Bildverarbeitung und Mustererkennung Image Processing and Pattern Recognition

Transcription

1 Bildverarbeitung und Mustererkennung Image Processing and Pattern Recognition 1. Image Pre-Processing - Image Denoising (cont.) - Inverse Filtering 1

2 Median Filter In probability theory (robust statistics), the median divides a probability distribution into the higher half and the lower half. In a discrete setting, the median is found by sorting the samples and selecting one Example: Samples: 3, 5, 2, 3, 99, 7, 9, 6, 5, 4, 3, 2, 8 Sorted: 2, 2, 3, 3, 3, 4, 5, 5, 6, 7, 8, 9, 99 Median 2

3 Properties does not blur across edges robust to large outliers Median Filter does not invent new gray values relatively fast to compute ( O(1) implementations exist ) Median filter is a special case of more general rank order filters Minimum filter: take maximum under mask Maximum filter: take minimum under mask Alpha-trimmed filter Samples: 3, 5, 2, 3, 99, 7, 9, 6, 5, 4, 3, 2, 8 Sorted: 2, 2, 3, 3, 3, 4, 5, 5, 6, 7, 8, 9, 99 compute mean 3

4 Example Original Median Filter 25 % salt & pepper Gaussian filter, sigma=3 5x5 Median 4

5 Iterative Median Filter Median Filter can also be applied iteratively 50 % salt & pepper noise 7x7 median filter 6 iterations of a 3x3 median filter 5

6 Bilateral Gaussian Filtering Nonlinear filtering technique introduced by C. Tomasi R. Manduchi, 1998 Extends the concept of Gaussian smoothing by using locally adaptive filter coefficients Filter coefficients are computed from the noisy input image Main advantage: does not smooth accross edges Has many applications in image processing 6

7 Blur Comes from Averaging across Edges input * output * * Same Gaussian kernel everywhere. 7

8 Bilateral Filter No Averaging across Edges input * output * * The kernel shape depends on the image content. 8

9 BF Bilateral Filter Definition: an Additional Edge Term Same idea: weighted average of pixels. new q S not new 1 [ I] = p Gσ r W s σ p new ( p q ) G ( I I ) p q I q normalization factor space weight range weight I 9

10 Illustration a 1D Image 1D image = line of pixels Better visualized as a plot pixel intensity pixel position 10

11 Gaussian blur Gaussian Blur and Bilateral Filter q p GB I = G [ ] p σ q S ( p q ) space I q Bilateral filter urich 95, Smith 97, Tomasi 98] q space p space range BF I] = 1 W normalization G ( p q ) G ( I I ) [ p σ s σ r p q S space range p q I q 11

12 Bilateral Filter on a Height Field BF I] = 1 W G ( p q ) G ( I I ) [ p σ s σ r p q S p q I q p p q output input reproduced from [Durand 1202]

13 Space and Range Parameters BF 1 [ I] = p G r W p q S ( p q ) G ( I I ) σ s σ p q I q space σ s : spatial extent of the kernel, size of the considered neighborhood. range σ r : minimum amplitude of an edge 13

14 Influence of Pixels Only pixels close in space and in range are considered. space range p 14

15 Exploring the Parameter Space σ r = 0.1 σ r = 0.25 σ r = (Gaussian blur) input σ s = 2 σ s = 6 σ s = 18 15

16 Varying the Range Parameter σ r = 0.1 σ r = 0.25 σ r = (Gaussian blur) input σ s = 2 σ s = 6 σ s = 18 16

17 input 17

18 σ r =

19 σ r =

20 σ r = (Gaussian blur) 20

21 Varying the Space Parameter σ r = 0.1 σ r = 0.25 σ r = (Gaussian blur) input σ s = 2 σ s = 6 σ s = 18 21

22 input 22

23 σ s = 2 23

24 σ s = 6 24

25 σ s = 18 25

26 How to Set the Parameters? Depends on the application. For instance: space parameter: proportional to image size e.g., 2% of image diagonal range parameter: proportional to edge amplitude e.g., mean or median of image gradients independent of resolution and exposure 26

27 Bilateral Filter Crosses Thin Lines Bilateral filter averages across features thinner than ~2σ s Desirable for smoothing: more pixels = more robust Different from diffusion that stops at thin lines close-up kernel 27

28 Bilateral Filtering Color Images For gray-level images 1 intensity difference BF [ I] p = Gσ ( p q ) G ( Ip Iq ) Iq W s σ r p q S scalar For color images 1 color difference BF [ I] p = Gσ ( p q ) G ( Cp Cq ) C r W s σ p q S q 3D vector (RGB, Lab) input output 28

29 Nonlinear Hard to Compute 1 BF [ I] p = G r W q S Complex, spatially varying kernels Cannot be precomputed, no FFT p ( p q ) G ( I I ) σ s σ p q I q Brute-force implementation is slow > 10min 29

30 Basic denoising Noisy input image 30

31 Basic denoising Bilateral filter Median 5x5 31

32 Iterating the Bilateral Filter ( n+ 1) [ I( n) I = BF Generate more piecewise-flat images Application: real-time movie abstraction ] 32

33 Fast Implementation Basic Implementation is very slow > 10 min per image Different approximations have been proposed to speedup the computation Doe not give the same results Still not realtime capable Recently O(1) implementations have been proposed Independent of spatial and range kernel size Additional speedup using GPUs 33

34 Realtime Bilateral Filtering 34

35 Filtering in the Frequency Domain Convolution in the spatial domain is equivalent to a multiplication in the frequency domain 35

36 Filtering in the Frequency Domain The fast fourier transform (FFT) yields a shifted spectrum. In order to obtain a cetralized spectrum, one can shift the spectrum or multiply the input image by (-1)^x. 36

37 Filtering in the Frequency Domain Input image Log spectrum Shifted log spectrum 37

38 Padding in 2D Padding is necessary, since the FFT assumes a periodical extension of the image data Minimum padding size for an image of size AxB and a filter maks of size CxD: P Q A + C 1 B + D 1 Padding of 256x256 image and a 256x256 filter mask: 38

39 Common Filters in the Frequency Domain Ideal Highpass Filter Butterworth Highpass Filter Gaussian Highpass Filter 39

40 Periodical Noise Affect a certain frequency spectrum Can be detected by analyzing the frequency spectrum Almost perfect restoration is often possible 40

41 Band eliminating Filter Suppresses a certain frequency band Ideal filter is given by H ( u, v) = if if if (, v) D u D 0 W 2 (, v) D u < > D D 0 0 W 2 (, v) D u + W 2 D 0 W + 2 D(u,v) is the distance from the origin D 0 is the distance of the centerline of the band from the origin W is the width of the band 41

42 42 More Band eliminating Filters Butterworth filter of order n Gaussian filter ( ) ( ) ( ) n D v u D W v D u v u H ,, 1 1, + = ( ) ( ) ( ) ,, 2 1 1, = W v u D D v u D e v u H

43 Example: Original corrupted by sinusoidal noise Fourier transform Result of filtering Butterworth filter 43

44 Inverse Filtering 1 First Model Spatial domain Frequency domain Properties Blur kernel H(u,v) has to be known in advance Model does not incorporate noise Inverse Filtering 44

45 Estimation of H(u,v) Estimation via a reference image Take a picture f(x,y) of a small light impulse Fourier transformation F(u,v) of the image Fourier transform of the impulse has a constant spectrum A 45

46 Practical Problems Spectrum of H(u,v) often contains values close to zero. Inverse filtering can lead to wrong results, in particular if the image contains high frequencies (noise) Soultion: Subsequent low pass filtering (smoothing) 46

47 Example: Original Without LP Blur+Noise With LP (sigma=2) 47

48 Inverse Filtering 2 Second model Spatial domain Frequency domain Properties Blur kernel has still to be known in advance Incorporates noise, but noise only known in a statistical sense 48

49 Wiener Filter Standard approach from Wiener, 1942 Tries to recover an image which has minimal squarred error to the true image Leads to the Wiener Filter 49

50 Example: 1 st column: Motion blur with different amount of noise 2 nd column: Inverse Filtering (no LP) 3 rd column: Wiener Filtering 50