Image Gradients and Gradient Filtering Computer Vision

Transcription

1 Image Gradients and Gradient Filtering Computer Vision

2 What is an image edge?

3 Recall that an image is a 2D function f(x)

4 edge edge How would you detect an edge? What kinds of filter would you use?

5 The Sobel filter a derivative filter (with some smoothing) Filter returns large response on vertical or horizontal lines?

6 The Sobel filter a derivative filter (with some smoothing) Filter returns large response on vertical or horizontal lines? Is the output always positive?

7 The Sobel filter a derivative filter (with some smoothing) Responds to horizontal lines Output can be positive or negative

8 Output of which Sobel filter? Output of which Sobel filter? How do you visualize negative derivatives/gradients?

9 Derivative in X direction Derivative in Y direction Visualize with scaled absolute value

10 The Sobel filter Where does this filter come?

11 Do you remember this from high school? f 0 (x) = lim h!0 f(x + h) f(x) h

12 Do you remember this from high school? The derivative of a function f at a point x is defined by the limit f 0 (x) = lim h!0 f(x + h) f(x) h Approximation of the derivative when h is small This definition is based on the forward difference but...

13 it turns out that using the central difference is more accurate f 0 (x) = lim h!0 f(x +0.5h) f(x 0.5h) h How do we compute the derivative of a discrete signal?

14 it turns out that using the central difference is more accurate f 0 (x) = lim h!0 f(x +0.5h) f(x 0.5h) h How do we compute the derivative of a discrete signal? f 0 (x) = f(x + 1) f(x 1) 2 = = D derivative filter

15 Decomposing the Sobel filter = Sobel What this?

16 Decomposing the Sobel filter = Sobel weighted average and scaling

17 Decomposing the Sobel filter What this? = Sobel weighted average and scaling

18 Decomposing the Sobel filter What this? = x-derivative Sobel weighted average and scaling

19 The Sobel filter only returns the x and y edge responses. How can you compute the image gradient?

20 How do you compute the image gradient? Choose a derivative filter S x = S y = What is this filter called? Run filter = S = S y f What are the dimensions? Image gradient rf What are the dimensions?

21 Matching that Gradient! (a) (1) rf = apple 0, f y (b) (2) rf = apple f x, 0 (c) (3) rf = apple f x, f y

22 Image Gradient Gradient in x only Gradient in y only Gradient in both x and y rf = apple f x, 0 rf = apple 0, f y rf = apple f x, f y Gradient direction Gradient magnitude??

23 Image Gradient Gradient in x only Gradient in y only Gradient in both x and y rf = apple f x, 0 rf = apple 0, f y rf = apple f x, f y Gradient direction = tan 1 f y / f x How does the gradient direction relate to the edge? Gradient magnitude f = s f x 2 + f y 2 What does a large magnitude look like in the image?

24 Common derivative filters Sobel Scharr Prewitt Roberts

25 Intensity plot How do you find the edge from this signal?

26 How do you find the edge from this signal? Intensity plot Use a derivative filter!

27 How do you find the edge from this signal? Intensity plot Use a derivative filter! Derivative plot What happened?

28 How do you find the edge from this signal? Intensity plot Use a derivative filter! Derivative plot Derivative filters are sensitive to noise

29 Input Gaussian Smoothed input Derivative Output Don t forget to smooth before running derivative filters!

30 Laplace filter A.K.A. Laplacian, Laplacian of Gaussian (LoG), Marr filter, Mexican Hat Function

31 Laplace filter A.K.A. Laplacian, Laplacian of Gaussian (LoG), Marr filter, Mexican Hat Function

32 Laplace filter A.K.A. Laplacian, Laplacian of Gaussian (LoG), Marr filter, Mexican Hat Function

33 first-order finite difference f 0 (x) = lim h!0 f(x +0.5h) f(x 0.5h) h derivative filter second-order finite difference Laplace filter???

34 first-order finite difference f 0 (x) = lim h!0 f(x +0.5h) f(x 0.5h) h derivative filter second-order finite difference Laplace filter 1-2 1

35 Input Laplacian Output?

36 Input Laplacian Output Zero crossings are more accurate at localizing edges Second derivative is noisy

37 2D Laplace filter D Laplace filter????????? 2D Laplace filter

38 2D Laplace filter D Laplace filter????????? 2D Laplace filter hint

39 2D Laplace filter D Laplace filter D Laplace filter If the Sobel filter approximates the first derivative, the Laplace filter approximates...?

40 Laplace filter Laplace filter with smoothing without smoothing

41 Laplace filter Sobel filter What s different between the two results?

42 zero-crossing peak Laplace Sobel Zero crossings are more accurate at localizing edges (but not very convenient)

43 Gaussian Derivative of Gaussian 2D Gaussian Filters Laplacian of Gaussian