Image Filtering. Slides, adapted from. Steve Seitz and Rick Szeliski, U.Washington
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1 Image Filtering Slides, adapted from Steve Seitz and Rick Szeliski, U.Washington
2 The power of blur All is Vanity by Charles Allen Gillbert ( ) Harmon LD & JuleszB (1973) The recognition of faces. Scientific American 229(5):71 82
3 The power of blur All is Vanity by Charles Allen Gillbert ( ) Harmon LD & JuleszB (1973) The recognition of faces. Scientific American 229(5):71 82
4 Image filtering applications Today: 1. Noise removal 2. Blur 3. Image gradient 4. Edge detection 5. Image pyramids
5 Images as functions We can think of an image as a function, f: R 2 -> R f( x, y ) = intensity at position/pixel ( x, y ) Realistically, we expect the image only to be defined over a rectangle, with a finite range: f: [a,b]x[c,d] -> [,1] A color image is just three functions pasted together. We will focus mostly on intensity images.
6 Images as functions
7 What is a digital image We usually work with digital (discrete) images: Sample the 2D space on a regular grid Quantize each sample (round to nearest integer)
8 What is a digital image We usually work with digital (discrete) images: Sample the 2D space on a regular grid Quantize each sample (round to nearest integer) The image can now be represented as a matrix of integer values
9 Filtering noise Image
10 Filtering noise Image
11 Filtering noise Mean image Image
12 Filtering noise Mean image Image What happens at boundaries?
13 Cross-correlation filtering Let s write this down as an equation. Assume the averaging window is (2k+1)x(2k+1): We can generalize this idea by allowing different weights for different neighboring pixels:
14 Cross-correlation filtering This is called a cross-correlation operation H is called the filter, kernel, or mask. Note: The above allows negative filter indices. When you implement need to use: H[u+k,v+k] instead of H[u,v]
15 Mean kernel What s the kernel for a 3x3 mean filter?
16 Mean kernel What s the kernel for a 3x3 mean filter?
17 Mean vs. Gaussian filtering = Blur!!!
18 Gaussian kernel A Gaussian kernel gives less weight to pixels further from the center of the window
19 Gaussian filtering This kernel is an approximation of a Gaussian function: What happens if you increase σ?
20 Mean vs. Gaussian filtering
21 Image filtering applications Today: 1. Noise removal 2. Blur 3. Image gradient 4. Edge detection 5. Image pyramids
22 Image gradient as cross-correlation How can we differentiate a digital image F[x,y]? Option 1: reconstruct a continuous image, image, f, then take gradient Option 2: take discrete derivative (finite difference)
23 Image gradient as cross-correlation How can we differentiate a digital image F[x,y]? Option 1: reconstruct a continuous image, image, f, then take gradient Option 2: take discrete derivative (finite difference) How would you implement this as a cross-correlation?
24 Image gradient as cross-correlation How can we differentiate a digital image F[x,y]? Option 1:reconstruct a continuous image, image, f, then take gradient Option 2: take discrete derivative (finite difference) How would you implement this as a cross-correlation?
25 Image gradient as cross-correlation How can we differentiate a digital image F[x,y]? Option 1: reconstruct a continuous image, image, f, then take gradient Option 2: take discrete derivative (finite difference) How would you implement this as a cross-correlation? Forward derivative
26 Image gradient as cross-correlation How can we differentiate a digital image F[x,y]? Option 1: reconstruct a continuous image, image, f, then take gradient Option 2: take discrete derivative (finite difference) How would you implement this as a cross-correlation? Central derivative
27 Image gradient
28 Image gradient The gradient direction is given by: how does this relate to the direction of the edge?
29 Image gradient The gradient direction is given by: how does this relate to the direction of the edge? The edge strength is given by the gradient magnitude
30 Convolution A convolution operation is a cross-correlation where the filter is flipped both horizontally and vertically before being applied to the image: It is written: Suppose H is a Gaussian or mean kernel. How does convolution differ from cross-correlation?
31 Convolution A convolution operation is a cross-correlation where the filter is flipped both horizontally and vertically before being applied to the image: It is written: Suppose H is a Gaussian or mean kernel. How does convolution differ from cross-correlation?
32 Image filtering applications Today: 1. Noise removal 2. Blur 3. Image gradient 4. Edge detection 5. Image pyramids
33 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?
34 Solution: smooth first Where is the edge? Look for peaks in
35 Derivative theorem of convolution This saves us one operation: Where is the edge? Maxima point
36 Laplacian of Gaussian Consider Laplacian of Gaussian operator Where is the edge? Zero-crossings of bottom graph
37 2D edge detection filters Gaussian derivative of Gaussian Laplacian of Gaussian is the Laplacian operator
38 Laplacian filter Gaussian delta function Laplacian of Gaussian (2 nd derivative operator)
39 Laplacian image smoothed (5x5 Gaussian) delta function Laplacian of Gaussian (2 nd derivative operator)
40 Laplacian image smoothed (5x5 Gaussian) original Laplacian of Gaussian (2 nd derivative operator)
41 Laplacian image smoothed (5x5 Gaussian) original Laplacian image smoothed original (scaled by 4, offset +128)
42 Laplacian image smoothed (5x5 Gaussian) original Laplacian image smoothed original (scaled by 4, offset +128) What happens as we change σ?
43 Scale space (Witkin 83) larger Gaussian filtered signal
44 Scale space: insights As the scale is increased edge position can change edges can disappear new edges are not created Bottom line need to consider information at different scales (or else know what scale you care about)
45 Image filtering applications Today: 1. Noise removal 2. Blur 3. Image gradient 4. Edge detection 5. Image pyramids
46 Image Pyramids
47 Image sub-sampling 1/8 1/4 Throw away every other row and column to create a ½ size image
48 Image sub-sampling 1/2 1/4 (2x zoom) 1/8(4x zoom) Why does this look so bad?
49 Even worse for synthetic images
50 Really bad in video
51 Gaussian pre-filtering G 1/8 G 1/4 Gaussian 1/2 Solution: filter the image, image, then subsample In video: filter also temporally
52 Subsampling with Gaussian pre-filtering Gaussian 1/2 Solution: filter the image, G 1/4 G 1/8
53 Compare with... 1/2 1/4 (2x zoom) 1/8(4x zoom)
54 Pyramid Creation
55 Octaves in the Spatial Domain Lowpass Images Bandpass Images
56 Pyramids Advantages of pyramids Faster than Fourier transform Avoids ringing artifacts Many applications small images faster to process good for multiresolution processing compression progressive transmission
57 Example: pyramid blending
58 Laplacian level 4 Laplacian level 2 Laplacian level left pyramid right pyramid blended pyramid
59 Pyramid Blending
60 Video Pyramids Same idea in 3 dimensions x,y,t Space-time pyramid: Temporal pyramid:
61 Example: action recognition
62 Recognizing Actions Use dense information Gradient orientation <--> space-time shape information (behavior) Gradient magnitude <--> photometric information (clothing, lighting, contrast, etc)
63 Behavior-Based Distance Measure Temporal pyramid: Distance measure = the χ2 distance between corresponding empirical distributions
64 Action recognition results
65 Next week:features and descriptors
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