Superresolution in MRI Imaging

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Superresolution in MRI Imaging

• Kenneth Neiss
• April 15th, 2004

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Introduction

• Motivation
• MRI Positive
• Detects abnormalities in living tissue
• Very good resolution in-plane
• MRI Negative
• Either
• Does not provide high resolution in all
  dimensions (slice-select) with ample acquisition
  time
• Acquires high resolution in all dimension (3-D)
  but requires much longer time to do so

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MRI resolution dimensionality example
Courtesy of Greenspan et. al.
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Introduction

• What is needed?
• Acquire high resolution images in all dimensions
• Keep acquisition time relatively low
• How to accomplish?
• Apply techniques used in image sequence
  superresolution (CCD and video) to MR images

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MRI Spatial Encoding

• Background
• Magnetic Resonance (MR)

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MRI Mapping Process
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Main Transforms in MRI

• There are two main steps in the MRI process as
  detailed previously
• the spin processing transform which encodes the
  data into the data space (k-space)
• the data processing transform which converts the
  k-space data into a viewable image

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Why are these steps important?

• The data processing transform is considered the
  unarguable method for image reconstruction
• The spin processing transform is the area by
  which the process has a direct affect on the
  ability for enhancing resolution by
  post-processing
• Thus, this latter encoding step will be further
  studied

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Spatial Encoding

• Activated MR signal takes the form of a complex
  exponential thus, the encoding can be done
  with
• Frequency encoding
• Phase encoding

See paper for more details and descriptions
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Properties of MRI Encoding

• Signal is frequency encoded in one dimension
  in-plane and phase encoded in the other in-plane
  dimension
• Upon application of gradients, a relationship
  between field strength and spatial location is
  identified
• The atomic spins will resonate (precess) at
  frequencies singular to their location, allowing
  for reconstruction of 2-D or 3-D images
• To fill the k-space (where the data is analyzed)
  requires N time-domain signals encoded with
  specific frequency and phase data to fill the NxN
  k-space

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k-space analysis (see paper for derivations)

• Frequency encoding gradient Gx maps a time signal
  to a k-space signal
• Visible FT properties
• As long as Gx ? 0, the frequency encoding
  gradient will uniquely encode the spatial
  information

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k-space continued

• When multiple gradients are used, the
  representation becomes in the form of a
  multidimensional FT, but S(k) is only present for
  a finite set of points for the k-space
• k-space trajectory is a straight line dependent
  upon values of G (not straight line if G values
  not constant)
• Phase encoding only affects the starting point of
  the trajectory and not the shape or form

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Problems with 3D MRI

• Acquiring 3-D voxels still might not provide the
  necessary resolution
• Reducing the voxel size during acquisition
  actually decreases the SNR per voxel (shown
  through experiment that a decrease in voxel size
  in all dimensions of factor of 2 leads to
  decreased SNR by factor of 8)
• Reducing voxel size leads to smaller volume
  coverage attainable without foldover effects
• Applying a stronger magnetic field leads to
  higher motion artifacts and large distortions at
  the boundaries

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2D MRI Slices

• In cases where 3-D MRI is unattainable or
  undesirable, 2-D stacks of slices are produced
• Problem does not yield best volume data when
  directly pieced together creates non-isotropic
  voxels with larger slice-select dimension than
  in-plane, resulting in higher resolution in-plane
  as compared with the slice-select dimension

Courtesy of Greenspan et. al.
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Superresolution

• Image sequence superresolution
• Makes use of a set of degraded low resolution
  shifted versions of an image to create a higher
  resolution filtered image of the object
• Interpolation first introduced by Tsai and Huang
• Kim, Bose, and Valenzuela imposed the idea of
  using noisy and blurred low resolution images as
  the input to interpolate and sequentially filter
  in order to generate a high resolution filtered
  image
• For simulation purposes, we need to reverse the
  process to extract a series of low resolution
  images to feed the system

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Imaging System
Initial High Resolution Image Scheme
Resulting Low Resolution Image Set
Geometric Transform
Blur
Downsampling
Noise
Model of Near Field Imaging System
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CCD and Video Image Characteristics

• Images are not Fourier-encoded such as in MRI
  images
• By overlaying various sub-pixel shifted versions
  of the object, one can extract extra Fourier data
  due to the shifts and piece together a higher
  resolution (in both in-plane dimensions)
  representation

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MRI Image Characteristics

• MRI data is acquired in 2-D stacks or 3-D slices
• Data is inherently Fourier-encoded (frequency and
  phase)
• The only new information that can be added with
  each successive slice or volume is the next piece
  in the stack (slice-select dimension)
• This and the fact that the in-plane data is
  bandlimited lead to the idea that only the
  slice-select dimension can be utilized to improve
  resolution
• Must take into account blurring by Point Spread
  Function (PSF)

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Main Difference

• Main difference between CCD/Video Camera Imaging
  and MRI Imaging
• In Fourier-encoded 2-D MRI, there can be no
  additional information interpolated regarding
  high frequencies in the in-plane dimension only
  in the slice-select dimension can additional
  frequency information be extracted for 2-D MRI
  datasets
• This is the basis for applying superresolution
  techniques in the slice-select direction for MRI
  images
• In CCD and video camera imaging, superresolution
  can be applied to all dimensions (all dimensions
  can be used to extract high frequency data)
• It has been shown that applying superresolution
  in MRI to the in-plane dimensions does not yield
  increased resolution and leads to similar results
  as if you zero-pad the temporal data shifting
  in-plane only provides a global phase shift in
  the time domain, not affecting the spatial
  frequency resolution

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Additional Differences

• Image registration
• In MRI, registration is known beforehand (assume
  minimal subject motion) due to predetermined
  single time shift between slices (motion only in
  slice-select dimension)
• MRI is similar to CCD, where there is a
  predetermined multisensor positioning resulting
  in subpixel shifts (all motion will occur
  in-plane, i.e. no panning, zooming, etc.)
• Video Camera imaging has motion in all
  dimensions, and one must account for this using
  projective transformations
• Since all of these imaging platforms have
  subpixel or subvoxel shifts, superresolution will
  be beneficial

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Main MRI Tradeoff

• Voxel Size vs. Acquisition Time
• The smaller the isotropic voxel size, the longer
  the acquisition time (higher costs)
• Solution
• Implement a post-processing superresolution
  algorithm

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Superresolution Framework

• Apply superresolution algorithm post-processing
  to the 2-D MRI datasets that creates higher
  resolution in the slice-select dimension, thus
  creating a high-resolution, 3-D image of the
  object
• First, a number of 2-D slice datasets are
  obtained, each shifted by a specified subpixel
  amount in the slice-select dimension relative to
  the other volume sets so as to create an
  isotropic volume resolution
• Then, merge data and implement superresolution
  algorithm, yielding a high resolution version
• Recursively implement algorithm until desired SNR
  is achieved

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Superresolution Framework
Courtesy P. Kornprobst
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Some Examples of Superresolution Algorithms

• Recursive Least Squares (RLS)
• Maximum Likelihood Restoration (ML)
• Maximize conditional pdf given the ideal image
• Maximum A Posteriori Estimator (MAP)
• Maximize conditional pdf of the ideal image given
  the measurements
• Irani-Peleg Iterative Back-Projection Algorithm
  (IBP)
• See paper for more descriptions

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IBP Framework
Is the MS difference between the actual
low-resolution images and the estimated
low-resolution images less than a desired
threshold?
Create the low resolution images
Hypothesize the high resolution image
End the Iterative Process
YES
NO
Update the best estimate of the
high-resolution image
n n 1

• Irani-Peleg back-projection flow diagram

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Image Sequence and MRI-Derived Superresolution
Results

• Image Sequence (CCD and Video Camera) and
  MRI-based are under different circumstances,
  parameters, and assumptions, so the results
  cannot be compared directly
• It will be shown in each cases associated
  reference frame that superresolution is beneficial

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Video Imaging Analysis

• Recursive Least Squares approach (Kim et al.)
• Initial low resolution data 16 images of 40x40
  pixels, each subpixel uniformly shifted
• Also, has additive white zero mean Gaussian noise
  of variance .001 (process also works for
  non-noisy case)

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Video Imaging Analysis
Input Image is lenna 16-40x40 pixel images each
subpixel shifted Additive Gaussian White Noise
with variance 0.001
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Video Imaging Analysis
Top Left Output of Initial High Resolution Image
(80x80 pixels) PSNR of 17.76 dB Right
Recursive Updates of High Resolution Images
(progressing top left to bottom right) PSNR of
13th update is 19.28 dB
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MRI Imaging Analysis

• Analyze Greenspans IBP method
• Comb-phantom and actual human brain results
• Comb-phantom
• Surrounded in doped water (doped with Gd-DTPA)
• Fast-spin echo
• 16 slices of resolution 1mm x 1mm x 3mm
• Slice-select (z) dimension parallel to teeth
• 3 sets of slices, each shifted by 1mm in the z
  dimension (high resolution output voxel 1mm x
  1mm x 1mm)

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Phantom Analysis
a) Original Low-Resolution Data b) Interpolation
through zero-padding c) Interleaving 3 sets
only d) Superresolution using Box-PSF e)
Superresolution using Gaussian-PSF
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Human Brain Analysis

• Actual Human Brain
• 3 sets of 22 slice-select shifted low resolution
  images (1.5mm x 1.5mm x 4.5mm)
• Each set shifted by 1.5mm
• Acquired high resolution (1.5mm x 1.5mm x 1.5mm)
  image for comparison purposes

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Human Brain Analysis
Top Left Low Resolution Version Top Right
Zero-padded interpolated version Bottom Left
Superresolution Image using Box-PSF Bottom Right
Superresolution Using Gaussian-PSF Bottom Actual
High Resolution Image (note image has been
expanded vertically for viewing purposes)
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Summary

• Goals
• To analyze whether superresolution could be
  applied to MRI, and if so, with what kinds of
  results
• Use normal video imaging as a baseline for
  comparison

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Summary

• Results
• Principle differences
• In normal imaging, superresolution can be applied
  to all dimensions
• In MRI Imaging, the properties of the encoding
  scheme only lead to superresolution being
  beneficial in the slice-select dimension
• MRI and CCD imaging do not require complex image
  registration, while video imaging does require a
  complex registration process projective
  transform

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Summary

• Additional MRI Superresolution Characteristics
• Can be accomplished either with 3-D stacks of
  data or 2-D slices
• Main necessity is subvoxel shifts in the
  slice-select dimension
• Having a blurring filter (PSF) that closely
  resembles the MRI imaging system along with the
  MRI image characteristics in the slice-select
  dimension yields a better process

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Benefits of MRI Superresolution

• Can acquire 2-D slices at a fraction of the time
  it takes to acquire 3-D volumes, yet still
  achieve the high resolution images through
  post-processing
• Reduces patient time, MRI ON time, and operator
  time
• Dramatically reduces associated costs
• Enables more people access to the machines over a
  given time period
• Same techniques can be applied to PET, CT

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Conclusion - Extension

• Limit on in-plane resolution enhancement due to
  Fourier-encoding
• If the signal is not Fourier-encoded, it may be
  possible to remove the limit and allow for
  resolution enhancement in-plane
• How to accomplish this encoding?
• One method could be to wavelet encode the
  temporal signals

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Wavelet Encoding (xy plane)

• Superresolution can be employed as long as there
  is sufficient subvoxel overlap, with wavelets as
  basis function for encoding
• Only a few processes currently where non-Fourier
  encoding is of use
• Diffusion-Weighted Imaging (DWI) high resolution
  images are difficult to acquire because of large
  phase variations due to slight patient motion
  during the gradient application process
• Diffusion-Tensor Imaging (DTI) variation of DWI
  in which at least 7 different images acquired for
  every slice, with at least 6 different directions
  of diffusion weighting

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Wavelet vs. Fourier Encoding

• Wavelets less sensitive to motion
• SNR and resolution of wavelet-encoded signal can
  be less than/equal to Fourier-encoded signal
  (based on choice of basis wavelet functions)
• Wavelets offer more flexibility (choice of basis
  based on application)

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Concluding Remarks

• Non-Fourier encoding is usually applied to
  slice-select dimension due to the amount of
  resolution enhancement possible
• One can theoretically increase the in-plane
  resolution (such as needed for very small subject
  or for process with great motion)
• These aformentioned processes lend themselves for
  the non-Fourier methods, but the tradeoff may be
  too great

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Thank You

• Any questions?
• Please come see me for references (if desired)