Image Processing. Daniel Danilov July 13, 2015

Transcription

1 Image Processing Daniel Danilov July 13, 2015

2 Overview 1. Principle of digital images and filters 2. Basic examples of filters 3. Edge detection and segmentation 1 / 25

3 Motivation For what image processing in medicine? Adaptation to human perception Adapt image s brightness Reduction of noise Contrast enhancement... Computer based image analysis Differentiate tissue (segmentation) 3D based diagnostics 2 / 25

4 Digital images Each image consist of N M picture elements (pixels) Every pixel displays a gray value P(x, y) P(x, y) is stored as an integer (8 bit) From: soonet.ca From: processing.org Images are N M matrices of integers. We can manipulate these integers! 3 / 25

5 Working principle of filters Define a Mask with central Pixel P(x, y) and n neighbor pixels M (2n+1) (2n+1) (x, y) Filtered image P arises from convolution with M n n P(x + i, y + j) M(i, j) i= n j= n Use Convolution theorem F( P) = F(P) F(M) From: H. Handels: Medizinische Bildverarbeitung 4 / 25

6 Working principle of filters Define a Mask with central Pixel P(x, y) and n neighbor pixels M (2n+1) (2n+1) (x, y) Filtered image P arises from convolution with M n n P(x + i, y + j) M(i, j) i= n j= n From: H. Handels: Medizinische Bildverarbeitung From: soonet.ca (edited) 4 / 25

7 Working principle of filters These filters take into account the local vicinity of the pixel: Local filters Point filters: Surrounding pixels are not considered e.g. inversion, darken/lighten From: med-ed.virginia.edu From: H. Handels: Medizinische Bildverarbeitung 5 / 25

8 Example: Mean filter From: H. Handels: Medizinische Bildverarbeitung 6 / 25

9 Example: Mean filter From: H. Handels: Medizinische Bildverarbeitung 6 / 25

10 Example: Mean filter From: H. Handels: Medizinische Bildverarbeitung Mean filter reduces inhomogeneities single outlier effects neighbor pixels Strong blurring effect (unsharp image) 7 / 25

11 Example: Median filter From: hindawi.com Salt and pepper noise Caused by defect detectors 8 / 25

12 Example: Median filter From: hindawi.com (Mean filter applied) Mean filter does not remove this noise Noise is smeared over neighbor pixels Use median filter! 8 / 25

13 Example: Median filter Consider the sequence S = {17, 5, 42, 2, 51} The median of S is the value in the middle of the ascending sorted sequence of S Ascending sorting: {2, 5, 17, 42, 51} Median value: 17 9 / 25

14 Example: Median filter Single pixel has outstanding gray value (salt and pepper) {0, 8, 11, 13, 17, 21, 24, 31, 255} 10 / 25

15 Example: Median filter Single pixel has outstanding gray value (salt and pepper) {0, 8, 11, 13, 17, 21, 24, 31, 255} 10 / 25

16 Example: Median filter Zoomed in on an edge (black-white transition) {33, 47, 65, 189, 205, 211, 247, 249, 255} 11 / 25

17 Example: Median filter From: hindawi.com (Median filter applied) Median filter removes reliably salt and pepper noise Preserves sharpness of the edges very well Very little blurring effect compared to mean filter 12 / 25

18 Edge detection: Motivation From: comp.nus.edu.sg From: keystonemedicalus.com How to teach a computer to distinguish between different tissues? 13 / 25

19 Edge detection: Motivation From: radiologie-ffm.de From: H. Handels Tissues are separated by steep changes of grey values Corresponds to steep gradient values Use differentiation filters! 14 / 25

20 Edge detection: Basics From: H. Handels On an edge... 1 st derivative maximal 2 nd derivative zero 15 / 25

21 Edge detection: Discrete differentiation Let f (x, y) be the grey values of the image at pixel x and y ) 2D derivative: f (x, y) = ( f x f y f x f y f (x,y) f (x 1,y) x (x 1) = f (x, y) f (x 1, y) = f (x, y) f (x, y 1) ( f (x, y) = f ) 2 ( ) 2 x + f y f (x,y) y tan Φ(x, y) = x f (x,y) From: H. Handels: Medizinische Bildverarbeitung 16 / 25

22 Edge detection: Gradient image Plot gradient magnitude f (x, y ) From: siemens.com (filtered) Better results via Sobel- and Prewitt-filter 17 / 25

23 Edge detection: Sobel- and Prewitt-filter Combination of filters possible Sobel- and Prewitt-filter: Mean filter + differentiation filter From: H. Handels: Medizinische Bildverarbeitung 18 / 25

24 Edge detection: Sobel- and Prewitt-filter From: siemens.com (filtered) Sobel-filter enhances edges But makes them also wider and slightly blurred 19 / 25

25 Segmentation: Difficulties Fully automatic segmentation does not exist due to... low image quality complex structures of human body Steering of segmentation precess by humans needed Goal: Try to minimize human intervention Leads e.g. to better comparability 20 / 25

26 Segmentation: Live wire technique Ansatz: Find optimal path between start node and further goal nodes From: Barrett, Mortensen: Interactive Live-Wire Boundary Extraction 21 / 25

27 Segmentation: Live wire technique What is the optimal path? Assign local cost L( p, q i ) to direct link from pixel p to its neighbours q i L( p, q i ) = ω G f G ( q i ) + ω Z f Z ( q i ) + ω D f D ( p, q i ) Path follows the direction q i p with lowest cost 22 / 25

28 Segmentation: Live wire technique L( p, q i ) = ω G f G ( q i ) + ω Z f Z ( q i ) + ω D f D ( p, q i ) Weights ω: ω G = ω Z = 0.43 f G ( q i ): Gradient magnitude (G) function f G ( q i ) = 1 G( q i) max(g) ω D = 0.14 (empirical!) f Z ( q i ): Laplacian zero-crossing function f Z ( q i ) = 0 for I( q i ) = 0 f Z = 1 otherwise f D ( p, q i ): Gradient direction function (prefer smooth curves) 23 / 25

29 Segmentation: Live wire technique User sets seed points Algorithm connects seed points via path of lowest cost Advantages Little human interaction needed Precise, fast and interactive Drawbacks Stronger near edges distract path Noise reduces precision From: Barrett, Mortensen: Interactive Live-Wire Boundary Extraction 24 / 25

30 Summary Filters convolute the image with predefined masks Examples: Mean and median filters for noise reduction Edge detection possible via differentiation masks Edges are basis for segmenting the image Segmentation via Live Wire Technique 25 / 25