Convolution

Basis beeldverwerking (8D040)

dr. Andrea Fusterdr. Anna VilanovaProf.dr.ir. Marcel Breeuwer

Convolution

Contents

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• Spatial filtering• Correlation• Convolution• Filters:

• Smoothing filters• Sharpening filters

• Borders

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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 (3x3 filter)

• 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-

Example

See blackboard ☺ (or figure 3.30 Gonzalez and Woods)

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Correlation vs. Convolution

• Correlation and convolution give the same result if the filter used is symmetric!

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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

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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 Tuesday 21!

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