# Convolution. 1D Formula: 2D Formula: Example on the web:

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1 Basic Filters (7) Convolution/correlation/Linear filtering Gaussian filters Smoothing and noise reduction First derivatives of Gaussian Second derivative of Gaussian: Laplacian Oriented Gaussian filters Steerability

2 Convolution 1D Formula: 2D Formula: Example on the web:

3

4 Kernel: 1/9 1/9 1/9 1/9 1/9 1/9 1/9 1/9 1/9

5 Kernel:

6 Kernel: 15 x 15 matrix of value 1/225

7 Basic Properties Commutes: f * g = g * f Associative: (f * g) * h = f * (g * h) Linear: (af + bg) * h = a f * h + b g * h Shift invariant: f t * h = (f * h) t Only operator both linear and shift invariant Differentiation:

8 Practicalities (discrete convolution/correlation) MATLAB: conv (1D) or conv2 (2D), corr Border issues: When applying convolution with a KxK kernel, the result is undefined for pixels closer than K pixels from the border of the image Options: Warp around Expand/Pad Crop K

9 1-D: 2-D: Slight abuse of notations: We ignore the normalization constant such that

10 ,! = 5 Kernel:

11 Simple Averaging Gaussian Smoothing

12 Image Noise

13 Gaussian Smoothing to Remove Noise No smoothing! = 2! =

14 Shape of Gaussian filter as function of!"! = 1! = 3! = 5

15 Note about Finite Kernel Support Gaussian function has infinite support In discrete filtering, we have finite kernel size

16 Increasing!"

17 Basic Properties Gaussian removes high-frequency components from the image! low pass filter Larger! remove more details Combination of 2 Gaussian filters is a Gaussian filter: Separable filter: Critical implication: Filtering with a NxN Gaussian kernel can be implemented as two convolutions of size N! reduction quadratic to linear! must be implemented that way

18 Oriented Gaussian Filters G! smoothes the image by the same amount in all directions If we have some information about preferred directions, we might want to smooth with some value! 1 in the direction defined by the unit vector [a b] and by! 2 in the direction defined by [c d] We can write this in a more compact form by using the standard multivariate Gaussian notation: The two (orthogonal) directions of filtering are given by the eigenvectors of #, the amount of smoothing is given by the square root of the corresponding eigenvalues of #.

19

20

21

22 Image Derivatives Image Derivatives Derivatives increase noise Derivative of Gaussian Laplacian of Gaussian (LOG)

23 Image Derivatives We want to compute, at each pixel (x,y) the derivatives: In the discrete case we could take the difference between the left and right pixels: Convolution of the image by Problem: Increases noise Difference between Actual image values True difference (derivative) Twice the amount of noise as in the original image

24 Original Image Noise Added Derivative in the horizontal direction

25 Smooth Derivatives Solution: First smooth the image by a Gaussian G! and then take derivatives: Applying the differentiation property of the convolution: Therefore, taking the derivative in x of the image can be done by convolution with the derivative of a Gaussian: Crucial property: The Gaussian derivative is also separable:

26 G G x

27 Derivative + Smoothing Without smoothing With smoothing Better but still blurs away edge information

28 I Applying the first derivative of Gaussian

29 Input Difference operator Derivative from difference operator Gaussian derivative operator Derivative from Gaussian derivative

30 There is ALWAYS a tradeoff between smoothing and good edge localization! Image with Edge Edge Location Image + Noise Derivatives detect edge and noise Smoothed derivative removes noise, but blurs edge

31 g Second derivatives: Laplacian g xx

32 DOG Approximation to LOG

33 Gaussian Separable, low-pass filter Derivatives of Gaussian Separable, output of convolution is gradient at scale!: Laplacian Directional Derivatives Not-separable, approximated by A difference of Gaussians. Output of convolution is Laplacian of image: Zero-crossings correspond to edges Output of convolution is magnitude of derivative in direction \$. Filter is linear combination of derivatives in x and y Oriented Gaussian Smooth with different scales in orthogonal directions

34 Edge Detection Edge Detection Gradient operators Canny edge detectors Laplacian detectors

35 Edge = discontinuity of intensity in some direction. Could be detected by looking for places where the derivatives of the image have large values. What is an edge?

36 \$ Edge pixels are at local maxima of gradient magnitude Gradient computed by convolution with Gaussian derivatives Gradient direction is always perpendicular to edge direction

37 Small sigma Large sigma

38 != 10!= 1 Large!! Good detection (high SNR) Poor localization Small!! Poor detection (low SNR) Good localization

39 Canny s Result Given a filter f, define the two objective functions: %(f) large if f produces good localization #(f) large if f produces good detection (high SNR) Problem: Find a family of filters f that maximizes the compromise criterion %(f)#(f) under the constraint that a single peak is generated by a step edge Solution: Unique solution, a close approximation is the Gaussian derivative filter! Canny Derivative of Gaussian

40 Next Steps The gradient magnitude enhances the edges but 2 problems remain: What threshold should we use to retain only the real edges? Even if we had a perfect threshold, we would still have poorly localized edges. How to extract optimally localize contours? Solution: Two standard tools: Non-local maxima suppression Hysteresis thresholding

41 Different thresholds applied to gradient magnitude

42 Input image

43 Different thresholds applied to gradient magnitude

44 Non-Local Maxima Suppression Gradient magnitude at center pixel is lower than the gradient magnitude of a neighbor in the direction of the gradient! Discard center pixel (set magnitude to 0) Gradient magnitude at center pixel is greater than gradient magnitude of all the neighbors in the direction of the gradient! Keep center pixel unchanged

45 T = 15 T = 5 Two thresholds applied to gradient magnitude

46 Hysteresis Thresholding Weak pixels but connected Weak pixels but isolated Very strong edge response. Let s start here Weaker response but it is connected to a confirmed edge point. Let s keep it. Continue. Note: Darker squares illustrate stronger edge response (larger M)

47 T=15 T=5 Hysteresis T h =15 T l = 5 Hysteresis thresholding

48

49 Summary Edges are discontinuities of intensity in images Correspond to local maxima of image gradient Gradient computed by convolution with derivatives of Gaussian General principle applies: Large!: Poor localization, good detection Small!: Good localization, poor detection Canny showed that Gaussian derivatives yield good compromise between localization and detection Edges correspond to zero-crossings of the second derivative (Laplacian in 2-D)

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