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

Contents Sharpening Spatial Filters 1 st order derivatives 2 nd order derivatives Laplacian Gaussian derivatives Laplacian of Gaussian (LoG) Unsharp masking 2

Sharpening spatial filters Image derivatives (1 st and 2 nd order) Define derivatives in terms of differences for the discrete domain How to define such differences? 3

1 st order derivatives Some requirements (1 st order): Zero in areas of constant intensity Nonzero at beginning of intensity step or ramp Nonzero along ramps 4

1 st order derivatives 5

2 nd order derivatives Requirements (2 nd order) Zero in constant areas Nonzero at beginning and end of intensity step or ramp Zero along ramps of constant slope 6

2 nd order derivatives 7

Image Derivatives 1 st order 2 nd order

9 1 st order2 nd order Zero crossing, locating edges

Edges are ramp-like transitions in intensity 1 st order derivative gives thick edges 2 nd order derivative gives double thin edge with zeros in between 2 nd order derivatives enhance fine detail much better 10

11 2 nd order Zero crossing, locating edges 1 st order

Filters related to first derivatives Recall: Prewitt filter, Sober filter (lecture 2 – 14/05/13) 12

Laplacian – second derivative Enhances edges Definition 13

Laplacian 14 Adding diagonal derivation Opposite sign for second order derivative

Laplacian Note: Laplacian filtering results in + and – pixel values Scale for image display - eqs. (2.6-10, ) Or: take absolute value or positive values 15

Line Detector 16 * scaled Laplacian Positive values Laplacian (figure 10.5 book)

Image sharpening - example 17 4-connected Laplacian8-connected LaplacianEnhanced + Laplacian x5 Enhanced + Laplacian x6 Enhanced + Laplacian x8 Better sharpening with 8-connected Laplacian (see figure 3.38 (d)-(e) book) C=+1 or -1

Filtering in frequency domain Basic steps: image f(x,y) Fourier transform F(u,v) filter H(u,v) H(u,v)F(u,v) inverse Fourier transform filtered image g(x,y) 18

Laplacian in the Fourier domain Spatial Fourier domain 19

Blur first, take derivative later Smoothing is a good idea to avoid enhancement of noise. Common smoothing kernel is a Gaussian. Scale of blurring

Gaussian Derivative Taking the derivative after blurring gives image g

Gaussian Derivative We can build a single kernel for both convolutions Use the associative property of the convolution

Laplacian of Gaussian (LoG) 23 LoG a.k.a. Mexican Hat

LoG applied to building 24

Sharpening with LoG 25 sharpening with LoG sharpening with Laplacian

Unsharp Masking / Highboost Filtering Subtraction of unsharp (smoothed) version of image from the original image. Blur the original image Subtract the blurred image from the original (results in image called mask) Add the mask to the original 26

Let denote the blurred image Obtain the mask Add weighted portion of mask to original image 27

If Unsharp masking If Highboost filtering 28 inputblurredunsharp masku.m. resulth.f. result (see also figure 3.40 book)

Unsharp masking Simple and often used sharpening method Poor result in the presence of noise – LoG performs better in this case 29