1. Effective Computation of Linear Filters The following properties are used to shorten the computation time f  g = g  f, f  (g  h) = (f  g)  h, f  (g + h) = f  g + f  h Example: Edge detection - Sobel Example: Edge detection - Laplace

2. Effective Computation of Linear Filters Example: Sharpening filter Example: Smoothing - Gaussian

3. Effective Computation of Linear Filters Example: Smoothing combined with edge detection (LoG)

4. Linear Filters in Frequency Domain Each linear filter in spatial domain (performed as convolution) has a corresponding filter in frequency domain (performed as multiplication) and vice versa. Example: Low-pass filters –Ideal low-pass filter 1D: rectangle in frequency, k · sinc ax (sin ax / ax) in spatial domain 2D: rectangular box in frequency, k · sinc ax · sinc by in spatial domain 2D: cylinder in frequency, k · J 1 (sqrt(x 2 +y 2 )) in spatial domain –Non-weighted averaging 1D: rectangle in spatial, k · sinc a  x in frequency domain 2D: rectangular box in spatial, k · sinc a  x · sinc b  y in frequency domain 2D: cylinder in spatial, k · J 1 (sqrt(  x 2 +  y 2 )) in frequency domain –Weighted Gaussian averaging 1D: a · exp (-b · x 2 ) in spatial, c · exp (-d ·  x 2 ) in frequency domain 2D: a · exp (-b · (x 2 +y 2 )) in spatial, c · exp (-d · (  x 2 +  y 2 )) in frequency

5. Linear Filters in Frequency Domain Example: High-pass filters –Sobel 1D: 1 st derivative in spatial, multiplication by -(i  x ) in frequency domain 2D: 1 st derivative in spatial, multiplication by -(i  d ) in frequency domain –Laplace 1D: 2 nd derivative in spatial, multiplication by -(  x 2 ) in frequency 2D: 2 nd derivative in spatial, multiplication by -(  x 2 +  y 2 ) in frequency