Dr' Claude C' Chibelushi

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Fac. of Comp., Eng. Tech. Staffordshire
University
Image Processing, Computer Vision, and Pattern
Recognition
Spatial Domain Filters
Dr. Claude C. Chibelushi
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Outline

• Introduction
• Filter pass band
• Spatial domain filters
• Convolution-based filters
• Mode and median filters
• Combination of convolution-based filters
• Summary

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Introduction
Rate of intensity change (i.e. intensity
gradient) gives information such as object
boundary, presence of noise, ...
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Introduction

• Filter transforms image such that
• certain type of information is retained /
  emphasized while others are rejected / attenuated
• based on rate of change of pixel values

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Introduction
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Introduction

• Filtering can be done in
• spatial domain
• i.e. applied to image pixels directly
• frequency domain
• i.e. applied to representation of image as sum of
  cyclic patterns

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Filter Pass Band

• Filters let through or attenuate specific ranges
  of image details
• e.g. slow/gradual or fast changes in pixel value
• Hence the classification of filters into
• low-pass filters
• high-pass filters
• and other types of filters
• band-pass filters, band-reject filters,

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Filter Pass Band

• Low-pass filter
• Attenuates fast changes in pixel value but lets
  through slow / gradual changes
• Typical application noise removal
• Tends to blur image

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Filter Pass Band

• High-pass filter
• Attenuates slow / gradual changes in pixel value
  but lets through fast changes
• Typical application edge enhancement
• Tends to be sensitive to noise
• Output values may be outside numerical range of
  pixel value
• shifting and scaling may be required (see notes
  on contrast enhancement)

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Spatial-Domain Filters

• Spatial-domain filters are based on moving
  window techniques e.g.
• convolution operation (e.g. Laplacian filter)
• sorting operation (e.g. median filter)
• pixel-counting operation (e.g. mode filter)

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Spatial-Domain Filters

• Moving window

• window repeatedly shifted over image
• calculations performed for each shift
• only pixels within window are considered

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Spatial-Domain Filters

• Convolution
• Many spatial-domain filters are based on
  convolution
• Convolution uses a moving window
• shift-multiply-add operation

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Spatial-Domain Filters
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Output value
(-1x9)
(-1x30)
(0x29)
(0x33)
(0x39)
(1x9)
(1x2)
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window

• Convolution (example)

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Spatial-Domain Filters

• Convolution
• Shift-multiply-add operation
• weighted sum of pixel values in neighbourhood
• weights specified in mask (kernel, template, ...)
• sum entered in output image at pixel location
  corresponding to mask anchor
• mask shifted by one pixel
• Note shift by more than 1 pixel is sometimes
  used (gives smaller output image)

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Spatial-Domain Filters

• Convolution
• Mathematical notation

Output at image row i, column j
for each mask row
for each mask column
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Spatial-Domain Filters

• Convolution pseudo code
• / at each image pixel /
• for each image row // downward window shifts
• for each image column // rightward window
  shifts
• / in image window covered by mask /
• for each mask row
• for each mask column
• update local cumulative weighted sum
• Note use output buffer

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Spatial-Domain Filters

• Convolution
• Warning
• Example given earlier shows common implementation
  of convolution
• but it actually uses formula for cross-correlation

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Spatial-Domain Filters

• Convolution-based filters
• Low-pass filter masks

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Spatial-Domain Filters

• Convolution-based filters
• High-pass filter
• change that corresponds to edge is measured using
  derivative
• 1st order derivative has high magnitude at edge
• gradient of image
• 2nd order derivative has zero-crossing at edge
  (but higher sensitivity to noise)
• Laplacian of image

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Spatial-Domain Filters

• Convolution-based filters
• High-pass filter

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Spatial-Domain Filters

• Convolution-based filters
• High-pass filter noise sensitivity of Laplacian

Zero-crossings of Laplacian
Original image
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Spatial-Domain Filters

• Convolution-based filters
• High-pass filter noise sensitivity of Laplacian
• possible solution pre-filtering with Gaussian LPF

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Spatial-Domain Filters

• Convolution-based filters
• Laplacian of Gaussian

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Spatial-Domain Filters

• Convolution-based filters
• High-pass filter masks
• gradient of discrete function can be estimated
  using finite difference
• e.g. gradient along x-axis
• other approximations Roberts masks, Sobel masks,
  Prewitt masks

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Spatial-Domain Filters

• Convolution-based filters
• High-pass filter masks

For general use combine outputs of vertical and
horizontal edge filters
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Spatial-Domain Filters
High-pass filtering (Sobel masks) note
directional sensitivity
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Spatial-Domain Filters

• Convolution-based filters
• High-pass filter masks

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Spatial-Domain Filters

• Convolution-based filters
• High-pass filter masks

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Spatial-Domain Filters

• Convolution-based filters
• High-pass filter masks
• Laplacian of discrete function can be estimated
  using finite difference

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Spatial-Domain Filters

• Convolution-based filters
• High-pass filter masks

Single mask detects vertical and horizontal edges
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Spatial-Domain Filters

• Mode and median filters
• Some low-pass filters not based on convolution,
  e.g.
• mode filter
• output pixel value (at anchor) most common
  value in neighbourhood requires local histogram
  
• median filter
• output pixel value (at anchor) median value
  in neighbourhood requires sorting

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Spatial-Domain Filters

• Mode filter pseudo code
• / at each image pixel /
• for each image row // downward window shifts
• for each image column // rightward window
  shifts
• / in current image window /
• for each window row
• for each window column
• update local histogram
• put local mode in output buffer

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Spatial-Domain Filters

• Median filter pseudo code
• / at each image pixel /
• for each image row // downward window shifts
• for each image column // rightward window
  shifts
• / in current image window /
• for each window row
• for each window column
• update local sort
• put local median in output buffer

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Spatial-Domain Filters

• Mode and median filters
• Less blurring of edges (compared to averaging
  filters)
• Computational requirements may be high sorting
  in the median filter

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Spatial-Domain Filters
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Spatial-Domain Filters

• Combination of convolution-based filters
• Several filtering passes may be applied to image
• passes may be parallel or sequential
• combined effect can be obtained by (one of
  following)
• combining results of individual passes
• combining masks into one single-pass mask

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Spatial-Domain Filters

• Combination of convolution-based filters
• Parallel passes
• e.g. combination of output of directional edge
  detectors (e.g. Sobel edge detector)
• each edge pixel considered as vector (with
  magnitude and direction)

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Spatial-Domain Filters

• Combination of convolution-based filters
• Parallel passes
• e.g. addition of output images

Single-pass equivalent add mask coefficients
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Spatial-Domain Filters

• Combination of convolution-based filters
• Parallel passes
• addition of output images (ctd.)
• relevant property of convolution
• distributivity with addition

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Spatial-Domain Filters

• Combination of convolution-based filters
• Parallel passes e.g. edge detection by
  substraction

Original image
-
Smoothed image (5x5 Gaussian mask)
Output values scaled by 4 and offset by 128
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Spatial-Domain Filters

• Combination of convolution-based filters
• Sequential passes

Single-pass equivalent convolve masks
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Spatial-Domain Filters

• Combination of convolution-based filters
• Sequential passes (ctd.)
• relevant property of convolution
• associativity

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Spatial-Domain Filters

• Combination of convolution-based filters
• Sometimes beneficial to blur image prior to edge
  detection
• e.g. in Canny edge detector

Increasing degree of blurring
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Summary

• Filters retain / reject image details based on
  rate of change of pixel values
• classes low-pass filter, high-pass filter, ...
• based on neighbourhood (window) operations
• e.g. convolution, sorting, counting operations,
• convolution-based filters use mask

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Summary

• Combination of convolution-based filters
• parallel e.g. for addition of filtered images
• add mask coefficients
• sequential convolve masks