  Fourier samples. Currently, samples are taken
  along many paths, resulting in a sufficiently
  dense set of samples. These sample points allow
  for a reasonable interpolation of
• on a lattice. The drawback is the time
  consuming.

6
Introduction

• Another challenge is the Gibbs oscillation. Since
  we can only use finitely many points, the Gibbs
  oscillation is inevitable, even when we do have a
  regular set of samples.
• In this paper, we propose a scheme for
  reconstruction of
• from irregular samples of . The
  scheme is based on Gaussian spectral mollifiers
  and the cubic-spline interpolation. This
  technique allows us to virtually eliminate the
  Gibbs oscillation in many cases.

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Fourier Frames and Reconstruction

• Frames and Tight Frames
• A set of vectors in a Hilbert space
  is called a frame if there exist
• such that for any , we have
• The constants are called
  lower frame bound and upper frame bound.

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Fourier Frames and Reconstruction

• A frame is called a tight frame if
  .
• Fourier frames for are frames of
  the form
• .
• Reconstruction
• Let be a tight frame for
  with the frame
• bound . Then
• If is an orthonormal basis, then
  .

9
DSC Technique and Special Mollifiers

• For simplicity, we incorporate the DSC scheme to
  the Fourier transform of Haar bases.
• Let and
  be given, where is
  inside the interval -K, K.
• Let
• We construct from the Fourier
  samples
• as follows

10
DSC Technique and Special Mollifiers

• Note that N is the resolution of the
  reconstruction and the selection of which depends
  on several factors and the actual application.
  (In MRI the reconstruction is typically 128x128.)
  We require that . Now instead of
  taking , where
  is the cubic spline interpolation of
  using the sample points , we
  apply the Gaussian mollifier by setting
  , where is the cubic
  spline interpolation of
  using the sample points

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DSC Technique and Special Mollifiers

• The Gaussian factor helps making the integrand
  decay faster. As shown in next section, the
  reconstruction is improved and the Gibbs
  oscillation in many cases are eliminated.

12
Examples

• For all examples, 900 irregular Fourier samples
  are used. The resolutions, i.e. the number of
  elements in our Haar bases, is 512. The value of
  0.0001 is used for the Gaussian mollifier.
• Example 1. .
• Figure 4.1 is the reconstruction of from
  the cubic spline interpolation without the
  Gaussian mollifier.
• Figure 4.2 is the reconstruction of
  from the cubic spline interpolation with the
  Gaussian mollifier
• .

13
Examples
14
Examples
15
Examples

• Example 2.
• Figure 4.3 is the reconstruction of form
  the cubic spline interpolation without the
  Gaussian mollifier .
• Figure 4.4 is the reconstruction of
  from the cubic spline interpolation with the
  Gaussian mollifier
• .

16
Examples
17
Examples
18
Examples

• Example 3.
• Figure 4.6 is the reconstruction of from
  the cubic spline interpolation without a Caussian
  mollifier.
• Figure 4.7 is the reconstruction of
  from the cubic spline interpolation with the
  Gaussian mollifier
• .

19
Examples
20
Examples