Basis beeldverwerking (8D040) dr. Andrea Fuster Prof.dr. Bart ter Haar Romeny dr. Anna Vilanova Prof.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 3 Basis beeldverwerking 8D040

Spatial filtering Filter operation More compact notation - filter 4 Basis beeldverwerking 8D040

Intuition to filtering 5 Basis beeldverwerking 8D040

Move filter over image 6 Basis beeldverwerking 8D040

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Correlation While moving the filter, at each position Multiply values of overlapping locations Sum all multiplication results 11 Basis beeldverwerking 8D040

Correlation vs. Convolution Discrete Correlation 2D Discrete Convolution 2D 12 Basis beeldverwerking 8D040 - 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. 13 convolutioncorrelation Basis beeldverwerking 8D040

Example See blackboard ☺ (or figure 3.30 Gonzalez and Woods) Basis beeldverwerking 8D040 14

Convolution – 1D cont. case Imagine a system with input signal transfer function output signal then 15 Basis beeldverwerking 8D040

Definition 16 output input system transfer function Basis beeldverwerking 8D040

Dirac delta function (unit impulse) Definition Constraint Sifting property Specifically for t=0 17 Basis beeldverwerking 8D040

Convolution Let 18 Basis beeldverwerking 8D040 We saw this already in the discrete case

Properties of convolution Commutative Associative Distributive 19 Basis beeldverwerking 8D040

Convolution is commutative Proof Let Q.E.D. 20 Basis beeldverwerking 8D040

Convolution is associative - 1 Proof 21 Basis beeldverwerking 8D040

Convolution is associative Basis beeldverwerking 8D040

Convolution is associative - 3 Let 23 Basis beeldverwerking 8D040

Convolution is associative - 4 Q.E.D. 24 Basis beeldverwerking 8D040

Convolution is distributive - 1 Proof 25 Basis beeldverwerking 8D040

Convolution is distributive - 2 Q.E.D. 26 Basis beeldverwerking 8D040

Discrete convolution 1D 2D 27 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) 28 Basis beeldverwerking 8D040

Filter Kernels 29 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 30

Smoothing filters Average intensities – result is blurred image, less details Response: (z’s image intensities) 31 Basis beeldverwerking 8D040 … NxN filter

Smoothing filters Note that: −Sum of filter coefficients is 1 (normalized filter) −Correlation = convolution (symmetric filter) −Filter size effect? Basis beeldverwerking 8D040 32

Smoothing filters - example 33 Basis beeldverwerking 8D040 Original 3x3 smoothing filter NxN filter (see figure 3.33 in Gonzalez and Woods!)

Effect of normalized smoothing kernel Basis beeldverwerking 8D 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 35 Basis beeldverwerking 8D040

Example 36 Basis beeldverwerking 8D040

Arbitrary angle derivative Given and 37 Basis beeldverwerking 8D040

Arbitrary angle derivative 38 Basis beeldverwerking 8D040

Prewitt gradient kernel Derivative in one direction, smoothing in the perpendicular direction 39 Basis beeldverwerking 8D040

Example 40 Basis beeldverwerking 8D Prewitt Basic derivative

Sobel kernel 41 Basis beeldverwerking 8D040

Example 42 (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 43

Borders Do you see any problems at image borders? Try position (0,0) 44 Basis beeldverwerking 8D040

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 45 Basis beeldverwerking 8D040

Zero padding 46 Basis beeldverwerking 8D040

Cyclic padding 47 Basis beeldverwerking 8D040

Padding Remember: padding is artificial! The values chosen outside the border influence the outcome image 48 Basis beeldverwerking 8D040

End of part 2 Thanks and see you Wednesday 16! Basis beeldverwerking 8D040 49