Spatial Filtering (Chapter 3)

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Spatial Filtering (Chapter 3)

• CS474/674 - Prof. Bebis

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Spatial Filtering Methods (or Mask Processing
Methods)
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Spatial Filtering

• The word filtering has been borrowed from the
  frequency domain.
• Filters are classified as
• Low-pass (i.e., preserve low frequencies)
• High-pass (i.e., preserve high frequencies)
• Band-pass (i.e., preserve frequencies within a
  band)
• Band-reject (i.e., reject frequencies within a
  band)

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Spatial Filtering (contd)

• Spatial filtering is defined by
• (1) A neighborhood
• (2) An operation that is performed on the pixels
  inside the neighborhood

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Spatial Filtering - Neighborhood

• Typically, the neighborhood is rectangular and
  its size
• is much smaller than that of f(x,y)
• - e.g., 3x3 or 5x5

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Spatial filtering - Operation
Assume the origin of the mask is the center of
the mask.
for a 3 x 3 mask
for a K x K mask
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Spatial Filtering - Example

• A filtered image is generated as the center of
  the mask moves to every pixel in the input image.

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Handling Pixels Close to Boundaries
pad with zeroes
0 0 0 .0
or
0 0 0 .0
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Linear vs Non-LinearSpatial Filtering Methods

• A filtering method is linear when the output is a
  weighted sum of the input pixels.
• Methods that do not satisfy the above property
  are called non-linear.
• e.g.,

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Linear Spatial Filtering Methods

• Two main linear spatial filtering methods
• Correlation
• Convolution

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Correlation
g(i,j)
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Correlation (contd)
Often used in applications where we need to
measure the similarity between images or parts of
images (e.g., pattern matching).
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Convolution

• Similar to correlation except that the mask is
  first flipped both horizontally and vertically.
• Note if w(x,y) is symmetric, that is
  w(x,y)w(-x,-y), then convolution is equivalent
  to correlation!

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Example
Correlation Convolution
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How do we choose the elements of a mask?

• Typically, by sampling certain functions.

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Filters

• Smoothing (i.e., low-pass filters)
• Reduce noise and eliminate small details.
• The elements of the mask must be positive.
• Sum of mask elements is 1 (after normalization)

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Filters (contd)

• Sharpening (i.e., high-pass filters)
• Highlight fine detail or enhance detail that has
  been blurred.
• The elements of the mask contain both positive
  and negative weights.
• Sum of the mask weights is 0 (after
  normalization)

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Smoothing Filters Averaging(Low-pass filtering)
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Smoothing Filters Averaging (contd)

• Mask size determines the degree of smoothing and
  loss of detail.

3x3
5x5
7x7
original
15x15
25x25
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Smoothing Filters Averaging (contd)
Example extract, largest, brightest objects
15 x 15 averaging
image thresholding
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Smoothing filters Gaussian

• The weights are samples of the Gaussian function

s 1.4
mask size is a function of s
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Smoothing filters Gaussian (contd)

• s controls the amount of smoothing
• As s increases, more samples must be obtained to
  represent
• the Gaussian function accurately.

s 3
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Smoothing filters Gaussian (contd)
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Averaging vs Gaussian Smoothing
Averaging
Gaussian
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Smoothing Filters Median Filtering(non-linear)

• Very effective for removing salt and pepper
  noise (i.e., random occurrences of black and
  white pixels).

median filtering
averaging
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Smoothing Filters Median Filtering (contd)

• Replace each pixel by the median in a
  neighborhood around the pixel.

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Sharpening Filters (High Pass filtering)

• Useful for emphasizing transitions in image
  intensity (e.g., edges).

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Sharpening Filters (contd)

• Note that the response of high-pass filtering
  might be negative.
• Values must be re-mapped to 0, 255

sharpened images
original image
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Sharpening Filters Unsharp Masking

• Obtain a sharp image by subtracting a lowpass
  filtered (i.e., smoothed) image from the original
  image

-

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Sharpening Filters High Boost

• Image sharpening emphasizes edges but details
  (i.e., low frequency components) might be lost.
• High boost filter amplify input image, then
  subtract a lowpass image.

(A-1)


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Sharpening Filters Unsharp Masking (contd)

• If A1, we get a high pass filter
• If Agt1, part of the original image is added back
  to the high pass filtered image.

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Sharpening Filters Unsharp Masking (contd)
A1.9
A1.4
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Sharpening Filters Derivatives

• Taking the derivative of an image results in
  sharpening the image.
• The derivative of an image can be computed using
  the gradient.

 
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Sharpening Filters Derivatives (contd)

• The gradient is a vector which has magnitude and
  direction

or
(approximation)
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Sharpening Filters Derivatives (contd)

• Magnitude provides information about edge
  strength.
• Direction perpendicular to the direction of the
  edge.

 
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Sharpening Filters Gradient Computation

• Approximate gradient using finite differences

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Sharpening Filters Gradient Computation (contd)
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Example
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Sharpening Filters Gradient Computation (contd)

• We can implement and using masks

(x1/2,y)
good approximation at (x1/2,y)
(x,y1/2)


good approximation at (x,y1/2)

• Example approximate gradient at z5

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Sharpening Filters Gradient Computation (contd)

• A different approximation of the gradient

good approximation
(x1/2,y1/2)

• We can implement and using the
  following masks

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Sharpening Filters Gradient Computation (contd)

• Example approximate gradient at z5

• Other approximations

Sobel
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Example
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Sharpening Filters Laplacian
The Laplacian (2nd derivative) is defined as
(dot product)
Approximate derivatives
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Sharpening Filters Laplacian (contd)
Laplacian Mask
detect zero-crossings
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Sharpening Filters Laplacian (contd)
Sobel
Laplacian