Image preprocessing in spatial domain

Transcription

1 Image preprocessing in spatial domain Sharpening, image derivatives, Laplacian, edges Revision: 1.2, dated: May 25, 2007 Tomáš Svoboda Czech Technical University, Faculty of Electrical Engineering Center for Machine Perception, Prague, Czech Republic

2 We have learned Spatial Filtering overview 2/35 smoothing remove noise pattern matching (normalised cross correlation)

3 We have learned Spatial Filtering overview 2/35 smoothing remove noise pattern matching (normalised cross correlation) We will learn today sharpening image derivatives edges

4 Sharpening Enhancing differences. So, the kernels involve differences combine positive and negative weights. 3/35 unsharp masking 1st and 2nd derivatives

5 Unsharp masking Often appears in Image manipulation packages (Gimp, ImageMagick) Quite powerful it cannot do miracles, though. Idea: Subtract out the blur. Procedure: 1. Blur the image 2. Subtract from original 3. Multiply by a weight 4. Combine (add to) with the original 4/35

6 Unsharp masking Mathematically 5/35 g = f + α(f f b ) f original image f b blurred image g sharpened result α controls the sharpening What is the unsharp mask?.

7 Unsharp masking Mathematically 5/35 g = f + α(f f b ) f original image f b blurred image g sharpened result α controls the sharpening What is the unsharp mask? g = 1 f + α(1 f B f).

8 Unsharp masking Mathematically 5/35 g = f + α(f f b ) f original image f b blurred image g sharpened result α controls the sharpening What is the unsharp mask? g = 1 f + α(1 f B f) = (1 + α(1 B)) f.

9 Unsharp masking Mathematically 5/35 g = f + α(f f b ) f original image f b blurred image g sharpened result α controls the sharpening What is the unsharp mask? g = 1 f + α(1 f B f) = (1 + α(1 B)) f = U f where U is the desired unsharp mask.

10 Unsharp masking Blur image 6/35

11 Unsharp masking Subtract from original 7/35 =

12 Unsharp masking Adding to the original 8/35 + =

13 Unsharp masking Result 9/35

14 Unsharp masking unsharp mask U 10/35 U = 1 + α(1 B) mask one mask one blurring mask

15 Unsharp masking unsharp mask U 11/35 U = 1 + α(1 B) unsharp mask We may combine only masks not the whole images!

16 Unsharp masking Subtract from original 12/35 =

17 Unsharp masking Adding to the original 13/35 + =

18 Unsharp masking Result 14/35

19 Unsharp masking Problems with noise 15/35

20 Unsharp masking Problems lossy JPG compression 16/35

21 Unsharp masking revisited Often appears in Image manipulation packages (Gimp, ImageMagick). It may help in practice. Low-cost lenses blur the image. Quite powerful it cannot do miracles, though. It also emphasises noise and JPG artifacts. 17/35

22 Image derivatives Measure local image geometry 18/35 Differential geometry a branch of mathematics built around We can use convolution to compute them

23 Image derivatives Measure local image geometry 18/35 Differential geometry a branch of mathematics built around We can use convolution to compute them First derivative local changes to the signal. (from physics: speed is derivative of a position with respect to time) Second derivative changes to change (from physics: acceleration is... )

24 Derivative reminder from calculus Consider a 1D signal f(x) 19/35 d f(x) = lim dx h 0 f(x + h) f(x) h

25 Derivative reminder from calculus Consider a 1D signal f(x) 19/35 d f(x) = lim dx h 0 f(x + h) f(x) h However, for sampled (discrete) signals, the smallest difference h is one. So, This called forward difference d f(x + 1) f(x) f(x) dx 1

26 Backward difference 20/35 Remind that the limit lim h 0 d f(x) = lim dx h 0 must exist for both lim h 0+ and lim h 0 f(x + h) f(x) h

27 Backward difference 20/35 Remind that the limit lim h 0 d f(x) = lim dx h 0 must exist for both lim h 0+ and lim h 0 So going from negative side of h d f(x) = lim dx h 0 f(x + h) f(x) h f(x) f(x h) h Sampled variant d f(x) f(x 1) f(x) dx 1

28 Kernels for derivatives Image is 2D function f(x, y). Derivatives may also be along y direction 21/35

29 Forward difference x direction 22/35

30 Backward difference x direction 23/35

31 Central difference x direction 24/35

32 Central difference x and y direction 25/35

33 Forward Backward Second derivatives d f(x) f(x + 1) f(x) dx d f(x) f(x) f(x 1) dx 26/35

34 Forward Backward Difference of differences Second derivatives d f(x) f(x + 1) f(x) dx d f(x) f(x) f(x 1) dx 26/35 d 2 dx2f(x) (f(x + 1) f(x)) (f(x) f(x 1)) = f(x + 1) 2f(x) + f(x 1)

35 Forward Backward Difference of differences Second derivatives d f(x) f(x + 1) f(x) dx d f(x) f(x) f(x 1) dx 26/35 d 2 dx2f(x) (f(x + 1) f(x)) (f(x) f(x 1)) = f(x + 1) 2f(x) + f(x 1) =

36 Second derivatives derivative of derivative 27/35

37 2D derivatives Differentiate in one dimension, ignore the other 28/35 x y x 2 y

38 2D derivatives with smoothing Differentiate in one dimension and smooth in the other = /35

39 2D derivatives with smoothing Differentiate in one dimension and smooth in the other = /35

40 The Gradient 30/35 f(x, y) = f x f y Magnitude is steepness in f(x, y) = ( f ) 2 + x ( ) 2 f, y direction ψ = atan ( f x, f ) y, A way to do the edge detection. Edge direction is perpendicular to ψ.

41 The Laplacian 31/35 2 f(x, y) = 2 f x f y 2 Sum of second derivatives in x and y directions. Sort of an overall curvature.

42 The Laplacian 31/35 2 f(x, y) = 2 f x f y 2 Sum of second derivatives in x and y directions. Sort of an overall curvature. With kernels: =

43 What is an edge? 32/35 x direction y direction edge profile/5 first derivative second derivative edge profile/5 first derivative second derivative

44 Partial derivatives 33/35 Extrema of partial derivatives are good candidates for edges.

45 Laplacian 34/35 Places where the Laplacian changes from positive to negative are also good potential edges.

46 Laplacian for sharpenning 35/35 original signal f first derivative f / x Second derivative Laplacian (scaled) C 2 f/ x improved signal f C 2 f/ x sharpened signal truncated to <0,255>

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58 mask one

59 blurring mask

60 unsharp mask

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87 x direction edge profile/5 first derivative second derivative

88 y direction edge profile/5 first derivative second derivative

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93 original signal f first derivative f / x Second derivative Laplacian (scaled) C 2 f/ x improved signal f C 2 f/ x sharpened signal truncated to <0,255>