KSpace Edge Detection

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K-Space Edge Detection

• Octavian Biris 09 EN250 Spring 2009

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Motivation

• The detection of tissue borders is of great
  importance in several MRI applications. Edge
  detection is typically performed as a
  post-processing step, using magnitude images that
  are reconstructed from fully-sampled k-space
  data. In dynamic imaging (e.g. of human speech,
  ventricular function, and joint kinematics),
  tissue borders often comprise the primary
  information of interest.
• Thus a fast and accurate method to detect tissue
  borders warrants the need to perform image
  processing prior to the MRI reconstruction step
  which adds unwanted image artifacts.

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Edge Detection Basics-Gradient Method

• Find the gradient of the image
• Take its magnitude
• Apply a threshold
• Thin it by applying non-maximal suppression
  along the direction of the gradient.

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How to find the gradient?

• Find its X and Y components
• Approximate the X derivative at a point (x,y) in
  the image as
• dI/dxI(x1,y)-I(x-1,y)/2.
• dI/dyI(x,y1)-I(x,y-1)/2
• This translates to convolving the image with
  -1/2 , 0 ,1/2 and -1/2, 0, 1/2 respectively

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Advantages and disadvantages

• Computationally cost-effective approximation on
  how the image varies in intensity.
• Crude approximation in regions of high frequency
  variation.
• Need to low-pass filter before applying.

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More complex derivative approximations

• Take a Gaussian kernel G(r). Take the
  differential operator del(r). Apply each one to
  the image I(r).
• J(r)del(r) º G(r) º I(r) ,where º stands for
  convolution.
• Convolution is associative so combine del(r) º
  G(r) in a single kernel
• Use this Kernel to compute the gradient more
  accurately since the high-frequencies are
  attenuated. The kernel is also the derivative of
  the Gaussian (DoG)

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Gradient computation methods.

• I have taken six total approaches in computing
  the gradient of the image. Three of them are
  standard spatial domain approaches while five of
  them are frequency domain methods.
• Convolve the image with the simple filter -0.5,
  0 ,0.5 and its transpose in the spatial domain
  and obtain the two derivatives.
• Convolve the image with the 3x3 Sobel operator
  filter and its transpose in the spatial domain
  and obtain the two derivatives.
• Convolve the image with the Derivative of the
  Gaussian operator. I chose a size of 5x5 and a
  standard deviation of 0.5 in each direction

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Fourier methods for determiningimage derivatices

• This approach finds the derivatives in each
  direction by using the differential property of
  the Continuous Time Fourier Transform

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Speed considerations

• Since most energy of the k-space is contained in
  certain samples one can do a partial sampling of
  the k-space that contain only the high energy
  samples that contain edge information.
• This will improve computation cost for large
  amounts of data.
• Also, since the derivative is just a measure of
  image intensity change, most image information
  does not contribute to its outcome.
• Thus I used a 2D hamming window to take only the
  high energy data.

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Simple derivative filter vs. CTFT derivative
approximation
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Sobel Derivative Filter vs. CTFT derivative
approximation
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Edge Detection Basics-Laplacian Method

• Find the Approximation of the Laplacian of the
  image. The Laplacian changes sign whenever the
  first derivative switches monotony. Usually that
  occurs at a local maximum.

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Edge Detection Basics-Laplacian Method(continued)

• Find the Laplacian of the image
• Iterate through every point and mark if there are
  zero crossings in its neighborhood (e.g. 5x5
  window). If so, keep the point, if not set it to
  zero.

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Future work

• Implement more Frequency domain methods.