Image gradients and edges
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1 Image gradients and edges April 7 th, 2015 Yong Jae Lee UC Davis
2 Announcements PS0 due this Friday Questions? 2
3 Last time Image formation Linear filters and convolution useful for Image smoothing, removing noise Box filter Gaussian filter Impact of scale / width of smoothing filter Separable filters more efficient Median filter: a non-linear filter, edge-preserving 3 Slide adapted from Kristen Grauman
4 Review Filter f = 1/9 x [ ] f*g=? original image g filtered 4
5 Review Filter f = 1/9 x [ ] T f*g=? original image g filtered 5
6 Review How do you sharpen an image? 6 Slide credit: Devi Parikh
7 Practice with linear filters Original Sharpening filter: accentuates differences with local average 7 Slide credit: David Lowe
8 Filtering examples: sharpening 8
9 Sharpening revisited What does blurring take away? = original Let s add it back: smoothed (5x5) detail + α = Slide credit: Svetlana Lazebnik original detail sharpened
10 Unsharp mask filter f + α( f f g) = (1 + α) f α f g = f ((1 + α) e g) image blurred image unit impulse (identity) unit impulse Gaussian Laplacian of Gaussian Slide credit: Svetlana Lazebnik
11 Review Median filter f: Is f(a+b) = f(a)+f(b)? Example: a = [ ] b = [ ] Is f linear? 11 Slide credit: Devi Parikh
12 Recall: Image filtering Compute a function of the local neighborhood at each pixel in the image Function specified by a filter or mask saying how to combine values from neighbors Uses of filtering: Enhance an image (denoise, resize, increase contrast, etc) Extract information (texture, edges, interest points, etc) Detect patterns (template matching), Adapted from Derek Hoiem 12
13 Edge detection Goal: map image from 2d array of pixels to a set of curves or line segments or contours. Why? Figure from J. Shotton et al., PAMI 2007 Main idea: look for strong gradients, post-process 13
14 What causes an edge? Reflectance change: appearance information, texture Depth discontinuity: object boundary Cast shadows Change in surface orientation: shape 14
15 Edges/gradients and invariance 15
16 Derivatives and edges An edge is a place of rapid change in the image intensity function. image intensity function (along horizontal scanline) first derivative Slide credit: Svetlana Lazebnik edges correspond to extrema of derivative 16
17 Derivatives with convolution For 2D function, f(x,y), the partial derivative is: For discrete data, we can approximate using finite differences: To implement above as convolution, what would be the associated filter? ε ε ε ), ( ), ( lim ), ( 0 y x f y x f x y x f + = 1 ), ( ) 1, ( ), ( y x f y x f x y x f + 17
18 Partial derivatives of an image f ( x, y) x f ( x, y) y ? or 1-1 Which shows changes with respect to x? 18 (showing filters for correlation)
19 Assorted finite difference filters >> My = fspecial( sobel ); >> outim = imfilter(double(im), My); >> imagesc(outim); >> colormap gray; 19
20 Image gradient The gradient of an image: The gradient points in the direction of most rapid change in intensity The gradient direction (orientation of edge normal) is given by: The edge strength is given by the gradient magnitude Slide credit: Steve Seitz 20
21 Effects of noise Consider a single row or column of the image Plotting intensity as a function of position gives a signal Where is the edge? 21 Slide credit: Steve Seitz
22 Effects of noise Difference filters respond strongly to noise Image noise results in pixels that look very different from their neighbors Generally, the larger the noise the stronger the response What can we do about it? Source: D. Forsyth
23 Solution: smooth first Where is the edge? Look for peaks in 23
24 Derivative theorem of convolution Differentiation property of convolution. 24 Slide credit: Steve Seitz
25 Derivative of Gaussian filters ( I g) h = I ( g h) [ ] [ 1 1 ] 25
26 Derivative of Gaussian filters x-direction y-direction 26 Slide credit: Svetlana Lazebnik
27 Laplacian of Gaussian Consider Laplacian of Gaussian operator Where is the edge? Slide credit: Steve Seitz Zero-crossings of bottom graph 27
28 2D edge detection filters Laplacian of Gaussian Gaussian derivative of Gaussian is the Laplacian operator: 28 Slide credit: Steve Seitz
29 Smoothing with a Gaussian Recall: parameter σ is the scale / width / spread of the Gaussian kernel, and controls the amount of smoothing. 29
30 Effect of σ on derivatives σ = 1 pixel σ = 3 pixels The apparent structures differ depending on Gaussian s scale parameter. Larger values: larger scale edges detected Smaller values: finer features detected 30 Slide credit: Yong Jae Lee, adapted from Kristen Grauman
31 So, what scale to choose? It depends what we re looking for. 31
32 Smoothing Values positive Mask properties Sum to 1 constant regions same as input Amount of smoothing proportional to mask size Remove high-frequency components; low-pass filter Derivatives signs used to get high response in regions of high contrast Sum to no response in constant regions High absolute value at points of high contrast 32
33 Seam carving: main idea [Shai & Avidan, SIGGRAPH 2007] 33
34 Seam carving: main idea Content-aware resizing Traditional resizing [Shai & Avidan, SIGGRAPH 2007] 34
35 Seam carving: main idea video 35
36 Seam carving: main idea Content-aware resizing Intuition: Preserve the most interesting content Prefer to remove pixels with low gradient energy To reduce or increase size in one dimension, remove irregularly shaped seams Optimal solution via dynamic programming. 36
37 Seam carving: main idea Energy( f ) = Want to remove seams where they won t be very noticeable: Measure energy as gradient magnitude Choose seam based on minimum total energy path across image, subject to 8-connectedness. 37
38 Seam carving: algorithm s 1 s 2 s 3 s 4 s 5 Energy( f ) = Let a vertical seam s consist of h positions that form an 8-connected path. Let the cost of a seam be: Cost(s) = Optimal seam minimizes this cost: Compute it efficiently with dynamic programming. h i= 1 Energy( f ( s i s* = min Cost( s) s )) 38
39 How to identify the minimum cost seam? How many possible seams are there? height h, width w First, consider a greedy approach: Energy matrix (gradient magnitude) Slide credit: Adapted from Kristen Grauman 39
40 Seam carving: algorithm Compute the cumulative minimum energy for all possible connected seams at each entry (i,j): M( i, j) = Energy( i, j) + min M( i 1, j 1), M( i 1, j), M( i 1, j + 1) ( ) row i-1 row i j-1 j j+1 j Energy matrix (gradient magnitude) M matrix: cumulative min energy (for vertical seams) Then, min value in last row of M indicates end of the minimal connected vertical seam. Backtrack up from there, selecting min of 3 above in M. 40
41 Example ( M( i 1, j 1), M( i 1, j), M( i 1, 1) ) M( i, j) = Energy( i, j) + min j Energy matrix (gradient magnitude) M matrix (for vertical seams) 41
42 Example ( M( i 1, j 1), M( i 1, j), M( i 1, 1) ) M( i, j) = Energy( i, j) + min j Energy matrix (gradient magnitude) M matrix (for vertical seams) 42
43 Real image example Original Image Energy Map Blue = low energy Red = high energy 43
44 Real image example 44
45 Other notes on seam carving Analogous procedure for horizontal seams Can also insert seams to increase size of image in either dimension Duplicate optimal seam, averaged with neighbors Other energy functions may be plugged in E.g., color-based, interactive, Can use combination of vertical and horizontal seams 45
46 Example results from students at UT Austin Results from Eunho Yang 46
47 Results from Suyog Jain 47
48 Conventional resize Original image Seam carving result Results from Martin Becker 48
49 Conventional resize Original image Seam carving result Results from Martin Becker 49
50 Conventional resize (399 by 599) Original image (599 by 799) Seam carving (399 by 599) Results from Jay Hennig 50
51 Removal of a marked object Results from Donghyuk Shin 51
52 Removal of a marked object Results from Eunho Yang 52
53 Failure cases with seam carving By Donghyuk Shin 53
54 Failure cases with seam carving By Suyog Jain 54
55 Questions? See you Thursday!
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