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Basis beeldverwerking (8D040 ) d r. Andrea Fuster Prof.dr . Bart ter Haar Romeny dr. Anna Vilanova Prof.dr.ir . Marcel Breeuwer. Convolution. Contents. Spatial filtering Correlation Convolution Filters: Smoothing filters Sharpening filters Borders. Spatial filtering. - PowerPoint PPT Presentation
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Basis beeldverwerking (8D040)
dr. Andrea FusterProf.dr. Bart ter Haar Romenydr. Anna VilanovaProf.dr.ir. Marcel Breeuwer
Convolution
Contents
2
• Spatial filtering• Correlation• Convolution• Filters:
• Smoothing filters• Sharpening filters
• Borders
Basis beeldverwerking 8D040
Spatial filtering
• Input image , use a filter to obtain processed image
• Filter consists of• Neighbourhood (rectangular)• Mostly odd dimensions• Predefined operation
• Create new pixel value in center of neighbourhood
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Spatial filtering
• Filter operation
• More compact notation - filter
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Intuition to filtering
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Move filter over image
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Correlation
• While moving the filter, at each position
• Multiply values of overlapping locations• Sum all multiplication results
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Correlation vs. Convolution
• Discrete Correlation 2D
• Discrete Convolution 2D
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- Equivalent to first rotate the filter 180 degrees and correlate-
Correlation vs. Convolution
• With convolution, first mirror the filter kernel with respect to its origin.
• With correlation, do not mirror the filter kernel with respect to its origin.
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convolution correlation
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Example
See blackboard ☺ (or figure 3.30 Gonzalez and Woods)
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Convolution – 1D cont. case
• Imagine a system with • input signal• transfer function• output signal
• then
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Definition
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output input
system transfer function
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Dirac delta function (unit impulse)
• Definition
• Constraint
• Sifting property
• Specifically for t=0
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Convolution
• Let
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We saw this already in the discrete case
Properties of convolution
• Commutative
• Associative
• Distributive19Basis beeldverwerking 8D040
Convolution is commutative• Proof
• Let
• Q.E.D.
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Convolution is associative - 1
• Proof
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Convolution is associative - 2
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Convolution is associative - 3
• Let
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Convolution is associative - 4
• Q.E.D.
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Convolution is distributive - 1
• Proof
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Convolution is distributive - 2
• Q.E.D.
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Discrete convolution
• 1D
• 2D
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Discrete convolution
• Formulas take summation from to• Filters have a limited size, e.g.,
• 1D 2a + 1
• 2D (2a + 1, 2b + 1)
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Filter Kernels
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Kernel
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Filters
• Idea: correlate or convolve image with different filters in order to obtain different results, i.e., processed images
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Smoothing filters
• Average intensities – result is blurred image, less details
• Response: (z’s image intensities)
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… NxN filter
Smoothing filters
• Note that:
− Sum of filter coefficients is 1 (normalized filter)− Correlation = convolution (symmetric filter)− Filter size effect?
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Smoothing filters - example
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Original 3x3 smoothing filter NxN filter (see figure 3.33 in
Gonzalez and Woods!)
Effect of normalized smoothing kernel
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normalized
non- normalized
Sharpening filters
• Enhance parts of the image where intensities change rapidly, such as edges • Basic derivative filters
• Measure change of intensity in x or y direction
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Example
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Arbitrary angle derivative
• Given and
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Arbitrary angle derivative
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Prewitt gradient kernel
• Derivative in one direction, smoothing in the perpendicular direction
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Example
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Prewitt
Basic derivative
Sobel kernel
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Example
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(Thanks to Wikipedia☺)
Derivative filters
• Note that coefficients in all of the previous filters sum to zero, i.e., zero response in area of constant intensity
• Also: gradient, Laplacian, …
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Borders
• Do you see any problems at image borders?
• Try position (0,0)
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Border problems
• How to handle?• No border handling
• Border is not filtered• Padding
• Put values outside image border • Cyclic padding
• Use values from the other side of the image
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Zero padding
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Cyclic padding
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Padding
• Remember: padding is artificial!• The values chosen outside the border influence the
outcome image
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End of part 2
Thanks and see you Wednesday 16!
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