Image Analysis. Edge Detection

Transcription

1 Image Analsis Image Analsis Edge Detection K. Buza Lars Schmidt-Thieme Inormation Sstems and Machine Learning Lab ISMLL Institute o Economics and Inormation Sstems & Institute o Computer Science Universit o Hildesheim K. Buza Lars Schmidt-Thieme Inormation Sstems and Machine Learning Lab ISMLL Universit o Hildesheim German Course on Image Analsis winter term 8/9

2 Image Analsis Edge Detection What is Edge Detection? Gradient or Edge Detection Convolution Marr-Hildreth Cann Hough Transormation K. Buza Lars Schmidt-Thieme Inormation Sstems and Machine Learning Lab ISMLL Universit o Hildesheim German Course on Image Analsis winter term 8/9

3 Image Analsis What is Edge Detection? Low level edge detection Which piels belong to an edge? What is the local orientation o those piels? High level edge detection Characterize edges in the picture For eample: lines b their oset and slope K. Buza Lars Schmidt-Thieme Inormation Sstems and Machine Learning Lab ISMLL Universit o Hildesheim German Course on Image Analsis winter term 8/9

4 Image Analsis Edge Detection What is Edge Detection? Gradient or Edge Detection Convolution Marr-Hildreth Cann Hough Transormation K. Buza Lars Schmidt-Thieme Inormation Sstems and Machine Learning Lab ISMLL Universit o Hildesheim German Course on Image Analsis winter term 8/9

5 Image Analsis Gradient or Edge Detection What is an edge? Signiicant change in intensit I we are given a unction how can we detect signiicant changes i.e. regions where the unction is steep? K. Buza Lars Schmidt-Thieme Inormation Sstems and Machine Learning Lab ISMLL Universit o Hildesheim German Course on Image Analsis winter term 8/9

6 Gradient or Edge Detection Derivation o two-dimensional unction One can derive the unction along a chosen dimension variable: Gradient: The magnitude o the gradient: Image Analsis K. Buza Lars Schmidt-Thieme Inormation Sstems and Machine Learning Lab ISMLL Universit o Hildesheim German Course on Image Analsis winter term 8/9 G mag

7 Gradient or Edge Detection Approimations or discrete unctions pictures: Image Analsis K. Buza Lars Schmidt-Thieme Inormation Sstems and Machine Learning Lab ISMLL Universit o Hildesheim German Course on Image Analsis winter term 8/9 G mag

8 Image Analsis Gradient or Edge Detection Edge detection: Calculate the gradient o an image unction gradientimage image { } or int =;<image.getwidth;++ { } or int =;<image.getheight;++ { } intensit = image.getintensit ; intensit = image.getintensit- ; intensit3 = image.getintensit -; gradient = sqrt intensit-intensit3*intensit-intensit3 + intensit-intensit*intensit-intensit; edge_image.setintensit gradient; return edge_image; K. Buza Lars Schmidt-Thieme Inormation Sstems and Machine Learning Lab ISMLL Universit o Hildesheim German Course on Image Analsis winter term 8/9

9 Image Analsis Gradient or Edge Detection K. Buza Lars Schmidt-Thieme Inormation Sstems and Machine Learning Lab ISMLL Universit o Hildesheim German Course on Image Analsis winter term 8/9

10 Image Analsis Edge Detection What is Edge Detection? Gradient or Edge Detection Convolution Marr-Hildreth Cann Hough Transormation K. Buza Lars Schmidt-Thieme Inormation Sstems and Machine Learning Lab ISMLL Universit o Hildesheim German Course on Image Analsis winter term 8/9

11 Can we ormulate gradient as Convolution? Image Analsis K. Buza Lars Schmidt-Thieme Inormation Sstems and Machine Learning Lab ISMLL Universit o Hildesheim German Course on Image Analsis winter term 8/9 K K? K grad G mag #

12 Sobel Edge Detector Calculate and components separatel b convolving the image with S and S and use them in the gradient ormula Image Analsis K. Buza Lars Schmidt-Thieme Inormation Sstems and Machine Learning Lab ISMLL Universit o Hildesheim German Course on Image Analsis winter term 8/9 ' S ' S * ' ' G mag

13 Image Analsis Sobel Edge Detector unction sobelimage image { int[][] S = { {-} {-} {-} }; int[][] S = { {---} {} {} }; } or int =;<image.getwidth-;++ { or int =;<image.getheight-;++ { int intensit_sum_= intensit_sum_=; or int i=-;i<=;i++ { or int j=-;j<=;j++ { int intensit = image.getintensit+i +j; intensit_sum_+=intensit*s[-j][-i]; intensit_sum_+=intensit*s[-j][-i]; // st inde o the arra S and S --> row o the kernel matri // nd inde o the arra S and S --> column o the kernel matri } } int new_intensit = sqrtintensit_sum_*intensit_sum_+intensit_sum_*intensit_sum_; edges_image.setintensitpiel new_intensit; } } return edges_image; K. Buza Lars Schmidt-Thieme Inormation Sstems and Machine Learning Lab ISMLL Universit o Hildesheim German Course on Image Analsis winter term 8/9

14 Image Analsis Sobel Edge Detector Better or worse than gradient? K. Buza Lars Schmidt-Thieme Inormation Sstems and Machine Learning Lab ISMLL Universit o Hildesheim German Course on Image Analsis winter term 8/9

15 Image Analsis Sobel Edge Detector under noise K. Buza Lars Schmidt-Thieme Inormation Sstems and Machine Learning Lab ISMLL Universit o Hildesheim German Course on Image Analsis winter term 8/9

16 Image Analsis Presence and absence o noise Edge detectors seen so ar work well i there is no noise In case o noise use noise ilter irst More robust edge detection net sections K. Buza Lars Schmidt-Thieme Inormation Sstems and Machine Learning Lab ISMLL Universit o Hildesheim German Course on Image Analsis winter term 8/9

17 Image Analsis Edge Detection What is Edge Detection? Gradient or Edge Detection Convolution Marr-Hildreth Cann Hough Transormation K. Buza Lars Schmidt-Thieme Inormation Sstems and Machine Learning Lab ISMLL Universit o Hildesheim German Course on Image Analsis winter term 8/9

18 Image Analsis Marr-Hildreth Edge Detection Gaussian ilter or noise reduction Edge Etrem value in the irst derivative o the intensit zero crossing in the second derivative Second derivative Laplacian o an image is: Laplacian is deined as the divergence o the gradient. Divergence: sum o partial derivatives scalar This is equivalent with the above ormula. The Laplacian consists o scalar values! Be careul: the gradient o an image consists o vectors not scalars! These vectors have some length called magnitude o course which is scalar. K. Buza Lars Schmidt-Thieme Inormation Sstems and Machine Learning Lab ISMLL Universit o Hildesheim German Course on Image Analsis winter term 8/9

19 Marr-Hildreth Edge Detection Laplacian o an image Image Analsis K. Buza Lars Schmidt-Thieme Inormation Sstems and Machine Learning Lab ISMLL Universit o Hildesheim German Course on Image Analsis winter term 8/9 K * K

20 Marr-Hildreth Edge Detection Image Analsis K. Buza Lars Schmidt-Thieme Inormation Sstems and Machine Learning Lab ISMLL Universit o Hildesheim German Course on Image Analsis winter term 8/9 K Second derivative in the direction Corresponding convolution kernel ' ' ''

21 Marr-Hildreth Edge Detection Image Analsis K. Buza Lars Schmidt-Thieme Inormation Sstems and Machine Learning Lab ISMLL Universit o Hildesheim German Course on Image Analsis winter term 8/9 K Second derivative in the direction Second derivative in the direction Convolution matri or the Laplacian L: K 4 K K L

22 Image Analsis Marr-Hildreth Edge Detection A possible gaussian noise ilter: G Appl Gaussian ilter irst then calculate the Laplacian o the image: G L Laplacian o the Gaussian LoG The both above operations as one convolution matri Homework: calculate the LoG matri K. Buza Lars Schmidt-Thieme Inormation Sstems and Machine Learning Lab ISMLL Universit o Hildesheim German Course on Image Analsis winter term 8/9

23 Image Analsis Marr-Hildreth Edge Detection K. Buza Lars Schmidt-Thieme Inormation Sstems and Machine Learning Lab ISMLL Universit o Hildesheim German Course on Image Analsis winter term 8/9

24 Image Analsis Marr-Hildreth Edge Detection Wh do things go wrong? For eample sk piels are not homogenous blue. I a piel is a little bit dierent than the others around it an edge will be detected What can be done? Reduce the number o gra levels used in the calculation o the LoG o the image Take onl drastic zero crossings into account when determining where are the edges K. Buza Lars Schmidt-Thieme Inormation Sstems and Machine Learning Lab ISMLL Universit o Hildesheim German Course on Image Analsis winter term 8/9

25 Image Analsis Marr-Hildreth Edge Detection K. Buza Lars Schmidt-Thieme Inormation Sstems and Machine Learning Lab ISMLL Universit o Hildesheim German Course on Image Analsis winter term 8/9

26 Image Analsis Marr-Hildreth Edge Detection K. Buza Lars Schmidt-Thieme Inormation Sstems and Machine Learning Lab ISMLL Universit o Hildesheim German Course on Image Analsis winter term 8/9

27 Image Analsis Marr-Hildreth Edge Detection in presence o noise K. Buza Lars Schmidt-Thieme Inormation Sstems and Machine Learning Lab ISMLL Universit o Hildesheim German Course on Image Analsis winter term 8/9

28 Image Analsis Marr-Hildreth Edge Detection in presence o noise K. Buza Lars Schmidt-Thieme Inormation Sstems and Machine Learning Lab ISMLL Universit o Hildesheim German Course on Image Analsis winter term 8/9

29 Image Analsis Edge Detection What is Edge Detection? Gradient or Edge Detection Convolution Marr-Hildreth Cann Hough Transormation K. Buza Lars Schmidt-Thieme Inormation Sstems and Machine Learning Lab ISMLL Universit o Hildesheim German Course on Image Analsis winter term 8/9

30 Image Analsis Cann Edge Detection Use the derivative o the Gaussian Speed-up edge detection process: instead o convolution with a matri two dimensional arra two convolutions with two vectors one dimensional arras Post-processing steps Non-maimal suppression Hsteresis sampling K. Buza Lars Schmidt-Thieme Inormation Sstems and Machine Learning Lab ISMLL Universit o Hildesheim German Course on Image Analsis winter term 8/9

31 Image Analsis Cann Edge Detection Derivative o the Gaussian G...4. G' Steps o the algorithm Input Image I. Convolve the image I with G I. Convolve the image I with G T I 3. Convolve I with G I 4. Convolve I with G T I 5. For each piel there is a vector: R = I I 6. Perorm Non-maimal suppression 7. Perorm Hsteresis Sampling K. Buza Lars Schmidt-Thieme Inormation Sstems and Machine Learning Lab ISMLL Universit o Hildesheim German Course on Image Analsis winter term 8/9

32 Image Analsis Non-maimal Suppression Denote the result o steps -5 with R. An edge at the piel in the image corresponds to a local maimum o R at. This local maimum is meant according to direction o R. For each piel: Select the net piels both in the direction o R and in the opposite directions?? linear Interpolation Check i there is a local maimum at and indicate non-local maima in R as non-edge piels i.e. set R= i is not a local maimum K. Buza Lars Schmidt-Thieme Inormation Sstems and Machine Learning Lab ISMLL Universit o Hildesheim German Course on Image Analsis winter term 8/9

33 Image Analsis Hsteresis sampling Given two thresholds T high and T low Mark all piels as non-edge i R< T low Mark all piels as edge i R> T high For the other piels: Mark the neighbors o edge piels as edge-piel iterativel K. Buza Lars Schmidt-Thieme Inormation Sstems and Machine Learning Lab ISMLL Universit o Hildesheim German Course on Image Analsis winter term 8/9

34 Image Analsis Cann Edge Detection K. Buza Lars Schmidt-Thieme Inormation Sstems and Machine Learning Lab ISMLL Universit o Hildesheim German Course on Image Analsis winter term 8/9

35 Image Analsis Edge Detection What is Edge Detection? Gradient or Edge Detection Convolution Marr-Hildreth Cann Hough Transormation K. Buza Lars Schmidt-Thieme Inormation Sstems and Machine Learning Lab ISMLL Universit o Hildesheim German Course on Image Analsis winter term 8/9

36 Image Analsis What is Edge Detection? Low level edge detection Which piels belong to an edge? What is the local orientation o those piels? Gradient Sobel Kirsch Marr-Hildreth Cann High level edge detection Characterize edges in the picture For eample: Lines b their oset and slope Circles b their center and radius Hough transorm K. Buza Lars Schmidt-Thieme Inormation Sstems and Machine Learning Lab ISMLL Universit o Hildesheim German Course on Image Analsis winter term 8/9

37 Image Analsis Hough Transormation: Detection o straight lines A line can be described b d and a. Make a catalog o all the possible lines create a counter or each o the lines Traverse through the piels o the image. For each piel o the oreground: This piel can belong the several lines. Increment the counters o ALL o these lines Finall the counters with highest values correspond to real lines d a a = 3 4 d= d= d= K. Buza Lars Schmidt-Thieme Inormation Sstems and Machine Learning Lab ISMLL Universit o Hildesheim German Course on Image Analsis winter term 8/9

38 Image Analsis Hough Transormation: Detection o straight lines A line can be described b d and a. Make a catalog o all the possible lines create a counter or each o the lines Traverse through the piels o the image. For each piel o the oreground: This piel can belong the several lines. Increment the counters o ALL o these lines Finall the counters with highest values correspond to real lines d d a a d 3 a = 3 4 d= d= d= K. Buza Lars Schmidt-Thieme Inormation Sstems and Machine Learning Lab ISMLL Universit o Hildesheim German Course on Image Analsis winter term 8/9

39 Image Analsis Hough Transormation: Detection o straight lines A line can be described b d and a. Make a catalog o all the possible lines create a counter or each o the lines Traverse through the piels o the image. For each piel o the oreground: This piel can belong the several lines. Increment the counters o ALL o these lines Finall the counters with highest values correspond to real lines a = 3 4 d= d= d= K. Buza Lars Schmidt-Thieme Inormation Sstems and Machine Learning Lab ISMLL Universit o Hildesheim German Course on Image Analsis winter term 8/9

40 Image Analsis Which lines belong to each piel? Traverse through the piels o the image. For each piel o the oreground: This piel can belong the several lines. Increment the counters o ALL o these lines This can be implemented the ollowing wa: For each piel o the image i is a oreground-piel For a...9 d = cos a + sin a incrementda d d a a d 3 a = 3 4 d= d= d= K. Buza Lars Schmidt-Thieme Inormation Sstems and Machine Learning Lab ISMLL Universit o Hildesheim German Course on Image Analsis winter term 8/9

41 Image Analsis Hough Transormation There are other was to describe a line or eample -oset and b Using some parameters one can describe other objects like circles ellipses The method seen beore can easil be adapted or these cases For eample one can describe a circle with its radius r and center For piels belonging to the circle: - +- =r To which circles can a oreground piel belong? or each oreground-piel or = ma or = ma r increment r K. Buza Lars Schmidt-Thieme Inormation Sstems and Machine Learning Lab ISMLL Universit o Hildesheim German Course on Image Analsis winter term 8/9 d a b

42 Image Analsis Hough Transormation b -oset horizontal ale vertical ale K. Buza Lars Schmidt-Thieme Inormation Sstems and Machine Learning Lab ISMLL Universit o Hildesheim German Course on Image Analsis winter term 8/9

43 Image Analsis Outlook Low level edge detection can be though as a preprocessing step or high level edge detection Hough transormation Suppose one wants to determine the size o the wheels o the car in this image automaticall First one can appl some low-level edge detection And then Hough transormation to detect circles d a b K. Buza Lars Schmidt-Thieme Inormation Sstems and Machine Learning Lab ISMLL Universit o Hildesheim German Course on Image Analsis winter term 8/9