Basis beeldverwerking (8D040) dr. Andrea Fuster dr. Anna Vilanova Prof Basis beeldverwerking (8D040) dr. Andrea Fuster dr. Anna Vilanova Prof.dr.ir. Marcel Breeuwer Convolution

Contents 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 Basis beeldverwerking 8D040

Spatial filtering Filter operation (3x3 filter) More compact notation - filter Basis beeldverwerking 8D040

Intuition to filtering Basis beeldverwerking 8D040

Move filter over image Basis beeldverwerking 8D040

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Correlation While moving the filter, at each position Multiply values of overlapping locations Sum all multiplication results Basis beeldverwerking 8D040

Correlation vs. Convolution Discrete Correlation 2D Discrete Convolution 2D - Equivalent to first rotate the filter 180 degrees and correlate- Basis beeldverwerking 8D040

Example See blackboard ☺ (or figure 3.30 Gonzalez and Woods) Basis beeldverwerking 8D040

Correlation vs. Convolution Correlation and convolution give the same result if the filter used is symmetric! Basis beeldverwerking 8D040 25-03-2009

Convolution – 1D cont. case Imagine a system with input signal transfer function output signal then Basis beeldverwerking 8D040

system transfer function Definition input output system transfer function Basis beeldverwerking 8D040

Dirac delta function (unit impulse) Definition Constraint Sifting property Specifically for t=0 Basis beeldverwerking 8D040

Convolution Let We saw this already in the discrete case Basis beeldverwerking 8D040

Properties of convolution Commutative Associative Distributive Basis beeldverwerking 8D040

Convolution is commutative Proof Let Q.E.D. Basis beeldverwerking 8D040

Convolution is associative - 1 Proof Basis beeldverwerking 8D040

Convolution is associative - 2 Basis beeldverwerking 8D040

Convolution is associative - 3 Let Basis beeldverwerking 8D040

Convolution is associative - 4 Q.E.D. Basis beeldverwerking 8D040

Convolution is distributive - 1 Proof Basis beeldverwerking 8D040

Convolution is distributive - 2 Q.E.D. Basis beeldverwerking 8D040

Discrete convolution 1D 2D Basis beeldverwerking 8D040

Discrete convolution Formulas take summation from to Filters have a limited size, e.g., 1D 2a + 1 2D (2a + 1, 2b + 1) Basis beeldverwerking 8D040

Filter Kernels Kernel Basis beeldverwerking 8D040

Filters Idea: correlate or convolve image with different filters in order to obtain different results, i.e., processed images Basis beeldverwerking 8D040

Smoothing filters Average intensities – result is blurred image, less details Response: (z’s image intensities) … NxN filter Basis beeldverwerking 8D040

Smoothing filters Note that: Sum of filter coefficients is 1 (normalized filter) Correlation = convolution (symmetric filter) Filter size effect? Basis beeldverwerking 8D040

Smoothing filters - example Original 3x3 smoothing filter NxN filter (see figure 3.33 in Gonzalez and Woods!) Basis beeldverwerking 8D040

Effect of normalized smoothing kernel non- normalized Basis beeldverwerking 8D040

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 Basis beeldverwerking 8D040

Example Basis beeldverwerking 8D040

Arbitrary angle derivative Given and Basis beeldverwerking 8D040

Arbitrary angle derivative Basis beeldverwerking 8D040

Prewitt gradient kernel Derivative in one direction, smoothing in the perpendicular direction Basis beeldverwerking 8D040

Example Prewitt Basic derivative Basis beeldverwerking 8D040 25-03-2009

Sobel kernel Basis beeldverwerking 8D040

Example (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, … Basis beeldverwerking 8D040

Borders Do you see any problems at image borders? Try position (0,0) Basis beeldverwerking 8D040

Border problems How to handle? No border handling Padding Border is not filtered Padding Put values outside image border Cyclic padding Use values from the other side of the image Basis beeldverwerking 8D040

Zero padding Basis beeldverwerking 8D040

Cyclic padding Basis beeldverwerking 8D040

Padding Remember: padding is artificial! The values chosen outside the border influence the outcome image Basis beeldverwerking 8D040

End of part 2 Thanks and see you Tuesday 21! Basis beeldverwerking 8D040